# Scientific Papers of Josiah Willard Gibbs, Volume 1/Chapter IIIb

The Conditions of Internal and External Equilibrium for Solids in contact with Fluids with regard to all possible States of Strain of the Solids.
In treating of the physical properties of a solid, it is necessary to consider its state of strain. A body is said to be strained when the relative position of its parts is altered, and by its state of strain is meant its state in respect to the relative position of its parts. We have hitherto considered the equilibrium of solids only in the case in which their state of strain is determined by pressures having the same values in all directions about any point. Let us now consider the subject without this limitation.
If $x',y',z'$ are the rectangular co-ordinates of a point of a solid body in any completely determined state of strain, which we shall call ​the state of reference, and $x,y,z$, the rectangular co-ordinates of the same point of the body in the state in which its properties are the subject of discussion, we may regard $x,y,z$ as functions of $x',y',z'$, the form of the functions determining the second state of strain. For brevity, we may sometimes distinguish the variable state, to which $x,y,z$ relate, and the constant state (state of reference) to which $x',y',z'$ relate, as the strained and unstrained states; but it must be remembered that these terms have reference merely to the change of form or strain determined by the functions which express the relations of $x,y,z$ and $x',y',z'$, and do not imply any particular physical properties in either of the two states, nor prevent their possible coincidence. The axes to which the co-ordinates $x,y,z$ and $x',y',z'$ relate will be distinguished as the axes of $X,Y,Z$ and $X',Y',Z'$. It is not necessary, nor always convenient, to regard these systems of axes as identical, but they should be similar, i.e., capable of superposition.
The state of strain of any element of the body is determined by the values of the differential coefficients of $x,y,$ and $z$ with respect to $x',y',$, and $z'$; for changes in the values of $x,y,z$, when the differential coefficients remain the same, only cause motions of translation of the body. When the differential coefficients of the first order do not vary sensibly except for distances greater than the radius of sensible molecular action, we may regard them as completely determining the state of strain of any element. There are nine of these differential coefficients, viz.,
| ${\frac {dx}{dx}},$ | ${\frac {dx}{dy}},$ | ${\frac {dx}{dz}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (354) |
| ${\frac {dy}{dx}},$ | ${\frac {dy}{dy}},$ | ${\frac {dy}{dz}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (354) |
| ${\frac {dz}{dx}},$ | ${\frac {dz}{dy}},$ | ${\frac {dz}{dz}}\cdot$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (354) |
It will be observed that these quantities determine the orientation of the element as well as its strain, and both these particulars must be given in order to determine the nine differential coefficients. Therefore, since the orientation is capable of three independent variations, which do not affect the strain, the strain of the element, considered without regard to directions in space, must be capable of six independent variations.
The physical state of any given element of a solid in any unvarying state of strain is capable of one variation, which is produced by addition or subtraction of heat. If we write $\epsilon _{V}$ and $\eta _{V}$ for the energy and entropy of the element divided by its volume in the state of reference, we shall have for any constant state of strain
* $\delta \epsilon _{V}=t\delta \eta _{V}.$
$\epsilon _{V'}$
$\eta _{V'}$
| $\delta \epsilon _{V'}=t\delta \eta _{V'}+X_{X'}\delta {\frac {dx}{dx'}}+X_{Y'}\delta {\frac {dx}{dy'}}+X_{Z'}\delta {\frac {dx}{dz'}}+Y_{X'}\delta {\frac {dy}{dx'}}+Y_{Y'}\delta {\frac {dy}{dy'}}+Y_{Z'}\delta {\frac {dy}{dz'}}+Z_{X'}\delta {\frac {dz}{dx'}}+Z_{Y'}\delta {\frac {dz}{dy'}}+Z_{Z'}\delta {\frac {dz}{dz'}},$ | (355) |
$X_{X'},...Z_{Z'}$
$\epsilon _{V'}$
${\frac {dx}{dx'}},...{\frac {dz}{dz'}}\cdot$
will be apparent, if we apply the formula to an element which in the state of reference is a right parallelepiped having the edges $dx',dy',dz',$, and suppose that in the strained state the face in which $x'$ has the smaller constant value remains fixed, while the opposite face is moved parallel to the axis of $X$. If we also suppose no heat to be imparted to the element, we shall have, on multiplying by $dx'dy'dz'$,
* $\delta \epsilon _{V'}dx'dy'dz'=X_{X'}\delta {\frac {dx}{dx'}}dx'dy'dz'.$
$X$
$\delta {\frac {dx}{dx'}}dx'$
$X_{X'}dy'dz'$
$X$
$x'$
$x'$
$X_{X'}$
$X$
$x'$
It will be convenient to use $\textstyle \sum$ and $\textstyle \sum '$ to denote summation with respect to quantities relating to the axes $X,Y,Z$, and to the axes $X',Y',Z'$, respectively. With this understanding we may write
| $\delta \epsilon _{V'}=t\delta \eta _{V'}+\textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)\cdot$ | (356) |
$\epsilon _{V'}$
$dx'dy'dz'$
* $\int \epsilon _{V'}\delta N'Ds'$
$Ds'$
$\delta N'$
| $\iiint t\delta \eta _{V'}dx'dy'dz'+\iiint \textstyle \sum \sum ^{\prime }\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)+\int \epsilon _{V'}\delta N'Ds'.$ | (357) |
To obtain the conditions of equilibrium for solid and fluid masses in contact, we should make the variation of the energy of the whole equal to or greater than zero. But since we have already examined the conditions of equilibrium for fluids, we need here only seek the conditions of equilibrium for the interior of a solid mass and for the surfaces where it comes in contact with fluids. For this it will be necessary to consider the variations of the energy of the fluids only so far as they are immediately connected with the changes in the solid. We may suppose the solid with so much of the fluid as is in close proximity to it to be enclosed in a fixed envelop, which is impermeable to matter and to heat, and to which the solid is firmly attached wherever they meet. We may also suppose that in the narrow space or spaces between the solid and the envelop, which are filled with fluid, there is no motion of matter or transmission of heat across any surfaces which can be generated by moving normals to the surface of the solid, since the terms in the condition of equilibrium relating to such processes may be cancelled on account of the internal equilibrium of the fluids. It will be observed that this method is perfectly applicable to the case in which a fluid mass is entirely enclosed in a solid. A detached portion of the envelop will then be necessary to separate the great mass of the fluid from the small portion adjacent to the solid, which alone we have to consider. Now the variation of the energy of the fluid mass will be, by equation (13),
| $\int ^{F}t\delta D\eta -\int ^{F}p\delta Dv+\textstyle \sum _{1}\displaystyle \int ^{F}\mu _{1}\delta Dm_{1},$ | (358) |
$\int ^{F}$
$\textstyle \sum _{1}$
If we wish to take account of gravity, we may suppose that it acts in the negative direction of the axis of $Z$. It is evident that the variation of the energy due to gravity for the whole mass considered is simply
| $\iiint g\Gamma '\delta z\delta x'\delta y'\delta z',$ | (359) |
$g$
$\Gamma '$
We have, then, for the general condition of equilibrium,
| $\iiint t\delta \eta _{V'}dx'dy'dz'+\iiint \textstyle \sum \sum \,'\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)dx'dy'dz'+\iiint g\Gamma '\delta z\delta x'\delta y'\delta z'+\int \epsilon _{V'}\delta N'Ds'+\int ^{F}t\delta D\eta -\int ^{F}p\delta Dv+\textstyle \sum _{1}\displaystyle \int ^{F}\mu _{1}\delta Dm_{1}\geqq 0.$ | (360) |
(1) that which expresses the constancy of the total entropy,
| $\iiint \delta \eta _{V'}\delta x'\delta y'\delta z'+\int \eta _{V'}\delta N'Ds'+\int ^{F}\delta D\eta =0;$ | (361) |
$\delta Dv$
| $\delta Dv=-(\alpha \delta x+\beta \delta y+\gamma \delta z)Ds-v_{V'}\delta N'Ds',$ | (362) |
$\alpha ,\beta ,\gamma$
$x,y,z$
$Ds$
$Ds'$
$v_{V'}$
(3) those which express how the values of $\delta Dm_{1},\delta Dm_{2}$, etc. for any element in the fluid are determined by the changes in the solid,
| $\delta Dm_{1}=-\Gamma _{1}'\delta N'Ds',$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$ (363) |
| $\delta Dm_{2}=-\Gamma _{2}'\delta N'Ds',$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$ (363) |
| etc., | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$ (363) |
where $\Gamma _{1}',\Gamma _{2}'$, etc. denote the separate densities of the several components in the solid in the state of reference.
Now, since the variations of entropy are independent of all the other variations, the condition of equilibrium (360), considered with regard to the equation of condition (361), evidently requires that throughout the whole system
| $t=const.$ | (364) |
​We may therefore use (361) to eliminate the fourth and fifth integrals from (360). If we multiply (362) by $p$, and take the integrals for the whole surface of the solid and for the fluid in contact with it, we obtain the equation {{MathForm2|(365)|$\int ^{F}p\delta Dv=-\int p(\alpha \delta x+\beta \delta y+\gamma \delta z)Ds-\int pv_{V'}\delta N'Ds',$ by means of which we may eliminate the sixth integral from (360). If we add equations (363) multiplied respectively by $\mu _{1},\mu _{2}$, etc., and take the integrals, we obtain the equation
| $\textstyle \sum _{1}\displaystyle \int ^{F}\mu _{1}\delta Dm_{1}=-\int \textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')\delta N'Ds',$ | (366) |
The condition of equilibrium is thus reduced to the form
| $\iiint \textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)dx'dy'dz'+\iiint g\gamma '\delta zdx'dy'dz'+\int \epsilon _{V'}\delta N'Ds'-\int t\eta _{V'}\delta N'Ds'+\int p(\alpha \delta x+\beta \delta y+\gamma \delta z)Ds+\int pv_{V'}\delta N'Ds'-\int \textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')\delta N'Ds'\geqq 0,$ | (367) |
$p$
$\mu _{1},\mu _{2}$
Now by the ordinary method of the calculus of variations, if we write $\alpha ',\beta ',\gamma ',$ for the direction cosines of the normal to the surface of the solid in the state of reference, we have
| $\iiint X_{X'}\delta {\frac {dx}{dx'}}dx'dy'dz'=\int \alpha 'X_{X'}\delta xDs'-\iiint {\frac {dX_{X'}}{dx'}}\delta xdx'dy'dz',$ | (368) |
| $-\iiint \textstyle \sum \sum '\displaystyle \left({\frac {dX_{X'}}{dx'}}\delta x\right)dx'dy'dz'+\iiint g\gamma '\delta zdx'dy'dz'+\int \textstyle \sum \sum '\displaystyle (\alpha 'X_{X'}\delta x)Ds'+\int p\textstyle \sum \displaystyle (\alpha \delta x)Ds+\int [\epsilon _{V'}-t\eta _{V'}+pv_{V'}-\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')]\delta N'Ds'\geqq 0.$ | (369) |
| $\textstyle \sum \sum '\displaystyle \left({\frac {dX_{X'}}{dx'}}\delta x\right)-g\Gamma '\delta z=0;$ | (370) |
| $Ds'\textstyle \sum \sum '\displaystyle (\alpha 'X_{X'}\delta x)+Dsp\textstyle \sum \displaystyle (\alpha \delta x)=0,$ | (371) |
| and$[\epsilon _{V'}-t\eta _{V'}+pv_{V'}-\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}')]\delta N'\geqq 0;$ | (372) |
Equation (370) expresses the mechanical conditions of internal equilibrium for a continuous solid under the influence of gravity. If we expand the first term, and set the coefficients of $\delta x,\delta y$, and $\delta z$ separately equal to zero, we obtain
| ${\frac {dX_{X'}}{dx'}}+{\frac {dX_{Y'}}{dy'}}+{\frac {dX_{Z'}}{dz'}}=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (374) |
| ${\frac {dY_{X'}}{dx'}}+{\frac {dY_{Y'}}{dy'}}+{\frac {dY_{Z'}}{dz'}}=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (374) |
| ${\frac {dZ_{X'}}{dx'}}+{\frac {dZ_{Y'}}{dy'}}+{\frac {dZ_{Z'}}{dz'}}=g\Gamma ',$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (374) |
The first member of any one of these equations multiplied by $dx'dy'dz'$ evidently represents the sum of the components parallel to one of the axes $X,Y,Z$ of the forces exerted on the six faces of the element $dx'dy'dz'$ by the neighboring elements.
As the state which we have called the state of reference is arbitrary, it may be convenient for some purposes to make it coincide with the state to which $x,y,z$ relate, and the axes $X',Y',Z'$ with the axes $X,Y,Z$. The values of $X_{X'},...Z_{Z'}$ on this particular supposition may be represented by the symbols $X_{X},...Z_{Z}$. Since
* $X_{Y'}={\frac {d\epsilon _{V'}}{d{\frac {dx}{dy'}}}},$and$Y_{X'}={\frac {d\epsilon _{V'}}{d{\frac {dy}{dx'}}}},$
$x,y,z$
$x',y',z'$
$X,Y,Z,$
$X',Y',Z',d{\frac {dx}{dy'}}$
$d{\frac {dy}{dx'}}$
| $X_{Y}=Y_{X},$ | (375) |
| $Y_{Z}=Z_{Y},$$Z_{X}=X_{Z}.$ | (376) |
​The six quantities $X_{X},Y_{Y},Z_{Z},X_{Y}$, or $Y_{X},Y_{Z}$ or $Z_{Y}$, and $Z_{X}$ or $X_{Z}$ are called the rectangular components of stress, the three first being the longitudinal stresses and the three last the shearing stresses. The mechanical conditions of internal equilibrium for a solid under the influence of gravity may therefore be expressed by the equations
| ${\frac {dX_{X}}{dx}}+{\frac {dX_{Y}}{dy}}+{\frac {dX_{Z}}{dz}}=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (377) |
| ${\frac {dY_{X}}{dx}}+{\frac {dY_{Y}}{dy}}+{\frac {dY_{Z}}{dz}}=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (377) |
| ${\frac {dZ_{X}}{dx}}+{\frac {dZ_{Y}}{dy}}+{\frac {dZ_{Z}}{dz}}=g\Gamma ,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (377) |
where $\Gamma$ denotes the density of the element to which the other symbols relate. Equations (375), (376) are rather to be regarded as expressing necessary relations (when $X_{X},...Z_{Z}$ are regarded as internal forces determined by the state of strain of the solid) than as expressing conditions of equilibrium. They will hold true of a solid which is not in equilibrium,—of one, for example, through which vibrations are propagated,—which is not the case with equations (377).
Equation (373) expresses the mechanical conditions of equilibrium for a surface of discontinuity within the solid. If we set the coefficients of $\delta x,\delta y,\delta z$, separately equal to zero we obtain
| $(\alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'})_{1}+(\alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'})_{2}=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (378) |
| $(\alpha 'Y_{X'}+\beta 'Y_{Y'}+\gamma 'Y_{Z'})_{1}+(\alpha 'Y_{X'}+\beta 'Y_{Y'}+\gamma 'Y_{Z'})_{2}=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (378) |
| $(\alpha 'Z_{X'}+\beta 'Z_{Y'}+\gamma 'Z_{Z'})_{1}+(\alpha 'Z_{X'}+\beta 'Z_{Y'}+\gamma 'Z_{Z'})_{2}=0.$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (378) |
Now when the $\alpha ',\beta ',\gamma '$ represent the direction-cosines of the normal in the state of reference on the positive side of any surface within the solid, an expression of the form
| $\alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'}$ | (379) |
$X$
$x'$
$y'$
$z'$
$X$
* $(\alpha ')_{1}=-(\alpha ')_{2},$$(\beta ')_{1}=-(\beta ')_{2},$$(\gamma ')_{1}=-(\gamma ')_{2},$
| $\alpha '(X_{X'})_{1}+\beta '(X_{Y'})_{1}+\gamma '(X_{Z'})_{1}=\alpha '(X_{X'})_{2}+\beta '(X_{Y'})_{2}+\gamma '(X_{Z'})_{2}$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (380) |
| etc., | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (380) |
where the signs of $\alpha ',\beta ',\gamma '$ may be determined by the normal on either side of the surface of discontinuity.
Equation (371) expresses the mechanical condition of equilibrium for a surface where the solid meets a fluid. It involves the separate equations
| $\alpha 'X_{X'}+\beta 'X_{Y'}+\gamma 'X_{Z'}=-\alpha p{\frac {Ds}{Ds'}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (381) |
| $\alpha 'Y_{X'}+\beta 'Y_{Y'}+\gamma 'Y_{Z'}=-\beta p{\frac {Ds}{Ds'}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (381) |
| $\alpha 'Z_{X'}+\beta 'Z_{Y'}+\gamma 'Z_{Z'}=-\gamma p{\frac {Ds}{Ds'}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (381) |
the fraction ${\frac {Ds}{Ds'}}$ denoting the ratio of the areas of the same element of the surface in the strained and unstrained states of the solid. These equations evidently express that the force exerted by the interior of the solid upon an element of its surface, and determined by the strain of the solid, must be normal to the surface and equal (but acting in the opposite direction) to the pressure exerted by the fluid upon the same element of surface.
If we wish to replace $\alpha$ and $Ds$ by $\alpha ',\beta ',\gamma ',$ and the quantities which express the strain of the element, we may make use of the following considerations. The product $\alpha Ds$ is the projection of the element $Ds$ on the Y-Z plane. Now since the ratio ${\frac {Ds}{Ds'}}$ is independent of the form of the element, we may suppose that it has any convenient form. Let it be bounded by the three surfaces $x'={\text{const.}}$, $y'={\text{const.}}$, $z'={\text{const.}}$, and let the parts of each of these surfaces included by the two others with the surface of the body be denoted by $L,M,$, and $N$, or by $L',M',$, and $N'$, according as we have reference to the strained or unstrained state of the body. The areas of $L',M',$, and $N'$ are evidently $\alpha 'Ds',\beta 'Ds',$, and $\gamma 'Ds'$; and the sum of the projections of $L,M,$, and $N$ upon any plane is equal to the projection of $Ds$ upon that plane, since $L,M,$, and $N$ with $Ds$ include a solid figure. (In propositions of this kind the sides of surfaces must be distinguished. If the normal to $Ds$ falls outward from the small solid figure, the normals to $L,M,$, and $N$ must fall inward, and vice versa.) Now $L'$ is a right-angled triangle of which the perpendicular sides may be called $dy'$ and $dz'$. ​The projection of $L$ on the Y-Z plane will be a triangle, the angular points of which are determined by the co-ordinates
* $y,z;$ $y+{\frac {dy}{dy'}}dy',$ $z+{\frac {dz}{dy'}}dy';$ $y+{\frac {dy}{dz'}}dz',$ $z+{\frac {dz}{dz'}}dz';$
* ${\tfrac {1}{2}}\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)dy'dz',$
${\tfrac {1}{2}}dy'dz'$
$L'$
* $\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)\alpha 'Ds'.$
$M$
$N$
$y',z'$
$\alpha '$
$z',x'$
$\beta '$
$x',y',$
$\gamma '$
$Ds$
We shall hereafter use $\textstyle \sum '$ to denote the sum of the three terms obtained by rotary substitutions of quantities relating to the axes $X',Y',Z'$ (i.e., by changing $x',y',z'$ into $y',z',x',$ and into $z',x',y',$ with similar changes in regard to $\alpha ',\beta ',\gamma ',$ and other quantities relating to these axes), and $\textstyle \sum$ to denote the sum of the three terms obtained by similar rotary changes of quantities relating to the axes $X,Y,Z$. This is only an extension of our previous use of these symbols.
With this understanding, equations (381) may be reduced to the form
| $\textstyle \sum '\displaystyle (\alpha 'X_{X'})+p\textstyle \sum '\displaystyle \left\lbrace \alpha '\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)\right\rbrace =0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (382) |
| etc. | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (382) |
The formula (372) expresses the additional condition of equilibrium which relates to the dissolving of the solid, or its growth without discontinuity. If the solid consists entirely of substances which are actual components of the fluid, and there are no passive resistances which impede the formation or dissolving of the solid, $\delta N'$ may have either positive or negative values, and we must have
| $\epsilon _{V'}-t\eta _{V'}-pv_{V'}=\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}').$ | (383) |
$\delta N'$
| $\epsilon _{V'}-t\eta _{V'}-pv_{V'}\leqq \textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}').$ | (384) |
$\epsilon _{V},\eta _{V},\Gamma _{1}$
$v_{V'}$
| $\epsilon _{V}+t\eta _{V}+p=\textstyle \sum _{1}\displaystyle (\mu _{1}\Gamma _{1}).$ | (385) |
$\mu _{1}$
$\Gamma$
$\Gamma '$
| $\epsilon _{V'}+t\eta _{V'}+pv_{V'}=\mu _{1}\Gamma ',$ | (386) |
| and$\epsilon _{V}+t\eta _{V}+p=\mu _{1}\Gamma .$ | (387) |
$\epsilon ,\eta ,v,$
$m$
| $\epsilon -t\eta +pv=\mu _{1}m.$ | (388) |
$\alpha ,\beta ,\gamma$
$X_{X}$
| $S=$ | $\alpha (\alpha X_{X}+\beta X_{Y}+\gamma X_{Z})$ |
| | $+\beta (\alpha Y_{X}+\beta Y_{Y}+\gamma Y_{Z})$ |
| | $+\gamma (\alpha Z_{X}+\beta Z_{Y}+\gamma Z_{Z}),$ |
or, by (375), (376),
| $S=\alpha ^{2}X_{X}+\beta ^{2}Y_{X}+\gamma ^{2}Z_{X}+2\alpha \beta X_{Y}+2\beta \gamma Y_{Z}+2\gamma \alpha Z_{X}.$ | (389) |
$X$
$S$
$Y$
$X$
$S$
$X$
${\frac {dS}{d\alpha }}$
${\frac {dS}{d\beta }}$
${\frac {dS}{d\gamma }}$
$\alpha ,\beta$
$\gamma$
* ${\frac {Ds}{d\alpha }}d\alpha +{\frac {dS}{d\beta }}d\beta +{\frac {dS}{d\gamma }}d\gamma =0,$
| when | $\alpha d\alpha +\beta d\beta +\gamma d\gamma =0,$ |
| and | $\alpha =1,$$\beta =0,$$\gamma =0,$ |
| That is, | ${\frac {dS}{d\beta }}=0,$and${\frac {dS}{d\gamma }}=0,$ |
| when | $\alpha =1,$$\beta =0,$$\gamma =0.$ |
| Hence | $X_{Y}=0,$and$Z_{X}=0.$ | (390) |
| Moreover, | ${\frac {dS}{d\beta }}d\beta +{\frac {dS}{d\gamma }}d\gamma =0,$ |
| when | $\alpha =0,$$d\alpha =0,$ |
| | $\beta d\beta +\gamma d\gamma =0,$ |
| and | $\beta =1,$$\gamma =0.$ |
| Hence | $Y_{Z}=0.$ | (391) |
Therefore, when the co-ordinate axes have the supposed directions, which are called the principal axes of stress, the rectangular components of the traction across any surface ($\alpha ,\beta ,\gamma$) are by (379)
| $\alpha X_{X},$ $\beta Y_{T},$ $\gamma Z_{Z}.$ | (392) |
(1), when the surface is perpendicular to a principal axis of stress;
(2), if two of the principal tractions $X_{X},Y_{Y},Z_{Z}$ are equal, when the surface is perpendicular to the plane containing the two corresponding axes (in this case the traction across any such surface is equal to the common value of the two principal tractions);
(3), if the principal tractions are all equal, the traction is normal and constant for all surfaces.
It will be observed that in the second and third cases the positions of the principal axes of stress are partially or wholly indeterminate (so that these cases may be regarded as included in the first), but the values of the principal tractions are always determinate, although not always different.
If, therefore, a solid which is homogeneous in nature and in state of strain is bounded by six surfaces perpendicular to the principal axes of stress, the mechanical conditions of equilibrium for these surfaces may be satisfied by the contact of fluids having the proper pressures ​(see (381)), which will in general be different for the different pairs of opposite sides, and may be denoted by $p',p'',p'''$. (These pressures are equal to the principal tractions of the solid taken negatively.) It will then be necessary for equilibrium with respect to the tendency of the solid to dissolve that the potential for the substance of the solid in the fluids shall have values $\mu _{1}',\mu _{1}'',\mu _{1}''',$ determined by the equations
| $\epsilon -t\eta +p'v=\mu _{1}'m,$ | (393) |
| $\epsilon -t\eta +p''v=\mu _{1}''m,$ | (394) |
| $\epsilon -t\eta +p'''v=\mu _{1}'''m.$ | (395) |
It may be interesting to compare one of these potentials, as $\mu _{1}'$, with the potential (for the same substance) in a fluid of the same temperature $t$ and pressure $p'$ which would be in equilibrium with the same solid subjected on all sides to the uniform pressure $p'$. If we write $[\epsilon ]_{p'},[\eta ]_{p'},[v]_{p'},$ and $[\mu _{1}]_{p'}$ for the values which $\epsilon ,\eta ,v,$ and $\mu _{1}$ would receive on this supposition, we shall have
| $[\epsilon ]_{p'}-t[\eta ]_{p'}+p'[v]_{p'}=[\mu _{1}]_{p'}m.$ | (396) |
| $\epsilon -[\epsilon ]_{p'}-t\eta +t[\eta ]_{p'}+p'v-p'[v]_{p'}=\mu _{1}m-[\mu _{1}]_{p'}m.$ | (397) |
$p'v-p'[v]_{p'}$
$p'$
$p'$
$p'$
$p',p'',p'''$
$p'=p''=p'''$
$\mu _{1}'$
$[\mu _{1}]_{p'}$
$\mu _{1}''$
$p''$
$\mu _{1}'''$
$p'''$
But in the case of a solid of continuous crystalline structure, subjected to distorting stresses and in contact with solutions satisfying the conditions deduced above, although crystals of hydrostatic stress would doubtless commence to form upon its surface (if the distorting stresses and consequent supersaturation of the fluid should be carried too far), before they would commence to be formed within the fluid or on the surface of most other bodies, yet within certain limits the relations expressed by equations (393)–(395) must admit of realization, especially when the solutions are such as can be easily supersaturated.
It may be interesting to compare the variations of $p$, the pressure in the fluid which determines in part the stresses and the state of strain of the solid, with other variations of the stresses or strains in the solid, with respect to the relation expressed by equation (388). To examine this point with complete generality, we may proceed in the following manner.
Let us consider so much of the solid as has in the state of reference the form of a cube, the edges of which are equal to unity, and parallel to the co-ordinate axes. We may suppose this body to be homogeneous in nature and in state of strain both in its state of ​reference and in its variable state. (This involves no loss of generality, since we may make the unit of length as small as we choose.) Let the fluid meet the solid on one or both of the surfaces for which $Z'$ is constant. We may suppose these surfaces to remain perpendicular to the axis of $Z$ in the variable state of the solid, and the edges in which $y'$ and $z'$ are both constant to remain parallel to the axis of $X$. It will be observed that these suppositions only fix the position of the strained body relatively to the co-ordinate axes, and do not in any way limit its state of strain.
It follows from the suppositions which we have made that
| ${\frac {dz}{dx'}}={\text{const.}}=0,$${\frac {dz}{dy'}}={\text{const.}}=0,$${\frac {dy}{dx'}}={\text{const.}}=0;$ | (398) |
| and$X_{Z'}=0,$$Y_{Z'}=0,$$Z_{Z'}=-p{\frac {dx}{dx'}}{\frac {dy}{dy'}}\cdot$ | (399) |
| $d\epsilon _{V'}=td\eta _{V'}+X_{X'}d{\frac {dx}{dx'}}+X_{Y'}d{\frac {dx}{dy'}}+Y_{Y'}d{\frac {dy}{dy'}}-p{\frac {dx}{dx'}}{\frac {dy}{dy'}}d{\frac {dz}{dz'}}\cdot$ | (400) |
| $d\epsilon =td\eta +\eta dt-pdv-vdp+md\mu _{1}.$ | (401) |
| $v={\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}},$ | (402) |
| and$dv={\frac {dy}{dy'}}{\frac {dz}{dz'}}d{\frac {dx}{dx'}}+{\frac {dz}{dz'}}{\frac {dx}{dx'}}d{\frac {dy}{dy'}}+{\frac {dx}{dx'}}{\frac {dy}{dy'}}d{\frac {dz}{dz'}}\cdot$ | (403) |
$\epsilon _{V'}$
$\eta _{V'}$
$\epsilon$
$\eta$
| $\eta dt-vdp+md\mu _{1}=\left(X_{X'}+p{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)d{\frac {dx}{dx'}}+X_{Y'}d{\frac {dx}{dy'}}+\left(Y_{Y'}+p{\frac {dz}{dz'}}{\frac {dx}{dx'}}\right)d{\frac {dy}{dy'}}\cdot$ | (404) |
$p$
${\frac {dy}{dy'}}{\frac {dz}{dz'}}$
$x'$
$p$
$X$
$X_{X'}$
$d{\frac {dy}{dy'}};$
$X_{Y'}$
$p$
We will first consider the case in which the fluid is identical in substance with the solid. We have then, by equation (97), for a mass of the fluid equal to that of the solid,
| $\eta _{F}dt-v_{F}dp+md\mu _{1}=0,$ | (405) |
$\eta _{F}$
$v_{F}$
| $-(\eta _{F}-\eta )dt+(v_{F}-v)dp=\left(X_{X'}+p{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)d{\frac {dx}{dx'}}+X_{Y'}d{\frac {dx}{dy'}}+\left(Y_{Y'}+p{\frac {dz}{dz'}}{\frac {dx}{dx'}}\right)d{\frac {dy}{dy'}}\cdot$ | (406) |
${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}$
| ${\frac {dy}{dp}}={\frac {v_{F}-v}{\eta _{F}-\eta }}=t{\frac {v_{F}-v}{Q}},$ | (407) |
$Q$
${\frac {dt}{dp}}$
$p$
$v$
$\eta$
$Q$
${\frac {dt}{dp}}$
${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}$
But if we wish to know how the temperature is affected if the pressure between the solid and fluid remains constant, but the strain of the solid is varied in any way consistent with this supposition, the differential coefficients of $t$ with respect to the quantities which express the strain are indicated by equation (406). These differential coefficients all vanish, when the pressures on all sides are normal and equal, but the differential coefficient ${\frac {dt}{dp}}$, when ${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}$ are ​constant, or when the pressures on all sides are normal and equal, vanishes only when the density of the fluid is equal to that of the solid.
The case is nearly the same when the fluid is not identical in substance with the solid, if we suppose the composition of the fluid to remain unchanged. We have necessarily with respect to the fluid
| $d\mu _{1}=\left({\frac {d\mu _{1}}{dt}}\right)_{p,m}^{(F)}dt+\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}dp,$ | (408) |
| $\left({\frac {d\mu _{1}}{dt}}\right)_{p,m}^{(F)}=-\left({\frac {d\eta }{dm_{1}}}\right)_{t,p,m}^{(F)}$,and$\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}=\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(F)}\cdot$ | (409) |
$m$
| $m\left({\frac {d\eta }{dm_{1}}}\right)_{t,p,m}^{(F)}-m\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(F)}dp+md\mu _{1}=0.$ | (410) |
$\eta _{F}$
$v_{F}$
* $m\left({\frac {d\eta }{dm_{1}}}\right)_{t,p,m}^{(F)}$and$m\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(F)}\cdot$
We may also wish to find the variations in the composition of the fluid which will be necessary for equilibrium when the pressure $p$ or the quantities ${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}$ are varied, the temperature remaining constant. If we know the value for the fluid of the quantity represented by $\xi$ on page 87 in terms of $t,p$, and the quantities of the several components $m_{1},m_{2},m_{3}$, etc., the first of which relates to the substance of which the solid is formed, we can easily find the value of $\mu _{1}$ in terms of the same variables. Now in considering variations in the composition of the fluid, it will be sufficient if we make all but one of the components variable. We may therefore give to $m_{1}$ a constant value, and making $t$ also constant, we shall have
* $d\mu _{1}=\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}dp+\left({\frac {d\mu _{1}}{dm_{2}}}\right)_{t,p,m}^{(F)}dm_{2}+\left({\frac {d\mu _{1}}{dm_{3}}}\right)_{t,p,m}^{(F)}dm_{3}+{\text{etc.}}$
$dt$
| $\left\{m\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}-v\right\}\quad dp+m\left({\frac {d\mu _{1}}{dm_{2}}}\right)_{t,p,m}^{(F)}dm_{2}+m\left({\frac {d\mu _{1}}{dm_{3}}}\right)_{t,p,m}^{(F)}dm_{3}+{\text{etc.}}=\left(X_{X'}+p{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)d{\frac {dx}{dz'}}+X_{Y'}d{\frac {dx}{dy'}}+\left(Y_{Y'}+p{\frac {dz}{dz'}}{\frac {dx}{dx'}}\right)d{\frac {dy}{dy'}}\cdot$ | (411) |
$p,$
${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dz}{dz'}}$
Fundamental Equations for Solids.
The principles developed in the preceding pages show that the solution of problems relating to the equilibrium of a solid, or at least their reduction to purely analytical processes, may be made to depend upon our knowledge of the composition and density of the solid at every point in some particular state, which we have called the state of reference, and of the relation existing between the quantities which have been represented by $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},{\frac {dx}{dy'}},...{\frac {dz}{dz'}},x',y',$ and $z'.$ When the solid is in contact with fluids, a certain knowledge of the properties of the fluids is also requisite, but only such as is necessary for the solution of problems relating to the equilibrium of fluids among themselves.
If in any state of which a solid is capable, it is homogeneous in its nature and in its state of strain, we may choose this state as the state of reference, and the relation between $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},{\frac {dx}{dy'}},...{\frac {dz}{dz'}}$ will be independent of $x',y',z'.$ But it is not always possible, even in the case of bodies which are homogeneous in nature, to bring all the elements simultaneously into the same state of strain. It would not be possible, for example, in the case of a Prince Rupert's drop.
If, however, we know the relation between $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},{\frac {dx}{dy'}},...{\frac {dz}{dz'}}$ for any kind of homogeneous solid, with respect to any given state of reference, we may derive from it a similar relation with respect to any other state as a state of reference. For if $x',y',z'$ denote the co-ordinates of points of the solid in the first state of reference, and ​$x'',y'',z''$ the co-ordinates of the same points in the second state of reference, we shall have necessarily
| ${\frac {dx}{dx'}}={\frac {dx}{dx''}}{\frac {dx''}{dx'}}+{\frac {dx}{dy''}}{\frac {dy''}{dx'}}+{\frac {dx}{dz''}}{\frac {dz''}{dx'}},$ etc. (nine equations), | (412) |
$R$
$(x'',y'',z'')$
$(x',y',z')$
| $R={\begin{vmatrix}{\frac {dx''}{dx'}}&{\frac {dx''}{dy'}}&{\frac {dx''}{dz'}}\\{\frac {dy''}{dx'}}&{\frac {dy''}{dy'}}&{\frac {dy''}{dz'}}\\{\frac {dz''}{dx'}}&{\frac {dz''}{dy'}}&{\frac {dz''}{dz'}}\end{vmatrix}},$ | (413) |
| $\epsilon _{V'}=R\epsilon _{V''},$$\eta _{V'}=R\eta _{V''}.$ | (414) |
$\epsilon _{V'}$
$\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}},$
$\epsilon _{V''}$
$\eta _{V''},{\frac {dx}{dx''}},...{\frac {dz}{dz''}},{\frac {dx''}{dx'}},...{\frac {dz''}{dz'}}$
We may apply this to the elements of a body which may be variable from point to point in composition and state of strain in a given state of reference $(x'',y'',z'')$, and if the body is fully described in that state of reference, both in respect to its composition and to the displacement which it would be necessary to give to a homogeneous solid of the same composition, for which $\epsilon _{V'}$ is known in terms of $\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}}$ and the quantities which express its composition, to bring it from the state of reference $(x',y',z')$ into a similar and similarly situated state of strain with that of the element of the non-homogeneous body, we may evidently regard ${\frac {dx''}{dx'}},...{\frac {dz''}{dz'}}$ as known for each element of the body, that is, as known in terms of $x'',y'',z''.$ We shall then have $\epsilon _{V''}$ in terms of $\eta _{V''},{\frac {dx}{dx''}},...{\frac {dz}{dz''}},x'',y'',z''$; and since the composition of the body is known in terms of $x'',y'',z''$, and the density, if not given directly, can be determined from the density of the homogeneous body in its state of reference $(x',y',z')$, this is sufficient for determining the equilibrium of any given state of the non-homogeneous solid.
An equation, therefore, which expresses for any kind of solid, and with reference to any determined state of reference, the relation ​between the quantities denoted by $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}}$, involving also the quantities which express the composition of the body, when that is capable of continuous variation, or any other equation from which the same relations may be deduced, may be called a fundamental equation for that kind of solid. It will be observed that the sense in which this term is here used, is entirely analogous to that in which we have already applied the term to fluids and solids which are subject only to hydrostatic pressure.
When the fundamental equation between $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}}$ is known, we may obtain by differentiation the values of $t,X_{X'},...Z_{Z'}$ in terms of the former quantities, which will give eleven independent relations between the twenty-one quantities
| $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}},t,X_{X'},...Z_{Z'},$ | (415) |
If we use the symbol $\psi _{V'}$ to denote the value of $\psi$ (as defined on page 89) for any element of a solid divided by the volume of the element in the state of reference, we shall have
| $\psi _{V'}=\epsilon _{V'}-t\eta _{V'}.$ | (416) |
| $\delta \psi _{V'}=-\eta _{V'}\delta t+\textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)\cdot$ | (417) |
$\psi _{V'}$
$t,{\frac {dx}{dx'}},...{\frac {dz}{dz'}},$
$\eta _{V'},X_{X'},...Z_{Z'},$
$\psi _{V'}$
$\epsilon _{V'}$
$\psi _{V'}$
$\psi _{V'}$
$t,{\frac {dx}{dx'}},...{\frac {dz}{dz'}}$
In the discussion of the conditions of equilibrium of a solid, we might have started with the principle that it is necessary and sufficient ​for equilibrium that the temperature shall be uniform throughout the whole mass in question, and that the variation of the force-function ($-\psi$) of the same mass shall be null or negative for any variation in the state of the mass not affecting its temperature. We might have assumed that the value of $\psi$ for any same element of the solid is a function of the temperature and the state of strain, so that for constant temperature we might write
* $\delta \psi _{V'}=\textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right),$
$X_{X'},...Z_{Z'}$
$X_{X'},...Z_{Z'}$
$X_{X'},...Z_{Z'}$
$\psi _{V'}$
$\epsilon _{V'}$
${\frac {dx}{dx'}},...{\frac {dz}{dz'}}$
$A,B,C,a,b,c$
| $A=\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2},$ | $B=\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2},$ | $C=\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}\right)^{2},$ | (418) |
| $a=\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dz'}}\right),$ | $b=\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dx'}}\right),$ | $c=\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\cdot$ | (419) |
The determination of the fundamental equation for a solid is thus reduced to the determination of the relation between $\epsilon _{V'},\eta _{V'},A,B,C,a,b,c,$ or of the relation between $\psi _{V'},t,A,B,C,a,b,c$.
In the case of isotropic solids, the state of strain of an element, so far as it can affect the relation of $\epsilon _{V'}$ and $\eta _{V'}$ or of $\psi _{V'}$ and $t$, is capable of only three independent variations. This appears most distinctly as a consequence of the proposition that for any given strain of an element there are three lines in the element which are at right angles to one another both in its unstrained and in its strained state. If the unstrained element is isotropic, the ratios of elongation for these three lines must with $\eta _{V'}$ determine the value $\epsilon _{V'}$, or with $t$ determine the value of $\psi _{V'}$.
To demonstrate the existence of such lines, which are called the principal axes of strain, and to find the relations of the elongations of such lines to the quantities ${\frac {dx}{dx'}},...{\frac {dz}{dz'}}$, we may proceed as follows. The ratio of elongation $r$ of any line of which $\alpha ',\beta ',\gamma '$ are the direction-cosines in the state of reference is evidently given by the equation
| $r^{2}$ | $=\left({\frac {dx}{dx'}}\alpha '+{\frac {dx}{dy'}}\beta '+{\frac {dx}{dz'}}\gamma '\right)^{2},$ |
| | $+\left({\frac {dy}{dx'}}\alpha '+{\frac {dy}{dy'}}\beta '+{\frac {dy}{dz'}}\gamma '\right)^{2},$ |
| | $+\left({\frac {dz}{dx'}}\alpha '+{\frac {dz}{dy'}}\beta '+{\frac {dz}{dz'}}\gamma '\right)^{2}\cdot$ |
Now the proposition to be established is evidently equivalent to this—that it is always possible to give such directions to the two systems of rectangular axes $X',Y',Z'$, and $X,Y,Z$, that
| ${\frac {dx}{dy'}}=0,$ | ${\frac {dx}{dz'}}=0,$ | ${\frac {dy}{dz'}}=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,$ (421) |
| ${\frac {dy}{dx'}}=0,$ | ${\frac {dz}{dx'}}=0,$ | ${\frac {dz}{dy'}}=0.$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,$ (421) |
We may choose a line in the element for which the value of $r$ is at least as great as for any other, and make the axes of $X$ and $X'$ parallel to this line in the strained and unstrained states respectively.
${\frac {dy}{dx'}}=0,$
${\frac {dz}{dx'}}=0.$
${\frac {d(r^{2})}{d\alpha '}},{\frac {d(r^{2})}{d\beta '}},{\frac {d(r^{2})}{d\gamma '}}$
$\alpha ',\beta ',\gamma '$
| | ${\frac {d(r^{2})}{d\alpha '}}+{\frac {d(r^{2})}{d\beta '}}+{\frac {d(r^{2})}{d\gamma '}}=0,$ |
| when | $\alpha 'd\alpha '+\beta 'd\beta '+\gamma 'd\gamma '=0,$ |
| and | $\alpha '=1,$$\beta '=0,$$\gamma '=0.$ |
| That is, | ${\frac {d(r^{2})}{d\beta '}}=0,$and${\frac {d(r^{2})}{d\gamma '}}=0,$ |
| when | $\alpha '=1,$$\beta '=0,$$\gamma '=0.$ |
| Hence, | ${\frac {dx}{dy'}}=0,$${\frac {dx}{dz'}}=0.$ |
Therefore a line of the element which in the unstrained state is perpendicular to $X'$ is perpendicular to $X$ in the strained state. Of all such lines we may choose one for which the value of $r$ is at least as great as for any other, and make the axes of $Y'$ and $Y$ parallel to this line in the unstrained and in the strained state respectively. Then
| ${\frac {dz}{dy'}}=0;$ | (424) |
| ${\frac {dy}{dz'}}=0.$ | (425) |
$X',Y'$
$Z'$
$X,Y$
$Z$
* ${\frac {dx}{dx'}},{\frac {dy}{dy'}},{\frac {dz}{dz'}}\cdot$
$r^{2}$
* $r^{2}={\frac {dx}{dx'}}\alpha ^{'2}+{\frac {dy}{dy'}}\beta ^{'2}+{\frac {dz}{dz'}}\gamma ^{'2}.$
${\frac {dx}{dx'}},...{\frac {dz}{dz'}}$
For any principal axis of strain we have
* ${\frac {d(r^{2})}{d\alpha '}}d\alpha '+{\frac {d(r^{2})}{d\beta '}}d\beta '+{\frac {d(r^{2})}{d\gamma '}}d\gamma '=0,$
* when$\alpha 'd\alpha '+\beta '\beta '+\gamma 'd\gamma '=0,$
| ${\frac {1}{\alpha '}}{\frac {d(r^{2})}{d\alpha '}}={\frac {1}{\beta '}}{\frac {d(r^{2})}{d\beta '}}={\frac {1}{\gamma '}}{\frac {d(r^{2})}{d\gamma '}}\cdot$ | (426) |
| ${\frac {\alpha '}{2}}{\frac {d(r^{2})}{d\alpha '}}+{\frac {\beta '}{2}}{\frac {d(r^{2})}{d\beta '}}+{\frac {\gamma '}{2}}{\frac {d(r^{2})}{d\gamma '}}=r^{2}.$ | (427) |
$\alpha ^{'2}+\beta ^{'2}+\gamma ^{'2}=1$
| ${\tfrac {1}{2}}{\frac {d(r^{2})}{d\alpha '}}=\alpha 'r^{2},$${\tfrac {1}{2}}{\frac {d(r^{2})}{d\beta '}}=\beta 'r^{2},$${\tfrac {1}{2}}{\frac {d(r^{2})}{d\gamma '}}=\gamma 'r^{2},$ | (428) |
| $\alpha '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}+\beta '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)+\gamma '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dz'}}\right)=\alpha 'r^{2},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (429) |
| $\alpha '\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dx'}}\right)+\beta '\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}+\gamma '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dz'}}\right)^{2}=\beta 'r^{2},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (429) |
| $\alpha '\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dx'}}\right)+\beta '\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dy'}}\right)+\gamma '\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}\right)^{2}=\gamma 'r^{2}.$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (429) |
If we eliminate $\alpha ',\beta ',\gamma '$ from these equations, we may write the result in the form,
| ${\begin{vmatrix}\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}-r^{2}&\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dz'}}\right)\\\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dx'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}-r^{2}&\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dz'}}\right)\\\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dx'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dy'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}\right)^{2}-r^{2}\end{vmatrix}}=0.$ | (430) |
| $-r^{5}+Er^{4}-Fr^{2}+G=0.$ | (431) |
| $E=\textstyle \sum '\sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}\cdot$ | (432) |
| $=\textstyle \sum '\displaystyle \left\{\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}-\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\right\}\quad$ |
| $=\textstyle \sum '\sum \displaystyle \left\{\left({\frac {dx}{dx'}}\right)^{2}\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}-{\frac {dx}{dx'}}{\frac {dx}{dy'}}\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\right\}\quad$ |
| $=\textstyle \sum '\sum \displaystyle \left\{\left({\frac {dx}{dx'}}\right)^{2}\left({\frac {dy}{dy'}}\right)^{2}+\left({\frac {dy}{dx'}}\right)^{2}\left({\frac {dz}{dy'}}\right)^{2}-{\frac {dx}{dx'}}{\frac {dx}{dy'}}{\frac {dy}{dx'}}{\frac {dy}{dy'}}-{\frac {dx}{dx'}}{\frac {dx}{dy'}}{\frac {dz}{dx'}}{\frac {dz}{dy'}}\right\}\quad$ |
| $=\textstyle \sum '\sum \displaystyle \left\{\left({\frac {dx}{dx'}}\right)^{2}\left({\frac {dy}{dy'}}\right)^{2}+\left({\frac {dy}{dx'}}\right)^{2}\left({\frac {dx}{dy'}}\right)^{2}-2{\frac {dx}{dx'}}{\frac {dx}{dy'}}{\frac {dy}{dx'}}{\frac {dy}{dy'}}\right\}\quad$ |
| $=\textstyle \sum '\sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dy}{dx'}}{\frac {dx}{dy'}}\right)^{2}\cdot$ | (433) |
This may also be written
| $F=\textstyle \sum '\sum \displaystyle {\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}\end{vmatrix}}^{2}\cdot$ | (434) |
$G$
$\textstyle \sum _{3+3}$
$x,y,z,$
$y,z,x;$
$z,x,y;$
$x,z,y;$
$y,x,z;$
$z,y,x$
$\textstyle \sum _{3-3}$
$\textstyle \sum _{3-3}'$
$x',y',z'$
${\text{x', y', z'}}$
$x',y',z'$
| $G=\textstyle \sum _{3-3}'\left\{\textstyle \sum \displaystyle \left({\frac {dx}{d{\text{x}}'}}{\frac {dx}{dx'}}\right)\textstyle \sum \displaystyle \left({\frac {dx}{d{\text{y}}'}}{\frac {dx}{dy'}}\right)\textstyle \sum \displaystyle \left({\frac {dx}{d{\text{z}}'}}{\frac {dx}{dz'}}\right)\right\}\quad \cdot$ | (435) |
$\textstyle \sum _{3-3}'$
$dx,dy,$
$dz$
| $G$ | $=\textstyle \sum _{3-3}'\sum _{3+3}\displaystyle \left({\frac {dx}{d{\text{x}}'}}{\frac {dx}{dx'}}{\frac {dy}{d{\text{y}}'}}{\frac {dy}{dy'}}{\frac {dz}{d{\text{z}}'}}{\frac {dz}{dz'}}\right)$ |
| | $=\textstyle \sum _{3+3}\displaystyle \left\{{\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\textstyle \sum _{3-3}'\displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)\right\}\quad$ |
| | $=\textstyle \sum _{3-3}\displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)\textstyle \sum _{3-3}'\displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)\cdot$ | (436) |
| $H={\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}&{\frac {dx}{dz'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}&{\frac {dy}{dz'}}\\{\frac {dz}{dx'}}&{\frac {dz}{dy'}}&{\frac {dz}{dz'}}\end{vmatrix}},$ | (437) |
| $G=H^{2}.$ | (438) |
$F$
$E$
Now we know by the theory of equations that equation (431) will be satisfied in general by three different values of $r^{2}$, which we may denote by $r_{1}^{2},r_{2}^{2},r_{3}^{2}$, and which must represent the squares of the ratios of elongation for the three principal axes of strain; also that $E,F,G$ are symmetrical functions of $r_{1}^{2},r_{2}^{2},r_{3}^{2}$, viz.,
| $E=r_{1}^{2}+r_{2}^{2}+r_{3}^{2},$ | $F=r_{1}^{2}r_{2}^{2}+r_{2}^{2}r_{3}^{2}+r_{3}^{2}r_{1}^{2},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (439) |
| | $G=r_{1}^{2}r_{2}^{2}r_{3}^{2}.$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (439) |
Hence, although it is possible to solve equation (431) by the use of trigonometrical functions, it will be more simple to regard $\epsilon _{V'}$ as a function of $\eta _{V'}$ and the quantities $E,F,G$ (or $H$), which we have expressed in terms of ${\frac {dx}{dx'}},...{\frac {dz}{dz'}}\cdot$ Since $\epsilon _{V'}$, is a single-valued function of $\eta _{V'}$ and $r_{1}^{2},r_{2}^{2},r_{3}^{2}$ (with respect to all the changes of which the body is capable), and a symmetrical function with respect to $r_{1}^{2},r_{2}^{2},r_{3}^{2}$, and since $r_{1}^{2},r_{2}^{2},r_{3}^{2}$ are collectively determined without ambiguity by the values of $E,F,$ and $H$, the quantity $\epsilon _{V'}$ must be a single-valued function of $\eta _{V'},E,F,$ and $H$. The determination of the fundamental equation for isotropic bodies is therefore reduced to the determination of this function, or (as appears from similar considerations) the determination of $\psi _{V'}$, as a function of $t,E,F,$ and $H$.
It appears from equations (439) that E represents the sum of the squares of the ratios of elongation for the principal axes of strain, that $F$ represents the sum of the squares of the ratios of enlargement for the three surfaces determined by these axes, and that $G$ represents the square of the ratio of enlargement of volume. Again, equation (432) shows that $E$ represents the sum of the squares of the ratios of elongation for lines parallel to $X',Y',$ and $Z'$; equation (434) shows that $F$ represents the sum of the squares of the ratios of enlargement for surfaces parallel to the planes ${\text{X'-Y', Y'-Z', Z'-X'}}$; and equation (438), like (439), shows that $G$ represents the square of the ratio of ​enlargement of volume. Since the position of the co-ordinate axes is arbitrary, it follows that the sum of the squares of the ratios of elongation or enlargement of three lines or surfaces which in the unstrained state are at right angles to one another, is otherwise independent of the direction of the lines or surfaces. Hence, ${\tfrac {1}{3}}E$ and ${\tfrac {1}{3}}F$ are the mean squares of the ratios of linear elongation and of superficial enlargement, for all possible directions in the unstrained solid.
There is not only a practical advantage in regarding the strain as determined by $E,F,$ and $H$, instead of $E,F,$ and $G$, because $H$ is more simply expressed in terms of ${\frac {dx}{dx'}},...{\frac {dz}{dz'}}$, but there is also a certain theoretical advantage on the side of $E,F,H$. If the systems of co-ordinate axes $X,Y,Z$, and $X',Y',Z'$, are either identical or such as are capable of superposition, which it will always be convenient to suppose, the determinant H will always have a positive value for any strain of which a body can be capable. But it is possible to give to $x,y,z$ such values as functions of $x',y',z'$ that $H$ shall have a negative value. For example, we may make
| $x=x',$$y=y',$$z=-z'.$ | (440) |
$H=-1$
| $x=x',$$y=y',$$z=z'.$ | (441) |
$H=1$
We may avoid all such confusion by using $E,F,$ and $H$ to represent a strain. Let us consider an element of the body strained which in the state $(x',y',z')$ is a cube with its edges parallel to the axes of $X',Y',Z'$, and call the edges $dx',dy',dz'$ according to the axes to which they are parallel, and consider the ends of the edges as positive for which the values of $x',y',$ or $z'$ are the greater. Whatever may be the nature of the parallelepiped in the state $(x,y,z)$ which corresponds to the cube $dx',dy',dz'$ and is determined by the quantities ${\frac {dx}{dx'}},...{\frac {dz}{dz'}}$, it may always be brought by continuous changes to the form of a cube and to a position in which the edges $dx',dy'$ shall be parallel to the axes of $X$ and $Y$, the positive ends of the edges toward the positive directions of the axes, and this may be done without giving the volume of the parallelepiped the value zero, and therefore without changing the sign of $H$. Now two cases are possible;—the positive end of the edge $dz'$ may be turned toward ​the positive or toward the negative direction of the axis of $Z$. In the first case, $H$ is evidently positive; in the second, negative. The determinant $H$ will therefore be positive or negative, we may say, if we choose, that the volume will be positive or negative, according as the element can or cannot be brought from the state $(x,y,z)$ to the state $(x',y',z')$ by continuous changes without giving its volume the value zero.
If we now recur to the consideration of the principal axes of strain and the principal ratios of elongation $r_{1}^{2},r_{2}^{2},r_{3}^{2}$ and denote by $U_{1},U_{2},U_{3}$ and $U_{1}',U_{2}',U_{3}'$ the principal axes of strain in the strained and unstrained element respectively, it is evident that the sign of $r_{1},$ for example, depends upon the direction in $U_{1}$ which we regard as corresponding to a given direction in $U_{1}'$. If we choose to associate directions in these axes so that $r_{1},r_{2},r_{3}$ shall all be positive, the positive or negative value of $H$ will determine whether the system of axes $U_{1},U_{2},U_{3}$ is or is not capable of superposition upon the system $U_{1}',U_{2}',U_{3}'$ so that corresponding directions in the axes shall coincide. Or, if we prefer to associate directions in the two systems of axes so that they shall be capable of superposition, corresponding directions coinciding, the positive or negative value of $H$ will determine whether an even or an odd number of the quantities $r_{1},r_{2},r_{3}$ are negative. In this case we may write
| $r_{1}r_{2}r_{3}=H={\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}&{\frac {dx}{dz'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}&{\frac {dy}{dz'}}\\{\frac {dz}{dx'}}&{\frac {dz}{dy'}}&{\frac {dz}{dz'}}\end{vmatrix}}.$ | (442) |
$r_{1},r_{2},r_{3}$
Whichever supposition we make with respect to the axes $U_{1},U_{2},U_{3}$, it is evident that the state of strain is completely determined by the values $E,F$, and $H$, not only when we limit ourselves to the consideration of such strains as are consistent with the idea of solidity, but also when we regard any values of ${\frac {dx}{dx'}},...{\frac {dz}{dz'}}$ as possible.
Approximative Formulæ.—For many purposes the value of $\epsilon _{V'}$ for an isotropic solid may be represented with sufficient accuracy by the formula
| $\epsilon _{V'}=i'+e'E'+f'F+h'H,$ | (443) |
$i',e',f',$
$h'$
$\eta _{V'}$
$\psi _{V'}$
| $\psi _{V'}=i+eE'+fF+hH,$ | (444) |
​where $i,e,f$, and $h$ denote functions of $t$. Let us first consider the second of these formulæ. Since $E,F$, and $H$ are symmetrical functions of $r_{1},r_{2},r_{3}$, if $\psi _{V'}$ is any function of $t,E,F,H$, we must have
| ${\frac {d\psi _{V'}}{dr_{1}}}={\frac {d\psi _{V'}}{dr_{2}}}={\frac {d\psi _{V'}}{dr_{3}}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (445) |
| ${\frac {d^{2}\psi _{V'}}{dr_{1}^{2}}}={\frac {d^{2}\psi _{V'}}{dr_{2}^{2}}}={\frac {d^{2}\psi _{V'}}{dr_{3}^{2}}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (445) |
| ${\frac {d^{2}\psi _{V'}}{dr_{1}dr_{2}}}={\frac {d^{2}\psi _{V'}}{dr_{2}dr_{3}}}={\frac {d^{2}\psi _{V'}}{dr_{3}dr_{1}}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (445) |
whenever $r_{1}=r_{2}=r_{3}$. Now $i,e,f$, and $h$ may be determined (as functions of $t$) so as to give to their proper values at every temperature for some isotropic state of strain, which may be determined by any desired condition. We shall suppose that they are determined so as to give the proper values to $\psi _{V'}$, etc., when the stresses in the solid vanish. If we denote by $r_{0}$ the common value of $r_{1},r_{2},r_{3}$ which will make the stresses vanish at any given temperature, and imagine the true value of $\psi _{V'}$, and also the value given by equation (444) to be expressed in terms of the ascending powers of
| $r_{1}-r_{0},$$r_{2}-r_{0},$$r_{3}-r_{0},$ | (446) |
$\psi _{V'}$
* ${\frac {d\psi _{V'}}{dr_{1}}},$${\frac {d\psi _{V'}}{dr_{2}}},$${\frac {d\psi _{V'}}{dr_{3}}},$
| ${\frac {d\psi _{V'}}{d{\frac {dx}{dx'}}}}={\frac {d\psi _{V'}}{dr_{1}}}{\frac {dr_{1}}{d{\frac {dx}{dx'}}}}+{\frac {d\psi _{V'}}{dr_{2}}}{\frac {dr_{2}}{d{\frac {dx}{dx'}}}}+{\frac {d\psi _{V'}}{dr_{3}}}{\frac {dr_{3}}{d{\frac {dx}{dx'}}}}$ | (447) |
$\psi _{V'}$
* ${\frac {dr_{1}}{d{\frac {dx}{dx'}}}},$${\frac {dr_{2}}{d{\frac {dx}{dx'}}}},$${\frac {dr_{3}}{d{\frac {dx}{dx'}}}}$
$X_{X'}$
$X_{Y'},Y_{Z'},Z_{X'}$
It will be interesting to see how the quantities $e,f,$ and $h$ are related to those which most simply represent the elastic properties of isotropic solids. If we denote by $V$ and $R$ the elasticity of volume and the rigidity (both determined under the condition of constant temperature and for states of vanishing stress), we shall have as definitions
| $V=-v\left({\frac {dp}{dv}}\right)_{t},$ when $v=r_{0}^{3}v',$ | (448) |
$p$
$v$
$v'$
| $r_{0}R={\frac {dX_{Y'}}{d{\frac {dx}{dy'}}}}={\frac {d^{2}\psi _{V'}}{\left(d{\frac {dx}{dy'}}\right)^{2}}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (449) |
| ${\frac {dx}{dx'}}={\frac {dx}{dy'}}={\frac {dz}{dz'}}=r_{0},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (449) |
| ${\frac {dx}{dy'}}={\frac {dx}{dz'}}={\frac {dy}{dz'}}={\frac {dy}{dx'}}={\frac {dz}{dx'}}={\frac {dz}{dy'}}=0.$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (449) |
Now when the solid is subject to uniform pressure on all sides, if we consider so much of it as has the volume unity in the state of reference, we shall have
| $r_{1}=r_{2}=r_{3}=v^{\tfrac {1}{3}},$ | (450) |
| $\psi _{V'}=i+3ev^{\tfrac {2}{3}}+3fv^{\tfrac {4}{3}}+hv.$ | (451) |
$\psi _{V'}$
$\psi$
| $-p=\left({\frac {d\psi }{dt}}\right)_{t}=2ev^{-{\tfrac {1}{3}}}+4fv^{\tfrac {1}{3}}+h,$ | (452) |
| $-v\left({\frac {dp}{dv}}\right)_{t}=-{\tfrac {2}{3}}ev^{-{\tfrac {1}{3}}}+{\tfrac {4}{3}}fv^{\tfrac {1}{3}};$ | (453) |
| $V=-{\tfrac {2}{3}}{\frac {e}{r_{0}}}+{\tfrac {4}{3}}fr_{0}.$ | (454) |
$R$
$E,F$
$H$
| $R={\frac {2e}{r_{0}}}+2fr_{0}.$ | (455) |
$p$
$v=r_{0}^{3}$
| $2e+4fr_{0}^{3}+hr_{0}=0.$ | (456) |
$e,f,h$
$r_{0},V$
$R$
| $e={\tfrac {1}{3}}r_{0}R-{\tfrac {1}{2}}r_{0}V,$$f={\frac {R+3V}{6r_{0}}}$$h=-{\tfrac {4}{3}}R-V.$ | (457) |
$r_{0}$
$R$
$V$
${\frac {d\log r_{0}}{dt}}$
It will not be necessary to discuss equation (443) at length, as the case is entirely analogous to that which has just been treated. (It must be remembered that $\eta _{V'}$, in the discussion of (443), will take the place everywhere of the temperature in the discussion of (444).) If we denote by $V'$ and $R'$ the elasticity of volume and the rigidity, both determined under the condition of constant entropy, (i.e., of no transmission of heat,) and for states of vanishing stress, we shall have the equations:—
| $V'=-{\frac {2e'}{3r_{0}}}+{\tfrac {4}{3}}f'r_{0},$ | (458) |
| $R'={\frac {2e'}{r_{0}}}+2f'r_{0},$ | (459) |
| $2e'+4f'r_{0}^{2}+h'r_{0}=0.$ | (460) |
| $e'={\tfrac {1}{3}}r_{0}R'-{\tfrac {1}{2}}r_{0}V',$$f'={\frac {R'+3V'}{6r_{0}}},$$h'=-{\tfrac {4}{3}}R'-V'.$ | (461) |
$r_{0},R'$
$V'$
$\eta _{V'}$
$\epsilon _{V'},\eta _{V'},\psi _{V'},H$
$a^{3}$
$E$
$a^{2}$
$F$
$a^{4}$
$a^{3}\psi _{V'},a^{2}E,a^{4}F,a^{3}H$
$\psi _{V'},E,F$
$H$
$a^{3}\eta _{V'}$
$\eta _{V'}$
$i',e',f'$
$h'$
$\eta _{V'}$
$\eta _{V'}$
Concerning Solids which absorb Fluids.
There are certain bodies which are solid with respect to some of their components, while they have other components which are fluid. In the following discussion, we shall suppose both the solidity and the fluidity to be perfect, so far as any properties are concerned which can affect the conditions of equilibrium,—i.e., we shall suppose that the solid matter of the body is entirely free from plasticity and that there are no passive resistances to the motion of the fluid components except such as vanish with the velocity of the motion,—leaving it to be determined by experiment how far and in what cases these suppositions are realized.
It is evident that equation (356) must hold true with regard to such a body, when the quantities of the fluid components contained in a given element of the solid remain constant. Let $\Gamma _{a}',\Gamma _{b}'$, etc., denote the quantities of the several fluid components contained in an element of the body divided by the volume of the element in the state of reference, or, in other words, let these symbols denote the densities which the several fluid components would have, if the body should be brought to the state of reference while the matter contained in each element remained unchanged. We may then say that equation (356) will hold true, when $\Gamma _{a}',\Gamma _{b}'$, etc., are constant. The complete value of the differential of $\epsilon _{V'}$ will therefore be given by an equation of the form
| $d\epsilon _{V'}=td\eta _{V'}+\textstyle \sum \sum '\displaystyle \left(X_{X'}d{\frac {dx}{dx'}}\right)+L_{a}d\Gamma _{a}'+L_{b}d\Gamma _{b}'+{\text{etc.}}$ | (462) |
$-pdv_{V'}$
$v_{V'}$
| $X_{X'}=-p\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right),$ | (463) |
* $\textstyle \sum \sum '\displaystyle \left(X_{X'}d{\frac {dx}{dx'}}\right)=-p\textstyle \sum \sum '\displaystyle \left\{\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)d{\frac {dx}{dx'}}\right\}\quad$
* $=-pd{\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}&{\frac {dx}{dz'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}&{\frac {dy}{dz'}}\\{\frac {dz}{dx'}}&{\frac {dz}{dy'}}&{\frac {dz}{dz'}}\end{vmatrix}}$
| $=-pdv_{V'}.$ | (464) |
​We have, therefore, for a state of hydrostatic stress,
| $d\epsilon _{V'}=td\eta _{V'}-pdv_{V'}+L_{a}d\Gamma _{a}'+L_{b}d\Gamma _{b}'+{\text{etc.,}}$ | (465) |
| $d\epsilon =td\eta -pdv+L_{a}dm_{a}+L_{b}dm_{b}+{\text{etc.,}}$ | (466) |
$\epsilon ,\eta ,v,m_{a},m_{b}$
But when the body is in a state of hydrostatic stress, and the solid matter is considered invariable, we have by equation (12)
| $d\epsilon =td\eta -pdv+\mu _{a}dm_{a}+\mu _{b}dm_{b}+{\text{etc.}}$ | (467) |
$L_{a},L_{b}$
$\mu _{a},\mu _{b}$
Since we have hitherto used the term potential solely with reference to bodies of hydrostatic stress, we may apply this term as we choose with regard to other bodies. We may therefore call the quantities $L_{a},L_{b}$, etc., the potentials for the several fluid components in the body considered, whether the state of the body is one of hydrostatic stress or not, since this use of the term involves only an extension of its former definition. It will also be convenient to use our ordinary symbol for a potential to represent these quantities. Equation (462) may then be written
| $d\epsilon _{V'}=td\eta _{V'}+\textstyle \sum \sum '\displaystyle \left(X_{X'}d{\frac {dx}{dx'}}\right)+\mu _{a}d\Gamma _{a}'+\mu _{b}d\Gamma _{b}'+{\text{etc.}}$ | (468) |
In regard to the conditions of equilibrium for a body of this kind, it is evident in the first place that if we make $\Gamma _{a}',\Gamma _{b}'$, etc., constant, we shall obtain from the general criterion of equilibrium ​all the conditions which we have obtained for ordinary solids, and which are expressed by the formulæ (364), (374), (380), (382)–(384). The quantities $\Gamma _{1}',\Gamma _{2}^{\prime }$, etc., in the last two formulæ include of course those which have just been represented by $\Gamma _{a}',\Gamma _{b}'$, etc., and which relate to the fluid components of the body, as well as the corresponding quantities relating to its solid components. Again, if we suppose the solid matter of the body to remain without variation in quantity or position, it will easily appear that the potentials for the substances which form the fluid components of the solid body must satisfy the same conditions in the solid body and in the fluids in contact with it, as in the case of entirely fluid masses. See eqs. (22).
The above conditions must however be slightly modified in order to make them sufficient for equilibrium. It is evident that if the solid is dissolved at its surface, the fluid components which are set free may be absorbed by the solid as well as by the fluid mass, and in like manner if the quantity of the solid is increased, the fluid components of the new portion may be taken from the previously existing solid mass. Hence, whenever the solid components of the solid body are actual components of the fluid mass, (whether the case is the same with the fluid components of the solid body or not,) an equation of the form (383) must be satisfied, in which the potentials $\mu _{a},\mu _{b}$, etc., contained implicitly in the second member of the equation are determined from the solid body. Also if the solid components of the solid body are all possible but not all actual components of the fluid mass, a condition of the form (384) must be satisfied, the values of the potentials in the second member being determined as in the preceding case.
The quantities
| $t,$$X_{X'},...Z_{Z'}$$\mu _{a}$$\mu _{b}$etc., | (469) |
$\epsilon _{V'}$
| $\eta _{V'}$${\frac {dx}{dx'}},...{\frac {dz}{dz'}},$$\Gamma _{a}',$$\Gamma _{b}',$etc., | (470) |
| ${\frac {dt}{d{\frac {dx}{dx'}}}}={\frac {dX_{X'}}{d\eta _{V'}}},$ etc. | (471) |
$\textstyle \sum \sum '$
$-\eta _{V'}dt$
$td\eta _{V'}$
$d(t\eta _{V'})$
$d$
| $\left({\frac {dX_{X'}}{d\Gamma _{a}'}}\right)_{\frac {dx}{dx'}}=\left({\frac {d\mu _{a}}{d{\frac {dx}{dx'}}}}\right)_{\Gamma _{a}'},$$\left({\frac {dX_{X'}}{d\mu _{a}}}\right)_{\frac {dx}{dx'}}=-\left({\frac {d\Gamma _{a}}{d{\frac {dx}{dx'}}}}\right)_{\mu _{a}},$ | (472) |
| $\left({\frac {d{\frac {dx}{dx'}}}{d\mu _{a}}}\right)_{X_{X'}}=\left({\frac {d\Gamma _{a}'}{dX_{X'}}}\right)_{\mu _{a}},$$\left({\frac {d{\frac {dx}{dx'}}}{d\Gamma _{a}'}}\right)_{X_{X'}}=-\left({\frac {d\mu _{a}}{dX_{X'}}}\right)_{\Gamma _{a}'},$ | (473) |
$t$
$\eta _{V'}$
$X_{Y'}$
${\frac {dx}{dy'}},...$
$Z_{Z'}$
${\frac {dz}{dz'}}$
$\mu _{b}$
$\Gamma _{b}'$
It will be observed that when the temperature is constant the conditions $\mu _{a}={\text{const.}},\mu _{b}={\text{const.}}$, represent the physical condition of a body in contact with a fluid of which the phase does not vary, and which contains the components to which the potentials relate. Also that when $\Gamma _{a}',\Gamma _{b}'$, etc., are constant, the heat absorbed by the body in any infinitesimal change of condition per unit of volume measured in the state of reference is represented by $td\eta _{V'}$. If we denote this quantity by $dQ_{V'}$, and use the suffix $_{Q}$ to denote the condition of no transmission of heat, we may write
| $\left({\frac {d\log t}{d{\frac {dx}{dx'}}}}\right)_{Q}=\left({\frac {dX_{X'}}{dQ_{V'}}}\right)_{\frac {dx}{dx'}},$$\left({\frac {d\log t}{dX_{X'}}}\right)_{Q}=-\left({\frac {d{\frac {dx}{dx'}}}{dQ_{V'}}}\right)_{X_{X'}},$ | (474) |
| $\left({\frac {dQ_{V'}}{dX_{X'}}}\right)_{t}=\left({\frac {d{\frac {dx}{dx'}}}{d\log t}}\right)_{X_{X'}},$$\left({\frac {dQ_{V'}}{d{\frac {dx}{dx'}}}}\right)_{t}=-\left({\frac {dX_{X'}}{d\log t}}\right)_{\frac {dx}{dx'}},$ | (475) |
$\Gamma _{a}',\Gamma _{b}'$
$X{Y'}$
${\frac {dx}{dy'}},...$
$Z_{Z'}$
${\frac {dz}{dz'}}$
Influence of Surfaces of Discontinuity upon the Equilibrium of Heterogeneous Masses.—Theory of Capillarity.
We have hitherto supposed, in treating of heterogeneous masses in contact, that they might be considered as separated by mathematical surfaces, each mass being unaffected by the vicinity of the others, so that it might be homogeneous quite up to the separating surfaces both with respect to the density of each of its various components and also with respect to the densities of energy and entropy. That such is not rigorously the case is evident from the consideration that if it were so with respect to the densities of the components it could not be so in general with respect to the density of energy, as the sphere of molecular action is not infinitely small. But we know from observation that it is only within very small distances of such a surface that any mass is sensibly affected by its vicinity,—a natural consequence of the exceedingly small sphere of sensible molecular action,—and this fact renders possible a simple method of taking account of the variations in the densities of the component substances and of energy and entropy, which occur in the vicinity of surfaces of discontinuity. We may use this term, for the sake of brevity, without implying that the discontinuity is absolute, or that the term distinguishes any surface with mathematical precision. It may be taken to denote the non-homogeneous film which separates homogeneous or nearly homogeneous masses.
Let us consider such a surface of discontinuity in a fluid mass which is in equilibrium and uninfluenced by gravity. For the precise measurement of the quantities with which we have to do, it will be convenient to be able to refer to a geometrical surface, which shall be sensibly coincident with the physical surface of discontinuity, but shall have a precisely determined position. For this end, let us take some point in or very near to the physical surface of discontinuity, and imagine a geometrical surface to pass through this point and all other points which are similarly situated with respect to the condition of the adjacent matter. Let this geometrical surface be called the dividing surface, and designated by the symbol $S$. It will be observed that the position of this surface is as yet to a certain extent arbitrary, but that the directions of its normals are already everywhere determined, since all the surfaces which can be formed in the manner described are evidently parallel to one another. Let us also imagine a closed surface cutting the surface $S$ and including a part of the homogeneous mass on each side. We will so far limit the form of this closed surface as to suppose that on each side of $S$, as far as there is any want of perfect homogeneity in the fluid masses, the closed surface is such as may be generated by a moving normal to $S$. ​Let the portion of $S$ which is included by the closed surface be denoted by $S$, and the area of this portion by a. Moreover, let the mass contained within the closed surface be divided into three parts by two surfaces, one on each side of $S$, and very near to that surface, although at such distance as to lie entirely beyond the influence of the discontinuity in its vicinity. Let us call the part which contains the surface $S$ (with the physical surface of discontinuity) $M$, and the homogeneous parts $M'$ and $M''$, and distinguish by $\epsilon ,\epsilon ',\epsilon '',\eta ,\eta ',\eta '',m_{1},m_{1}',m_{1}'',m_{2},m_{2}',m_{2}''$, etc., the energies and entropies of these masses, and the quantities which they contain of their various components.
It is necessary, however, to define more precisely what is to be understood in cases like the present by the energy of masses which are only separated from other masses by imaginary surfaces. A part of the total energy which belongs to the matter in the vicinity of the separating surface, relates to pairs of particles which are on different sides of the surface, and such energy is not in the nature of things referable to either mass by itself. Yet, to avoid the necessity of taking separate account of such energy, it will often be convenient to include it in the energies which we refer to the separate masses. When there is no break in the homogeneity at the surface, it is natural to treat the energy as distributed with a uniform density. This is essentially the case with the initial state of the system which we are considering, for it has been divided by surfaces passing in general through homogeneous masses. The only exception—that of the surface which cuts at right angles the non-homogeneous film—(apart from the consideration that without any important loss of generality we may regard the part of this surface within the film as very small compared with the other surfaces) is rather apparent than real, as there is no change in the state of the matter in the direction perpendicular to this surface. But in the variations to be considered in the state of the system, it will not be convenient to limit ourselves to such as do not create any discontinuity at the surfaces bounding the masses $M,M',M''$; we must therefore determine how we will estimate the energies of the masses in case of such infinitesimal discontinuities as may be supposed to arise. Now the energy of each mass will be most easily estimated by neglecting the discontinuity, i.e., if we .estimate the energy on the supposition that beyond the bounding surface the phase is identical with that within the surface. This will evidently be allowable, if it does not affect the total amount of energy. To show that it does not affect this quantity, we have only to observe that, if the energy of the mass on one side of a surface where there is an infinitesimal discontinuity of phase is greater as determined by this rule than if determined by ​any other (suitable) rule, the energy of the mass on the other side must be less by the same amount when determined by the first rule than when determined by the second, since the discontinuity relative to the second mass is equal but opposite in character to the discontinuity relative to the first.
If the entropy of the mass which occupies any one of the spaces considered is not in the nature of things determined without reference to the surrounding masses, we may suppose a similar method to be applied to the estimation of entropy.
With this understanding, let us return to the consideration of the equilibrium of the three masses $M,M'$, and $M''$. We shall suppose that there are no limitations to the possible variations of the system due to any want of perfect mobility of the components by means of which we express the composition of the masses, and that these components are independent, i.e., that no one of them can be formed out of the others.
With regard to the mass $M$, which includes the surface of discontinuity, it is necessary for its internal equilibrium that when its boundaries are considered constant, and when we consider only reversible variations (i.e., those of which the opposite are also possible), the variation of its energy should vanish with the variations of its entropy and of the quantities of its various components. For changes within this mass will not affect the energy or the entropy of the surrounding masses (when these quantities are estimated on the principle which we have adopted), and it may therefore be treated as an isolated system. For fixed boundaries of the mass $M$, and for reversible variations, we may therefore write
| $\delta \epsilon =A_{0}\delta \eta +A_{1}\delta m_{1}+A_{2}\delta m_{2}+{\text{etc.,}}$ | (476) |
$A_{0},A_{1},A_{2}$
$A_{0}$
$A_{1},A_{2}$
$A_{1},A_{2}$
| $\delta \epsilon =t\delta \eta +\mu _{1}\delta m_{1}+\mu _{2}\delta m_{2}+{\text{etc.}}$ | (477) |
​If we substitute $\geqq$ for $=$ in this equation, the formula will hold true of all variations whether reversible or not; for if the variation of energy could have a value less than that of the second member of the equation, there must be variation in the condition of $M$ in which its energy is diminished without change of its entropy or of the quantities of its various components.
It is important, however, to observe that for any given values of $\delta \eta ,\delta m_{1},\delta m_{2}$, etc., while there may be possible variations of the nature and state of $M$ for which the value of $\delta \epsilon$ is greater than that of the second member of (477), there must always be possible variations for which the value of $\delta \epsilon$ is equal to that of the second member. It will be convenient to have a notation which will enable us to express this by an equation. Let ${\mathfrak {d}}\epsilon$ denote the smallest value (i.e., the value nearest to $\infty$) of $\delta \epsilon$ consistent with given values of the other variations, then
| ${\mathfrak {d}}\epsilon =t\delta \eta +\mu _{1}\delta m_{1}+\mu _{2}m_{2}+{\text{etc.}}$ | (478) |
$M,M',M''$
| $\delta \epsilon +\delta \epsilon '+\delta \epsilon ''\geqq 0$ | (479) |
$M,M'$
$M''$
* $t\delta \eta +\mu _{1}\delta m_{1}+\mu _{2}\delta m_{2}+{\text{etc.}}$
* $+t'\delta \eta '+\mu _{1}'\delta m_{1}'+\mu _{2}'\delta m_{2}'+{\text{etc.}}$
| $+t''\delta \eta ''+\mu _{1}''\delta m_{1}''+\mu _{2}''\delta m_{2}''+{\text{etc.}}\geqq 0,$ | (480) |
​the variations being subject to the equations of condition
| $\delta \eta +\delta \eta '+\delta \eta ''=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,$ (481) |
| $\delta m_{1}+\delta m_{1}'+\delta m_{1}''=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,$ (481) |
| $\delta m_{2}+\delta m_{2}'+\delta m_{2}''=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,$ (481) |
| etc. | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,$ (481) |
It may also be the case that some of the quantities | $\delta m_{1}',\delta m_{1}'',\delta m_{2}',\delta m_{2}''$, etc., are incapable of negative values or can only have the value zero. This will be the case when the substances to which these quantities relate are not actual or possible components of $M'$ or $M''$. (See page 64.) To satisfy the above condition it is necessary and sufficient that
| $t=t'=t'',$ | (482) |
| $\mu _{1}'\delta m_{1}'\geqq \mu _{1}\delta m_{1}',$$\mu _{2}'\delta m_{2}'\geqq \mu _{2}\delta m_{2}',$etc., | (483) |
| $\mu _{1}''\delta m_{1}''\geqq \mu _{1}\delta m_{1}'',$$\mu _{2}''\delta m_{2}''\geqq \mu _{2}\delta m_{2}'',$etc. | (484) |
$\mu _{1}$
$\mu _{1}=\mu _{1}'=\mu _{1}''.$
$\mu _{1}=\mu _{1}'.$
$\mu _{1}\geqq \mu _{1}''.$
$\mu _{1}$
It appears, therefore, that the particular conditions of equilibrium relating to temperature and the potentials which we have before obtained by neglecting the influence of the surfaces of discontinuity (pp. 65, 66, 74) are not invalidated by the influence of such discontinuity in their application to homogeneous parts of the system bounded like $M'$ and $M''$ by imaginary surfaces lying within the limits of homogeneity,—a condition which may be fulfilled by surfaces very near to the surfaces of discontinuity. It appears also that similar conditions will apply to the non-homogeneous films like $M$, which separate such homogeneous masses. The properties of such films, which are of course different from those of homogeneous masses, require our farther attention.
The volume occupied by the mass $M$ is divided by the surface ${\mathsf {s}}$ into two parts which we will call $v'''$ and $v'''',v'''$ lying next to $M'$, and $v''''$ to $M''$. Let us imagine these volumes filled by masses having throughout the same temperature, pressure and potentials, and the same densities of energy and entropy, and of the various components, ​as the masses $M'$ and $M''$ respectively. We shall then have, by equation (12), if we regard the volumes as constant,
| $\delta \epsilon '''=t'\delta \eta '''+\mu _{1}'\delta m_{1}'''+\mu _{2}'\delta m_{2}'''+{\text{etc.,}}$ | (485) |
| $\delta \epsilon ''''=t''\delta \eta ''''+\mu _{1}''\delta m_{1}''''+\mu _{2}''\delta m_{2}''''+{\text{etc.;}}$ | (486) |
| $\delta \epsilon '''=t\delta \eta '''+\mu _{1}\delta m_{1}'''+\mu _{2}\delta m_{2}'''+{\text{etc.,}}$ | (487) |
| $\delta \epsilon ''''=t\delta \eta ''''+\mu _{1}\delta m_{1}''''+\mu _{2}\delta m_{2}''''+{\text{etc.}}$ | (488) |
| $\delta (\epsilon -\epsilon '''-\epsilon '''')=t\delta (\eta -\eta '''-\eta '''')+\mu _{1}\delta (m_{1}-m_{1}'''-m_{1}'''')+\mu _{2}\delta (m_{2}-m_{2}'''-m_{2}'''')+{\text{etc.}}$ | (489) |
| $\epsilon ^{S}=\epsilon -\epsilon '''-\epsilon '''',$$\eta ^{S}=\eta -\eta '''-\eta ''''$ | (490) |
| $m_{1}^{S}=m_{1}-m_{1}'''-m_{1}'''',$$m_{2}^{S}=m_{2}-m_{2}'''-m_{2}'''',$ etc., | (491) |
| $\delta \epsilon ^{S}=t\delta \eta ^{S}+\mu _{1}\delta m_{1}^{S}+\mu _{2}\delta m_{2}^{S}+{\text{etc.}}$ | (492) |
$\epsilon ^{S}$
$S$
$\eta ^{S},m_{1}^{S},m_{2}^{S}$
$\epsilon ^{S}$
$\eta ^{S}$
${\frac {\epsilon ^{S}}{s}}$
${\frac {\eta ^{S}}{s}}$
${\frac {m_{1}^{S}}{s}},{\frac {m_{2}^{S}}{s}}$
Now these quantities ($\epsilon ^{S},\eta ^{S},m_{1}^{S}$, etc.) are determined partly by the state of the physical system which we are considering, and partly by the various imaginary surfaces by means of which these quantities have been defined. The position of these surfaces, it will be remembered, has been regarded as fixed in the variation of the system. It is evident, however, that the form of that portion of these surfaces which lies in the region of homogeneity on either side of the surface of discontinuity cannot affect the values of these quantities. To obtain the complete value of $\delta \epsilon ^{S}$ for reversible variations, we have ​therefore only to regard variations in the position and form of the limited surface ${\mathsf {s}}$, as this determines all of the surfaces in question lying within the region of non-homogeneity. Let us first suppose the form of ${\mathsf {s}}$ to remain unvaried and only its position in space to vary, either by translation or rotation. No change in (492) will be necessary to make it valid in this case. For the equation is valid if ${\mathsf {s}}$ remains fixed and the material system is varied in position; also, if the material system and ${\mathsf {s}}$ are both varied in position, while their relative position remains unchanged. Therefore, it will be valid if the surface alone varies its position.
But if the form of ${\mathsf {s}}$ be varied, we must add to the second member of (492) terms which shall represent the value of
* $\delta \epsilon ^{S}-t\delta \eta ^{S}-\mu _{1}\delta m_{1}^{S}-\mu _{2}\delta m_{2}^{S}-{\text{etc.}}$
${\mathsf {s}}$
${\mathsf {s}}$
$\delta s$
$\delta c_{1}$
$\delta c_{2}$
| ${\begin{alignedat}{2}\delta \epsilon ^{S}&=t\delta \eta ^{S}+\mu _{1}\delta m_{1}^{S}+\mu _{2}\delta m_{2}^{S}+{\text{etc.}}\\&+\sigma \delta s+C_{1}\delta c_{1}+C_{2}\delta c_{2},\\\end{alignedat}}$ | (493) |
| ${\begin{alignedat}{2}\delta \epsilon ^{S}&=t\delta \eta ^{S}+\mu _{1}\delta m_{1}^{S}+\mu _{2}\delta m_{2}^{S}+{\text{etc.}}\\&+\sigma \delta s+{\tfrac {1}{2}}(C_{1}+C_{2})\delta (c_{1}+c_{2})+{\tfrac {1}{2}}(C_{1}-C_{2})\delta (c_{1}-c_{2}),\\\end{alignedat}}$ | (494) |
$\sigma ,C_{1},$
$C_{2}$
${\mathsf {s}}$
$\epsilon ^{S}$
${\mathsf {s}}$
$C_{1}+C_{2}$
To show this, it will be convenient to write the equation in the longer form {see (490), (491)}
| $\delta \epsilon -t\delta \eta ^{S}-\mu _{1}\delta m_{1}^{S}-\mu _{2}\delta m_{2}^{S}-{\text{etc.}}$ |
| $\delta \epsilon '''-t\delta \eta '''-\mu _{1}\delta m_{1}'''-\mu _{2}\delta m_{2}'''-{\text{etc.}}$ |
| $\delta \epsilon ''''-t\delta \eta ''''-\mu _{1}\delta m_{1}''''-\mu _{2}\delta m_{2}''''-{\text{etc.}}$ |
| $=\sigma \delta s+{\tfrac {1}{2}}(C_{1}+C_{2})\delta (c_{1}+c_{2})+{\tfrac {1}{2}}(C_{1}-C_{2})\delta (c_{1}-c_{2}),$ | (495) |
i.e., by (482)–(484) and (12),
| ${\begin{alignedat}{2}\delta \epsilon -t\delta \eta &-\mu _{1}\delta m_{1}-\mu _{2}\delta m_{2}+{\text{etc.}}+p'\delta v'''+p''\delta v''''\\&=\sigma \delta s+{\tfrac {1}{2}}(C_{1}+C_{2})\delta (c_{1}+c_{2})+{\tfrac {1}{2}}(C_{1}-C_{2})\delta (c_{1}-c_{2}).\\\end{alignedat}}$ | (496) |
${\mathsf {s}}$
$M$
* $\delta \epsilon -t\delta \eta -\mu _{1}\delta m_{1}-\mu _{2}\delta m_{2}-{\text{etc.}}=0.$
$\delta s=0,\delta (c_{1}+c_{2})=0,\delta (c_{1}-c_{2})=0,$
$\delta v'''=-\delta v''''.$
$p'=p''$
Let us now examine the effect of different positions of the surface ${\mathsf {s}}$ in the same material system upon the value of $C_{1}+C_{2}$, supposing at first that in the initial state of the system the surface of discontinuity is plane. Let us give the surface ${\mathsf {s}}$ some particular position. In the initial state of the system this surface will of course be plane like the physical surface of discontinuity, to which it is parallel. In the varied state of the system, let it become a portion of a spherical surface having positive curvature; and at sensible distances from this surface let the matter be homogeneous and with the same phases as in the initial state of the system; also at and about the surface let the state of the matter so far as possible be the same as at and about the plane surface in the initial state of the system. (Such a variation in the system may evidently take place negatively as well as positively, as the surface may be curved toward either side. But whether such a variation is consistent with the maintenance of equilibrium is of no consequence, since in the preceding equations only the initial state is supposed to be one of equilibrium.) Let the surface ${\mathsf {s}}$, placed as supposed, whether in the initial or the varied state of the surface, be distinguished by the symbol ${\mathsf {s}}'$. Without changing either the initial or the varied state of the material system, let us make another supposition with respect to the imaginary surface ${\mathsf {s}}$. In the unvaried system let it be parallel to its former position but removed from it a distance $\lambda$ on the side on which lie the centers of positive curvature. In the varied state of the system, let it be spherical and concentric with ${\mathsf {s}}'$, and separated from it by the same distance $\lambda$. It will of course lie on the same side of ${\mathsf {s}}'$ as in the unvaried system. Let the surface ${\mathsf {s}}$, placed in accordance with this second supposition, be distinguished by the symbol ${\mathsf {s}}''$. Both in the initial and the varied state, let the perimeters of ${\mathsf {s}}'$ and ${\mathsf {s}}''$ be traced by a common normal. Now the value of
* $\delta \epsilon -t\delta \eta -\mu _{1}\delta m_{1}-\mu _{2}\delta m_{2}-{\text{etc.}}$
${\mathsf {s}}$
$M$
$p'\delta v''',p''\delta v''''$
$p'\delta (v'''+v''''),v'''+v''''$
$M$
${\mathsf {s}}'$
${\mathsf {s}}''$
$\delta (c_{1}-c_{2})=0$
${\mathsf {s}}'$
${\mathsf {s}}''$
${\mathsf {s}}'$
${\mathsf {s}}''$
* $\sigma '\delta s'+{\tfrac {1}{2}}(C_{1}'+C_{2}')\delta (c_{1}'+c_{2}')=\sigma ''+\delta s''{\tfrac {1}{2}}(C_{1}''+C_{2}'')\delta (c_{1}''+c_{2}'').$
* Now if we make$\delta s''=0,$
* $\delta s'=s\lambda \delta (c_{1}''+c_{2}'').$
* $\sigma 's'\lambda \delta (c_{1}''+c_{2}'')+{\tfrac {1}{2}}(C_{1}'+C_{2}')\delta (c_{1}'+c_{2}')={\tfrac {1}{2}}(C_{1}''+C_{2}'')\delta (c_{1}''+c_{2}'').$
* But$\delta (c_{1}'+c_{2}')=\delta (c_{1}''+c_{2}'').$
* Therefore,$C_{1}'+C_{2}'+2\sigma 's\lambda =C_{1}''+C_{2}''.$
$C_{1}''+C_{2}''$
${\mathsf {s}}''$
${\mathsf {s}}'$
${\mathsf {s}}$
$C_{1}+C_{2}$
Now we may easily convince ourselves by equation (493) that if ${\mathsf {s}}$ is placed within the non-homogeneous film, and $s=1$, the quantity $\sigma$ is of the same order of magnitude as the values of $\epsilon ^{S},\eta ^{S},m_{1}^{S},m_{2}^{S}$, etc., while the values of $C_{1}$ and $C_{2}$ are of the same order of magnitude as the changes in the values of the former quantities caused by increasing the curvature of ${\mathsf {s}}$ by unity. Hence, on account of the thinness of the non-homogeneous film, since it can be very little affected by such a change of curvature in ${\mathsf {s}}$, the values of $C_{1}$ and $C_{2}$ must in general be very small relatively to $\sigma$. And hence, if ${\mathsf {s}}'$ be placed within the non-homogeneous film, the value of $\lambda$ which will make $C_{1}''+C_{2}''$ vanish must be very small (of the same order of magnitude as the thickness of the non-homogeneous film). The position of ${\mathsf {s}}$, therefore, which will make $C_{1}+C_{2}$ in (494) vanish, will in general be sensibly coincident with the physical surface of discontinuity.
We shall hereafter suppose, when the contrary is not distinctly indicated, that the surface ${\mathsf {s}}$, in the unvaried state of the system, has such a position as to make $C_{1}+C_{2}=0$. It will be remembered that the surface s is a part of a larger surface $S$, which we have called the dividing surface, and which is coextensive with the physical surface of discontinuity. We may suppose that the position of the dividing surface is everywhere determined by similar considerations. This ​is evidently consistent with the suppositions made on page 219 with regard to this surface.
We may therefore cancel the term
* ${\tfrac {1}{2}}(C_{1}+C_{2})\delta (c_{1}+c_{2})$
$C_{1}$
$C_{2}$
$c_{1}=c_{2}$
* ${\tfrac {1}{2}}(C_{1}-C_{2})\delta (c_{1}-c_{2}).$
| $\delta \epsilon ^{S}=t\delta \eta ^{S}+\sigma \delta s+\mu _{1}\delta m_{1}^{S}+\mu _{2}\delta m_{2}^{S}+{\text{etc.}}$ | (497) |
$\delta \epsilon ^{S}$
The above equation is of fundamental importance in the theory of capillarity. It expresses a relation with regard to surfaces of discontinuity analogous to that expressed by equation (12) with regard to homogeneous masses. From the two equations may be directly deduced the conditions of equilibrium of heterogeneous masses in contact, subject or not to the action of gravity, without disregard of the influence of the surfaces of discontinuity. The general problem, including the action of gravity, we shall take up hereafter; at present we shall only consider, as hitherto, a small part of a surface of discontinuity with a part of the homogeneous mass on either side, in order to deduce the additional condition which may be found when we take account of the motion of the dividing surface.
We suppose as before that the mass especially considered is bounded by a surface of which all that lies in the region of non-homogeneity is such as may be traced by a moving normal to the dividing surface. But instead of dividing the mass as before into four parts, it will be sufficient to regard it as divided into two parts by the dividing surface. The energy, entropy, etc., of these parts, estimated on the supposition that its nature (including density of energy, etc.) is uniform quite up to the dividing surface, will be denoted by $\epsilon ',\eta '$, etc., $\epsilon '',\eta ''$, etc. Then the total energy will be $\epsilon ^{S}+\epsilon '+\epsilon ''$, and the general condition of internal equilibrium will be that
| $\delta \epsilon ^{S}+\delta \epsilon '+\delta \epsilon ''\geqq 0,$ | (498) |
​when the bounding surface is fixed, and the total entropy and total quantities of the various components are constant. We may suppose $\eta ^{S},\eta ',\eta '',m_{1}^{S},m_{1}',m_{1}'',m_{2}^{S},m_{2}',m_{2}'',$ etc., to be all constant. Then by (497) and (12) the condition reduces to
| $\sigma \delta s-p'\delta v'-p''\delta v''=0.$ | (499) |
$\geqq$
$\sigma$
$\sigma$
$\delta N$
* $\delta s=(c_{1}+c_{2})s\delta N,$$\delta v'=s\delta N,$$\delta v''=-s\delta N;$
$\sigma (c_{1}+c_{2})=p'-p'',$
$p'$
Fundamental Equations for Surfaces of Discontinuity between Fluid Masses.
In equation (497) the initial state of the system is supposed to be one of equilibrium. The only limitation with respect to the varied state is that the variation shall be reversible, i.e., that an opposite variation shall be possible. Let us now confine our attention to variations in which the system remains in equilibrium. To distinguish this case, we may use the character $d$ instead of $\delta$, and write
| $d\epsilon ^{S}=td\eta ^{S}+\sigma ds+\mu _{1}dm_{1}^{S}+\mu _{2}dm_{2}^{S}+{\text{etc.}}$ | (501) |
If we integrate this equation, supposing the area s to increase from zero to any finite value s, while the material system to a part of which the equation relates remains without change, we obtain
| $\epsilon ^{S}=t\eta ^{S}+\sigma s+\mu _{1}m_{1}^{S}+\mu _{2}m_{2}^{S}+{\text{etc,}}$ | (502) |
​which may be applied to any portion of any surface of discontinuity (in equilibrium) which is of the same nature throughout, or throughout which the values of $t,\sigma ,\mu _{1},\mu _{2}$, etc., are constant.
If we differentiate this equation, regarding all the quantities as variable, and compare the result with (501), we obtain
| $\eta ^{S}dt+sd\sigma +m_{1}^{S}d\mu _{1}+m_{2}^{S}d\mu _{2}+{\text{etc.}}=0.$ | (503) |
$\epsilon _{S},\eta _{S},\Gamma _{1},\Gamma _{2}$
| $\epsilon _{S}={\frac {\epsilon ^{S}}{s}},$$\eta _{s}={\frac {\eta ^{S}}{s}},$ | (504) |
| $\Gamma _{1}={\frac {m_{1}^{S}}{s}},$$\Gamma _{2}={\frac {m_{2}^{S}}{s}},$ etc., | (505) |
| $d\epsilon _{S}=td\eta _{S}+\mu _{1}d\Gamma _{1}+\mu _{2}d\Gamma _{2}+{\text{etc.,}}$ | (506) |
| $\epsilon _{S}=t\eta _{S}+\mu _{1}\Gamma _{1}+\mu _{2}\Gamma _{2}+{\text{etc.,}}$ | (507) |
| $d\sigma =-\eta _{S}dt-\Gamma _{1}d\mu _{1}-\Gamma _{2}d\mu _{2}-{\text{etc.}}$ | (508) |
$\mu _{1},\mu _{2},$
$t,\mu _{1},\mu _{2},$
$\sigma$
$n+1$
$n$
$\eta _{S},\Gamma _{1},\Gamma _{2}$
$n+3$
$2n+4$
$n+1$
$n+3$
$2n+5$
$n+2$
​An equation, therefore, between
| $\sigma ,t,\mu _{1},\mu _{2},{\text{etc.,}}$ | (509) |
| $\epsilon ^{S},\eta ^{S},s,m_{1}^{S},m_{2}^{S},{\text{etc}}.,$ | (510) |
$\epsilon _{S},\eta _{s},\Gamma _{1},\Gamma _{2},{\text{etc.}}$
$n+2$
$n+1$
$n+1$
$n+2$
$2n+5$
The same designation may be applied to any equation from which, by differentiation and the aid only of general principles and relations, $n+3$ independent relations between the same $2n+5$ quantities may be obtained.
$\sigma ^{S}=\epsilon ^{S}-t\eta ^{S},$
| $d\psi ^{S}=-\eta dt+\sigma ds+\mu _{1}dm_{1}^{S}+\mu _{2}dm_{2}^{S}+{\text{etc.}}$ | (513) |
$\psi ^{S},t,s,m_{1}^{S},m_{2}^{S}$
The reader will not fail to notice the analogy between these fundamental equations, which relate to surfaces of discontinuity, and those relating to homogeneous masses, which have been described on pages 85–89.
On the Experimental Determination of Fundamental Equations for Surfaces of Discontinuity between Fluid Masses.
When all the substances which are found at a surface of discontinuity are components of one or the other of the homogeneous masses, the potentials $\mu _{1},\mu _{2}$, etc., as well as the temperature, may be determined from these homogeneous masses. The tension $\sigma$ may ​be determined by means of the relation (500). But our measurements are practically confined to cases in which the difference of the pressures in the homogeneous masses is small; for with increasing differences of pressure the radii of curvature soon become too small for measurement. Therefore, although the equation $p'=p''$ (which is equivalent to an equation between $t,\mu _{1},\mu _{2}$, etc., since $p'$ and $p''$ are both functions of these variables) may not be exactly satisfied in cases in which it is convenient to measure the tension, yet this equation is so nearly satisfied in all the measurements of tension which we can make, that we must regard such measurements as simply establishing the values of $\sigma$ for values of $t,\mu _{1},\mu _{2}$, etc., which satisfy the equation $p'=p''$, but not as sufficient to establish the rate of change in the value of $\sigma$ for variations of $t,\mu _{1},\mu _{2}$, etc., which are inconsistent with the equation $p'=p''$.
To show this more distinctly, let $t,\mu _{2},m_{3}$, etc., remain constant, then by (508) and (98)
* $d\sigma =-\Gamma _{1}d\mu _{1},$ $dp'=\gamma _{1}'d\mu _{1},$ $dp''=\gamma _{1}''d\mu _{1},$
$\gamma _{1}'$
$\gamma _{1}''$
${\frac {m_{1}'}{v'}}$
${\frac {m_{1}''}{v''}}\cdot$
* $dp'-dp''=(\gamma _{1}'-\gamma _{1}'')d\mu _{1},$
$\Gamma _{1}d(p'-p'')=(\gamma _{1}''-\gamma _{1}')d\sigma .$
* $(c_{1}+c_{2})d\sigma +\sigma d(c_{1}+c_{2})=d(p'-p'').$
* $\Gamma _{1}(c_{1}+c_{2})d\sigma +\Gamma _{1}\sigma d(c_{1}+c_{2})=d(p'-p'').$
${\gamma _{1}''-\gamma _{1}'-\Gamma _{1}(c_{1}+c_{2})}d\sigma =\Gamma _{1}\sigma d(c_{1}+c_{2}).$
$\Gamma _{1}(c_{1}+c_{2})$
$\gamma _{1}''-\gamma _{1}'.$
* ${\frac {d\sigma }{\sigma }}={\frac {\Gamma _{1}}{\gamma _{1}''-\gamma _{1}'}}d(c_{1}+c_{2}).$
$\Gamma _{1},\gamma _{1}',\gamma _{1}''$
* $\log {\frac {\sigma }{\sigma '}}={\frac {\Gamma _{1}}{\gamma _{1}''-\gamma _{1}'}}(c_{1}+c_{2}),$
$\sigma '$
$\sigma$
$\sigma$
Accordingly, we cannot in general expect to determine the superficial density $\Gamma _{1}$ from its value $-\left({\frac {d\sigma }{d\mu _{1}}}\right)_{t,\mu }$ by measurements of superficial tensions. The case will be the same with $\Gamma _{2},\Gamma _{3}$, etc., and also with $\eta _{S}$, the superficial density of entropy.
The quantities $\epsilon _{S},\eta _{S},\Gamma _{1},\Gamma _{2}$, etc., are evidently too small in general to admit of direct measurement. When one of the components, however, is found only at the surface of discontinuity, it may be more easy to measure its superficial density than its potential. But except in this case, which is of secondary interest, it will generally be easy to determine $\sigma$ in terms of $t,\mu _{1},\mu _{2}$, etc., with considerable accuracy for plane surfaces, and extremely difficult or impossible to determine the fundamental equation more completely.
Fundamental Equations for Plane Surfaces of Discontinuity between Fluid Masses.
An equation giving $\sigma$ in terms of $t,\mu _{1},\mu _{2}$, etc., which will hold true only so long as the surface of discontinuity is plane, may be called a fundamental equation for a plane surface of discontinuity.
It will be interesting to see precisely what results can be obtained from such an equation, especially with respect to the energy and entropy and the quantities of the component substances in the vicinity of the surface of discontinuity.
These results can be exhibited in a more simple form, if we deviate to a certain extent from the method which we have been following. The particular position adopted for the dividing surface (which determines the superficial densities) was chosen in order to make the term ${\tfrac {1}{2}}(C_{1}+C_{2})\delta (c_{1}+c_{2})$ in (494) vanish. But when the curvature of the surface is not supposed to vary, such a position of the dividing surface is not necessary for the simplification of the formula. It is evident that equation (501) will hold true for plane surfaces (supposed to remain such) without reference to the position of the dividing surface, except that it shall be parallel to the surface of discontinuity. We are therefore at liberty to choose such a position for the dividing surface as may for any purpose be convenient.
None of the equations (502)–(513), which are either derived from (501), or serve to define new symbols, will be affected by such a ​change in the position of the dividing surface. But the expressions $\epsilon ^{S},\eta ^{S},m_{1}^{S},m_{2}^{S}$, etc., as also $\epsilon _{S},\eta _{S},\Gamma _{1},\Gamma _{2}$, etc., and $\psi ^{S}$, will of course have different values when the position of that surface is changed. The quantity $\sigma$, however, which we may regard as defined by equations (501), or, if we choose, by (502) or (507), will not be affected in value by such a change. For if the dividing surface be moved a distance $\lambda$ measured normally and toward the side to which $v''$ relates, the quantities
* $\epsilon _{S},$$\eta _{S},$$\Gamma _{1},$$\Gamma _{2},$etc.,
* $\lambda (\epsilon _{V}''-\epsilon _{V}'),$$\lambda (\eta _{V}''-\eta _{V}'),$$\lambda (\gamma _{1}''-\gamma _{1}'),$$\lambda (\gamma _{2}''-\gamma _{2}'),$etc.,
$\epsilon _{V}',\epsilon _{V}'',\eta _{V}',\eta _{V}''$
$\sigma$
* $\lambda (\epsilon _{V}''-\epsilon _{V}')-t\lambda (\eta _{1}''-\eta _{1}')-\mu _{1}\lambda (\gamma _{1}''-\gamma _{1}')-\mu _{2}\lambda (\gamma _{2}''-\gamma _{2}')-{\text{etc.}}$
* $-p''=\epsilon _{V}''-t\eta _{V}''-\mu _{1}\gamma _{1}''-\mu _{2}\gamma _{2}''-{\text{etc.,}}$ $-p'=\epsilon _{V}'-t\eta _{V}'-\mu _{1}\gamma _{1}'-\mu _{2}\gamma _{2}'-{\text{etc.}}$
$p'=p''$
$\sigma$
$\sigma$
It is generally possible to place the dividing surface so that the total quantity of any desired component in the vicinity of the surface of discontinuity shall be the same as if the density of that component were uniform on each side quite up to the dividing surface. In other words, we may place the dividing surface so as to make any one of the quantities $\Gamma _{1},\Gamma _{2}$, etc., vanish. The only exception is with regard to a component which has the same density in the two homogeneous masses. With regard to a component which has very nearly the same density in the two masses such a location of the dividing surface might be objectionable, as the dividing surface might fail to coincide sensibly with the physical surface of discontinuity. Let us suppose that $\gamma _{1}'$ is not equal (nor very nearly equal) to $\gamma _{1}''$, and that the dividing surface is so placed as to make $\Gamma _{1}=0$. Then equation (508) reduces to
| $d\sigma =-\eta _{S(1)}dt-\Gamma _{2(1)}d\mu _{2}-\Gamma _{3(1)}d\mu _{3}-{\text{etc.,}}$ | (514) |
​where the symbols $\eta _{S(1)},\Gamma _{2(1)}$, etc., are used for greater distinctness to denote the values of $\eta _{S},\Gamma _{2}$, etc., as determined by a dividing surface placed so that $\Gamma _{1}=0$. Now we may consider all the differentials in the second member of this equation as independent, without violating the condition that the surface shall remain plane, i.e., that $dp'=dp''$. This appears at once from the values of $dp'$ and $dp''$ given by equation (98). Moreover, as has already been observed, when the fundamental equations of the two homogeneous masses are known, the equation $p'=p''$ affords a relation between the quantities $t,\mu _{1},\mu _{2}$, etc. Hence, when the value of $\sigma$ is also known for plane surfaces in terms of $t,\mu _{1},\mu _{2}$, etc., we can eliminate $\mu _{1}$ from this expression by means of the relation derived from the equality of pressures, and obtain the value of $\sigma$ for plane surfaces in terms of $t,\mu _{2},\mu _{3}$, etc. From this, by differentiation, we may obtain directly the values of $\eta _{S(1)},\Gamma _{2(1)},\Gamma _{3(1)}$, etc., in terms of $t,\mu _{2},\mu _{3}$, etc. This would be a convenient form of the fundamental equation. But, if the elimination of $p',p''$, and $\mu _{1}$ from the finite equations presents algebraic difficulties, we can in all cases easily eliminate $dp',dp'',d\mu _{1}$ from the corresponding differential equations and thus obtain a differential equation from which the values of $\eta _{S(1)},\Gamma _{2(1)},\Gamma _{3(1)}$, etc., in terms of $t,\mu _{1},\mu _{2}$, etc., may be at once obtained by comparison with (514). ​ The same physical relations may of course be deduced without giving up the use of the surface of tension as a dividing surface, but the formulæ which express them will be less simple. If we make $t,\mu _{3},\mu _{4}$, etc., constant, we have by (98) and (508)
* $dp'=\gamma _{1}'d\mu _{1}+\gamma _{2}'d\mu _{2},$ $dp''=\gamma _{1}''d\mu _{1}+\gamma _{2}''d\mu _{2},$ $d\sigma =-\Gamma _{1}d\mu _{1}-\Gamma _{2}d\mu _{2},$
$\Gamma _{1}$
$\Gamma _{2}$
$dp'=dp''$
* $(\gamma _{1}'-\gamma _{1}'')d\mu _{1}+(\gamma _{2}'-\gamma _{2}'')d\mu _{2}=0,$
* $d\sigma =\Gamma _{1}{\frac {\sigma _{2}'-\sigma _{2}''}{\sigma _{1}'-\sigma _{1}''}}d\mu _{2}-\Gamma _{2}d\mu _{2}.$
| $\left({\frac {d\sigma }{d\mu _{2}}}\right)_{p'-p'',t,\mu _{3},\mu _{4},etc.}=-\Gamma _{2}+\Gamma _{1}{\frac {\gamma _{2}'-\gamma _{2}''}{\gamma _{1}'-\gamma _{1}''}}\cdot$ | (515) |
${\frac {\Gamma _{1}}{\gamma _{1}'-\gamma _{1}''}}$
$\Gamma _{1}=0$
$-\Gamma _{2(1)}$
If any component substance has the same density in the two homogeneous masses separated by a plane surface of discontinuity, the value of the superficial density for that component is independent of the position of the dividing surface. In this case alone we may derive the value of the superficial density of a component with reference to the surface of tension from the fundamental equation for plane surfaces alone. Thus in the last equation, when $\gamma _{2}'=\gamma _{2}''$, the second member will reduce to $-\Gamma _{2}$. It will be observed that to ​make $p'-p'',t,\mu _{3},\mu _{4}$, etc. constant is in this case equivalent to making $t,\mu _{1},\mu _{3},\mu _{4}$, etc. constant.
Substantially the same is true of the superficial density of entropy or of energy, when either of these has the same density in the two homogeneous masses.
Concerning the Stability of Surfaces of Discontinuity between Fluid Masses.
We shall first consider the stability of a film separating homogeneous masses with respect to changes in its nature, while its position and the nature of the homogeneous masses are not altered. For this purpose, it will be convenient to suppose that the homogeneous masses are very large, and thoroughly stable with respect to the possible formation of any different homogeneous masses out of their components, and that the surface of discontinuity is plane and uniform.
Let us distinguish the quantities which relate to the actual components of one or both of the homogeneous masses by the suffixes $_{a},_{b}$, etc., and those which relate to components which are found only at the surface of discontinuity by the suffixes $_{g},_{h}$ etc., and consider the variation of the energy of the whole system in consequence of a given change in the nature of a small part of the surface of discontinuity, while the entropy of the whole system and the total quantities of the several components remain constant, as well as the volume of each of the homogeneous masses, as determined by the surface of tension. This small part of the surface of discontinuity in its changed state is supposed to be still uniform in nature, and such as may subsist in equilibrium between the given homogeneous masses, which will evidently not be sensibly altered in nature or thermodynamic state.
The remainder of the surface of discontinuity is also supposed to ​remain uniform, and on account of its infinitely greater size to be infinitely less altered in its nature than the first part. Let $\Delta \epsilon ^{S}$ denote the increment of the superficial energy of this first part, $\Delta \eta ^{S},\Delta m_{a}^{S},\Delta m_{b}^{S}$, etc., $\Delta m_{g}^{S},\Delta m_{h}^{S}$, etc., the increments of its superficial entropy and of the quantities of the components which we regard as belonging to the surface. The increments of entropy and of the various components which the rest of the system receive will be expressed by
* $-\Delta \eta ^{S},$$-\Delta m_{a}^{S},$$-\Delta m_{b}^{S},$etc.,$-\Delta m_{g}^{S},$$-\Delta m_{h}^{S},$etc.,
* $-t\Delta \eta ^{S}-\mu _{a}\Delta m_{a}^{S}-\mu _{b}\Delta m_{b}^{S}-{\text{etc.}}-\mu _{g}\Delta m_{g}^{S}-\mu _{h}m_{h}^{S}-{\text{etc.}}$
| $\Delta \epsilon ^{S}-t\Delta \eta ^{S}$ | $-\mu _{a}\Delta m_{a}^{S}-\mu _{b}\Delta m_{b}^{S}-{\text{etc.}}$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (516) |
| | $-\mu _{g}\Delta m_{g}^{S}-\mu _{h}\Delta m_{h}^{S}-{\text{etc.}}$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (516) |
If the value of this expression is necessarily positive, for finite changes as well as infinitesimal in the nature of the part of the film to which $\Delta \epsilon ^{S}$, etc. relate, the increment of energy of the whole system will be positive for any possible changes in the nature of the film, and the film will be stable, at least with respect to changes in its nature, as distinguished from its position. For, if we write
* $D\epsilon ^{S},$$D\eta ^{S},$$Dm_{a}^{S},$$Dm_{b}^{S},$etc.,$Dm_{g}^{S},$$Dm_{h}^{S},$etc.,
| $\Delta D\epsilon ^{S}-t\Delta D\eta ^{S}$ | $-\mu _{a}\Delta Dm_{a}^{S}-\mu _{b}\Delta Dm_{b}^{S}-{\text{etc.}}$ |
| | $-\mu _{g}\Delta Dm_{g}^{S}-\mu _{h}\Delta Dm_{h}^{S}-{\text{etc.}}>0;$ |
and integrating for the whole surface, since
* $\Delta \int Dm_{g}^{S}=0,$$\Delta \int Dm_{h}^{S}=0,$etc.,
| $\Delta \int D\epsilon ^{S}-t\Delta \int D\eta ^{S}-\mu _{a}\Delta \int Dm_{a}^{S}-\mu _{b}\Delta \int Dm_{b}^{S}-{\text{etc.}}\geqq 0.$ | (519) |
$\Delta \int D\eta ^{S}$
$-\Delta \int D\eta ^{S}$
$-\Delta \int Dm_{a}^{S},-\Delta \int Dm_{b}^{S}$
* $-t\Delta \int D\eta ^{S}-\mu _{a}\Delta \int Dm_{a}^{S}-\mu _{b}\Delta \int Dm_{b}^{S}-{\text{etc.}}$
$\Delta \int D\epsilon ^{S}$
If the expression (516) is capable of a negative value for an infinitesimal change in the nature of the part of the film to which the symbols relate, the film is obviously unstable.
If the expression is capable of a negative value, but only for finite and not for infinitesimal changes in the nature of this part of the film, the film is practically unstable i.e., if such a change were made in a small part of the film, the disturbance would tend to increase. But it might be necessary that the initial disturbance should also have a finite magnitude in respect to the extent of surface in which it occurs; for we cannot suppose that the thermodynamic relations of an infinitesimal part of a surface of discontinuity are independent of the adjacent parts. On the other hand, the changes which we have been considering are such that every part of the film remains in equilibrium with the homogeneous masses on each side; and if the energy of the system can be diminished by a finite change satisfying this condition, it may perhaps be capable of diminution by an infinitesimal change which does not satisfy the same condition. We must therefore leave it undetermined whether the film, which in this case is practically unstable, is or is not unstable in the strict mathematical sense of the term.
Let us consider more particularly the condition of practical stability, in which we need not distinguish between finite and infinitesimal changes. To determine whether the expression (516) is capable of a negative value, we need only consider the least value of which it is capable. Let us write it in the fuller form
| $\epsilon ^{S''}-\epsilon ^{S'}-t(\eta ^{S''}-\eta ^{S'})$ | $-\mu _{a}(m_{a}^{S''}-m_{a}^{S'})-\mu _{b}(m_{b}^{S''}-m_{b}^{S'})-{\text{etc.}}$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (519) |
| | $-\mu _{g}(m_{g}^{S''}-m_{a}^{S'})-\mu _{h}(m_{h}^{S''}-m_{h}^{S'})-{\text{etc.,}}$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (519) |
where the single and double accents distinguish the quantities which relate to the first and second states of the film, the letters without accents denoting those quantities which have the same value in both states. The differential of this expression when the quantities distinguished by double accents are alone considered variable, and the area of the surface is constant, will reduce by (501) to the form
* $(\mu _{g}''-\mu _{g}')dm_{g}^{S''}+(\mu _{h}''-\mu _{h}')dm_{h}^{S''}+{\text{etc.}}$
* $\mu _{g}''=\mu _{g}',$unless$m_{g}^{S''}=0,$ $\mu _{h}''=\mu _{h}',$unless$m_{h}^{S''}=0.$
* $\sigma ''s-\sigma 's,$
| $\sigma ''-\sigma '$ | (520) |
That is, if the tension of the film is less than that of any other film of the same components which can exist between the same homogeneous masses (which has therefore the same values of $t,\mu _{a},\mu _{b}$, etc.), and which moreover has the same values of the potentials $\mu _{g},\mu _{h}$, etc., so far as it contains the substances to which these relate, then the first film will be stable. But the film will be practically unstable, if any other such film has a less tension. (Compare the expression (141), by which the practical stability of homogeneous masses is tested.)
It is, however, evidently necessary for the stability of the surface of discontinuity with respect to deformation, that the value of the superficial tension should be positive. Moreover, since we have by (502) for the surface of discontinuity
* $\epsilon ^{S}-t\eta ^{S}-\mu _{a}m_{a}^{S}-\mu _{b}m_{b}^{S}-{\text{etc.}}-\mu _{g}m_{g}^{S}-\mu _{h}m_{h}^{S}-{\text{etc.}}=\sigma s,$
* $\epsilon '-t\eta '+pv'-\mu _{a}m_{a}'-\mu _{b}m_{b}'-{\text{etc.}}=0,$ $\epsilon ''-t\eta ''+pv''-\mu _{a}m_{a}''-\mu _{b}m_{b}''-{\text{etc.}}=0,$
* $\epsilon ,$$\eta ,$$v,$$m_{a},$$m_{b},$etc.,$m_{g},$$m_{h},$etc.,
* $\epsilon -t\eta +pv-\mu _{a}m_{a}-\mu _{b}m_{b}-{\text{etc.}}-\mu _{g}m_{g}-\mu _{h}m_{h}-{\text{etc.}}=\sigma s.$
$\sigma$
$s$
$v$
* $t,$$p,$$\mu _{a},$$\mu _{b},$etc.,$\mu _{g},$$\mu _{h},$etc.,
* $\epsilon ,$$\eta ,$$m_{a},$$m_{b},$etc.,$m_{g},$$m_{h},$etc.,
$v$
$\sigma$
The body determining $\epsilon ,\eta$, etc. which will give this least value to this expression will evidently be sensibly homogeneous. With respect to the formation of such a body, the system consisting of the two homogeneous masses and the surface of discontinuity with the negative tension is by (53) (see also page 79) at least practically unstable, if the surface of discontinuity is very large, so that it can afford the requisite material without sensible alteration of the values of the potentials. (This limitation disappears, if all the component substances are found in the homogeneous masses.) Therefore, in a system satisfying the conditions of practical stability with respect to the possible formation of all kinds of homogeneous masses, negative tensions of the surfaces of discontinuity are necessarily excluded.
Let us now consider the condition which we obtain by applying (516) to infinitesimal changes. The expression may be expanded as before to the form (519), and then reduced by equation (502) to the form
* $s(\sigma ''-\sigma ')+m_{g}^{S''}(\mu _{g}''-\mu _{g}')+m_{h}^{S''}(\mu _{h}''-\mu _{h}')+{\text{etc.}}$
$\mu _{g},\mu _{h}$
* $-s\Delta \sigma -m_{g}^{S}\Delta \mu _{g}-m_{h}^{S}\Delta \mu _{h}-{\text{etc.}}>0,$
| $\Delta \sigma <-\Gamma _{g}\Delta \mu _{g}-\Gamma _{h}\Delta \mu _{h}-{\text{etc.}}$ | (521) |
In the foregoing discussion of stability, the surface of discontinuity is supposed plane. In this case, as the tension is supposed positive, there can be no tendency to a change of form of the surface. We now pass to the consideration of changes consisting in or connected with motion and change of form of the surface of tension, which we shall at first suppose to be and to remain spherical and uniform throughout.
In order that the equilibrium of a spherical mass entirely surrounded by an indefinitely large mass of different nature shall be neutral with respect to changes in the value of r, the radius of the sphere, it is evidently necessary that equation (500), which in this case may be written
| $2\sigma =r(p'-p''),$ | (522) |
$r$
* $2d\sigma =(p'-p'')dr+rdp'$
$d\sigma ,dp'$
$dr$
| $r{\frac {dp'}{dr}}<2{\frac {d\sigma }{dr}}-p'+p'',$ | (523) |
| $r{\frac {dp'}{dr}}>2{\frac {d\sigma }{dr}}-p'+p'',$ | (524) |
| $r{\frac {dp'}{dr}}=2{\frac {d\sigma }{dr}}-p'+p'',$ | (525) |
When the interior mass and the surface of discontinuity are formed entirely of substances which are components of the external mass, $p'$ and $\sigma$ cannot vary, and condition (524) being satisfied the equilibrium is unstable.
But if either the interior homogeneous mass or the surface of discontinuity contains substances which are not components of the enveloping mass, the equilibrium may be stable. If there is but one such substance, and we denote its densities and potential by $\gamma _{1}',\Gamma _{1}$, and $\mu _{1}$, the condition of stability (523) will reduce to the form
* $\left(r{\frac {dp'}{d\mu _{1}}}-2{\frac {d\sigma }{d\mu _{1}}}\right){\frac {d\mu _{1}}{dr}}<p''-p',$
| $(r\gamma _{1}'+2\Gamma _{1}){\frac {d\mu _{1}}{dr}}<p''-p'.$ | (526) |
$\mu _{2},\mu _{3}$
* $\gamma _{1}'v'+\Gamma _{1}s,$
* ${\tfrac {4}{3}}\pi r^{3}\gamma _{1}'+4\pi r^{2}\Gamma _{1}.$
$4\pi$
* $(r^{2}\gamma _{1}'+2r\Gamma _{1})dr+{\tfrac {1}{3}}r^{3}d\gamma _{1}'+r^{2}d\Gamma _{1}=0,$
$\gamma _{1}'$
$\Gamma _{1}$
$\mu _{1}$
| $(r\gamma _{1}'+2\Gamma _{1})dr+\left({\frac {r^{2}}{3}}{\frac {d\gamma _{1}'}{d\mu _{1}}}+r{\frac {d\Gamma _{1}}{d\mu _{1}}}\right)d\mu _{1}=0.$ | (527) |
| ${\frac {(r\gamma _{1}'+2\Gamma _{1})^{2}}{{\frac {r^{2}}{3}}{\frac {d\gamma _{1}'}{d\mu _{1}}}+r{\frac {d\Gamma _{1}}{d\mu _{1}}}}}>p'-p''.$ | (528) |
$r$
| ${\frac {\left({\frac {\gamma _{1}'}{p'-p''}}+{\frac {\Gamma _{1}}{\sigma }}\right)^{2}}{{\frac {1}{3(p'-p'')}}{\frac {d\gamma _{1}'}{d\mu _{1}}}+{\frac {1}{2\sigma }}{\frac {d\Gamma _{1}}{d\mu _{1}}}}}<1.$ | (529) |
$p'$
$\sigma$
$t,\mu _{1},\mu _{2}$
$p''$
$\mu _{1}$
If the component to which $\gamma _{1}'$ and $\Gamma _{1}$ relate is found only at the surface of discontinuity, the condition of stability reduces to
| ${\frac {\Gamma _{1}^{2}}{\sigma }}{\frac {d\mu _{1}}{d\Gamma _{1}}}<{\frac {1}{2}}\cdot$ | (530) |
$\Gamma _{1}=-{\frac {d\sigma }{d\mu _{1}}},$
| ${\frac {\Gamma _{1}}{\sigma }}{\frac {d\sigma }{d\Gamma _{1}}}<-{\frac {1}{2}},$or${\frac {d\log \sigma }{d\log \Gamma _{1}}}<-{\frac {1}{2}}$ | (531) |
$\Gamma _{1}=0$
${\frac {d\Gamma _{1}}{d\mu _{1}}}=0$
| ${\frac {3\gamma _{1}^{'2}}{p'-p''}}{\frac {d\mu _{1}}{d\gamma _{1}'}}>1.$ | (532) |
$\gamma _{1}'={\frac {dp'}{d\mu _{1}}},$
| ${\frac {\gamma _{1}'}{p'-p''}}{\frac {dp'}{d\gamma _{1}'}}>{\frac {1}{3}}$or${\frac {d\log(p'-p'')}{d\log \gamma _{1}'}}>{\frac {1}{3}}\cdot$ | (533) |
$r$
$\Gamma _{1}$
$\gamma _{1}'$
Similar conditions of stability may be found when there are more substances than one in the inner mass or the surface of discontinuity, which are not components of the enveloping mass. In this case, we have instead of (526) a condition of the form
| $(r\gamma _{1}'+2\Gamma _{1}){\frac {d\mu _{1}}{dr}}+(r\gamma _{1}'+2\Gamma _{1}){\frac {2\mu _{2}}{dr}}+{\text{etc.}}<p''-p',$ | (534) |
${\frac {d\mu _{1}}{dr}},{\frac {d\mu _{2}}{dr}}$
* $\gamma _{1}'v'+\Gamma _{1}s,$$\gamma _{2}'v'+\Gamma _{2}s,$etc.
Nearly the same method may be applied to the following problem. Two different homogeneous fluids are separated by a diaphragm having a circular orifice, their volumes being invariable except by the motion of the surface of discontinuity, which adheres to the edge of the orifice;—to determine the stability or instability of this surface when in equilibrium. ​The condition of stability derived from (522) may in this case be written
| $r{\frac {d(p'-p'')}{dv'}}<2{\frac {d\sigma }{dv'}}-(p'-p''){\frac {dr}{dv'}},$ | (535) |
If both the masses are infinitely large, or if one which contains all the components of the system is infinitely large, $p'-p''$ and $\sigma$ will be constant, and the condition reduces to
* ${\frac {dr}{dv'}}<0.$
To return to the general problem:—if we denote by $x$ the part of the axis of the circular orifice intercepted between the center of the orifice and the surface of tension, by $R$ the radius of the orifice, and by $V'$ the value of $v'$ when the surface of tension is plane, we shall have the geometrical relations
* $R^{2}=2rx-x^{2},$
| and | ${\begin{aligned}v'&=V'+{\tfrac {2}{3}}\pi r^{3}x-{\tfrac {1}{3}}\pi R^{2}(r-x)\\&=V'+\pi rx^{2}-{\tfrac {1}{3}}\pi x^{3}.\\\end{aligned}}$ |
* $(r-x)dx+xdr=0,$
| and | $dv'=\pi x^{2}dr+(2\pi rx-\pi x^{2})dx;$ |
| whence | $(r-x)dv'=-\pi rx^{2}dr.$ |
| ${\frac {dp'}{dv'}}-{\frac {dp''}{dv'}}-{\frac {2}{r}}{\frac {d\sigma }{dv'}}<(p'-p''){\frac {r-x}{\pi r^{2}x^{2}}}\cdot$ | (537) |
$\mu _{1}$
| $\left(\gamma _{1}'-\gamma _{1}''+{\frac {2\Gamma _{1}}{r}}\right){\frac {d\mu _{1}}{dv'}}<(p'-p''){\frac {r-x}{\pi r^{2}x^{2}}}\cdot$ | (538) |
$\gamma _{1}'v'+\gamma _{1}''v''+\Gamma _{1}s$
* $dv''=-dv',$and$ds={\frac {2}{r}}dv',$
| $\left(v'{\frac {d\gamma _{1}'}{d\mu _{1}}}+v''{\frac {d\gamma _{1}''}{d\mu _{1}}}+s{\frac {d\Gamma _{1}}{d\mu _{1}}}\right)+\left(\gamma _{1}'-\gamma _{1}''+{\frac {2\Gamma _{1}}{r}}\right)dv'=0.$ | (538) |
| ${\frac {\left(\gamma _{1}'-\gamma _{1}''+{\frac {2\Gamma _{1}}{r}}\right)^{2}}{v'{\frac {d\gamma _{1}'}{d\mu _{1}}}+v''{\frac {d\gamma _{1}''}{d\mu _{1}}}+s{\frac {d\Gamma _{1}}{d\mu _{1}}}}}>(p'-p''){\frac {x-r}{\pi x^{2}r^{2}}}\cdot$ | (540) |
${\frac {2\Gamma _{1}}{r}}$
$s{\frac {d\Gamma _{1}}{d\mu _{1}}}$
$\gamma _{1}',\gamma _{1}'',v'{\frac {d\gamma _{1}'}{d\mu _{1}}},{\frac {d\gamma _{1}''}{d\mu _{1}}}$
In the cases just discussed, the problem of the stability of certain surfaces of tension has been solved by considering the case of neutral equilibrium,—a condition of neutral equilibrium affording the equation of the limit of stability. This method probably leads as directly as any to the result, when that consists in the determination of the value of a certain quantity at the limit of stability, or of the relation which exists at that limit between certain quantities specifying the state of the system. But problems of a more general character may require a more general treatment.
Let it be required to ascertain the stability or instability of a fluid system in a given state of equilibrium with respect to motion of the surfaces of tension and accompanying changes. It is supposed that the conditions of internal stability for the separate homogeneous masses are satisfied, as well as those conditions of stability for the surfaces of discontinuity which relate to small portions of these surfaces with the adjacent masses. (The conditions of stability which are here supposed to be satisfied have been already discussed in part and will be farther discussed hereafter.) The fundamental equations for all the masses and surfaces occurring in the system are supposed to be known. In applying the general criteria of stability which are given on page 57, we encounter the following difficulty.
The question of the stability of the system is to be determined by the consideration of states of the system which are slightly varied ​from that of which the stability is in question. These varied states of the system are not in general states of equilibrium, and the relations expressed by the fundamental equations may not hold true of them. More than this,—if we attempt to describe a varied state of the system by varied values of the quantities which describe the initial state, if these varied values are such as are inconsistent with equilibrium, they may fail to determine with precision any state of the system. Thus, when the phases of two contiguous homogeneous masses are specified, if these phases are such as satisfy all the conditions of equilibrium, the nature of the surface of discontinuity (if without additional components) is entirely determined; but if the phases do not satisfy all the conditions of equilibrium, the nature of the surface of discontinuity is not only undetermined, but incapable of determination by specified values of such quantities as we have employed to express the nature of surfaces of discontinuity in equilibrium. For example, if the temperatures in contiguous homogeneous masses are different, we cannot specify the thermal state of the surface of discontinuity by assigning to it any particular temperature. It would be necessary to give the law by which the temperature passes over from one value to the other. And if this were given, we could make no use of it in the determination of other quantities, unless the rate of change of the temperature were so gradual that at every point we could regard the thermodynamic state as unaffected by the change of temperature in its vicinity. It is true that we are also ignorant in respect to surfaces of discontinuity in equilibrium of the law of change of those quantities which are different in the two phases in contact, such as the densities of the components, but this, although unknown to us, is entirely determined by the nature of the phases in contact, so that no vagueness is occasioned in the definition of any of the quantities which we have occasion to use with reference to such surfaces of discontinuity.
It may be observed that we have established certain differential equations, especially (497), in which only the initial state is necessarily one of equilibrium. Such equations may be regarded as establishing certain properties of states bordering upon those of equilibrium. But these are properties which hold true only when we disregard quantities proportional to the square of those which express the degree of variation of the system from equilibrium. Such equations are therefore sufficient for the determination of the conditions of equilibrium, but not sufficient for the determination of the conditions of stability.
We may, however, use the following method to decide the question of stability in such a case as has been described.
Beside the real system of which the stability is in question, it will be convenient to conceive of another system, to which we shall ​attribute in its initial state the same homogeneous masses and surfaces of discontinuity which belong to the real system. We shall also suppose that the homogeneous masses and surfaces of discontinuity of this system, which we may call the imaginary system, have the same fundamental equations as those of the real system. But the imaginary system is to differ from the real in that the variations of its state are limited to such as do not violate the conditions of equilibrium relating to temperature and the potentials, and that the fundamental equations of the surfaces of discontinuity hold true for these varied states, although the condition of equilibrium expressed by equation (500) may not be satisfied.
Before proceeding farther, we must decide whether we are to examine the question of stability under the condition of a constant external temperature, or under the condition of no transmission of heat to or from external bodies, and in general, to what external influences we are to regard the system as subject. It will be convenient to suppose that the exterior of the system is fixed, and that neither matter nor heat can be transmitted through it. Other cases may easily be reduced to this, or treated in a manner entirely analogous.
Now if the real system in the given state is unstable, there must be some slightly varied state in which the energy is less, but the entropy and the quantities of the components the same as in the given state, and the exterior of the system unvaried. But it may easily be shown that the given state of the system may be made stable by constraining the surfaces of discontinuity to pass through certain fixed lines situated in the unvaried surfaces. Hence, if the surfaces of discontinuity are constrained to pass through corresponding fixed lines in the surfaces of discontinuity belonging to the varied state just mentioned, there must be a state of stable equilibrium for the system thus constrained which will differ infinitely little from the given state of the system, the stability of which is in question, and will have the same entropy, quantities of components, and exterior, but less energy. The imaginary system will have a similar state, since the real and imaginary systems do not differ in respect to those states which satisfy all the conditions of equilibrium for each surface of discontinuity. That is, the imaginary system has a state, differing infinitely little from the given state, and with the same entropy, quantities of components, and exterior, but with less energy.
Conversely, if the imaginary system has such a state as that just described, the real system will also have such a state. This may be shown by fixing certain lines in the surfaces of discontinuity of the imaginary system in its state of less energy and then making the energy a minimum under the conditions. The state thus determined ​will satisfy all the conditions of equilibrium for each surface of discontinuity, and the real system will therefore have a corresponding state, in which the entropy, quantities of components, and exterior will be the same as in the given state, but the energy less.
We may therefore determine whether the given system is or is not unstable, by applying the general criterion of instability (7) to the imaginary system.
If the system is not unstable, the equilibrium is either neutral or stable. Of course we can determine which of these is the case by reference to the imaginary system, since the determination depends upon states of equilibrium, in regard to which the real and imaginary systems do not differ. We may therefore determine whether the equilibrium of the given system is stable, neutral, or unstable, by applying the criteria (3)–(7) to the imaginary system.
The result which we have obtained may be expressed as follows:—In applying to a fluid system which is in equilibrium, and of which all the small parts taken separately are stable, the criteria of stable, neutral, and unstable equilibrium, we may regard the system as under constraint to satisfy the conditions of equilibrium relating to temperature and the potentials, and as satisfying the relations expressed by the fundamental equations for masses and surfaces, even when the condition of equilibrium relating to pressure {equation (500)} is not satisfied.
It follows immediately from this principle, in connection with equations (501) and (86), that in a stable system each surface of tension must be a surface of minimum area for constant values of the volumes which it divides, when the other surfaces bounding these volumes and the perimeter of the surface of tension are regarded as fixed; that in a system in neutral equilibrium each surface of tension will have as small an area as it can receive by any slight variations under the same limitations; and that in seeking the remaining conditions of stable or neutral equilibrium, when these are satisfied, it is only necessary to consider such varied surfaces of tension as have similar properties with reference to the varied volumes and perimeters.
We may illustrate the method which has been described by applying it to a problem but slightly different from one already (pp. 244, 245) discussed by a different method. It is required to determine the conditions of stability for a system in equilibrium, consisting of two different homogeneous masses meeting at a surface of discontinuity, the perimeter of which is invariable, as well as the exterior of the whole system, which is also impermeable to heat.
To determine what is necessary for stability in addition to the condition of minimum area for the surface of tension, we need only ​consider those varied surfaces of tension which satisfy the same condition. We may therefore regard the surface of tension as determined by $v'$, the volume of one of the homogeneous masses. But the state of the system would evidently be completely determined by the position of the surface of tension and the temperature and potentials, if the entropy and the quantities of the components were variable; and therefore, since the entropy and the quantities of the components are constant, the state of the system must be completely determined by the position of the surface of tension. We may therefore regard all the quantities relating to the system as functions of $v'$, and the condition of stability may be written
* ${\frac {d\epsilon }{dv'}}dv'+{\frac {1}{2}}{\frac {d^{2}\epsilon }{dv^{'2}}}dv^{'2}+{\text{etc.}}>0,$
$\epsilon$
* ${\frac {d\epsilon }{dv'}}=0.$
| ${\frac {d^{2}\epsilon }{dv^{'2}}}>0.$ | (541) |
$\epsilon ',\epsilon '',\epsilon ^{S}$
* $d\epsilon =d\epsilon '+d\epsilon ''+d\epsilon ^{S}=-p'dv'-p''dv''+\sigma ds,$
$dv''=-dv'$
| ${\frac {d\epsilon }{dv'}}=-p'+p''+\sigma {\frac {ds}{dv'}}\cdot$ | (542) |
| ${\frac {d^{2}\epsilon }{dv^{'2}}}=-{\frac {dp'}{dv'}}+{\frac {dp''}{dv'}}+{\frac {d\sigma }{dv'}}{\frac {ds}{dv'}}+\sigma {\frac {d^{2}s}{dv^{'2}}},$ | (543) |
| $\sigma {\frac {d^{2}s}{dv^{'2}}}>{\frac {dp'}{dv'}}-{\frac {dp''}{dv'}}-{\frac {d\sigma }{dv'}}{\frac {ds}{dv'}}\cdot$ | (544) |
| $\sigma {\frac {d^{2}s}{dv^{'2}}}>\left(\gamma _{1}'-\gamma _{1}''+\Gamma _{1}{\frac {ds}{dv'}}\right){\frac {d\mu _{1}}{dv'}}\cdot$ | (545) |
$_{1}$
* $\gamma _{1}'v'+\gamma _{1}''v''+\Gamma _{1}s.$
| $\left(\gamma _{1}'-\gamma _{1}''+\Gamma _{1}{\frac {ds}{dv'}}\right)dv'+\left(v'{\frac {d\gamma _{1}'}{d\mu _{1}}}+v''{\frac {d\gamma _{1}''}{d\mu _{1}}}+s{\frac {d\Gamma _{1}}{d\mu _{1}}}\right)d\mu _{1}=0.$ | (546) |
| $\sigma {\frac {d^{2}s}{dv^{'2}}}>-{\frac {\left(\gamma _{1}'-\gamma _{1}''-\Gamma _{1}{\frac {ds}{dv'}}\right)^{2}}{v'{\frac {d\gamma _{1}'}{d\mu _{1}}}+v''{\frac {d\gamma _{1}''}{d\mu _{1}}}+s{\frac {d\Gamma _{1}}{d\mu _{1}}}}},$ | (547) |
${\frac {ds}{dv'}}$
${\frac {d^{2}s}{dv'}}$
The condition of stability with reference to motion of surfaces of discontinuity admits of a very simple expression when we can treat the temperature and potentials as constant. This will be the case when one or more of the homogeneous masses, containing together all the component substances, may be considered as indefinitely large, the surfaces of discontinuity being finite. For if we write $\textstyle \sum \displaystyle \Delta \epsilon$ for the sum of the variations of the energies of the several homogeneous masses, and $\textstyle \sum \displaystyle \Delta \epsilon ^{S}$ for the sum of the variations of the energies of the several surfaces of discontinuity, the condition of stability may be written
| $\textstyle \sum \displaystyle \Delta \epsilon +\textstyle \sum \displaystyle \Delta \epsilon ^{S}>0,$ | (548) |
* $\Delta \epsilon =t\Delta \eta -p\Delta v+\mu _{1}\Delta m_{1}+\mu _{2}\Delta m_{2}+{\text{etc.,}}$
* $\Delta \epsilon ^{S}=t\Delta \eta ^{S}+\sigma \Delta s+\mu _{1}\Delta m_{1}^{S}+\mu _{2}\Delta m_{2}^{S}+{\text{etc.}}$
$t,p,\sigma ,\mu _{1},\mu _{2}$
| $\textstyle -\sum \displaystyle (p\Delta v)+\textstyle -\sum \displaystyle (\sigma \Delta s)>0,$ | (549) |
$\textstyle \sum \displaystyle (p\Delta v)-\textstyle -\sum \displaystyle (\sigma \Delta s)<0.$
This is a purely geometrical condition, since the pressures and tensions are constant. This condition is of interest, because it is always sufficient for stability with reference to motion of surfaces of discontinuity. For any system may be reduced to the kind described by putting certain parts of the system in communication (by means of fine tubes if necessary) with large masses of the proper temperatures and potentials. This may be done without introducing any new movable surfaces of discontinuity. The condition (549) when applied to the altered system is therefore the same as when applied to the original system. But it is sufficient for the stability of the altered system, and therefore sufficient for its stability if we diminish its freedom by breaking the connection between the original system and the additional parts, and therefore sufficient for the stability of the original system.
On the Possibility of the Formation of a Fluid of different Phase within any Homogeneous Fluid.
The study of surfaces of discontinuity throws considerable light upon the subject of the stability of such homogeneous fluid masses as have a less pressure than others formed of the same components (or some of them) and having the same temperature and the same potentials for their actual components.
In considering this subject, we must first of all inquire how far our method of treating surfaces of discontinuity is applicable to cases in which the radii of curvature of the surfaces are of insensible magnitude. That it should not be applied to such cases without limitation is evident from the consideration that we have neglected the term ${\tfrac {1}{2}}(C{1}-C_{2})\delta (c_{1}-c_{2})$ in equation (494) on account of the magnitude of the radii of curvature compared with the thickness of the non-homogeneous film. (See page 228.) When, however, only spherical masses are considered, this term will always disappear, since $C_{1}$ and $C_{2}$ will necessarily be equal. ​Again, the surfaces of discontinuity have been regarded as separating homogeneous masses. But we may easily conceive that a globular mass (surrounded by a large homogeneous mass of different nature) may be so small that no part of it will be homogeneous, and that even at its center the matter cannot be regarded as having any phase of matter in mass. This, however, will cause no difficulty, if we regard the phase of the interior mass as determined by the same relations to the exterior mass as in other cases. Beside the phase of the exterior mass, there will always be another phase having the same temperature and potentials, but of the general nature of the small globule which is surrounded by that mass and in equilibrium with it. This phase is completely determined by the system considered, and in general entirely stable and perfectly capable of realization in mass, although not such that the exterior mass could exist in contact with it at a plane surface. This is the phase which we are to attribute to the mass which we conceive as existing within the dividing surface.
With this understanding with regard to the phase of the fictitious interior mass, there will be no ambiguity in the meaning of any of the symbols which we have employed, when applied to cases in which the surface of discontinuity is spherical, however small the radius may be. Nor will the demonstration of the general theorems require any material modification. The dividing surface which determines the value of $\epsilon ^{S},\eta ^{S},m_{1}^{S},m_{2}^{S}$, etc. is as in other cases to be placed so as to make the term ${\tfrac {1}{2}}(C_{1}+C_{2})\delta (c_{1}+c_{2})$ in equation (494) vanish, i.e., so as to make equation (497) valid. It has been shown on pages 225–227 that when thus placed it will sensibly coincide with the physical surface of discontinuity, when this consists of a non-homogeneous film separating homogeneous masses, and having radii of curvature which are large compared with its thickness. But in regard to globular masses too small for this theorem to have any application, it will be worth while to examine how far we may be certain that the radius of the dividing surface will have a real and positive value, since it is only then that our method will have any natural application. The value of the radius of the dividing surface, supposed spherical, of any globule in equilibrium with a surrounding homogeneous fluid may be most easily obtained by eliminating $\sigma$ from equations (500) and (502), which have been derived from (497), and contain the radius implicitly. If we write r for this radius, equation (500) may be written
| $2\sigma =(p'-p'')r,$ | (550) |
​the single and double accents referring respectively to the interior and exterior masses. If we write $[\epsilon ],[\eta ],[m_{1}],[m_{2}]$, etc., for the excess of the total energy, entropy, etc., in and about the globular mass above what would be in the same space if it were uniformly filled with matter of the phase of the exterior mass, we shall have necessarily with reference to the whole dividing surface
| $\epsilon ^{S}=[\epsilon ]-v'(\epsilon _{V'}-\epsilon _{V''}),$ | $\eta ^{S}=[\eta ]-v'(\eta _{V'}-\eta _{V''}),$ |
| $m_{1}^{S}=[m_{1}]-v'(\gamma _{1}'-\gamma _{1}''),$ | $m_{2}^{S}=[m_{2}]-v'(\gamma _{2}'-\gamma _{2}''),$ | etc., |
$\epsilon _{V'},\epsilon _{V''},\eta _{V'},\eta _{V''},\gamma _{1}',\gamma _{1}''$
| ${\begin{aligned}\sigma s=[\epsilon ]&-v'(\epsilon _{V'}-\epsilon _{V''})-t[\eta ]+tv'(\eta _{V'}-\eta _{V''})\\&-\mu _{1}[m_{1}]+\mu _{1}v'(\gamma _{1}'-\gamma _{1}'')-\mu _{2}[m_{2}]+\mu _{2}v'(\gamma _{2}'-\gamma _{2}'')-{\text{etc.}}\\\end{aligned}}$ | (551) |
* $p'=-\epsilon _{V'}+t\eta _{V'}+\mu _{1}\gamma _{1}'+\mu _{2}\gamma _{2}'+{\text{etc.}}$ $p''=-\epsilon _{V''}+t\eta _{V''}+\mu _{1}\gamma _{1}''+\mu _{2}\gamma _{2}''+{\text{etc.}}$
| $W=[\epsilon ]-t[\eta ]-\mu _{1}[m_{1}]-\mu _{2}[m_{2}]-{\text{etc.}}$ | (552) |
$W$
| $\sigma s=W+v'(p'-p''),$ | (553) |
$s$
$v'$
$r$
| $4\pi ^{2}\sigma =W+{\tfrac {4}{3}}\pi r^{3}(p'-p''),$ | (554) |
$\sigma$
| ${\tfrac {2}{3}}\pi r^{3}(p'-p'')=W,$ | (555) |
| $r=\left({\frac {3W}{2\pi (p'-p'')}}\right)^{\tfrac {1}{3}}\cdot$ | (556) |
$r$
$\sigma$
| ${\frac {16\pi \sigma ^{3}}{3(p'-p'')^{2}}}=W,$ | (557) |
| $\sigma =\left({\frac {3W(p'-p'')^{2}}{16\pi }}\right)^{\tfrac {1}{3}}\cdot$ | (558) |
$\sigma$
$\sigma$
$p'=p''$
$\sigma ,p'-p''$
$r$
$W$
$W$
$r$
$p'=p''$
$p'-p''$
$W,r$
$\sigma$
$r$
It may easily be shown that the quantity denoted by $W$ is the work which would be required to form (by a reversible process) the heterogeneous globule in the interior of a very large mass having initially the uniform phase of the exterior mass. For this work is equal to the increment of energy of the system when the globule is formed without change of the entropy or volume of the whole system or of the quantities of the several components. Now $[\eta ],[m_{1}],[m_{2}]$, etc. denote the increments of entropy and of the components in the space where the globule is formed. Hence these quantities with the negative sign will be equal to the increments of entropy and of the components in the rest of the system. And hence, by equation (86),
* $-t[\eta ]-\mu _{1}[m_{1}]-\mu _{2}[m_{2}]-{\text{etc.}}$
$[\epsilon ]$
$W$
The conclusions which may be drawn from these considerations with respect to the stability of the homogeneous mass of the pressure $p''$ (supposed less than $p'$, the pressure belonging to a different phase of the same temperature and potentials) are very obvious. Within those limits within which the method used has been justified, the mass in question must be regarded as in strictness stable with respect to the growth of a globule of the kind considered, since $W$, the work ​required for the formation of such a globule of a certain size (viz., that which would be in equilibrium with the surrounding mass), will always be positive. Nor can smaller globules be formed, for they can neither be in equilibrium with the surrounding mass, being too small, nor grow to the size of that to which $W$ relates. If, however, by any external agency such a globular mass (of the size necessary for equilibrium) were formed, the equilibrium has already (page 243) been shown to be unstable, and with the least excess in size, the interior mass would tend to increase without limit except that depending on the magnitude of the exterior mass. We may therefore regard the quantity W as affording a kind of measure of the stability of the phase to which $p''$ relates. In equation (557) the value of $W$ is given in terms of $\sigma$ and $p'-p''$. If the three fundamental equations which give $\sigma ,p'$, and $p''$ in terms of the temperature and the potentials were known, we might regard the stability $(W)$ as known in terms of the same variables. It will be observed that when $p'=p''$ the value of $W$ is infinite. If $p'-p''$ increases without greater changes of the phases than are necessary for such increase, $W$ will vary at first very nearly inversely as the square of $p'-p''$. If $p'-p''$ continues to increase, it may perhaps occur that $W$ reaches the value zero; but until this occurs the phase is certainly stable with respect to the kind of change considered. Another kind of change is conceivable, which initially is small in degree but may be great in its extent in space. Stability in this respect or stability in respect to continuous changes of phase has already been discussed (see page 105), and its limits determined. These limits depend entirely upon the fundamental equation of the homogeneous mass of which the stability is in question. But with respect to the kind of changes here considered, which are initially small in extent but great in degree, it does not appear how we can fix the limits of stability with the same precision. But it is safe to say that if there is such a limit it must be at or beyond the limit at which $\sigma$ vanishes. This latter limit is determined entirely by the fundamental equation of the surface of discontinuity between the phase of which the stability is in question and that of which the possible formation is in question. We have already seen that when $\sigma$ vanishes, the radius of the dividing surface and the work $W$ vanish with it. If the fault in the homogeneity of the mass vanishes at the same time (it evidently cannot vanish sooner), the phase becomes unstable at this limit. But if the fault in the homogeneity of the physical mass does not vanish with $r,\sigma$ or and $W$,—and no sufficient reason appears why this should not be considered as the general case,—although the amount of work necessary to upset the equilibrium of the phase is infinitesimal, this is not enough to make the phase unstable. ​It appears therefore that $W$ is a somewhat one-sided measure of stability.
It must be remembered in this connection that the fundamental equation of a surface of discontinuity can hardly be regarded as capable of experimental determination, except for plane surfaces (see pp. 231–233), although the relation for spherical surfaces is in the nature of things entirely determined, at least so far as the phases are separately capable of existence. Yet the foregoing discussion yields the following practical results. It has been shown that the real stability of a phase extends in general beyond that limit (discussed on pages 103–105), which may be called the limit of practical stability, at which the phase can exist in contact with another at a plane surface, and a formula has been deduced to express the degree of stability in such cases as measured by the amount of work necessary to upset the equilibrium of the phase when supposed to extend indefinitely in space. It has also been shown to be entirely consistent with the principles established that this stability should have limits, and the manner in which the general equations would accommodate themselves to this case has been pointed out.
By equation (553), which may be written
| $W=\sigma s-(p'-p'')v',$ | (559) |
$W$
sign, is always equal to two-thirds of the first, as appears from equation (550) and the geometrical relation $v'={\tfrac {1}{3}}rs$. We may therefore write
| $W={\tfrac {1}{3}}\sigma s={\tfrac {1}{2}}(p'-p'')v'.$ | (560) |
On the Possible Formation at the Surface where two different Homogeneous Fluids meet of a Fluid of different Phase from either.
Let $A,B$, and $C$ be three different fluid phases of matter, which satisfy all the conditions necessary for equilibrium when they meet at plane surfaces. The components of $A$ and $B$ may be the same or different, but $C$ must have no components except such as belong to $A$ or $B$. Let us suppose masses of the phases $A$ and $B$ to be separated by a very thin sheet of the phase $C$. This sheet will not necessarily be plane, but the sum of its principal curvatures must be zero. We may treat such a system as consisting simply of masses of the phases $A$ and $B$ with a certain surface of discontinuity, for in our previous discussion there has been nothing to limit the thickness or the nature of the film separating homogeneous masses, except that its thickness has generally been supposed to be small in comparison with its radii of curvature. The value of the superficial tension for such a film will be $\sigma _{AC}+\sigma _{BC}$, if we denote by these symbols the tensions of the surfaces of contact of the phases $A$ and $C$, and $B$ and $C$, respectively.
This not only appears from evident mechanical considerations, but may also be easily verified by equations (502) and (93), the first of which may be regarded as defining the quantity or. This value will not be affected by diminishing the thickness of the film, until the limit is reached at which the interior of the film ceases to have the properties of matter in mass. Now if $\sigma _{AC}+\sigma _{BC}$ is greater than $\sigma _{AB}$, the tension of the ordinary surface between $A$ and $B$, such a film will be at least practically unstable. (See page 240.) We cannot suppose that $\sigma _{AB}>\sigma _{AC}+\sigma _{BC}$, for this would make the ordinary surface between $A$ and $B$ unstable and difficult to realize. If $\sigma _{AB}=\sigma _{AC}+\sigma _{BC}$ we may assume, in general, that this relation is not accidental, and that the ordinary surface of contact for $A$ and $B$ is of the kind which we have described.
Let us now suppose the phases $A$ and $B$ to vary, so as still to satisfy the conditions of equilibrium at plane contact, but so that the pressure of the phase $C$ determined by the temperature and potentials ​of $A$ and $B$ shall become less than the pressure of $A$ and $B$. A system consisting of the phases $A$ and $B$ will be entirely stable with respect to the formation of any phase like $C$. (This case is not quite identical with that considered on page 104, since the system in question contains two different phases, but the principles involved are entirely the same.)
With respect to variations of the phases $A$ and $B$ in the opposite direction we must consider two cases separately. It will be convenient to denote the pressures of the three phases by $p_{A},p_{B},p_{C}$, and to regard these quantities as functions of the temperature and potentials.
If $\sigma _{AB}=\sigma _{AC}+\sigma _{BC}$ for values of the temperature and potentials which make $p_{A}=p_{B}=p_{C}$, it will not be possible to alter the temperature and potentials at the surface of contact of the phases $A$ and $B$ so that $p_{A}=p_{b}$, and $p_{C}>p_{A}$, for the relation of the temperature and potentials necessary for the equality of the three pressures will be preserved by the increase of the mass of the phase $C$. Such variations of the phases $A$ and $B$ might be brought about in separate masses, but if these were brought into contact, there would be an immediate formation of a mass of the phase $C$, with reduction of the phases of the adjacent masses to such as satisfy the conditions of equilibrium with that phase.
$\sigma _{AB}<\sigma _{AC}+\sigma _{BC}$
$p_{A}=p_{B}$
$p_{C}>p_{A}$
$C$
$C$
$A$
$B$
$\sigma _{AC}+\sigma _{BC}-\sigma _{AB}$
$C$
$A$
$B$
$A$
$B$
$C$
$C$
$C$
$\sigma _{AB}<\sigma _{AC}+\sigma _{BC}$
It will be convenient to consider first a lentiform mass of phase $C$ in equilibrium between masses of phases $A$ and $B$ which meet in a plane surface. Let figure 10 represent a section of such a system through the centers of the spherical surfaces, the mass of phase $A$ lying on the left of $DEH'FG$, and that of phase $B$ on the right of $DEH''FG$. Let the line joining the centers cut the spherical surfaces in $H'$ and $H''$, and the plane of the surface of contact of $A$ and $B$ in $I$. Let the radii of $EH'F$ and $EH''F$ be denoted by $r',r''$, and the segments $IH',IH''$, by $x',x''$. Also let $IE$, the radius of the circle in which the spherical surfaces intersect, be denoted by $R$. By a suitable application of the general condition of equilibrium we may easily obtain the equation
| $\sigma _{AC}{\frac {r'-x'}{r'}}+\sigma _{BC}{\frac {r''-x''}{r''}}=\sigma _{AB},$ | (561) |
$EF$
$\sigma _{AC}$
$\sigma _{BC}$
$\sigma _{AB}$
$W$
$A$
$B$
| $W=M-N,$ | (562) |
$M$
$A$
$B$
$A$
$C$
$B$
$C$
$N$
$A$
$B$
$C$
| $M=\sigma _{AC}s_{AC}+\sigma _{BC}s_{BC}-\sigma _{AB}s_{AB},$ | (563) |
$s_{AC},s_{BC},s_{AB}$
| $N=V'(p_{C}-p_{A})+V''(p_{C}-p_{B}),$ | (564) |
$V'$
$V''$
$A$
$B$
| $p_{C}-p_{A}={\frac {2\sigma _{AC}}{r'}}$,and$p_{C}-p_{B}={\frac {2\sigma _{BC}}{r''}}\cdot$ | (565) |
| $V'={\tfrac {2}{3}}\pi r'^{2}x'-{\tfrac {1}{3}}\pi R^{2}(r'-x'),$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ |
| $V''={\tfrac {2}{3}}\pi r''^{2}x''-{\tfrac {1}{3}}\pi R^{2}(r''-x'').$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ |
By substitution we obtain
| ${\begin{aligned}N={\tfrac {4}{3}}\pi \sigma _{AC}r'x'&-{\tfrac {2}{3}}\pi R^{2}\sigma _{AC}{\frac {r'-x'}{r'}}\\&+{\tfrac {4}{3}}\pi \sigma _{BC}r''x''-{\tfrac {2}{3}}\pi R^{2}\sigma _{BC}{\frac {r''-x''}{r''}},\\\end{aligned}}$ | (567) |
| $N={\tfrac {4}{3}}\pi \sigma _{AC}r'x'+{\tfrac {4}{3}}\pi \sigma _{BC}r''x''-{\tfrac {2}{3}}\pi R^{2}\sigma _{AB}.$ | (568) |
* $2\pi r'x'=s_{AC},$$2\pi r''x''=s_{BC}$$\pi R^{2}=s_{AB},$
| $N={\tfrac {2}{3}}(\sigma _{AC}s_{AC}+\sigma _{BC}s_{BC}-\sigma _{AB}s_{AB}).$ | (569) |
$M$
$N$
| $W={\tfrac {1}{3}}(\sigma _{AC}s_{AC}+\sigma _{BC}s_{BC}-\sigma _{AB}s_{AB}).$ | (570) |
* $\sigma _{AC}+\sigma _{bc}>\sigma _{AB},$
| since | $s_{AC}>s_{AB}$,and$s_{BC}>s_{AB}.$ |
we see by (561) that
* $s_{AC}=s_{AB},$and$s_{BC}=s_{AB},$
| and therefore | $W=0.$ |
We may conclude that after we pass the limit at which $p_{C}$ becomes greater than $p_{A}$ and $p_{B}$ (supposed equal) lentiform masses of phase $C$ will not be formed until either $\sigma _{AB}=\sigma _{AC}+\sigma _{BC}$, or $p_{C}-p_{A}$ becomes so great that the lentiform mass which would be in equilibrium is one ​of insensible magnitude. {The diminution of the radii with increasing values of $p_{C}-p_{A}$ is indicated by equation (565).} Hence, no mass of phase $C$ will be formed until one of these limits is reached. Although the demonstration relates to a plane surface between $A$ and $B$, the result must be applicable whenever the radii of curvature have a sensible magnitude, since the effect of such curvature may be disregarded when the lentiform mass is sufficiently small.
The equilibrium of the lentiform mass of phase $C$ is easily proved to be unstable, so that the quantity $W$ affords a kind of measure of the stability of plane surfaces of contact of the phases $A$ and $B$.
Essentially the same principles apply to the more general problem in which the phases $A$ and $B$ have moderately different pressures, so that their surfaces of contact must be curved, but the radii of curvature have a sensible magnitude.
In order that a thin film of the phase $C$ may be in equilibrium between masses of the phases $A$ and $B$, the following equations must be satisfied:—
* $\sigma _{AC}(c_{1}+c_{2})=p_{A}-p_{C},$ $\sigma _{BC}(c_{1}+c_{2})=p_{C}-p_{B},$
$c_{1}$
$c_{2}$
$A$
$c_{1}+c_{2}$
* $\sigma _{BC}(p_{A}-p_{C})=\sigma _{AC}(p_{C}-p_{B}),$
$p_{C}={\frac {\sigma _{BC}p_{A}+\sigma _{AC}p_{B}}{\sigma _{BC}+\sigma _{AC}}}\cdot$
$p_{C}$
$p_{C}$
$A$
$B$
$A$
$B$
$p_{C}$
$\sigma _{AC}+\sigma _{BC}$
$A$
$B$
$\sigma _{AB}$
$\sigma _{AB}=\sigma _{AC}+\sigma _{BC}$
$C$
$A$
$B$
$C$
$\sigma _{AB}<\sigma _{AC}+\sigma _{BC}$
$A$
$B$
$C$
The work which must be expended in order to form on the surface between indefinitely large masses of phases $A$ and $B$ a lentiform mass of phase $C$ in equilibrium, may evidently be represented by the formula
| ${\begin{aligned}W&=\sigma _{AC}S_{AC}+\sigma _{BC}S_{BC}-\sigma _{AB}S_{AB}\\&-p_{C}V_{C}+p_{A}V_{A}+p_{B}V_{B},\\\end{aligned}}$ | (573) |
$S_{AC},S_{BC}$
$A$
$C$
$B$
$C$
$S_{AB}V_{A},V_{B}$
$A$
$B$
$V_{C}$
$C$
$V_{A},V_{B}$
$A$
$B$
$\sigma _{AC},\sigma _{BC},\sigma _{AB},p_{A},p_{B}$
$A$
$B$
$p_{C}$
$W$
| ${\begin{aligned}dW&=\sigma _{AC}ds_{AC}+\sigma _{BC}-\sigma _{AB}S_{AB}\\&-p_{C}dV_{c}+p_{A}dV_{A}+p_{B}dV_{B}-V_{C}dp_{c}.\\\end{aligned}}$ | (574) |
| ${\begin{aligned}\sigma _{AC}dS_{AC}&+\sigma _{BC}-\sigma _{AB}S_{AB}\\&-p_{C}dV_{c}+p_{A}dV_{A}+p_{B}dV_{B}=0.\\\end{aligned}}$ | (575) |
​Hence,
| $dW=-V_{C}dp_{C}.$ | (576) |
$V_{C}$
$p_{C}$
$p_{C}$
$V_{C}$
| $W''-W'={\text{a negative quantity,}}$ | (577) |
$W'$
$W''$
$W$
$W''=0$
$W$
$W$
$A$
$B$
$p_{C}-p_{A},p_{C}-p_{B}$
$A$
$B$
$C$
$C$
$A$
$B$
$W$
Substitution of Pressures for Potentials in Fundamental Equations for Surfaces.
The fundamental equation of a surface which gives the value of the tension in terms of the temperature and potentials seems best adapted to the purposes of theoretical discussion, especially when the number of components is large or undetermined. But the experimental determination of the fundamental equations, or the application of any result indicated by theory to actual cases, will be facilitated by the use of other quantities in place of the potentials, which shall be capable of more direct measurement, and of which the numerical expression (when the necessary measurements have been made) shall depend upon less complex considerations. The numerical value of a potential depends not only upon the system of units employed, but also upon the arbitrary constants involved in the definition of the energy and entropy of the substance to which the potential relates, or, it may be, of the elementary substances of which that substance is formed. (See page 96.) This fact and the want of means of direct measurement may give a certain vagueness to the idea of the ​potentials, and render the equations which involve them less fitted to give a clear idea of physical relations.
Now the fundamental equation of each of the homogeneous masses which are separated by any surface of discontinuity affords a relation between the pressure in that mass and the temperature and potentials. We are therefore able to eliminate one or two potentials from the fundamental equation of a surface by introducing the pressures in the adjacent masses. Again, when one of these masses is a gas-mixture which satisfies Dalton's law as given on page 155, the potential for each simple gas may be expressed in terms of the temperature and the partial pressure belonging to that gas. By the introduction of these partial pressures we may eliminate as many potentials from the fundamental equation of the surface as there are simple gases in the gas-mixture.
An equation obtained by such substitutions may be regarded as a fundamental equation for the surface of discontinuity to which it relates, for when the fundamental equations of the adjacent masses are known, the equation in question is evidently equivalent to an equation between the tension, temperature, and potentials, and we must regard the knowledge of the properties of the adjacent masses as an indispensable preliminary, or an essential part, of a complete knowledge of any surface of discontinuity. It is evident, however, that from these fundamental equations involving pressures instead of potentials we cannot obtain by differentiation (without the use of the fundamental equations of the homogeneous masses) precisely the same relations as by the differentiation of the equations between the tensions, temperatures, and potentials. It will be interesting to inquire, at least in the more important cases, what relations may be obtained by differentiation from the fundamental equations just described alone.
If there is but one component, the fundamental equations of the two homogeneous masses afford one relation more than is necessary for the elimination of the potential. It may be convenient to regard the tension as a function of the temperature and the difference of the pressures. Now we have by (508) and (98)
* $d\sigma =-\eta _{S}dt-\Gamma d\mu _{1},$ $d(p'-p'')=(\eta _{V}'-\eta _{V}'')dt+(\gamma '-\gamma '')d\mu _{1}.$
| $d\sigma =-\left(\eta _{S}-{\frac {\Gamma }{\gamma '-\gamma ''}}(\eta _{V}'-\eta _{V}'')\right)dt-{\frac {\Gamma }{\gamma '-\gamma ''}}(p'-p''),$ | (578) |
$\sigma$
$t$
$p'-p''$
${\frac {\Gamma }{\gamma '-\gamma ''}}$
$dt$
$\eta _{S(1)}$
When there are two components, neither of which is confined to the surface of discontinuity, we may regard the tension as a function of the temperature and the pressures in the two homogeneous masses. The values of the differential coefficients of the tension with respect to these variables may be represented in a simple form if we choose such substances for the components that in the particular state considered each mass shall consist of a single component. This will always be possible when the composition of the two masses is not identical, and will evidently not affect the values of the differential coefficients. We then have
* $d\sigma =-\eta _{S}dt-\Gamma _{'}d\mu _{'}-\Gamma _{''}d\mu _{''}$ $dp'=\eta _{V}'dt+\gamma 'd\mu _{'},$ $dp''=\eta _{V}''dt+\gamma ''d\mu _{''},$
$_{'}$
$_{''}$
$_{1}$
$_{2}$
$d\mu _{'}$
$d\mu _{''}$
| $d\sigma =-\left(\eta _{S}-{\frac {\Gamma _{'}}{\gamma '}}\eta _{V}'-{\frac {\Gamma _{''}}{\gamma ''}}\eta _{V}''\right)dt-{\frac {\Gamma _{'}}{\gamma '}}dp'-{\frac {\Gamma _{''}}{\gamma ''}}dp''.$ | (579) |
$p'$
$p''$
| $d\sigma =-\left(\eta _{S}-{\frac {\Gamma _{'}}{\gamma '}}\eta _{V}'-{\frac {\Gamma _{''}}{\gamma ''}}\eta _{V}''\right)dt-\left({\frac {\Gamma _{'}}{\gamma '}}+{\frac {\Gamma _{''}}{\gamma ''}}\right)dp.$ | (580) |
${\frac {\Gamma _{'}}{\gamma '}}$
${\frac {\Gamma _{'}}{\gamma ''}}$
$\Gamma _{'}$
$\Gamma _{''}$
${\frac {\Gamma _{'}}{\gamma '}}+{\frac {\Gamma _{''}}{\gamma ''}}$
$dt$
The form in which the values of $\left({\frac {d\sigma }{dt}}\right)_{p}$ and $\left({\frac {d\sigma }{dp}}\right)_{t}$ are given in equation (580) is adapted to give a clear idea of the relations of these quantities to the particular state of the system for which they are to be determined, but not to show how they vary with the state of the system. For this purpose it will be convenient to have the values of these differential coefficients expressed with reference to ordinary components. Let these be specified as usual by $_{1}$ and $_{2}$.
If we eliminate $d\mu _{1}$ and $d\mu _{2}$ from the equations
* $-d\sigma =\eta _{S}dt+\Gamma _{1}d\mu _{1}+\Gamma _{2}d\mu _{2},$ $dp=\eta _{V'}dt+\gamma _{1}'d\mu _{1}+\gamma _{2}'d\mu _{2}$ $dp=\eta _{V''}dt+\gamma _{1}''d\mu _{1}+\gamma _{2}''d\mu _{2}$
| $d\sigma ={\frac {B}{A}}dt+{\frac {C}{A}}dp,$ | (581) |
| $A=\gamma _{1}''\gamma _{2}'-\gamma _{1}'\gamma _{2}'',$ | (582) |
| $B={\begin{vmatrix}\eta _{S}&\Gamma _{1}&\Gamma _{2}\\\eta _{V}'&\gamma _{1}'&\gamma _{2}'\\\eta _{V}''&\gamma _{1}''&\gamma _{2}''\end{vmatrix}},$ | (583) |
| $C=\Gamma _{1}(\gamma _{2}''-\gamma _{2}')+\Gamma _{2}(\gamma _{1}'-\gamma _{1}'').$ | (584) |
$A$
$B$
$C$
$A$
$_{1}$
$\left({\frac {d\sigma }{dt}}\right)_{p}$
$\left({\frac {d\sigma }{dp}}\right)_{t}$
With regard to gas-mixtures which conform to Dalton's law, we shall only consider the fundamental equation for plane surfaces, and shall suppose that there is not more than one component in the liquid which does not appear in the gas-mixture. We have already seen that in limiting the fundamental equation to plane surfaces we can get rid of one potential by choosing such a dividing surface that the superficial density of one of the components vanishes. Let this be done with respect to the component peculiar to the liquid, if such there is; if there is no such component, let it be done with respect to one of the gaseous components. Let the remaining potentials be eliminated by means of the fundamental equations of the simple gases. We may thus obtain an equation between the superficial tension, the temperature, and the several pressures of the simple gases in the gas-mixture or all but one of these pressures. Now, if we eliminate $d\mu _{2},d\mu _{3}$, etc. from the equations
* $d\sigma =-\eta _{S(1)}dt-\Gamma _{2(1)}d\mu _{2}-\Gamma _{3(1)}d\mu _{3}-{\text{etc.,}}$ $dp_{2}=\eta _{V2}dt+\gamma _{2}d\mu _{2},$ $dp_{3}=\eta _{V3}dt+\gamma _{3}d\mu _{3},$ ${\text{etc.,}}$
$_{1}$
$\gamma _{2},\gamma _{3}$
$p_{2},p_{3}$
$\eta _{V2},\eta _{V3}$
| $-\left(\eta _{S(1)}-{\frac {\Gamma _{2(1)}}{\gamma _{2}}}-{\frac {\Gamma _{3(1)}}{\gamma _{3}}}\eta _{V3}-{\text{etc.}}\right)dt-{\frac {\Gamma _{2(1)}}{\gamma _{2}}}dp_{2}-{\frac {\Gamma _{3(1)}}{\gamma _{3}}}dp_{3}-{\text{etc.}}$ | (585) |
$\sigma$
$t,p_{2},p_{3}$
Thermal and Mechanical Relations pertaining to the Extension of a Surface of Discontinuity.
The fundamental equation of a surface of discontinuity with one or two component substances, besides its statical applications, is of use to determine the heat absorbed when the surface is extended under certain conditions.
Let us first consider the case in which there is only a single component substance. We may treat the surface as plane, and place the dividing surface so that the surface density of the single component vanishes. (See page 234.) If we suppose the area of the surface to be increased by unity without change of temperature or of the quantities of liquid and vapor, the entropy of the whole will be increased by $\eta _{S(1)}$. Therefore, if we denote by $Q$ the quantity of heat which must be added to satisfy the conditions, we shall have
| $Q=t\eta _{S(1)},$ | (586) |
| $Q=-t{\frac {d\sigma }{dt}}=-{\frac {d\sigma }{d\log t}},$ | (587) |
Again, if the surface is extended without application of heat, while the pressure in the liquid and vapor remains constant, the temperature will evidently be maintained constant by condensation of the vapor. If we denote by $M$ the mass of vapor condensed per unit of surface formed, and by $\eta _{M'}$ and $\eta _{M''}$ the entropies of the liquid and vapor per unit of mass, the condition of no addition of heat will require that
| $M(\eta _{M}''-\eta _{M}')=\eta _{S(1)}.$ | (588) |
​The increase of the volume of liquid will be
| ${\frac {\eta _{S(1)}}{\gamma '(\eta _{M}''-\eta _{M}')}},$ | (589) |
| ${\frac {\eta _{S(1)}}{\gamma ''(\eta _{M}''-\eta _{M}')}}\cdot$ | (590) |
| $W={\frac {p\eta _{S(1)}}{\eta _{M}''-\eta _{M}'}}\left({\frac {1}{\gamma ''}}-{\frac {1}{\gamma '}}\right),$ | (591) |
| $W=-p{\frac {d\sigma }{dt}}{\frac {dt}{dp}}=-p{\frac {d\sigma }{dp}}=-{\frac {d\sigma }{d\log p}}\cdot$ | (592) |
$\sigma$
Let us now consider the case in which there are two component substances, neither of which is confined to the surface. Since we cannot make the superficial density of both these substances vanish by any dividing surface, it will be best to regard the surface of tension as the dividing surface. We may, however, simplify the formula by choosing such substances for components that each homogeneous mass shall consist of a single component. Quantities relating to these components will be distinguished as on page 266. If the surface is extended until its area is increased by unity, while heat is added at the surface so as to keep the temperature constant, and the pressure of the homogeneous masses is also kept constant, the phase of these masses will necessarily remain unchanged, but the quantity of one will be diminished by $\Gamma _{'}$, and that of the other by $\Gamma _{''}$. Their entropies will therefore be diminished by ${\frac {\Gamma _{'}}{\gamma '}}\eta _{V}'$ and ${\frac {\Gamma _{''}}{\gamma ''}}\eta _{V}''$, respectively. Hence, since the surface receives the increment of entropy $\eta _{S}$, the total quantity of entropy will be increased by
* $\eta _{S}-{\frac {\Gamma _{'}}{\gamma '}}\eta _{V}'-{\frac {\Gamma _{''}}{\gamma ''}}\eta _{V}'',$
* $-\left({\frac {d\sigma }{dt}}\right)_{p}\cdot$
$Q$
| $Q=-t\left({\frac {d\sigma }{dt}}\right)_{p}=-\left({\frac {d\sigma }{d\log t}}\right)_{p}\cdot$ | (593) |
​We must notice the difference between this formula and (587). In (593) the quantity of heat $Q$ is determined by the condition that the temperature and pressures shall remain constant. In (587) these conditions are equivalent and insufficient to determine the quantity of heat. The additional condition by which $Q$ is determined may be most simply expressed by saying that the total volume must remain constant. Again, the differential coefficient in (593) is defined by considering $p$ as constant; in the differential coefficient in (587) $p$ cannot be considered as constant, and no condition is necessary to give the expression a definite value. Yet, notwithstanding the difference of the two cases, it is quite possible to give a single demonstration which shall be applicable to both. This may be done by considering a cycle of operations after the method employed by Sir William Thomson, who first pointed out these relations.
The diminution of volume (per unit of surface formed) will be
| $V={\frac {\Gamma _{'}}{\gamma '}}+{\frac {\Gamma _{''}}{\gamma ''}}=-\left({\frac {d\sigma }{dp}}\right)_{t};$ | (594) |
| $W=-p\left({\frac {d\sigma }{dp}}\right)_{t}=-\left({\frac {d\sigma }{d\log p}}\right)_{t}\cdot$ | (595) |
The values of $Q$ and $W$ may also be expressed in terms of quantities relating to the ordinary components. By substitution in (593) and (595) of the values of the differential coefficients which are given by (581), we obtain
| $Q=-t{\frac {B}{A}},$$W=-p{\frac {C}{A}},$ | (596) |
$A,B$
$C$
$Q$
$W$
$A$
$Q$
$W$
If, instead of considering the amount of heat necessary to keep the phases from altering while the surface of discontinuity is extended, we consider the variation of temperature caused by the extension of the surface while the pressure remains constant, it appears that this variation of temperature changes sign with $\gamma _{1}''\gamma _{2}'-\gamma _{1}'\gamma _{2}''$, but vanishes with this quantity, i.e., vanishes when the composition of the phases becomes the same. This may be inferred from the statements on page 99, or from a consideration of the figure on page 125. When the composition of the homogeneous masses is initially absolutely identical, the effect on the temperature of a finite extension or contraction of the surface of discontinuity will be the same,—either of the two will lower or raise the temperature according as the temperature is a maximum or minimum for constant pressure.
The effect of the extension of a surface of discontinuity which is most easily verified by experiment is the effect upon the tension before complete equilibrium has been reëstablished throughout the adjacent masses. A fresh surface between coexistent phases may be regarded in this connection as an extreme case of a recently extended surface. When sufficient time has elapsed after the extension of a surface originally in equilibrium between coexistent phases, the superficial tension will evidently have sensibly its original value, unless there are substances at the surface which are either not found at all in the adjacent masses, or are found only in quantities comparable to those in which they exist at the surface. But a surface newly formed or extended may have a very different tension.
This will not be the case, however, when there is only a single component substance, since all the processes necessary for equilibrium are confined to a film of insensible thickness, and will require no appreciable time for their completion.
When there are two components, neither of which is confined to the surface of discontinuity, the reëstablishment of equilibrium after the extension of the surface does not necessitate any processes reaching into the interior of the masses except the transmission of heat between the surface of discontinuity and the interior of the masses. It appears from equation (593) that if the tension of the surface diminishes with a rise of temperature, heat must be supplied to the surface to maintain the temperature uniform when the surface is extended, i.e., the effect of extending the surface is to cool it; but if the tension of any surface increases with the temperature, the ​effect of extending the surface will be to raise its temperature. In either case, it will be observed, the immediate effect of extending the surface is to increase its tension. A contraction of the surface will of course have the opposite effect. But the time necessary for the reestablishment of sensible thermal equilibrium after extension or contraction of the surface must in most cases be very short.
In regard to the formation or extension of a surface between two coexistent phases of more than two components, there are two extreme cases which it is desirable to notice. When the superficial density of each of the components is exceedingly small compared with its density in either of the homogeneous masses, the matter (as well as the heat) necessary for the formation or extension of the normal surface can be taken from the immediate vicinity of the surface without sensibly changing the properties of the masses from which it is taken. But if any one of these superficial densities has a considerable value, while the density of the same component is very small in each of the homogeneous masses, both absolutely and relatively to the densities of the other components, the matter necessary for the formation or extension of the normal surface must come from a considerable distance. Especially if we consider that a small difference of density of such a component in one of the homogeneous masses will probably make a considerable difference in the value of the corresponding potential {see eq. (217)}, and that a small difference in the value of the potential will make a considerable difference in the tension (see eq. (508)}, it will be evident that in this case a considerable time will be necessary after the formation of a fresh surface or the extension of an old one for the reestablishment of the normal value of the superficial tension. In intermediate cases, the reestablishment of the normal tension will take place with different degrees of rapidity.
But whatever the number of component substances, provided that it is greater than one, and whether the reestablishment of equilibrium is slow or rapid, extension of the surface will generally produce increase and contraction decrease of the tension. It would evidently be inconsistent with stability that the opposite effects should be produced. In general, therefore, a fresh surface between coexistent phases has a greater tension than an old one. By the use of fresh surfaces, in experiments in capillarity, we may sometimes avoid the effect of minute quantities of foreign substances, which may be ​present without our knowledge or desire, in the fluids which meet at the surface investigated.
When the establishment of equilibrium is rapid, the variation of the tension from its normal value will be manifested especially during the extension or contraction of the surface, the phenomenon resembling that of viscosity, except that the variations of tension arising from variations in the densities at and about the surface will be the same in all directions, while the variations of tension due to any property of the surface really analogous to viscosity would be greatest in the direction of the most rapid extension.
Impermeable Films.
We have so far supposed, in treating of surfaces of discontinuity, that they afford no obstacle to the passage of any of the component substances from either of the homogeneous masses to the other. The case, however, must be considered, in which there is a film of matter at the surface of discontinuity which is impermeable to some or all of the components of the contiguous masses. Such may be the case, for example, when a film of oil is spread on a surface of water, even when the film is too thin to exhibit the properties of the oil in mass. In such cases, if there is communication between the contiguous masses through other parts of the system to which they belong, such that the components in question can pass freely from one mass to the other, the impossibility of a direct passage through the film may be regarded as an immaterial circumstance, so far as states of equilibrium are concerned, and our formulas will require no change. But when there is no such indirect communication, the potential for any component for which the film is impermeable may have entirely different values on opposite sides of the film, and the case evidently requires a modification of our usual method.
A single consideration will suggest the proper treatment of such cases. If a certain component which is found on both sides of a film cannot pass from either side to the other, the fact that the part of the component which is on one side is the same kind of matter with the part on the other side may be disregarded. All the general relations must hold true, which would hold if they were really different substances. We may therefore write $\mu _{1}$ for the potential of the component on one side of the film, and $\mu _{2}$ for the potential of the same substance (to be treated as if it were a different substance) on the other side; $m_{1}^{S}$ for the excess of the quantity of the substance on the first side of the film above the quantity which would be on that side of the dividing surface (whether this is determined by the surface of tension or otherwise) if the density of the substance were the same near the dividing surface as at a distance, and $m_{2}^{S}$ for a similar quantity relating to the other side of the film and dividing ​surface. On the same principle, we may use $\Gamma _{1}$ and $\Gamma _{2}$ to denote the values of $m_{1}^{S}$ and $m_{2}^{S}$ per unit of surface, and $m_{1}',m_{2}'',\gamma _{1}',\gamma _{2}''$ to denote the quantities of the substance and its densities in the two homogeneous masses.
With such a notation, which may be extended to cases in which the film is impermeable to any number of components, the equations relating to the surface and the contiguous masses will evidently have the same form as if the substances specified by the different suffixes were all really different. The superficial tension will be a function of $\mu _{1}$ and $\mu _{2}$, with the temperature and the potentials for the other components, and $-\Gamma _{1},-\Gamma _{2}$ will be equal to its differential coefficients with respect to $\mu _{1}$ and $\mu _{2}$. In a word, all the general relations which have been demonstrated may be applied to this case, if we remember always to treat the component as a different substance according as it is found on one side or the other of the impermeable film.
When there is free passage for the component specified by the suffixes $_{1}$ and $_{2}$ through other parts of the system (or through any flaws in the film), we shall have in case of equilibrium $\mu _{1}=\mu _{2}$. If we wish to obtain the fundamental equation for the surface when satisfying this condition, without reference to other possible states of the surface, we may set a single symbol for $\mu _{1}$ and $\mu _{2}$ in the more general form of the fundamental equation. Cases may occur of an impermeability which is not absolute, but which renders the transmission of some of the components exceedingly slow. In such cases, it may be necessary to distinguish at least two different fundamental equations, one relating to a state of approximate equilibrium which may be quickly established, and another relating to the ultimate state of complete equilibrium. The latter may be derived from the former by such substitutions as that just indicated.
The Conditions of Internal Equilibrium for a System of Heterogeneous Fluid Masses without neglect of the Influence of the Surfaces of Discontinuity or of Gravity.
Let us now seek the complete value of the variation of the energy of a system of heterogeneous fluid masses, in which the influence of gravity and of the surfaces of discontinuity shall be included, and deduce from it the conditions of internal equilibrium for such a system. In accordance with the method which has been developed, the intrinsic energy (i.e. the part of the energy which is independent of gravity), the entropy, and the quantities of the several components must each be divided into two parts, one of which we regard as belonging to the surfaces which divide approximately homogeneous ​masses, and the other as belonging to these masses. The elements of intrinsic energy, entropy, etc., relating to an element of surface $Ds$ will be denoted by $D\epsilon ^{S},D\eta ^{S},Dm_{1}^{S},Dm_{2}^{S}$, etc., and those relating to an element of volume $Dv$, by $D\epsilon ^{V},D\eta ^{V},Dm_{1}^{V},Dm_{2}^{V}$, etc. We shall also use $Dm^{S}$ or $\Gamma Ds$ and $Dm^{V}$ or $\gamma Dv$ to denote the total quantities of matter relating to the elements $Ds$ and $Dv$ respectively. That is,
| $Dm^{S}=\Gamma Ds=Dm_{1}^{S}+Dm_{2}^{S}+{\text{etc.,}}$ | (597) |
| $Dm^{V}=\gamma Ds=Dm_{1}^{V}+Dm_{2}^{V}+{\text{etc.}}$ | (598) |
$Dm^{S}$
$Dm^{V}$
| $\delta \int D\epsilon ^{V}+\delta \int \epsilon ^{S}+\delta \int gzDm^{V}+\delta \int gzDm^{S},$ | (599) |
$g$
$z$
It will be convenient to limit ourselves at first to the consideration of reversible variations. This will exclude the formation of new masses or surfaces. We may therefore regard any infinitesimal variation in the state of the system as consisting of infinitesimal variations of the quantities relating to its several elements, and bring the sign of variation in the preceding formula after the sign of integration. If we then substitute for $\delta D\epsilon ^{V},\delta D\epsilon ^{S},\delta Dm^{V},\delta Dm^{S}$, the values given by equations (13), (497), (597), (598), we shall have for the condition of equilibrium with respect to reversible variations of the internal state of the system
| ${\begin{aligned}\int t\delta D\eta ^{V}&-\int p\delta Dv+\int \mu _{1}\delta Dm_{1}^{V}+\int \mu _{2}\delta Dm_{2}^{V}+{\text{etc.}}\\+\int t\delta D\eta ^{S}&+\int \sigma \delta Ds+\int \mu _{1}\delta Dm_{1}^{S}+\int \mu _{2}\delta Dm_{2}^{S}+{\text{etc.}}\\&+\int g\delta zDm^{V}+\int gz\delta Dm_{1}^{V}+\int gz\delta Dm_{2}^{V}+{\text{etc.}}\\&+\int g\delta zDm^{S}+\int gz\delta Dm_{1}^{S}+\int gz\delta Dm_{2}^{S}+{\text{etc.}}&=0.\\\end{aligned}}$ | (600) |
The variations in (600) are subject to the conditions which arise from the nature of the system and from the supposition that the changes in the system are not such as to affect external bodies. This supposition is necessary, unless we are to consider the variations in the state of the external bodies, and is evidently allowable in seeking the conditions of equilibrium which relate to the interior of the system. But before we consider the equations of condition in detail, we may divide the condition of equilibrium (600) into the three conditions
| $\int t\delta D\eta ^{V}+\int t\delta D\eta ^{S}=0,$ | (601) |
| $-\int p\delta Dv+\int \sigma \delta Ds+\int g\delta zDm^{V}+\int g\delta zDm^{S}=0,$ | (602) |
| $\int \mu _{1}\delta Dm_{1}^{V}+\int \mu _{1}\delta Dm_{1}^{S}+\int gz\delta Dm_{1}^{V}+\int gz\delta Dm_{1}^{S}$ $+\int \mu _{2}\delta Dm_{2}^{V}+\int \mu _{2}\delta Dm_{2}^{S}+\int gz\delta Dm_{2}^{V}+\int gz\delta Dm_{2}^{S}$ $+{\text{etc.}}=0.$ | (603) |
The variations in condition (601) are subject to the condition that the entropy of the whole system shall remain constant. This may be expressed by the equation
| $\int \delta D\eta ^{V}+\int \delta D\eta ^{S}=0.$ | (604) |
| $t={\text{const.}}$ | (605) |
The conditions of mechanical equilibrium, or those that relate to the possible deformation of the system, are contained in (602), which may also be written
| $-\int p\delta Dv+\int \sigma \delta Ds+\int g\gamma \delta zDv+\int g\Gamma \delta zDs=0.$ | (606) |
​It will be observed that this condition has the same form as if the different fluids were separated by heavy and elastic membranes without rigidity and having at every point a tension uniform in all directions in the plane of the surface. The variations in this formula, beside their necessary geometrical relations, are subject to the conditions that the external surface of the system, and the lines in which the surfaces of discontinuity meet it, are fixed. The formula may be reduced by any of the usual methods, so as to give the particular conditions of mechanical equilibrium. Perhaps the following method will lead as directly as any to the desired result.
It will be observed the quantities affected by $\delta$ in (606) relate exclusively to the position and size of the elements of volume and surface into which the system is divided, and that the variations $\delta p$ and $\delta \sigma$ do not enter into the formula either explicitly or implicitly. The equations of condition which concern this formula also relate exclusively to the variations of the system of geometrical elements, and do not contain either $\delta p$ or $\delta \sigma$. Hence, in determining whether the first member of the formula has the value zero for every possible variation of the system of geometrical elements, we may assign to $\delta p$ and $\delta \sigma$ any values whatever which may simplify the solution of the problem, without inquiring whether such values are physically possible.
Now when the system is in its initial state, the pressure $p$, in each of the parts into which the system is divided by the surfaces of tension, is a function of the co-ordinates which determine the position of the element $Dv$, to which the pressure relates. In the varied state of the system, the element $Dv$ will in general have a different position. Let the variation $\delta p$ be determined solely by the change in position of the element $Dv$. This may be expressed by the equation
| $\delta p={\frac {dp}{dx}}\delta x+{\frac {dp}{dy}}\delta y+{\frac {dp}{dz}}\delta z,$ | (607) |
${\frac {dp}{dx}},{\frac {dp}{dy}},{\frac {dp}{dz}}$
$\delta x,\delta y,\delta z$
$Dv$
Again, in the initial state of the system the tension $\sigma$, in each of the different surfaces of discontinuity, is a function of two co-ordinates $\omega _{1},\omega _{2}$, which determine the position of the element $Ds$. In the varied state of the system, this element will in general have a different position. The change of position may be resolved into a component lying in the surface and another normal to it. Let the variation $\delta \sigma$ be determined solely by the first of these components of the motion of $Ds$. This may be expressed by the equation
| $\delta \sigma ={\frac {d\sigma }{d\omega _{1}}}\delta \omega _{1}+{\frac {d\sigma }{d\omega _{2}}}\delta \omega _{2},$ | (608) |
​in which ${\frac {d\sigma }{d\omega _{1}}},{\frac {d\sigma }{d\omega _{2}}}$ are determined by the function mentioned, and $\delta \omega _{1},\delta \omega _{2}$, by the component of the motion of $Ds$ which lies in the plane of the surface.
With this understanding, which is also to apply to $\delta p$ and $\delta \sigma$ when contained implicitly in any expression, we shall proceed to the reduction of the condition (606).
With respect to any one of the volumes into which the system is divided by the surfaces of discontinuity, we may write
* $\int p\delta Dv=\delta \int pDv-\int \delta pDv.$
* $\delta \int pDv=\int p\delta NDs,$
$\delta N$
* $\int p\delta Dv=\int p\delta NDs-\int \delta pDv.$
| $\int p\delta Dv=\int (p'-p'')\delta NDs-\int \delta pDv,$ | (609) |
$p'$
$p''$
$Ds$
$\delta N$
Again, for each of the surfaces of discontinuity, taken separately,
* $\int \sigma \delta Ds=\delta \int \sigma Ds-\int \delta \sigma Ds,$
* $\delta \int \sigma Ds=\int \sigma (c_{1}+c_{2})\delta NDs+\int \sigma \delta TDl,$
$c_{1}$
$c_{2}$
$\delta N$
$Dl$
$\delta T$
| $\int \sigma \delta Ds=\int \sigma (c_{1}+c_{2})\delta N+Ds+\int \textstyle \sum \displaystyle (\sigma \delta T)Dl-\int \delta \sigma Ds,$ | (610) |
$\textstyle \sum$
By equations (609) and (610), the general condition of mechanical equilibrium is reduced to the form
* $-\int (p'-p'')\delta N+\int \delta pDv+\int \sigma (c_{1}+c_{2})\delta NDs+\int \textstyle \sum \displaystyle (\sigma \delta T)Dl-\int \delta \sigma Ds+\int g\gamma \delta zDv+\int g\Gamma \delta zDs=0.$
| $\int (g\gamma +dp)Dv+\int [(p''-p')\delta N+\sigma (c_{1}+c_{2})\delta N+g\Gamma \delta z-\delta \sigma ]Ds+\int \textstyle \sum \displaystyle (\sigma \delta T)Dl=0.$ | (611) |
$Dv,Ds$
$Dl$
In order that the coefficient of $Dv$ shall vanish, it is necessary and sufficient that in each of the masses into which the system is divided by the surfaces of tension, $p$ shall be a function of $z$ alone, such that
| ${\frac {dp}{dz}}=-g\gamma .$ | (612) |
$Ds$
* $\delta \sigma =0,$and$\delta z=\cos \theta \delta N,$
$\theta$
| $p'-p''=\sigma (c_{1}+c_{2})+g\Gamma \cos \theta ,$ | (613) |
$\sigma$
$z$
| ${\frac {d\sigma }{dz}}=g\Gamma .$ | (614) |
$Dl$
| $\textstyle \sum \displaystyle (\sigma \delta T)=0.$ | (615) |
In condition (603), the variations which relate to any component are to be regarded as having the value zero in any part of the system in which that substance is not an actual component. The same is true ​with respect to the equations of condition, which are of the form
| $\int \delta Dm_{1}^{V}+\int \delta Dm_{2}^{V}=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (616) |
| $\int \delta Dm_{1}^{V}+\int \delta Dm_{2}^{V}=0,$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (616) |
| etc., | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ (616) |
(It is here supposed that the various components are independent, i.e., that none can be formed out of others, and that the parts of the system in which any component actually occurs are not entirely separated by parts in which it does not occur.) To satisfy the condition (603), subject to these equations of condition, it is necessary and sufficient that the conditions
| $\mu _{1}+gz=M_{1},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$ (617) |
| $\mu _{2}+gz=M_{2},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$ (617) |
| etc., | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,$ (617) |
($M_{1},M_{2}$, etc. denoting constants,) shall each hold true in those parts of the system in which the substance specified is an actual component. We may here add the condition of equilibrium relative to the possible absorption of any substance (to be specified by the suffix $_{a}$) by parts of the system of which it is not an actual component, viz., that the expression $\mu _{a}+gz$ must not have a less value in such parts of the system than in a contiguous part in which the substance is an actual component.
From equation (613) with (605) and (617) we may easily obtain the differential equation of a surface of tension (in the geometrical sense of the term), when $p',p''$, and $\sigma$ are known in terms of the temperature and potentials. For $c_{1}+c_{2}$ and $\theta$ may be expressed in terms of the first and second differential coefficients of $z$ with respect to the horizontal co-ordinates, and $p',p'',\sigma$, and $\Gamma$ in terms of the temperature and potentials. But the temperature is constant, and for each of the potentials we may substitute—$gz$ increased by a constant. We thus obtain an equation in which the only variables are $z$ and its first and second differential coefficients with respect to the horizontal co-ordinates. But it will rarely be necessary to use so exact a method. Within moderate differences of level, we may regard $\gamma ',\gamma ''$, and $\sigma$ as constant. We may then integrate the equation {derived from (612)}
* $d(p'-p'')=g(\gamma ''-\gamma ')dz,$
| $p'-p''=g(\gamma ''-\gamma ')z,$ | (618) |
$z$
$p'=p''$
$\Gamma$
| $c_{1}+c_{2}={\frac {g(\gamma ''-\gamma ')}{\sigma }}z,$ | (619) |
​where the coefficient of $z$ is to be regarded as constant. Now the value of $z$ cannot be very large, in any surface of sensible dimensions, unless $\gamma ''-\gamma '$ is very small. We may therefore consider this equation as practically exact, unless the densities of the contiguous masses are very nearly equal. If we substitute for the sum of the curvatures its value in terms of the differential coefficients of $z$ with respect to the horizontal rectangular co-ordinates, $x$ and $y$, we have
| ${\frac {\left(1+{\frac {dz^{2}}{dy^{2}}}\right){\frac {d^{2}z}{dx^{2}}}-2{\frac {dz}{dx}}{\frac {dz}{dy}}{\frac {d^{2}z}{dxdy}}+\left(1+{\frac {dz^{2}}{dx^{2}}}\right){\frac {d^{2}z}{dy^{2}}}}{\left(1+{\frac {dz^{2}}{dx^{2}}}+{\frac {dz^{2}}{dy^{2}}}\right)^{\tfrac {3}{2}}}}={\frac {g(\gamma ''-\gamma ')}{\sigma }}z.$ | (620) |
The particular conditions of equilibrium which are given in the last paragraph but one may be regarded in general as the conditions of chemical equilibrium between the different parts of the system, since they relate to the separate components. But such a designation is not entirely appropriate unless the number of components is greater than one. In no case are the conditions of mechanical equilibrium entirely independent of those which relate to temperature and the potentials. For the conditions (612) and (614) may be regarded as consequences of (605) and (617) in virtue of the necessary relations (98) and (508).
​When a surface of discontinuity has more components than one which do not occur in the contiguous masses, the adjustment of the potentials for these components in accordance with equations (617) may take place very slowly, or not at all, for want of sufficient mobility in the components of the surface. But when this surface has only one component which does not occur in the contiguous masses, and the temperature and potentials in these masses satisfy the conditions of equilibrium, the potential for the component peculiar to the surface will very quickly conform to the law expressed in (617), since this is a necessary consequence of the condition of mechanical equilibrium (614) in connection with the conditions relating to temperature and the potentials which we have supposed to be satisfied. The necessary distribution of the substance peculiar to the surface will be brought about by expansions and contractions of the surface. If the surface meets a third mass containing this component and no other which is foreign to the masses divided by the surface, the potential for this component in the surface will of course be determined by that in the mass which it meets.
The particular conditions of mechanical equilibrium (612)–(615), which may be regarded as expressing the relations which must subsist between contiguous portions of a fluid system in a state of mechanical equilibrium, are serviceable in determining whether a given system is or is not in such a state. But the mechanical theorems which relate to finite parts of the system, although they may be deduced from these conditions by integration, may generally be more easily obtained by a suitable application of the general condition of mechanical equilibrium (606), or by the application of ordinary mechanical principles to the system regarded as subject to the forces indicated by this equation.
It will be observed that the conditions of equilibrium relating to temperature and the potentials are not affected by the surfaces of discontinuity. {Compare (228) and (234).} Since a phase cannot vary continuously without variations of the temperature or the potentials, it follows from these conditions that the phase at any point in a fluid system which has the same independently variable components throughout, and is in equilibrium under the influence of gravity, must be one of a certain number of phases which are completely determined by the phase at any given point and the difference of level of the two points considered. If the phases ​throughout the fluid system satisfy the general condition of practical stability for phases existing in large masses (viz., that the pressure shall be the least consistent with the temperature and potentials), they will be entirely determined by the phase at any given point and the differences of level. (Compare page 149, where the subject is treated without regard to the influence of the surfaces of discontinuity.)
Conditions of equilibrium relating to irreversible changes.—The conditions of equilibrium relating to the absorption, by any part of the system, of substances which are not actual components of that part have been given on page 282. Those relating to the formation of new masses and surfaces are included in the conditions of stability relating to such changes, and are not always distinguishable from them. They are evidently independent of the action of gravity. We have already discussed the conditions of stability with respect to the formation of new fluid masses within a homogeneous fluid and at the surface when two such masses meet (see pages 252–264), as well as the condition relating to the possibility of a change in the nature of a surface of discontinuity. (See pages 237–240, where the surface considered is plane, but the result may easily be extended to curved surfaces.) We shall hereafter consider, in some of the more important cases, the conditions of stability with respect to the formation of new masses and surfaces which are peculiar to lines in which several surfaces of discontinuity meet, and points in which several such lines meet.
Conditions of stability relating to the whole system.—Besides the conditions of stability relating to very small parts of a system, which are substantially independent of the action of gravity, and are discussed elsewhere, there are other conditions, which relate to the whole system or to considerable parts of it. To determine the question of the stability of a given fluid system under the influence of gravity, when all the conditions of equilibrium are satisfied as well as those conditions of stability which relate to small parts of the system taken separately, we may use the method described on page 249, the demonstration of which (pages 247, 248) will not require any essential modification on account of gravity.
When the variations of temperature and of the quantities $M_{1},M_{2}$, etc. {see (617)} involved in the changes considered are so small that they may be neglected, the condition of stability takes a very simple form, as we have already seen to be the case with respect to a system uninfluenced by gravity. (See page 251.)
We have to consider a varied state of the system in which the total entropy and the total quantities of the various components are unchanged, and all variations vanish at the exterior of the system,— ​in which, moreover, the conditions of equilibrium relating to temperature and the potentials are satisfied, and the relations expressed by the fundamental equations of the masses and surfaces are to be regarded as satisfied, although the state of the system is not one of complete equilibrium. Let us imagine the state of the system to vary continuously in the course of time in accordance with these conditions and use the symbol d to denote the simultaneous changes which take place at any instant. If we denote the total energy of the system by $E$, the value of $dE$ may be expanded like that of $\delta E$ in (599) and (600), and then reduced (since the values of $t,\mu _{1}+gz,\mu _{2}+gz$, etc., are uniform throughout the system, and the total entropy and total quantities of the several components are constant) to the form
| ${\begin{aligned}dE&=-\int pdDv+\int gdzDm^{V}+\int \sigma dDs+\int gdzDm^{S}\\&=-\int pdDv+\int g\gamma dzDv+\int \sigma dDs+\int g\Gamma dzDs,\\\end{aligned}}$ | (621) |
$D$
$p$
$\sigma$
$t$
$M_{1},M_{2}$
$p$
$\sigma$
* $dp=-g\gamma dz,$$d\sigma =g\Gamma dz;$
| ${\begin{aligned}dE&=-\int pdDv-\int dpDv+\int \sigma dDs+\int d\sigma Ds\\&=-d\int pDv+d\int \sigma Ds.\\\end{aligned}}$ | (622) |
$d$
| $\Delta E=-\Delta \int pDv+\Delta \int \sigma Ds,$ | (623) |
$\Delta$
| $\Delta \int pDv-\Delta \int \sigma Ds<0,$ | (624) |
| $\int pDv-\int \sigma Ds$ | (625) |
$p$
$\sigma$
$z$
$\sigma$
$p$
$z$
$t,M_{1},M_{2}$
On the Possibility of the Formation of a New Surface of Discontinuity where several Surfaces of Discontinuity meet.
When more than three surfaces of discontinuity between homogeneous masses meet along a line, we may conceive of a new surface being formed between any two of the masses which do not meet in a surface in the original state of the system. The condition of stability with respect to the formation of such a surface may be easily obtained by the consideration of the limit between stability and instability, as exemplified by a system which is in equilibrium when a very small surface of the kind is formed.
$A,B,C$
$D$
${\text{A-B, B-C, C-D}}$
${\text{D-A}}$
$L$
$L$
$O$
$A$
$C$
${\text{A-C}}$
$L$
$L'$
$L''$
$O'$
$O''$
$\alpha \beta ,\beta \gamma ,\gamma \delta ,\delta \alpha$
${\text{A-B, B-C, C-D, D-A}}$
$O$
$O'$
$O''$
${\text{A-B}}$
${\text{B-C}}$
$O'$
${\text{C-D}}$
${\text{D-A}}$
$O''$
$O$
$\alpha \beta ,\beta \gamma$
$O'$
$\gamma \delta ,\delta \alpha$
$O''$
$\alpha \gamma$
${\text{A-C}}$
$A$
$C$
$\alpha \gamma$
The result which we have obtained may be generalized as follows. When more than three surfaces of discontinuity in a fluid system meet in equilibrium along a line, with respect to the surfaces and masses immediately adjacent to any point of this line, we may form a polygon of which the angular points shall correspond in order to the different masses separated by the surfaces of discontinuity, and ​the sides to these surfaces, each side being perpendicular to the corresponding surface, and equal to its tension. With respect to the formation of new surfaces of discontinuity in the vicinity of the point especially considered, the system is stable, if every diagonal of the polygon is less, and practically unstable, if any diagonal is greater, than the tension which would belong to the surface of discontinuity between the corresponding masses. In the limiting case, when the diagonal is exactly equal to the tension of the corresponding surface, the system may often be determined to be unstable by the application of the principle enunciated to an adjacent point of the line in which the surfaces of discontinuity meet. But when, in the polygons constructed for all points of the line, no diagonal is in any case greater than the tension of the corresponding surface, but a certain diagonal is equal to the tension in the polygons constructed for a finite portion of the line, farther investigations are necessary to determine the stability of the system. For this purpose, the method described on page 249 is evidently applicable.
A similar proposition may be enunciated in many cases with respect to a point about which the angular space is divided into solid angles by surfaces of discontinuity. If these surfaces are in equilibrium, we can always form a closed solid figure without re-entrant angles of which the angular points shall correspond to the several masses, the edges to the surfaces of discontinuity, and the sides to the lines in which these edges meet, the edges being perpendicular to the corresponding surfaces, and equal to their tensions, and the sides being perpendicular to the corresponding lines. Now if the solid angles in the physical system are such as may be subtended by the sides and bases of a triangular prism enclosing the vertical point, or can be derived from such by deformation, the figure representing the tensions will have the form of two triangular pyramids on opposite sides of the same base, and the system will be stable or practically unstable with respect to the formation of a surface between the masses which only meet in a point, according as the tension of a surface between such masses is greater or less than the diagonal joining the corresponding angular points of the solid representing the tensions. This will easily appear on consideration of the case in which a very small surface between the masses would be in equilibrium.
The Conditions of Stability for Fluids relating to the Formation of a New Phase at a Line in which Three Surfaces of Discontinuity meet.
With regard to the formation of new phases there are particular conditions of stability which relate to lines in which several surfaces ​of discontinuity meet. We may limit ourselves to the case in which there are three such surfaces, this being the only one of frequent occurrence, and may treat them as meeting in a straight line. It will be convenient to commence by considering the equilibrium of a system in which such a line is replaced by a filament of a different phase.
Let us suppose that three homogeneous fluid masses, $A,B$, and $C$ are separated by cylindrical (or plane) surfaces, ${\text{A-B, B-C, C-A}}$, which at first meet in a straight line, each of the surface-tensions $\sigma _{AB},\sigma _{BC},\sigma _{CA}$ being less than the sum of the other two. Let us suppose that the system is then modified by the introduction of a fourth fluid mass $D$, which is placed between $A,B$, and $C$, and is separated from them by cylindrical surfaces ${\text{D-A, D-B, D-C}}$ meeting ${\text{A-B, B-C}}$, and ${\text{C-A}}$ in
${\text{A-B, B-C, C-A}}$
$v_{D}$
$D$
$abc$
$\sigma _{AB},\sigma _{BC},\sigma _{CA}$
$D$
$v_{D}$
$D$
${\text{A-B, B-C}}$
${\text{C-A}}$
${\text{D-A, D-B, D-C}}$
$D$
$D$
${\text{A-B, B-C, C-A}}$
$D$
$D$
$D$
${\text{A-B, B-C, C-D}}$
It will be observed that all the angles at $a,b,c$, and $d$ in figure 14 are entirely determined by the six surface-tensions $\sigma _{AB},\sigma _{BC},\sigma _{CA},\sigma _{DA},\sigma _{DB},\sigma _{DC}$. (See (615).) The angles may be derived from the tensions by the following construction, which will also indicate some important properties. If we form a triangle $\alpha \beta \gamma$ (figure 15 or 16) having sides equal to $\sigma _{AB},\sigma _{BC},\sigma _{CA}$, the angles of the triangle will be supplements of the angles at $d$. To fix our ideas, we may suppose the sides of the triangle to be perpendicular to the surfaces at $d$. Upon $\beta \gamma$ we may then construct (as in figure 16) a triangle $\beta \gamma \delta '$ having sides equal to $\sigma _{BC},\sigma _{DC},\sigma _{DB}$, upon $\gamma \alpha$ a triangle $\gamma \alpha \delta ''$ having sides equal to $\sigma _{CA},\sigma _{DA},\sigma _{DC}$ and upon $\alpha \beta$ a triangle $\alpha \beta \delta '''$ having sides equal to $\sigma _{AB},\sigma _{DB},\sigma _{DA}$. These triangles are to be on the same sides of the lines $\beta \gamma ,\gamma \alpha ,\alpha \beta$, respectively, as the triangle $\alpha \beta \gamma$. The angles of these triangles will be supplements of the angles of the surfaces of discontinuity at $a,b$, and $c$. Thus $\beta \gamma \delta '=dab$, and $\alpha \gamma \delta ''=dba$. Now if $\delta '$ and $\delta ''$ fall together in a single point $\delta$ within the triangle $\alpha \beta \gamma$, $\delta '''$ will fall in the same point, as in figure 15. In this case we shall have $\beta \gamma \delta +\alpha \gamma \delta =\alpha \gamma \beta$, and the three angles of the curvilinear triangle $adb$ will be together equal to two right angles. The same will be true of the three angles of each of the triangles $bdc,cda$, and hence of the three angles of the triangle $abc$. But if $\delta ',\delta '',\delta '''$ do not fall together in the same point within the triangle $\alpha \beta \gamma$, it is either possible to bring these points to coincide within the triangle by increasing some or all of the tensions $\sigma _{DA},\sigma _{DB},\sigma _{DC}$, or to effect the same result by diminishing some or all of these tensions. (This will easily appear when one of the points $\delta ',\delta '',\delta ''$ falls within the triangle, if we let the two tensions which determine this point remain constant, and the third tension vary. When all the points $\delta ',\delta '',\delta ''$ fall without the triangle $\alpha \beta \gamma$, we may suppose the greatest of the tensions $\sigma _{DA},\sigma _{DB},\sigma _{DC}$—the two greatest, when these are equal, and all three when they all are equal—to diminish until one of the points $\delta ',\delta '',\delta ''$ is brought within the triangle $\alpha \beta \gamma$.) In the first case we may say that the tensions of the new surfaces are too small to be represented by the distances of an internal point from the vertices of the triangle representing the tensions of the original surfaces (or, for brevity, that they are too small to be represented as in figure 15); in the second case we may say that they are too great to be thus represented. In the first case, the sum of the angles in each of the triangles $adb,bdc,cda$ is less than two right angles (compare figures 14 and 16); ​in the second case, each pair of the triangles $\alpha \beta \delta ''',\beta \gamma \delta ',\gamma \alpha \delta ''$ will overlap, at least when the tensions $\sigma _{DA},\sigma _{DB},\sigma _{DC}$ are only a little too great to be represented as in figure 15, and the sum of the angles of each of the triangles adb, bdc, cda will be greater than two right angles.
Let us denote by $v_{A},v_{B},v_{C}$ the portions of $v_{D}$ which were originally occupied by the masses $A,B,C$, respectively, by $s_{DA},s_{DB},s_{DC}$, the areas of the surfaces specified per unit of length of the mass $D$, and by $s_{AB},s_{BC},s_{CA}$, the areas of the surfaces specified which were replaced by the mass $D$ per unit of its length. In numerical value, $v_{A},v_{B},v_{C}$ will be equal to the areas of the curvilinear triangles $bcd,cad,abd$; and $s_{DA},s_{DB},s_{DC},s_{AB},s_{BC},s_{CA}$ to the lengths of the lines $bc,ca,ab,cd,ad,bd$. Also let
| $W_{S}=\sigma _{DA}s_{DA}+\sigma _{DB}s_{DB}+\sigma _{DC}s_{DC}-\sigma _{AB}s_{AB}-\sigma _{BC}s_{BC}-\sigma _{CA}s_{CA},$ | (626) |
$W_{V}=p_{D}v_{D}-p_{A}v_{A}-p_{B}v_{B}-p_{C}v_{C}.$
| $\textstyle \sum \displaystyle (\sigma \delta s)-\textstyle \sum \displaystyle (p\delta v)=0.$ | (628) |
$A,B$
$C$
$D$
| $\sigma _{DA}\delta s_{DA}+\sigma _{DB}\delta s_{DB}+\sigma _{DC}\delta s_{DC}-\sigma _{AB}\delta s_{AB}-\sigma _{BC}\delta s_{BC}-\sigma _{CA}\delta s_{CA}-p_{D}\delta v_{D}+p_{A}\delta v_{A}+p_{B}\delta v_{B}+p_{C}\delta v_{C}=0.$ | (629) |
| $dW_{S}-dW_{V}=s_{DA}d\sigma _{DA}+s_{DB}d\sigma _{DB}+s_{DC}d\sigma _{DC}-s_{AB}d\sigma _{AB}-s_{BC}d\sigma _{BC}-s_{CA}d\sigma _{CA}.$ | (630) |
$s_{DA},\sigma _{DA},s_{DB},\sigma _{DB}$
| $s_{DA}d\sigma _{DA}={\tfrac {1}{2}}d(\sigma _{DA}s_{DA}),$$s_{DB}d\sigma _{DB}={\tfrac {1}{2}}d(\sigma _{DB}s_{DB}),$etc. | (631) |
| $W_{S}-W_{V}={\tfrac {1}{2}}(\sigma _{DA}s_{DA}+\sigma _{DB}s_{DB}+\sigma _{DC}s_{DC}-\sigma _{AB}s_{AB}-\sigma _{BC}s_{BC}-\sigma _{CA}s_{CA}),$ | (632) |
| $W_{S}=2W_{V}.$ | (633) |
​The condition of stability for the system when the pressures and tensions are regarded as constant, and the position of the surfaces ${\text{A-B, B-C, C-A}}$ as fixed, is that $W_{S}-W_{V}$ shall be a minimum under the same conditions. (See (549).) Now for any constant values of the tensions and of $p_{A},p_{B},p_{C}$, we may make $v_{D}$ so small that when it varies, the system remaining in equilibrium (which will in general require a variation of $p_{D}$), we may neglect the curvatures of the lines $da,db,dc$, and regard the figure $abcd$ as remaining similar to itself. For the total curvature (i.e., the curvature measured in degrees) of each of the lines $ab,bc,ca$ may be regarded as constant, being equal to the constant difference of the sum of the angles of one of the curvilinear triangles $adb,bdc,cda$ and two right angles. Therefore, when V D is very small, and the system is so deformed that equilibrium would be preserved if $v_{D}$ had the proper variation, but this pressure as well as the others and all the tensions remain constant, $W_{S}$ will vary as the lines in the figure $abcd$, and $W_{V}$ as the square of these lines. Therefore, for such deformations,
* $W_{V}\propto W_{S}^{2}.$
$W_{V}$
$W_{V}$
$W_{S}-W_{V}$
$W_{V}$
$W_{S}-W_{V}$
$W_{S}-W_{V}$
$v_{D}$
$p_{D}$
These results will enable us to determine the most important questions relating to the stability of a line along which three homogeneous fluids $A,B,C$ meet, with respect to the formation of a different fluid $D$. The components of $D$ must of course be such as are found in the surrounding bodies. We shall regard $p_{D}$ and $\sigma _{DA},\sigma _{DB},\sigma _{DC}$ as determined by that phase of $D$ which satisfies the conditions of equilibrium with the other bodies relating to temperature and the potentials. These quantities are therefore determinable, by means of the fundamental equations of the mass $D$ and of the surfaces ${\text{D-A, D-B, D-C}}$, from the temperature and potentials of the given system.
Let us first consider the case in which the tensions, thus determined, can be represented as in figure 15, and $p_{D}$ has a value consistent with the equilibrium of a small mass such as we have been considering. It appears from the preceding discussion that ​when $v_{D}$ is sufficiently small the figure abed may be regarded as rectilinear, and that its angles are entirely determined by its tensions. Hence the ratios of $v_{A},v_{B},v_{C},v_{D}$, for sufficiently small values of $v_{D}$, are determined by the tensions alone, and for convenience in calculating these ratios, we may suppose $p_{A},p_{B},p_{C}$ to be equal, which will make the figure $abcd$ absolutely rectilinear, and make $p_{D}$ equal to the other pressures, since it is supposed that this quantity has the value necessary for equilibrium. We may obtain a simple expression for the ratios of $v_{A},v_{B},v_{C},v_{D}$ in terms of the tensions in the following manner. We shall write $[{\text{DBC}}],[{\text{DCA}}]$, etc., to denote the areas of triangles having sides equal to the tensions of the surfaces between the masses specified.
* $::{\text{triangle}}\,\,bdc:{\text{triangle}}\,\,adc$
* $::bc\sin bcd:ac\sin acd$
* $::\sin bac\sin bcd:\sin abc\sin acd$
* $::\sin \gamma \delta \beta \sin \delta \alpha \beta :\sin \gamma \delta \alpha \sin \delta \beta \alpha$
* $::\sin {\gamma \delta \beta }\,\,\delta \beta :\sin {\gamma \delta \alpha }\,\,\delta \alpha$
* $::{\text{triangle}}\,\,\gamma \delta \beta :{\text{triangle}}\,\,\gamma \delta \alpha$
* $::[{\text{DBC}}]:[{\text{DCA}}].$
Hence,
| $v_{A}:v_{B}:v_{C}:v_{D}::[{\text{DBC}}]:[{\text{DCA}}]:[{\text{DAB}}]:[{\text{ABC}}],$ | (634) |
* ${\tfrac {1}{4}}{\sqrt}[(\sigma _{AB}+\sigma _{BC}+\sigma _{CA})(\sigma _{AB}+\sigma _{BC}-\sigma _{CA})(\sigma _{BC}+\sigma _{CA}-\sigma _{AB})(\sigma _{CA}+\sigma _{AB}-\sigma _{BC})]$
$[{\text{ABC}}]$
${\sqrt}$
$v_{D}$
$p_{A}=p_{B}=p_{C},$
$p_{D}$
$W_{V}=0$
$W_{S}=0$
$v_{D}$
$W_{S}$
$v_{D}$
$v_{D}$
$p_{A},p_{B},p_{C}$
| $0=W_{S}=W_{V}=p_{D}v_{D}-p_{A}v_{A}-p_{B}v_{B}-p_{C}v_{C}.$ | (635) |
| $p_{D}={\frac {[{\text{DBC}}]p_{A}+[{\text{DCA}}]p_{B}+[{\text{DAB}}]p_{C}}{[{\text{DBC}}]+[{\text{DCA}}]+[{\text{DAB}}]}}\cdot$ | (636) |
$p_{D}$
$v_{D}$
$p_{D}$
$A,B,C$
$D$
$p_{D}$
$D$
Let us next consider the case in which the tensions of the new surfaces are too small to be represented as in figure 15. If the pressures and tensions are consistent with equilibrium for any very small value of $v_{D}$, the angles of each of the curvilinear triangles $adb,bdc,cda$ will be together less than two right angles, and the lines $ab,be,ca$ will be convex toward the mass $D$. For given values of the pressures and tensions, it will be easy to determine the magnitude of $v_{D}$. For the tensions will give the total curvatures (in degrees) of the lines $ab,bc,ca$; and the pressures will give the radii of curvature. These lines are thus completely determined. In order that $v_{D}$ shall be very small it is evidently necessary that $p_{D}$ shall be less than the other pressures. Yet if the tensions of the new surfaces are only a very little too small to be represented as in figure 15, $v_{D}$ may be quite small when the value of $p_{D}$ is only a little less than that given by equation (636). In any case, when the tensions of the new surfaces are too small to be represented as in figure 15, and $v_{D}$ is small, $W_{V}$ is negative, and the equilibrium of the mass $D$ is stable. Moreover, $W_{S}-W_{V}$, which represents the work necessary to form the mass D with its surfaces in place of the other masses and surfaces, is negative.
With respect to the stability of a line in which the surfaces ${\text{A-B, B-C, C-A}}$ meet, when the tensions of the new surfaces are too small to be represented as in figure 15, we first observe that when the pressures and tensions are such as to make $v_{D}$ moderately small but not so small as to be neglected (this will be when $p_{D}$ is somewhat smaller than the second member of (636),—more or less smaller according as the tensions differ more or less from such as are represented in figure 15), the equilibrium of such a line as that supposed (if it is capable of existing at all) is at least practically unstable. For greater values of $p_{D}$ (with the same values of the other pressures and the tensions) the same will be true. For somewhat smaller values of $p_{D}$, the mass of the phase $D$ which will be formed will be so small, that we may neglect this mass and regard the surfaces ${\text{A-B, B-C, C-A}}$ as meeting in a line in stable equilibrium. For still smaller values of $p_{D}$, we may likewise regard the surfaces ${\text{A-B, B-C, C-A}}$ as capable of meeting in stable equilibrium. It may be observed that when $v_{D}$, as determined by our equations, becomes quite insensible, the conception of a small mass $D$ having the properties deducible from our equations ceases to be accurate, since the matter in the vicinity of a line where these surfaces of discontinuity meet must be ​in a peculiar state of equilibrium not recognized by our equations. But this cannot affect the validity of our conclusion with respect to the stability of the line in question.
The case remains to be considered in which the tensions of the new surfaces are too great to be represented as in figure 15. Let us suppose that they are not very much too great to be thus represented. When the pressures are such as to make $v_{D}$ moderately small (in case of equilibrium) but not so small that the mass $D$ to which it relates ceases to have the properties of matter in mass (this will be when $p_{D}$ is somewhat greater than the second member of (636),—more or less greater according as the tensions differ more or less from such as are represented in figure 15), the line where the surfaces ${\text{A-B, B-C, C-A}}$ meet will be in stable equilibrium with respect to the formation of such a mass as we have considered, since $W_{S}-W_{V}$ will be positive. The same will be true for less values of $p_{D}$. For greater values of $p_{D}$, the value of $W_{S}-W_{V}$, which measures the stability with respect to the kind of change considered, diminishes. It does not vanish, according to our equations, for finite values of $p_{D}$. But these equations are not to be trusted beyond the limit at which the mass $D$ ceases to be of sensible magnitude.
But when the tensions are such as we now suppose, we must also consider the possible formation of a mass $D$ within a closed figure in which the surfaces ${\text{D-A, D-B, D-C}}$ meet together (with the surfaces ${\text{A-B, B-C, C-A}}$) in two opposite points. If such a figure is to be in equilibrium, the six tensions must be such as can be represented by the six distances of four points in space (see pages 288, 289), a condition which evidently agrees with the supposition which we have made. If we denote by $w_{V}$ the work gained in forming the mass $D$ (of such size and form as to be in equilibrium) in place of the other masses, and by $w_{S}$ the work expended in forming the new surfaces in place of the old, it may easily be shown by a method similar to that used on page 292 that $w_{S}={\tfrac {3}{2}}w_{V}$. From this we obtain $w_{S}-w_{V}={\tfrac {1}{2}}w_{V}$. This is evidently positive when $p_{D}$ is greater than the other pressures. But it diminishes with increase of $p_{D}$, as easily appears from the ​equivalent expression ${\tfrac {1}{3}}w_{S}$. Hence the line of intersection of the surfaces of discontinuity ${\text{A-B, B-C, C-A}}$ is stable for values of greater than the other pressures (and therefore for all values of so long as our method is to be regarded as accurate, which will be so long as the mass $D$ which would be in equilibrium has a sensible size.
In certain cases in which the tensions of the new surfaces are much too large to be represented as in figure 15, the reasoning of the two last paragraphs will cease to be applicable. These are cases in which the six tensions cannot be represented by the sides of a tetrahedron. It is not necessary to discuss these cases, which are distinguished by the different shape which the mass $D$ would take if it should be formed, since it is evident that they can constitute no exception to the results which we have obtained. For an increase of the values of $\sigma _{DA},\sigma _{DB},\sigma _{DC}$ cannot favor the formation of $D$, and hence cannot impair the stability of the line considered, as deduced from our equations. Nor can an increase of these tensions essentially affect the fact that the stability thus demonstrated may fail to be realized when $p_{D}$ is considerably greater than the other pressures, since the a priori demonstration of the stability of any one of the surfaces ${\text{A-B, B-C, C-A}}$, taken singly, is subject to the limitation mentioned. (See pages 261, 262.)
The Condition of Stability for Fluids relating to the Formation of a New Phase at a Point where the Vertices of Four Different Masses meet.
Let four different fluid masses $A,B,C,D$ meet about a point, so as to form the six surfaces of discontinuity ${\text{A-B, B-C, C-A, D-A, D-B, D-C}}$, which meet in the four lines ${\text{A-B-C, B-C-D, C-D-A, D-A-B}}$, these lines meeting in the vertical point. Let us suppose the system stable in other respects, and consider the conditions of stability for the vertical point with respect to the possible formation of a different fluid mass $E$.
If the system can be in equilibrium when the vertical point has been replaced by a mass $E$ against which the four masses $A,B,C,D$ abut, being truncated at their vertices, it is evident that $E$ will have four vertices, at each of which six surfaces of discontinuity meet. (Thus at one vertex there will be the surfaces formed by $A,B,C$, and $E$.) The tensions of each set of six surfaces (like those of the six surfaces formed by $A,B,C$, and $D$) must therefore be such that they can be represented by the six edges of a tetrahedron. When the tensions do not satisfy these relations, there will be no particular condition of stability for the point about which $A,B,C$, and $D$ meet, since if a mass $E$ should be formed, it would distribute itself along some of the lines or surfaces which meet at the vertical point, and it ​is therefore sufficient to consider the stability of these lines and surfaces. We shall suppose that the relations mentioned are satisfied.
If we denote by $W_{V}$ the work gained in forming the mass $E$ (of such size and form as to be in equilibrium) in place of the portions of the other masses which are suppressed, and by $W_{S}$ the work expended in forming the new surfaces in place of the old, it may easily be shown by a method similar to that used on page 292 that
| $W_{S}={\tfrac {3}{2}}W_{V},$ | (637) |
$W_{S}-W_{V}={\tfrac {1}{2}}W_{V};$
$E$
$E$
$W_{S}$
$W_{V}$
A critical relation for the tensions is that which makes equilibrium possible for the system of the five masses $A,B,C,D,E$, when all the surfaces are plane. The ten tensions may then be represented in magnitude and direction by the ten distances of five points in space $\alpha ,\beta ,\gamma ,\delta ,\epsilon$, viz., the tension of ${\text{A-B}}$ and the direction of its normal by the line $\alpha \beta$, etc. The point $\epsilon$ will lie within the tetrahedron formed by the other points. If we write $v_{E}$ for the volume of $E$, and $v_{A},v_{B},v_{C},v_{D}$ for the volumes of the parts of the other masses which are suppressed to make room for $E$, we have evidently
| $W_{V}=p_{E}v_{E}-p_{A}v_{A}-p_{B}v_{B}-p_{C}v_{C}-p_{D}v_{D}.$ | (639) |
$W_{V}=0$
$W_{S}=0$
$v_{E}$
$p_{A},p_{B},p_{C},p_{D}$
$W_{S}=0$
$p_{A}=p_{B}=p_{C}=p_{D}$
$v_{E}$
$W_{S}$
$p_{A},p_{B},p_{C},p_{D}$
$W_{S}=0$
$W_{V}=0$
$v_{E}$
| $p_{E}={\frac {v_{A}p_{A}+v_{B}p_{B}+v_{C}p_{C}+v_{D}p_{D}}{v_{E}}}\cdot$ | (640) |
$v_{A},v_{B},v_{C},v_{D},v_{E}$
$E$
${\text{a, b, c, d}}$
$v_{A},v_{B},v_{C},v_{D}$
${\text{e}}$
$\alpha \beta \gamma \delta \epsilon$
${\text{ab}}$
${\text{C-D, D-E, E-C}}$
$\gamma \delta ,\delta \epsilon ,\epsilon \gamma$
$\gamma \delta \epsilon$
$\gamma ,\delta$
$\epsilon$
${\text{a}}$
${\text{b}}$
${\text{abc}}$
${\text{D-E}}$
$\delta \epsilon$
${\text{tetr abcd}}$
${\text{trian abc}}$
$\sin {({\text{ab, bc}})},\sin {({\text{abc, dbc}})},\sin {({\text{abc, ad}})}$
$[{\text{BCDE}}],[{\text{CDEA}}]$
${\tfrac {1}{3}}$
${\tfrac {2}{3}}$
* $::{\text{tetr bcde}}:{\text{tetr acde}}$
* $::{\text{bc}}\,\,\sin {({\text{ab, bc}})}:{\text{ac}}\,\,\sin {({\text{ac, cde}})}$
* $::\sin {({\text{ba, ac}})}\sin {({\text{bc, cde}})}:{\text{ac}}\,\,\sin {({\text{ac, cde}})}$
* $::\sin {(\gamma \delta \epsilon ,\,\beta \delta \epsilon )}\sin {(\alpha \delta \epsilon ,\,\alpha \beta )}:\,\sin {(\gamma \delta \epsilon ,\,\alpha \delta \epsilon )}\,\sin {(\beta \delta \epsilon ,\,\alpha \beta )}$
* $::{\frac {{\text{tetr }}\alpha \beta \delta \epsilon \,{\text{tetr }}\beta \alpha \delta \epsilon }{{\text{trian }}\beta \delta \epsilon \,{\text{trian }}\alpha \delta \epsilon }}\,\,:\,\,{\frac {{\text{tetr }}\gamma \alpha \delta \epsilon \,{\text{tetr }}\alpha \beta \delta \epsilon }{{\text{trian }}\alpha \delta \epsilon \,{\text{trian }}\beta \delta \epsilon }}$
* $::{\text{tetr }}\gamma \beta \delta \epsilon \,\,:\,\,{\text{tetr }}\gamma \alpha \delta \epsilon$
* $::[{\text{BCDE}}]\,\,:\,\,[{\text{CDEA}}].$
Hence,
| $v_{A}:v_{B}:v_{C}:v_{D}::[{\text{BCDE}}]:[{\text{CDEA}}]:[{\text{DEAB}}]:[{\text{EABC}}],$ | (641) |
| $p_{E}={\frac {[{\text{BCDE}}]p_{A}+[{\text{CDEA}}]p_{B}+[{\text{DEAB}}]p_{C}+[{\text{EABC}}]p_{D}}{[{\text{BCDE}}]+[{\text{CDEA}}]+[{\text{DEAB}}]+[{\text{EABC}}]}}\cdot$ | (642) |
$p_{E}$
$A,B,C,D$
$E$
$p_{E}$
$A,B,C,D$
When the tensions of the new surfaces are too small to satisfy the critical relation with the other tensions, these surfaces will be convex toward $E$; when their tensions are too great for that relation, the surfaces will be concave toward $E$. In the first case, $W_{V}$ is negative, and the equilibrium of the five masses $A,B,C,D,E$ is stable, but the equilibrium of the four masses $A,B,C,D$ meeting at a point is impossible or at least practically unstable. This is subject to the limitation that when $p_{E}$ is sufficiently small the mass $E$ which will form will be so small that it may be neglected. This will only be the case when $p_{E}$ is smaller—in general considerably smaller—than the second member of (642). In the second case, the equilibrium of the five masses $A,B,C,D,E$ will be unstable, but the equilibrium ​of the four masses $A,B,C,D$ will be stable unless $v_{E}$ (calculated for the case of the five masses) is of insensible magnitude. This will only be the case when $p_{E}$ is greater—in general considerably greater than the second member of (642).
Liquid Films.
When a fluid exists in the form of a thin film between other fluids, the great inequality of its extension in different directions will give rise to certain peculiar properties, even when its thickness is sufficient for its interior to have the properties of matter in mass. The frequent occurrence of such films, and the remarkable properties which they exhibit, entitle them to particular consideration. To fix our ideas, we shall suppose that the film is liquid and that the contiguous fluids are gaseous. The reader will observe our results are not dependent, so far as their general character is concerned, upon this supposition.
Let us imagine the film to be divided by surfaces perpendicular to its sides into small portions of which all the dimensions are of the same order of magnitude as the thickness of the film,—such portions to be called elements of the film,—it is evident that far less time will in general be required for the attainment of approximate equilibrium between the different parts of any such element and the other fluids which are immediately contiguous, than for the attainment of equilibrium between all the different elements of the film. There will accordingly be a time, commencing shortly after the formation of the film, in which its separate elements may be regarded as satisfying the conditions of internal equilibrium, and of equilibrium with the contiguous gases, while they may not satisfy all the conditions of equilibrium with each other. It is when the changes due to this want of complete equilibrium take place so slowly that the film appears to be at rest, except so far as it accommodates itself to any change in the external conditions to which it is subjected, that the characteristic properties of the film are most striking and most sharply defined.
​Let us first consider the properties which will belong to each element of the film under the conditions mentioned. Let us suppose the element extended, while the temperature and the potentials which are determined by the contiguous gas-masses are unchanged. If the film has no components except those of which the potentials are maintained constant, there will be no variation of tension in its surfaces. The same will be true when the film has only one component of which the potential is not maintained constant, provided that this is a component of the interior of the film and not of its surface alone. If we regard the thickness of the film as determined by dividing surfaces which make the surface-density of this component vanish, the thickness will vary inversely as the area of the element of the film, but no change will be produced in the nature or the tension of its surfaces. If, however, the single component of which the potential is not maintained constant is confined to the surfaces of the film, an extension of the element will generally produce a decrease in the potential of this component, and an increase of tension. This will certainly be true in those cases in which the component shows a tendency to distribute itself with a uniform superficial density.
When the film has two or more components of which the potentials are not maintained constant by the contiguous gas-masses, they will not in general exist in the same proportion in the interior of the film as on its surfaces, but those components which diminish the tensions will be found in greater proportion on the surfaces. When the film is extended, there will therefore not be enough of these substances to keep up the same volume- and surface-densities as before, and the deficiency will cause a certain increase of tension. The value of the elasticity of the film (i.e., the infinitesimal increase of the united tensions of its surfaces divided by the infinitesimal increase of area in a unit of surface) may be calculated from the quantities which specify the nature of the film, when the fundamental equations of the interior mass, of the contiguous gas-masses, and of the two surfaces of discontinuity are known. We may illustrate this by a simple example.
Let us suppose that the two surfaces of a plane film are entirely alike, that the contiguous gas-masses are identical in phase, and that they determine the potentials of all the components of the film except two. Let us call these components $S_{1}$ and $S_{2}$, the latter denoting that which occurs in greater proportion on the surface than in the interior of the film. Let us denote by $\gamma _{1}$ and $\gamma _{2}$ the densities of these components in the interior of the film, by $\lambda$ the thickness of the film determined by such dividing surfaces as make the surface-density of $S_{1}$ vanish (see page 234), by $\Gamma _{2(1)}$ the surface-density of the other component as determined by the same ​surfaces, by $\sigma$ and $s$ the tension and area of one of these surfaces, and by $E$ the elasticity of the film when extended under the supposition that the total quantities of $S_{1}$ and $S_{2}$ in the part of the film extended are invariable, as also the temperature and the potentials of the other components. From the definition of $E$ we have
| $2d\sigma =E{\frac {ds}{s}},$ | (643) |
| ${\frac {ds}{s}}=-{\frac {d(\lambda \gamma _{1})}{\lambda \gamma _{1}}}=-{\frac {d(\lambda \gamma _{2}+2\Gamma _{2(1)})}{\lambda \gamma _{2}+2\Gamma _{2(1)}}}\cdot$ | (644) |
* $\lambda \gamma _{1}{\frac {ds}{s}}=-\gamma _{1}d\lambda -\lambda d\gamma _{1},$ $(\lambda \gamma _{2}+2\Gamma _{2(1)}){\frac {ds}{s}}=-\gamma _{2}d\lambda -\lambda d\gamma _{2}-2d\Gamma _{2(1)},$
$d\lambda$
| $2\gamma _{1}\Gamma _{2(1)}{\frac {ds}{s}}=-\lambda \gamma _{1}d\gamma _{2}+\lambda \gamma _{2}d\gamma _{1}-2\gamma _{1}d\Gamma _{2(1)}.$ | (645) |
$r={\frac {\gamma _{2}}{\gamma _{1}}},$
$dr={\frac {\gamma _{1}d\gamma _{2}-\gamma _{2}d\gamma _{1}}{\gamma _{1}}},$
$2\Gamma _{2(1)}{\frac {ds}{s}}=-\lambda \gamma _{1}dr-2d\Gamma _{2(1)}.$
$d\sigma$
* $d\sigma =-\Gamma _{2(1)}d\mu _{2}.$
| $4\Gamma _{2(1)}^{2}d\mu _{2}=E(\lambda \gamma _{1}dr+2d\Gamma _{2(1)}),$ | (649) |
| ${\frac {4\Gamma _{2(1)}^{2}}{E}}=\lambda \gamma _{1}{\frac {dr}{d\mu _{2}}}+2{\frac {d\Gamma _{2(1)}}{d\mu _{2}}},$ | (650) |
$\mu _{1}$
$\mu _{2}$
| $(\gamma _{1}-\gamma _{1}')d\mu _{1}+(\gamma _{2}-\gamma _{2}')d\mu _{2}=0,$ | (651) |
$\gamma _{1}'$
$\gamma _{2}'$
$S_{2}$
$S_{2}$
$E$
$\lambda$
If we write $G_{1}$ and $G_{2}$ for the total quantities of $S_{1}$ and $S_{2}$ per unit of area of the film, we have
| $G_{1}=\lambda \gamma _{1},$ | (652) |
| $G_{2}=\lambda \gamma _{2}+2\Gamma _{2(1)}.$ | (653 |
* $G_{2}=G_{1}r+2\Gamma _{2(1)},$
| $\left({\frac {dG_{2}}{d\mu _{2}}}\right)_{G_{1}}=\lambda \gamma _{1}{\frac {dr}{d\mu _{2}}}+2{\frac {d\Gamma _{2(1)}}{d\mu _{2}}},$ | (654) |
$G_{1}$
* ${\frac {4\Gamma _{2(1)}^{2}}{E}}=\left({\frac {dG_{2}}{d\mu _{2}}}\right)_{G_{1}},$
| $E=4\Gamma _{2(1)}^{2}\left({\frac {dG_{2}}{d\mu _{2}}}\right)_{G_{1}},$ | (655) |
$E$
These equations give the elasticity of any element of the film when the temperature and the potentials for the substances which are found in the contiguous gas-masses are regarded as constant, and the potentials for the other components, $\mu _{1}$ and $\mu _{2}$, have had time to equalize themselves throughout the element considered. The increase of tension immediately after a rapid extension will be greater than that given by these equations.
The existence of this elasticity, which has thus been established from a priori considerations, is clearly indicated by the phenomena which liquid films present. Yet it is not to be demonstrated simply by comparing the tensions of films of different thickness, even when they are made from the same liquid, for difference of thickness does not necessarily involve any difference of tension. When the phases within the films as well as without are the same, and the surfaces of the films are also the same, there will be no difference of tension. Nor will the tension of the same film be altered, if a part of the interior drains away in the course of time, without affecting the surfaces. In case the thickness of the film is reduced by evaporation, the tension may be either increased or diminished. (The evaporation of the substance $S_{1}$ in the case we have just considered, would diminish the tension.) Yet it may easily be shown that ​extension increases the tension of a film and contraction diminishes it. When a plane film is held vertically, the tension of the upper portions must evidently be greater than that of the lower. The tensions in every part of the film may be reduced to equality by turning it into a horizontal position. By restoring the original position we may restore the original tensions, or nearly so. It is evident that the same element of the film is capable of supporting very unequal tensions. Nor can this be always attributed to viscosity of the film. For in many cases, if we hold the film nearly horizontal, and elevate first one side and then another, the lighter portions of the film will dart from one side to the other, so as to show a very striking mobility in the film. The differences of tension which cause these rapid movements are only a very small fraction of the difference of tension in the upper and lower portions of the film when held vertically.
If we account for the power of an element of the film to support an increase of tension by viscosity, it will be necessary to suppose that the viscosity offers a resistance to a deformation of the film in which its surface is enlarged and its thickness diminished, which is enormously great in comparison with the resistance to a deformation in which the film is extended in the direction of one tangent and contracted in the direction of another, while its thickness and the areas of its surfaces remain constant. This is not to be readily admitted as a physical explanation, although to a certain extent the phenomena resemble those which would be caused by such a singular viscosity. (See page 274.) The only natural explanation of the phenomena is that the extension of an element of the film, which is the immediate result of an increase of external force applied to its perimeter, causes an increase of its tension, by which it is brought into true equilibrium with the external forces.
The phenomena to which we have referred are such as are apparent to a very cursory observation. In the following experiment, which is described by M. Plateau, an increased tension is manifested in a film while contracting after a previous extension. The warmth of a finger brought near to a bubble of soap-water with glycerine, which is thin enough to show colors, causes a spot to appear indicating a diminution of thickness. When the finger is removed, the spot returns to its original color. This indicates a contraction, which would be resisted by any viscosity of the film, and can only be due to an excess of tension in the portion stretched, on the return of its original temperature.
We have so far supposed that the film is thick enough for its ​interior to have the properties of matter in mass. Its properties are then entirely determined by those of the three phases and the two surfaces of discontinuity. From these we can also determine, in part at least, the properties of a film at the limit at which the interior ceases to have the properties of matter in mass. The elasticity of the film, which increases with its thinness, cannot of course vanish at that limit, so that the film cannot become unstable with respect to extension and contraction of its elements immediately after passing that limit.
Yet a certain kind of instability will probably arise, which we may here notice, although it relates to changes in which the condition of the invariability of the quantities of certain components in an element of the film is not satisfied. With respect to variations in the distribution of its components, a film will in general be stable, when its interior has the properties of matter in mass, with the single exception of variations affecting its thickness without any change of phase or of the nature of the surfaces. With respect to this kind of change, which may be brought about by a current in the interior of the film, the equilibrium is neutral. But when the interior ceases to have the properties of matter in mass, it is to be supposed that the equilibrium will generally become unstable in this respect. For it is not likely that the neutral equilibrium will be unaffected by such a change of circumstances, and since the film certainly becomes unstable when it is sufficiently reduced in thickness, it is most natural to suppose that the first effect of diminishing the thickness will be in the direction of instability rather than in that of stability. (We are here considering liquid films between gaseous masses. In certain other cases, the opposite supposition might be more natural, as in respect to a tilm of water between mercury and air, which would certainly become stable when sufficiently reduced in thickness.)
Let us now return to our former suppositions—that the film is thick enough for the interior to have the properties of matter in mass, and that the matter in each element is invariable, except with respect to those substances which have their potentials determined by the contiguous gas-masses—and consider what conditions are necessary for equilibrium in such a case.
In consequence of the supposed equilibrium of its several elements, such a film may be treated as a simple surface of discontinuity between the contiguous gas-masses (which may be similar or different), whenever its radius of curvature is very large in comparison with its thickness,—a condition which we shall always suppose to be fulfilled. With respect to the film considered in this light, the mechanical conditions of equilibrium will always be satisfied, or very nearly so, as soon as a state of approximate rest is attained, except in those ​cases in which the film exhibits a decided viscosity. That is, the relations (618), (614), (615) will hold true, when by or we understand the tension of the film regarded as a simple surface of discontinuity (this is equivalent to the sum of the tensions of the two surfaces of the film), and by $\Gamma$ its mass per unit of area diminished by the mass of gas which would occupy the same space if the film should be suppressed and the gases should meet at its surface of tension. This surface of tension of the film will evidently divide the distance between the surfaces of tension for the two surfaces of the film taken separately, in the inverse ratio of their tensions. For practical purposes, we may regard F simply as the mass of the film per unit of area. It will be observed that the terms containing $\Gamma$ in (613) and (614) are not to be neglected in our present application of these equations.
But the mechanical conditions of equilibrium for the film regarded us an approximately homogeneous mass in the form of a thin sheet bounded by two surfaces of discontinuity are not necessarily satisfied when the film is in a state of apparent rest. In fact, these conditions cannot be satisfied (in any place where the force of gravity has an appreciable intensity) unless the film is horizontal. For the pressure in the interior of the film cannot satisfy simultaneously condition (612), which requires it to vary rapidly with the height $z$, and condition (613) applied separately to the different surfaces, which makes it a certain mean between the pressures in the adjacent gas-masses. Nor can these conditions be deduced from the general condition of mechanical equilibrium (606) or (611), without supposing that the interior of the film is free to move independently of the surfaces, which is contrary to what we have supposed.
Moreover, the potentials of the various components of the film will not in general satisfy conditions (617), and cannot (when the temperature is uniform) unless the film is horizontal. For if these conditions were satisfied, equation (612) would follow as a consequence. (See page 283.)
We may here remark that such a film as we are considering cannot form any exception to the principle indicated on page 284,—that when a surface of discontinuity which satisfies the conditions mechanical equilibrium has only one component which is not found in the contiguous masses, and these masses satisfy all the conditions of equilibrium, the potential for the component mentioned must satisfy the law expressed in (617), as a consequence of the condition of mechanical equilibrium (614). Therefore, as we have just seen that it is impossible that all the potentials in a liquid film which is not horizontal should conform to (617) when the temperature is uniform it follows that if a liquid film exhibits any persistence which is ​not due to viscosity, or to a horizontal position, or to differences of temperature, it must have more than one component of which the potential is not determined by the contiguous gas- masses in accordance with (617).
The difficulties of the quantitative experimental verification of the properties which have been described would be very great, even in cases in which the conditions we have imagined were entirely fulfilled. Yet the general effect of any divergence from these conditions will be easily perceived, and when allowance is made for such divergence, the general behavior of liquid films will be seen to agree with the requirements of theory.
The formation of a liquid film takes place most symmetrically when a bubble of air rises to the top of a mass of the liquid. The motion of the liquid, as it is displaced by the bubble, is evidently such as to stretch the two surfaces in which the liquid meets the air, where these surfaces approach one another. This will cause an increase of tension, which will tend to restrain the extension of the surfaces. The extent to which this effect is produced will vary with the nature of the liquid. Let us suppose that the case is one in which the liquid contains one or more components which, although constituting but a very small part of its mass, greatly reduce its tension. Such components will exist in excess on the surfaces of the liquid. In this case the restraint upon the extension of the surfaces will be considerable, and as the bubble of air rises above the general level of the liquid, the motion of the latter will consist largely of a running out from between the two surfaces. But this running out of the liquid will be greatly retarded by its viscosity as soon as it is reduced to the thickness of a film, and the effect of the extension of the surfaces in increasing their tension will become greater and more permanent as the quantity of liquid diminishes which is available for supplying the substances which go to form the increased surfaces.
We may form a rough estimate of the amount of motion which is possible for the interior of a liquid film, relatively to its exterior, by calculating the descent of water between parallel vertical planes at which the motion of the water is reduced to zero. If we use the coefficient of viscosity as determined by Helmholtz and Piotrowski, we obtain
| $V=581D^{2},$ | (656) |
$V$
$D$
* $V=337D^{2}$
| $V=899D^{2}$ | (657) |
$1000D^{2}$
$V$
A little consideration will show that the phenomenon is essentially of the same nature when films are formed in any other way, as by dipping a ring or the mouth of a cup in the liquid and then withdrawing it. When the film is formed in the mouth of a pipe, it may sometimes be extended so as to form a large bubble. Since the elasticity (i.e., the increase of the tension with extension) is greater in the thinner parts, the thicker parts will be most extended, and the effect of this process (so far as it is not modified by gravity) will be to diminish the ratio of the greatest to the least thickness of the film. During this extension, as well as at other times, the increased elasticity due to imperfect communication of heat, etc., will serve to protect the bubble from fracture by shocks received from the air or the pipe. If the bubble is now laid upon a suitable support, the condition (613) will be realized almost instantly. The bubble will ​then tend toward conformity with condition (614), the lighter portions rising to the top, more or less slowly, according to the viscosity of the film. The resulting difference of thickness between the upper and the lower parts of the bubble is due partly to the greater tension to which the upper parts are subject, and partly to a difference in the matter of which they are composed. When the film has only two components of which the potentials are not determined by the contiguous atmosphere, the laws which govern the arrangement of the elements of the film may be very simply expressed. If we call these components $S_{1}$ and $S_{2}$, the latter denoting (as on page 301) that which exists in excess at the surface, one element of the film will tend toward the same level with another, or a higher, or a lower level, according as the quantity of $S_{2}$ bears the same ratio to the quantity of $S_{1}$ in the first element as in the second, or a greater, or a less ratio.
When a film, however formed, satisfies both the conditions (613) and (614), its thickness being sufficient for its interior to have the properties of matter in mass, the interior will still be subject to the slow current which we have already described, if it is truly fluid, however great its viscosity may be. It seems probable, however, that this process is often totally arrested by a certain gelatinous consistency of the mass in question, in virtue of which, although practically fluid in its behavior with reference to ordinary stresses, it may have the properties of a solid with respect to such very small stresses as those which are caused by gravity in the interior of a very thin film which satisfies the conditions (613) and (614).
​The energy of the suction which produces these effects may be inferred from the following considerations. The pressure in the slender liquid mass which encircles the film is of course variable, being greater in the lower portions than in the upper, but it is everywhere less than the pressure of the atmosphere. Let us take a point where the pressure is less than that of the atmosphere by an amount represented by a column of the liquid one centimeter in height. (It is probable that much greater differences of pressure occur.) At a point near by in the interior of the film the pressure is that of the atmosphere. Now if the difference of pressure of these two points were distributed uniformly through the space of one centimeter, the intensity of its action would be exactly equal to that of gravity. But since the change of pressure must take place very suddenly (in a small fraction of a millimeter), its effect in producing a current in a limited space must be enormously great compared with that of gravity.
Since the process just described is connected with the descent of the liquid in the mass encircling the film, we may regard it as another example of the downward tendency of the interior of the film. There is a third way in which this descent may take place, when the principal component of the interior is volatile, viz., through the air. Thus, in the case of a film of soap-water, if we suppose the atmosphere to be of such humidity that the potential for water at a level mid- way between the top and bottom of the film has the same value in the atmosphere as in the film, it may easily be shown that evaporation will take place in the upper portions and condensation in the lower. These processes, if the atmosphere were otherwise undisturbed, would occasion currents of diffusion and other currents, the general effect of which would be to carry the moisture downward. Such a precise adjustment would be hardly attainable, and the processes described would not be so rapid as to have a practical importance.
But when the potential for water in the atmosphere differs considerably from that in the film, as in the case of a film of soap-water in a dry atmosphere, or a film of soap- water with glycerine in a moist atmosphere, the effect of evaporation or condensation is not to be neglected. In the first case, the diminution of the thickness of the film will be accelerated, in the second, retarded. In the case of the film containing glycerine, it should be observed that the water condensed cannot in all respects replace the fluid carried down by the internal current but that the two processes together will tend to wash out the glycerine from the film.
But when a component which greatly diminishes the tension of the film, although forming but a small fraction of its mass (therefore ​existing in excess at the surface), is volatile, the effect of evaporation and condensation may be considerable, even when the mean value of the potential for that component is the same in the film as in the surrounding atmosphere. To illustrate this, let us take the simple case of two components $S_{1}$ and $S_{2}$, as before. (See page 301.) It appears from equation (508) that the potentials must vary in the film with the height $z$, since the tension does, and from (98) that these variations must (very nearly) satisfy the relation
| $\gamma _{1}{\frac {d\mu _{1}}{dz}}+\gamma _{2}{\frac {d\mu _{2}}{dz}}=0,$ | (658) |
$\gamma _{1}$
$\gamma _{2}$
$S_{1}$
$S_{2}$
$S_{1}$
$S_{2}$
${\frac {d\mu _{1}}{dz}}$
${\frac {d\mu _{2}}{dz}}$
$S_{1}$
$S_{2}$
$S_{2}$
$S_{1}$
$S_{2}$
$S_{1}$
But apart from any action of the atmosphere, we have seen that a film which is truly fluid in its interior is in general subject to a continual diminution of thickness by the internal currents due to gravity and the suction at its edge. Sooner or later, the interior will somewhere cease to have the properties of matter in mass. The film will then probably become unstable with respect to a flux of the interior (see page 305), the thinnest parts tending to become still more thin (apart from any external cause) very much as if there were an attraction between the surfaces of the film, insensible at greater distances, but becoming sensible when the thickness of the film is sufficiently reduced. We should expect this to determine the rupture of the film, and such is doubtless the case with most liquids. In a film of soap-water, however, the rupture does not take place, and the processes ​which go on can be watched. It is apparent even to a very superficial observation that a film of which the tint is approaching the black exhibits a remarkable instability. The continuous change of tint is interrupted by the breaking out and rapid extension of black spots. That in the formation of these black spots a separation of different substances takes place, and not simply an extension of a part of the film, is shown by the fact that the film is made thicker at the edge of these spots.
This is very distinctly seen in a plane vertical film, when a single black spot breaks out and spreads rapidly over a considerable area which was before of a nearly uniform tint approaching the black. The edge of the black spot as it spreads is marked as it were by a string of bright beads, which unite together on touching, and thus becoming larger, glide down across the bands of color below. Under favorable circumstances, there is often quite a shower of these bright spots. They are evidently small spots—very much thicker apparently many times thicker—than the part of the film out of which they are formed. Now if the formation of the black spots were due to a simple extension of the film, it is evident that no such appearance would be presented. The thickening of the edge of the film cannot be accounted for by contraction. For an extension of the upper portion of the film and contraction of the lower and thicker portion, with descent of the intervening portions, would be far less resisted by viscosity, and far more favored by gravity than such extensions and contractions as would produce the appearances described. But the rapid formation of a thin spot by an internal current would cause an accumulation at the edge of the spot of the material forming the interior of the film, and necessitate a thickening of the film in that place.
That which is most difficult to account for in the formation of the black spots is the arrest of the process by which the film grows thinner. It seems most natural to account for this, if possible, by passive resistance to motion due to a very viscous or gelatinous condition of the film. For it does not seem likely that the film, after becoming unstable by the flux of matter from its interior, would become stable (without the support of such resistance) by a continuance of the same process. On the other hand, gelatinous properties are very marked in soap-water which contains somewhat more soap than is best for the formation of films, and it is entirely natural that, even when such properties are wanting in the interior of a mass or thick film of a liquid, they may still exist in the immediate vicinity of the surface (where we know that the soap or some of its components exists in excess), or throughout a film which is so thin that the interior has ceased to have the properties of matter ​in mass. But these considerations do not amount to any a priori probability of an arrest of the tendency toward an internal current between adjacent elements of a black spot which may differ slightly in thickness, in time to prevent rupture of the film. For, in a thick film, the increase of the tension with the extension, which is necessary for its stability with respect to extension, is connected with an excess of the soap (or of some of its components) at the surface as compared with the interior of the film. With respect to the black spots, although the interior has ceased to have the properties of matter in mass, and any quantitative determinations derived from the surfaces of a mass of the liquid will not be applicable, it is natural to account for the stability with reference to extension by supposing that the same general difference of composition still exists. If therefore we account for the arrest of internal currents by the increasing density of soap or some of its components in the interior of the film, we must still suppose that the characteristic difference of composition in the interior and surface of the film has not been obliterated.
The preceding discussion relates to liquid films between masses of gas. Similar considerations will apply to liquid films between other liquids or between a liquid and a gas, and to films of gas between masses of liquid. The latter may be formed by gently depositing a liquid drop upon the surface of a mass of the same or a different liquid. This may be done (with suitable liquids) so that the continuity of the air separating the liquid drop and mass is not broken, but a film of air is formed, which, if the liquids are similar, is a counterpart of the liquid film which is formed by a bubble of air rising to the top of a mass of the liquid. (If the bubble has the same volume as the drop, the films will have precisely the same form, as well as the rest of the surfaces which bound the bubble and the drop.) Sometimes, when the weight and momentum of the drop carry it through the surface of the mass on which it falls, it appears surrounded by a complete spherical film of air, which is the counterpart on a small scale of a soap-bubble hovering in air. Since, however, the substance to which the necessary differences of ​tension in the film are mainly due is a component of the liquid masses on each side of the air film, the necessary differences of the potential of this substance cannot be permanently maintained, and these films have little persistence compared with films of soap-water in air. In this respect, the case of these air-films is analogous to that of liquid films in an atmosphere containing substances by which their tension is greatly reduced. Compare pages 310, 311.
Surfaces of Discontinuity between Solids and Fluids.
We have hitherto treated of surfaces of discontinuity on the supposition that the contiguous masses are fluid. This is by far the most simple case for any rigorous treatment, since the masses are necessarily isotropic both in nature and in their state of strain. In this case, moreover, the mobility of the masses allows a satisfactory experimental verification of the mechanical conditions of equilibrium. On the other hand, the rigidity of solids is in general so great, that any tendency of the surfaces of discontinuity to variation in area or form may be neglected in comparison with the forces which are produced in the interior of the solids by any sensible strains, so that it is not generally necessary to take account of the surfaces of discontinuity in determining the state of strain of solid masses. But we must take account of the nature of the surfaces of discontinuity between solids and fluids with reference to the tendency toward solidification or dissolution at such surfaces, and also with reference to the tendencies of different fluids to spread over the surfaces of solids.
Let us therefore consider a surface of discontinuity between a fluid and a solid, the latter being either isotropic or of a continuous crystalline structure, and subject to any kind of stress compatible with a state of mechanical equilibrium with the fluid. We shall not exclude the case in which substances foreign to the contiguous masses are present in small quantities at the surface of discontinuity, but we shall suppose that the nature of this surface (i.e., of the non-homogeneous film between the approximately homogeneous masses) is entirely determined by the nature and state of the masses which it separates, and the quantities of the foreign substances which may be present. The notions of the dividing surface, and of the superficial densities of energy, entropy, and the several components, which we have used with respect to surfaces of discontinuity between fluids (see pages 219 and 224), will evidently apply without modification to the present case. We shall use the suffix $_{1}$ with reference to the substance of the solid, and shall suppose the dividing surface to be determined so as to make the superficial density of this substance vanish. The superficial densities of energy, of entropy, and of the ​other component substances may then be denoted by our usual symbols (see page 235),
* $\epsilon _{S(1)},$$\eta _{S(1)}$$\Gamma _{2(1)}$$\Gamma _{3(1)},$etc.
| $\sigma =\epsilon _{S(1)}-t\eta _{S(1)}-\mu _{2}\Gamma _{2(1)}-\mu _{3}\Gamma _{3(1)}-{\text{etc.,}}$ | (659) |
$t$
$\mu _{2},\mu _{3}$
$\sigma$
​With these preliminary notions, we now proceed to discuss the condition of equilibrium which relates to the dissolving of a solid at the surface where it meets a fluid, when the thermal and mechanical conditions of equilibrium are satisfied. It will be necessary for us to consider the case of isotropic and of crystallized bodies separately, since in the former the value of $\sigma$ is independent of the direction of the surface, except so far as it may be influenced by the state of strain of the solid, while in the latter the value of $\sigma$ varies greatly with the direction of the surface with respect to the axes of crystallization, and in such a manner as to have a large number of sharply defined minima. This may be inferred from the phenomena which crystalline bodies present, as will appear more distinctly in the following discussion. Accordingly, while a variation in the direction of an element of the surface may be neglected (with respect to its effect on the value of $\sigma$) in the case of isotropic solids, it is quite otherwise with crystals. Also, while the surfaces of equilibrium between fluids and soluble isotropic solids are without discontinuities of direction, being in general curved, a crystal in a state of equilibrium with a fluid in which it can dissolve is bounded in general by a broken surface consisting of sensibly plane portions.
For isotropic solids, the conditions of equilibrium may be deduced as follows. If we suppose that the solid is unchanged, except that an infinitesimal portion is dissolved at the surface where it meets the fluid, and that the fluid is considerable in quantity and remains homogeneous, the increment of energy in the vicinity of the surface will be represented by the expression
* $\int [\epsilon _{V}'-\epsilon _{V}''+(c_{1}+c_{2})\epsilon _{S(1)}]\delta NDs$
$Ds$
$\delta N$
$c_{1}$
$c_{2}$
$\epsilon _{S(1)}$
$\epsilon _{V}'$
$\epsilon _{V}''$
$Ds$
* $\int [\eta _{V}'-\eta _{V}''+(c_{1}+c_{2})\eta _{S(1)}]\delta NDs,$ $\int [\gamma _{1}'-\gamma _{2}']\delta NDs,$ $\int [-\gamma _{2}''+(c_{1}+c_{2})\Gamma _{2(1)}]\delta NDs,$ etc.
$-t$
$-\mu _{1}'',-\mu _{2}$
* $p''=-\epsilon _{V}''+t\eta _{V}''+\mu _{1}''\gamma _{1}''+\mu _{2}\gamma _{2}''+{\text{etc.}}$
| $\int [\epsilon _{V}'-t\eta _{V}'-\mu _{1}''\gamma _{1}'+p''+(c_{1}+c_{2})\sigma ]\delta NDs.$ | (660) |
$\delta N$
$\delta NDs$
| $\mu _{1}''={\frac {\epsilon _{V}'-t\eta _{V}'+p''+(c_{1}+c_{2})\sigma }{\gamma _{1}'}}\cdot$ | (661) |
$\sigma$
We may also observe that when the solid has no stresses except an isotropic pressure, if the quantity represented by $\sigma$ is equal to the true tension of the surface, $p''+(c_{1}+c_{2})\sigma$ will represent the pressure in the interior of the solid, and the second member of the equation will represent (see equation (93)) the value of the potential in the solid for the substance of which it consists. In this case, therefore, the equation reduces to
* $\mu _{1}''=\mu _{1}',$
Moreover, the compressibility of all solids is so small that, although or may not represent the true tension of the surface, nor $p''+(c_{1}+c_{2})\sigma$ the true pressure in the solid when its stresses are isotropic, the quantities $\epsilon _{V}'$ and $\eta _{V}'$ if calculated for the pressure $p''+(c_{1}+c_{2})\sigma$ with the actual temperature will have sensibly the same values as if calculated for the true pressure of the solid. Hence, the second member of equation (661), when the stresses of the solid are sensibly isotropic, is sensibly equal to the potential of the same body at the same temperature but with the pressure $p''+(c_{1}+c_{2})\sigma$ and the condition of equilibrium with respect to dissolving for a solid of isotropic stresses may be expressed with sufficient accuracy by saying that the potential for the substance of the solid in the fluid must have this value. In like manner, when the solid is not in a state of isotropic stress, the difference of the two pressures in question will not sensibly affect the values of $\epsilon _{V}'$ and $\eta _{V}'$, and the value of the second member of the equation may be calculated as if $p''+(c_{1}+c_{2})\sigma$ represented the true pressure in the solid in the direction of the normal to the surface. Therefore, if we had taken for granted that the quantity or represents the tension of a surface between a solid and a fluid, as it does when both masses are fluid, this assumption would not have led us into any practical error in determining the value of the potential $\mu _{1}''$ which is necessary for equilibrium. On the other hand, if in the case of any amorphous body the value of or differs notably from the true surface-tension, the latter quantity substituted for $\sigma$ in (661) will make the second member of the equation equal to the true value of $\mu _{1}'$, when the stresses are isotropic, but this will not be equal to the value of $\mu _{1}''$ in case of equilibrium, unless $c_{1}+c_{2}=0$.
When the stresses in the solid are not isotropic, equation (661) may be regarded as expressing the condition of equilibrium with respect to the dissolving of the solid, and is to be distinguished from the condition of equilibrium with respect to an increase of solid matter, since the new matter would doubtless be deposited in a state of isotropic stress. (The case would of course be different with crystalline bodies, which are not considered here.) The value of $\mu _{1}''$ necessary for equilibrium with respect to the formation of new matter is a little less than that necessary for equilibrium with respect to the dissolving of the solid. In regard to the actual behavior of the solid and fluid, all that the theory enables us to predict with certainty is that the solid will not dissolve if the value of the potential $\mu _{1}''$ is greater than that given by the equation for the solid with its distorting stresses, and that new matter will not be formed if the value of $\mu _{1}''$ is less than the same equation would give for the case of ​the solid with isotropic stresses. It seems probable, however, that if the fluid in contact with the solid is not renewed, the system will generally find a state of equilibrium in which the outermost portion of the solid will be in a state of isotropic stress. If at first the solid should dissolve, this would supersaturate the fluid, perhaps until a state is reached satisfying the condition of equilibrium with the stressed solid, and then, if not before, a deposition of solid matter in a state of isotropic stress would be likely to commence and go on until the fluid is reduced to a state of equilibrium with this new solid matter.
The action of gravity will not affect the nature of the condition of equilibrium for any single point at which the fluid meets the solid, but it will cause the values of $p''$ and $\mu _{1}''$ in (661) to vary according to the laws expressed by (612) and (617). If we suppose that the outer part of the solid is in a state of isotropic stress, which is the most important case, since it is the only one in which the equilibrium is in every sense stable, we have seen that the condition (661) is at least sensibly equivalent to this:—that the potential for the substance of the solid which would belong to the solid mass at the temperature $t$ and the pressure $p''+(c_{1}+c_{2})\sigma$ must be equal to $\mu _{1}''$. Or, if we denote by $(p')$ the pressure belonging to solid with the temperature $t$ and the potential equal to $\mu _{1}''$, the condition may be expressed in the form
| $(p')=p''+(c_{1}+c_{2})\sigma .$ | (662) |
$\gamma ''$
* $dp''=-g\gamma ''dz.$
$d(p')=\gamma _{1}'d\mu _{1}'',$
$d\mu _{1}''=-gdz;$
$d(p')=-g\gamma _{1}dz.$
* $d(p')-dp''=g(\gamma ''-\gamma _{1}')dz,$
* $(p')-p''=g(\gamma ''-\gamma _{1}')z,$
$z$
$(p')=p''$
| $c_{1}+c_{2}={\frac {g(\gamma ''-\gamma _{1}')}{\sigma }}z,$ | (663) |
​precisely as if both masses were fluid, and $\sigma$ denoted the tension of their common surface, and $(p')$ the true pressure in the mass specified. (Compare (619).)
The obstacles to an exact experimental realization of these relations are very great, principally from the want of absolute uniformity in the internal structure of amorphous solids, and on account of the passive resistances to the processes which are necessary to bring about a state satisfying the conditions of theoretical equilibrium, but it may be easy to verify the general tendency toward diminution of surface, which is implied in the foregoing equations.
Let us apply the same method to the case in which the solid is a crystal. The surface between the solid and fluid will now consist of plane portions, the directions of which may be regarded as invariable. If the crystal grows on one side a distance $\delta N$, without other change, the increment of energy in the vicinity of the surface will be
* $(\epsilon _{V}'-\epsilon _{V}'')s\delta N+\textstyle \sum '\displaystyle (\epsilon _{S(1)}'l'\csc \omega '-\epsilon _{S(1)}l'\cot \omega ')\delta N,$
​where $\epsilon _{V}'$ and $\epsilon _{V}''$ denote the volume-densities of energy in the crystal and fluid respectively, $s$ the area of the side on which the crystal grows, $\epsilon _{S(1)}$ the surface-density of energy on that side, $\epsilon _{S(1)}'$ the surface-density of energy on an adjacent side, $\omega '$ the external angle of these two sides, $l'$ their common edge, and the symbol $\textstyle \sum '$ a summation with respect to the different sides adjacent to the first. The increments of entropy and of the quantities of the several components will be represented by analogous formulae, and if we deduce as on pages 316, 317 the expression for the increase of energy in the whole system due to the growth of the crystal without change of the total entropy or volume, and set this expression equal to zero, we shall obtain for the condition of equilibrium
| $(\epsilon _{V}'-t\eta _{V}'-\mu _{1}''\gamma _{1}+p'')s\,\delta N+\textstyle \sum '\displaystyle (\sigma 'l'\csc \omega '-\sigma l'\cot \omega ')\delta N=0,$ | (664) |
$\sigma$
$\sigma '$
$\epsilon _{S(1)}$
$\epsilon _{S(1)}'$
| $\mu _{1}''={\frac {\epsilon _{V}'-t\eta _{V}'+p''}{\gamma _{1}'}}+{\frac {\textstyle \sum '\displaystyle (\sigma 'l'\csc \omega '-\sigma l'\cot \omega ')}{s\gamma _{1}'}}\cdot$ | (665) |
$\mu _{1}''$
Since a similar equation must hold true with respect to every other side of the crystal of which the equilibrium is not affected by meeting some other body, the condition of equilibrium for the crystalline form (when unaffected by gravity) is that the expression
| ${\frac {\textstyle \sum '\displaystyle (\sigma 'l'\csc \omega '-\sigma l'\cot \omega ')}{s}}$ | (666) |
$\sigma$
$s$
If we write $v$ for the volume of a crystal, and $\textstyle \sum \displaystyle (\sigma s)$ for the sum of the areas of all its sides multiplied each by the corresponding value of $\sigma$, the numerator and denominator of the fraction (666), multiplied each by $\delta N$, may be represented by $\delta \textstyle \sum \displaystyle (\sigma s)$ and $\delta v$ ​respectively. The value of the fraction is therefore equal to that of the differential coefficient
* ${\frac {d\textstyle \sum \displaystyle (\sigma s)}{dv}}$
$\textstyle \sum \displaystyle (\sigma s)$
When there are no foreign substances at the surfaces of the crystal, and the surrounding fluid is indefinitely extended, the quantity $\textstyle \sum \displaystyle (\sigma s)$ represents the work required to form the surfaces of the crystal, and the coefficient of $s\,\,\delta N$ in (664) with its sign reversed represents the work gained in forming a mass of volume unity like the crystal but regarded as without surfaces. We may denote the work required to form the crystal by
* $W_{S}-W_{V},$
$W_{S}$
$\textstyle \sum \displaystyle (\sigma s)$
$W_{V}$
| $-\delta W_{V}+\textstyle \sum \displaystyle (\sigma \delta s)=0.$ | (667) |
$\sigma$
$W_{S}$
* $dW_{S}=d\textstyle \sum \displaystyle (\sigma \delta s)={\tfrac {3}{2}}\textstyle \sum \displaystyle (\sigma \delta s),$
$W_{V}$
* $dW_{S}={\tfrac {3}{2}}dW_{V},$
$W_{S}$
$W_{V}$
* $W_{S}={\tfrac {3}{2}}W_{V},$
| $W_{S}-W_{V}={\tfrac {1}{3}}W_{S}={\tfrac {1}{2}}W_{V},$ | (668) |
​To take account of the influence of gravity, we must give to $\mu _{1}''$ and $p''$ in (665) their average values in the side considered. These coincide (when the fluid is in a state of internal equilibrium) with their values at the center of gravity of the side. The values of $\gamma _{1}',\epsilon _{V}',\eta _{V}'$, may be regarded as constant, so far as the influence of gravity is concerned. Now since by (612) and (617)
* $dp''=-g\gamma ''dz,$
* $d\mu _{1}''=-gdz,$
* $d(\gamma _{1}'\mu _{1}''-p'')=g(\gamma ''-\gamma _{1}')dz.$
In the preceding paragraph the fluid is regarded as in a state of internal equilibrium. If we suppose the composition and temperature of the fluid to be uniform, the condition which will make the effect of gravity vanish will be that
* ${\frac {d(\gamma _{1}'\mu _{1}''-p'')}{dz}}=0,$
* $\left({\frac {d\mu _{1}''}{dp}}\right)_{t,m}''={\frac {1}{\gamma _{1}'}},$
| $\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}''={\frac {1}{\gamma _{1}'}}\cdot$ | (669) |
Again, we may suppose the composition of the fluid and its entropy per unit of mass to be uniform. The temperature will then vary with the pressure, that is, with $z$. We may also suppose the temperature of different crystals or different parts of the same crystal to be determined by the fluid in contact with them. These conditions express a state which may perhaps be realized when the fluid is gently stirred. Owing to the differences of temperature we cannot regard $\epsilon _{V}'$ and $\eta _{V}'$ ​in (664) as constant, but we may regard their variations as subject to the relation $d\epsilon _{V}'=td\eta _{V}'$. Therefore, if we make $\eta _{V}'=0$ for the mean temperature of the fluid (which involves no real loss of generality), we may treat $\epsilon _{V}'-\eta _{V}'$ as constant. We shall then have for the condition that the effect of gravity shall vanish
* ${\frac {d(\gamma _{1}'\mu _{1}''-p'')}{dz}}=0,$
* $\left({\frac {d\mu _{1}}{dp}}\right)_{\eta ,m}''={\frac {1}{\gamma _{1}'}},$
| $\left({\frac {dv}{dm_{1}}}\right)_{\eta ,p,m}''={\frac {1}{\gamma _{1}'}}\cdot$ | (670) |
The manner in which crystals actually grow or dissolve is often principally determined by other differences of phase in the surrounding fluid than those which have been considered in the preceding paragraph. This is especially the case when the crystal is growing or dissolving rapidly. When the great mass of the fluid is considerably supersaturated, the action of the crystal keeps the part immediately contiguous to it nearer the state of exact saturation. The farthest projecting parts of the crystal will therefore be most exposed to the action of the supersaturated fluid, and will grow most rapidly. The same parts of a crystal will dissolve most rapidly in a fluid considerably below saturation.
$\textstyle \sum \displaystyle (\sigma s)$
$\delta N$
​ Let us now examine the special condition of equilibrium which relates to a line at which three different masses meet, when one or more of these masses is solid. If we apply the method of pages 316, 317 to a system containing such a line, it is evident that we shall obtain in the expression corresponding to (660), beside the integral relating to the surfaces, a term of the form
* $\int \textstyle \sum \displaystyle (\sigma \delta T)Dl$
$\sigma$
| $\textstyle \sum \displaystyle (\sigma \delta T)\geqq 0$ | (671) |
The simplest case is when two masses are fluid and the third is solid and insoluble. Let us denote the solid by $S$, the fluids by $A$ and $B$, and the angles filled by these fluids by $\alpha$ and $\beta$ respectively. If the surface of the solid is continuous at the line where it meets the two fluids, the condition of equilibrium reduces to
| $\sigma _{AB}\cos \alpha =\sigma _{BS}-\sigma _{AS}.$ | (672) |
| | $\sigma _{AB}\cos \alpha \leqq \sigma _{BS}-\sigma _{AS};$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (673) |
| and | $\sigma _{AB}\cos \beta \leqq \sigma _{AS}-\sigma _{BS};$ | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ (673) |
which reduces to the preceding when $\alpha +\beta =\pi$. Since the displacement of the line can take place by a purely mechanical process, this ​condition is capable of a more satisfactory experimental verification than those conditions which relate to processes of solidification and dissolution. Yet the fractional resistance to a displacement of the line is enormously greater than in the case of three fluids, since the relative displacements of contiguous portions of matter are enormously greater. Moreover, foreign substances adhering to the solid are not easily displaced, and cannot be distributed by extensions and contractions of the surface of discontinuity, as in the case of fluid masses. Hence, the distribution of such substances is arbitrary to a greater extent than in the case of fluid masses (in which a single foreign substance in any surface of discontinuity is uniformly distributed, and a greater number are at least so distributed as to make the tension of the surface uniform), and the presence of these substances will modify the conditions of equilibrium in a more irregular manner. If one or more of three surfaces of discontinuity which meet in a line divides an amorphous solid from a fluid in which it is soluble, such a surface is to be regarded as movable, and the particular conditions involved in (671) will be accordingly modified. If the soluble solid is a crystal, the case will properly be treated by the method used on pages 320, 321. The condition of equilibrium relating to the line will not in this case be entirely separable from those relating to the adjacent surfaces, since a displacement of the line will involve a displacement of the whole side of the crystal which is terminated at this line. But the expression for the total increment of energy in the system due to any internal changes not involving any variation in the total entropy or volume will consist of two parts, of which one relates to the properties of the masses of the system, and the other may be expressed in the form
* $\textstyle \sum \displaystyle (\sigma S),$
$\textstyle \sum \displaystyle (\sigma S)$
​General Relations.—For any constant state of strain of the surface of the solid we may write
| $d\epsilon _{S(1)}=td\eta _{S(1)}+\mu _{2}d\Gamma _{2(1)}+\mu _{3}d\Gamma _{3(1)}+{\text{etc.,}}$ | (674) |
| $d\sigma =-\eta _{S(1)}dt-\Gamma _{2(1)}d\mu _{2}-\Gamma _{3(1)}d\mu _{2}-{\text{etc.,}}$ | (675) |
$\sigma$
Another method and notation.—We have so far supposed that we have to do with a non-homogeneous film of matter between two homogeneous (or very nearly homogeneous) masses, and that the nature and state of this film is in all respects determined by the nature and state of these masses together with the quantities of the foreign substances which may be present in the film. (See page 314.) Problems relating to processes of solidification and dissolution seem hardly capable of a satisfactory solution, except on this supposition, which appears in general allowable with respect to the surfaces produced by these processes. But in considering the equilibrium of fluids at the surface of an unchangeable solid, such a limitation is neither necessary nor convenient. The following method of treating the subject will be found more simple and at the same time more general.
Let us suppose the superficial density of energy to be determined by the excess of energy in the vicinity of the surface over that which would belong to the solid, if (with the same temperature and state of strain) it were bounded by a vacuum in place of the fluid, and to the fluid, if it extended with a uniform volume-density of energy just up to the surface of the solid, or, if in any case this does not sufficiently define a surface, to a surface determined in some definite way by the exterior particles of the solid. Let us use the symbol $(\epsilon _{S})$ to denote the superficial energy thus defined. Let us suppose a superficial density of entropy to be determined in a manner entirely analogous, and be denoted by $(\eta _{S})$. In like manner also, for all the components of the fluid, and for all foreign fluid substances which may be present at the surface, let the superficial densities be determined, and denoted by $(\Gamma _{2}),(\Gamma _{3})$, etc. These superficial densities of the fluid components relate solely to the matter which is fluid or movable. All matter which is immovably attached to the solid mass is to be regarded as a part of the same. Moreover, let y be defined by the equation
| $\varsigma =(\epsilon _{S})-t(\eta _{S})-\mu _{2}(\Gamma _{2})-\mu _{3}(\Gamma _{3})-{\text{etc.}}$ | (676) |
​These quantities will satisfy the following general relations:—
| $d(\epsilon _{S})=td(\eta _{S})+\mu _{2}d(\Gamma _{2})+\mu _{3}d(\Gamma _{3})+{\text{etc.,}}$ | (677) |
| $\varsigma =-(\eta _{S})dt-(\Gamma _{2})d\mu _{2}-(\Gamma _{3})d\mu _{3}={\text{etc.}}$ | (678) |
$\varsigma ,(\epsilon _{S})$
$\sigma \epsilon _{S(1)}$
The quantity $\varsigma$ evidently represents the tendency to contraction in that portion of the surface of the fluid which is in contact with the solid. It may be called the superficial tension of the fluid in contact with the solid. Its value may be either positive or negative.
It will be observed for the same solid surface and for the same temperature but for different fluids the values of $\sigma$ (in all cases to which the definition of this quantity is applicable) will differ from those of $\varsigma$ by a constant, viz., the value of $\sigma$ for the solid surface in a vacuum.
For the condition of equilibrium of two different fluids at a line on the surface of the solid, we may easily obtain
| $\sigma _{AB}\cos \alpha =\varsigma _{BS}-\varsigma _{AS},$ | (679) |
It must also be regarded as a condition of theoretical equilibrium at the line considered (subject, like (679), to limitation on account of passive resistances to motion), that if there are any foreign substances in the surfaces ${\text{A-S}}$ and ${\text{B-S}}$, the potentials for these substances shall have the same value on both sides of the line; or, if any such substance is found only on one side of the line, that the potential for that substance must not have a less value on the other side; and that the potentials for the components of the mass $A$, for example, must have the same values in the surface ${\text{B-C}}$ as in the mass $A$, or, if they are not actual components of the surface ${\text{B-C}}$, a value not less than in $A$. Hence, we cannot determine the difference of the surface-tensions of two fluids in contact with the same solid, by bringing them together upon the surface of the solid, unless these conditions are satisfied, as well as those which are necessary to prevent the mixing of the fluid masses.
The investigation on pages 276–282 of the conditions of equilibrium for a fluid system under the influence of gravity may easily be extended to the case in which the system is bounded by or includes solid masses, when these can be treated as rigid and incapable of ​dissolution. The general condition of mechanical equilibrium would be of the form
| $-\int p\delta Dv+\int g\gamma \,\,\delta z\,\,Dv+\int \sigma \,\,\delta Ds+\int g\Gamma \,\,\delta z\,\,Ds+\int g\,\,\delta {\text{z}}\,\,Dm+\int \varsigma \,\,\delta Ds+\int g(\Gamma )\delta z\,\,Ds=0,$ | (680) |
$(\Gamma )$
$(\Gamma _{2}),(\Gamma _{3})$
$\delta {\text{z}}$
$\delta z$
From this equation, among other particular conditions of equilibrium, we may derive the following:—
| $d\varsigma =g(\Gamma )dz$ | (681) |
By applying equation (680) to the case of a vertical cylindrical tube containing two different fluids, we may easily obtain the well-known theorem that the product of the perimeter of the internal surface by the difference $\varsigma '-\varsigma ''$ of the superficial tensions of the upper and lower fluids in contact with the tube is equal to the excess of weight of the matter in the tube above that which would be there, if the boundary between the fluids were in the horizontal plane at which their pressures would be equal. In this theorem, we may either include or exclude the weight of a film of fluid matter adhering to the tube. The proposition is usually applied to the column of fluid in mass between the horizontal plane for which $p'=p''$ and the actual boundary between the two fluids. The superficial tensions $\varsigma '$ and $\varsigma ''$ are then to be measured in the vicinity of this column. But we may also include the weight of a film adhering to the internal surface of the tube. For example, in the case of water in equilibrium with its own vapor in a tube, the weight of all the water-substance in the tube above the plane $p'=p''$, diminished by that of the water-vapor which would fill the same space, is equal to the perimeter multiplied by the difference in the values of $\varsigma$ at the top of the tube and at the plane $p'=p''$. If the height of the tube is infinite, the value of $\varsigma$ at the top vanishes, and the weight of the film of water adhering to the tube and of the mass of liquid water above the plane $p'=p''$ diminished by the weight of vapor which would fill the same space is equal in numerical value but of opposite sign to the product of the perimeter of the internal ​surface of the tube multiplied by $\varsigma ''$, the superficial tension of liquid water in contact with the tube at the pressure at which the water and its vapor would be in equilibrium at a plane surface. In this sense, the total weight of water which can be supported by the tube per unit of the perimeter of its surface is directly measured by the value of $-\varsigma$ for water in contact with the tube.
Modification of the Conditions of Equilibrium by Electromotive Force.—Theory of a Perfect Electro-Chemical Apparatus.
We know by experience that in certain fluids (electrolytic conductors) there is a connection between the fluxes of the component substances and that of electricity. The quantitative relation between these fluxes may be expressed by an equation of the form
| $De={\frac {Dm_{\text{a}}}{a_{\text{a}}}}+{\frac {Dm_{b}}{a_{\text{b}}}}+{\text{etc.}}-{\frac {Dm_{\text{g}}}{a_{\text{g}}}}-{\frac {Dm_{\text{h}}}{a_{\text{h}}}}-{\text{etc.,}}$ | (682) |
$De,Dm_{\text{a}}$
$a_{\text{a}},a_{\text{b}}$
$a_{\text{g}},a_{\text{h}}$
$Dm_{\text{a}},Dm_{\text{b}}$
$Dm_{\text{g}},Dm_{\text{h}}$
Since the motion of the fluid as a whole will not involve any electrical current, the densities of the components specified by the suffixes must satisfy the relation
| ${\frac {\gamma _{\text{a}}}{a_{\text{a}}}}+{\frac {\gamma _{\text{b}}}{a_{\text{b}}}}+{\text{etc.}}={\frac {\gamma _{\text{g}}}{a_{\text{g}}}}+{\frac {\gamma _{\text{h}}}{a_{\text{h}}}}+{\text{etc.}}$ | (683) |
$a_{\text{a}},a_{\text{b}}$
$a_{\text{g}},a_{\text{h}}$
$a_{\text{a}},a_{\text{h}}$
​It will be observed that the choice of the substances which we regard as the components of the fluid is to some extent arbitrary, and that the same physical relations may be expressed by different equations of the form (682), in which the fluxes are expressed with reference to different sets of components. If the components chosen are such as represent what we believe to be the actual molecular constitution of the fluid, those of which the fluxes appear in the equation of the form (682) are called the ions, and the constants of the equation are called their electro-chemical equivalents. For our present purpose, which has nothing to do with any theories of molecular constitution, we may choose such a set of components as may be convenient, and call those ions, of which the fluxes appear in the equation of the form (682), without farther limitation.
Now, since the fluxes of the independently variable components of an electrolytic fluid do not necessitate any electrical currents, all the conditions of equilibrium which relate to the movements of these components will be the same as if the fluid were incapable of the electrolytic process. Therefore all the conditions of equilibrium which we have found without reference to electrical considerations, will apply to an electrolytic fluid and its independently variable components. But we have still to seek the remaining conditions of equilibrium, which relate to the possibility of electrolytic conduction.
For simplicity, we shall suppose that the fluid is without internal surfaces of discontinuity (and therefore homogeneous except so far as it may be slightly affected by gravity), and that it meets metallic conductors (electrodes) in different parts of its surface, being otherwise bounded by non-conductors. The only electrical currents which it is necessary to consider are those which enter the electrolyte at one electrode and leave it at another.
If all the conditions of equilibrium are fulfilled in a given state of the system, except those which relate to changes involving a flux of electricity, and we imagine the state of the system to be varied by the passage from one electrode to another of the quantity of electricity $\delta e$ accompanied by the quantity $\delta m_{\text{a}}$ of the component specified, without any flux of the other components or any variation in the total entropy, the total variation of energy in the system will be represented by the expression
* $(V''-V')\delta e+(\mu _{\text{a}}''-\mu _{\text{a}}')\delta m_{\text{a}}+(\Upsilon '-\Upsilon '')\delta m_{\text{a}},$
$V',V''$
$\Upsilon ',\Upsilon ''$
$\mu _{\text{a}}'',\mu _{\text{a}}''$
* $\delta m_{\text{a}}=a_{\text{a}}\delta \epsilon .$
| $V''-V'+a_{\text{a}}(\mu _{\text{a}}''-\mu _{\text{a}}'-\Upsilon ''+\Upsilon ')=0.$ | (684) |
| $V'+a_{\text{a}}(\mu _{\text{a}}'-\Upsilon ')=V''+a_{\text{a}}(\mu _{\text{a}}''-\Upsilon '')=V''+a_{\text{a}}(\mu _{\text{a}}'''-\Upsilon ''')={\text{etc.}}$ | (685) |
| $V'-a_{\text{g}}(\mu _{\text{g}}'-\Upsilon ')=V''-a_{\text{g}}(\mu _{\text{g}}''-\Upsilon '')=V'''-a_{\text{g}}(\mu _{\text{g}}'''-\Upsilon ''')={\text{etc.}}$ | (686) |
| $V''-V'=a_{\text{a}}(\mu _{\text{a}}'-\mu _{\text{a}}''),$ | (687) |
| $V''-V'=a_{\text{g}}(\mu _{\text{g}}'-\mu _{\text{g}}'')$ | (688) |
$V''-V'$
When all the conditions of equilibrium are fulfilled in a galvanic or electrolytic cell, the electromotive force is equal to the difference in the values of the potential for any ion or apparent ion at the surfaces of the electrodes multiplied by the electro-chemical equivalent of that ion, the greater potential of an anion being at the same electrode as the greater electrical potential, and the reverse being true of a cation.
Let us apply this principle to different cases.
(I.) If the ion is an independently variable component of an electrode, or by itself constitutes an electrode, the potential for the ion (in any case of equilibrium which does not depend upon passive resistances to change) will have the same value within the electrode as on its surface, and will be determined by the composition of the electrode with its temperature and pressure. This might be illustrated by a cell with electrodes of mercury containing certain quantities of zinc in solution (or with one such electrode and the other of pure zinc) and an electrolytic fluid containing a salt of zinc, but not capable of dissolving the mercury. We may regard ​a cell in which hydrogen acts as an ion between electrodes of palladium charged with hydrogen as another illustration of the same principle, but the solidity of the electrodes and the consequent resistance to the diffusion of the hydrogen within them (a process which cannot be assisted by convective currents as in a liquid mass) present considerable obstacles to the experimental verification of the relation.
(II.) Sometimes the ion is soluble (as an independently variable component) in the electrolytic fluid. Of course its condition in the fluid when thus dissolved must be entirely different from its condition when acting on an ion, in which case its quantity is not independently variable, as we have already seen. Its diffusion in the fluid in this state of solution is not necessarily connected with any electrical current, and in other relations its properties may be entirely changed. In any discussion of the internal properties of the fluid (with respect to its fundamental equation, for example), it would be necessary to treat it as a different substance. (See page 63.) But if the process by which the charge of electricity passes into the electrode, and the ion is dissolved in the electrolyte is reversible, we may evidently regard the potentials for the substance of the ion in (687) or (688) as relating to the substance thus dissolved in the electrolyte. In case of absolute equilibrium, the density of the substance thus dissolved would of course be uniform throughout the fluid (since it can move independently of any electrical current), so that by the strict application of our principle we only obtain the somewhat barren result that if any of the ions are soluble in the fluid without their electrical charges, the electromotive force must vanish in any case of absolute equilibrium not dependent upon passive resistances. Nevertheless, cases in which the ion is thus dissolved in the electrolytic fluid only to a very small extent, and its passage from one electrode to the other by ordinary diffusion is extremely slow, may be regarded as approximating to the case in which it is incapable of diffusion. In such cases, we may regard the relations (687), (688) as approximately valid, although the condition of equilibrium relating to the diffusion of the dissolved ion is not satisfied. This may be the case with hydrogen and oxygen as ions (or apparent ions) between electrodes of platinum in some of its forms.
(III.) The ion may appear in mass at the electrode. If it be a conductor of electricity, it may be regarded as forming an electrode, as soon as the deposit has become thick enough to have the properties of matter in mass. The case therefore will not be different from that first considered. When the ion is a non-conductor, a continuous thick deposit on the electrode would of course prevent the possibility of an ​electrical current. But the case in which the ion being a non-conductor is disengaged in masses contiguous to the electrode but not entirely covering it, is an important one. It may be illustrated by hydrogen appearing in bubbles at a cathode. In case of perfect equilibrium, independent of passive resistances, the potential of the ion in (687) or (688) may be determined in such a mass. Yet the circumstances are quite unfavorable for the establishment of perfect equilibrium, unless the ion is to some extent absorbed by the electrode or electrolytic fluid, or the electrode is fluid. For if the ion must pass immediately into the non-conducting mass, while the electricity passes into the electrode, it is evident that the only possible terminus of an electrolytic current is at the line where the electrode, the non-conducting mass, and the electrolytic fluid meet, so that the electrolytic process is necessarily greatly retarded, and an approximate ceasing of the current cannot be regarded as evidence that a state of approximate equilibrium has been reached. But even a slight degree of solubility of the ion in the electrolytic fluid or in the electrode may greatly diminish the resistance to the electrolytic process, and help toward producing that state of complete equilibrium which is supposed in the theorem we are discussing. And the mobility of the surface of a liquid electrode may act in the same way. When the ion is absorbed by the electrode, or by the electrolytic fluid, the case of course comes under the heads which we have already considered, yet the fact that the ion is set free in mass is important, since it is in such a mass that the determination of the value of the potential will generally be most easily made.
(IV.) When the ion is not absorbed either by the electrode or by the electrolytic fluid, and is not set free in mass, it may still be deposited on the surface of the electrode. Although this can take place only to a limited extent (without forming a body having the properties of matter in mass), yet the electro-chemical equivalents of all substances are so small that a very considerable flux of electricity may take place before the deposit will have the properties of matter in mass. Even when the ion appears in mass, or is absorbed by the electrode or electrolytic fluid, the non-homogeneous film between the electrolytic fluid and the electrode may contain an additional portion of it. Whether the ion is confined to the surface of the electrode or not, we may regard this as one of the cases in which we have to recognize a certain superficial density of substances at surfaces of discontinuity, the general theory of which we have already considered.
The deposit of the ion will affect the superficial tension of the electrode if it is liquid, or the closely related quantity which we have denoted by the same symbol $\sigma$ (see pages 314–331) if the electrode is solid. The effect can of course be best observed in the case of a liquid ​electrode. But whether the electrodes are liquid or solid, if the external electromotive force $V'-V''$ applied to an electrolytic combination is varied, when it is too weak to produce a lasting current, and the electrodes are thereby brought into a new state of polarization in which they make equilibrium with the altered value of the electromotive force, without change in the nature of the electrodes or of the electrolytic fluid, then by (508) or (675)
* $d\sigma '=-\Gamma _{\text{a}}'d\mu _{\text{a}}',$ $d\sigma ''=-\Gamma _{\text{a}}''d\mu _{\text{a}}'';$
* $d(V'-V'')=-a_{\text{a}}(d\mu _{\text{a}}'-d\mu _{\text{a}}'').$
| $d(V'-V'')={\frac {a_{\text{a}}}{\Gamma _{\text{a}}'}}d\sigma '-{\frac {a_{\text{a}}}{\Gamma _{\text{a}}''}}d\sigma ''.$ | (689) |
| $d\sigma '={\frac {\Gamma _{\text{a}}'}{a_{\text{a}}}}d(V'-V'').$ | (690) |
This principle has been applied by M. Lippmann to the construction of the electrometer which bears his name. In applying equations (689) and (690) to dilute sulphuric acid between electrodes of mercury, as in a Lippmann's electrometer, we may suppose that the suffix refers to hydrogen. It will be most convenient to suppose the dividing surface to be so placed as to make the surface-density of mercury zero. (See page 234.) The matter which exists in excess or deficiency at the surface may then be expressed by the surface-densities of sulphuric acid, of water, and of hydrogen. The value of the last may be determined from equation (690). According to M. Lippmann's determinations, it is negative when the surface is in its natural state (i.e., the state to which it tends when no external electromotive force is applied), since $\sigma '$ increases with $V''-V'$. When $V''-V'$ is equal to nine-tenths of the electromotive force of a Daniell's cell, the electrode to which $V''$ relates remaining in its natural state, the tension $\sigma '$ of the surface of the other electrode has a maximum value, and there is no excess or deficiency of hydrogen at that surface. This is the condition toward which a surface tends when it is extended while no flux of electricity takes place. The flux of electricity per unit of new surface formed, which will maintain a surface in a ​constant condition while it is extended, is represented by ${\frac {\Gamma _{\text{a}}'}{a_{\text{a}}}}$ in numerical value, and its direction, when $\Gamma _{\text{a}}'$ is negative, is from the mercury into the acid.
We have so far supposed, in the main, that there are no passive resistances to change, except such as vanish with the rapidity of the processes which they resist. The actual condition of things with respect to passive resistances appears to be nearly as follows. There does not appear to be any passive resistance to the electrolytic process by which an ion is transferred from one electrode to another, except such as vanishes with the rapidity of the process. For, in any case of equilibrium, the smallest variation of the externally applied electro-motive force appears to be sufficient to cause a (temporary) electrolytic current. But the case is not the same with respect to the molecular changes by which the ion passes into new combinations or relations, as when it enters into the mass of the electrodes, or separates itself in mass, or is dissolved (no longer with the properties of an ion) in the electrolytic fluid. In virtue of the passive resistance to these processes, the external electromotive force may often vary within wide limits, without creating any current by which the ion is transferred from one of the masses considered to the other. In other words, the value of $V'-V''$ may often differ greatly from that obtained from (687) or (688) when we determine the values of the potentials for the ion as in cases I, II, and III. We may, however, regard these equations as entirely valid, when the potentials for the ions are determined at the surface of the electrodes with reference to the ion in the condition in which it is brought there or taken away by an electrolytic current, without any attendant irreversible processes. But in a complete discussion of the properties of the surface of an electrode it may be necessary to distinguish (both in respect to surface-densities and to potentials) between the substance of the ion in this condition and the same substance in other conditions into which it cannot pass (directly) without irreversible processes. No such distinction, however, is necessary when the substance of the ion can pass at the surface of the electrode by reversible processes from any one of the conditions in which it appears to any other.
General Properties of a Perfect Electro-chemical Apparatus.
When an electrical current passes through a galvanic or electrolytic cell, the state of the cell is altered. If no changes take place in the cell except during the passage of the current, and all changes which accompany the current can be reversed by reversing the current, the cell may be called a perfect electro-chemical apparatus. The electromotive force of the cell may be determined by the equations which have just been given. But some of the general relations to which such an apparatus is subject may be conveniently stated in a form in which the ions are not explicitly mentioned.
In the most general case, we may regard the cell as subject to external action of four different kinds. (1) The supply of electricity at one electrode and the withdrawal of the same quantity at the other. (2) The supply or withdrawal of a certain quantity of heat. (3) The action of gravity. (4) The motion of the surfaces enclosing the apparatus, as when its volume is increased by the liberation of gases.
The increase of the energy in the cell is necessarily equal to that which it receives from external sources. We may express this by the equation
| $d\epsilon =(V'-V'')de+dQ+dW_{\text{G}}+dW_{\text{P}}$ | (691) |
$d\epsilon$
$de$
$V'$
$V''$
$dQ$
$dW_{\text{G}}$
$dW_{\text{P}}$
The conditions under which we suppose the processes to take place are such that the increase of the entropy of the apparatus is equal to the entropy which it receives from external sources. The only external source of entropy is the heat which is communicated to the cell by the surrounding bodies. If we write $d\eta$ for the increment of entropy in the cell, and $t$ for the temperature, we have
| $d\eta ={\frac {dQ}{t}}\cdot$ | (692) |
$dQ$
| $d\epsilon =(V'-V'')de+td\eta +dW_{\text{G}}+dW_{\text{P}},$ | (693) |
| $V''-V'=-{\frac {d\epsilon }{de}}+t{\frac {d\eta }{de}}+{\frac {dW_{\text{G}}}{de}}+{\frac {dW_{\text{P}}}{de}}\cdot$ | (694) |
| $d\eta \geqq {\frac {dQ}{t}},$ | (695) |
| $(V''-V')de\leqq -d\epsilon +td\eta +dW_{\text{G}}+dW_{\text{P}}.$ | (696) |
$d\eta$
The quantities expressed by the terms containing $dQ$ and $d\eta$ in (691), (693), (694), and (696) are frequently neglected in the consideration of cells of which the temperature is supposed to remain constant. In other words, it is frequently assumed that neither heat nor cold is produced by the passage of an electrical current through a perfect electro-chemical combination (except that heat which may be indefinitely diminished by increasing the time in which a given quantity of electricity passes), and that only heat can be produced in any cell, unless it be by processes of a secondary nature, which are not immediately or necessarily connected with the process of electrolysis.
It does not appear that this assumption is justified by any sufficient reason. In fact, it is easy to find a case in which the electromotive force is determined entirely by the term $t{\frac {d\eta }{de}}$ in (694), all the other terms in the second member of the equation vanishing. This is true of a Grove's gas battery charged with hydrogen and nitrogen. In this case, the hydrogen passes over to the nitrogen,—a process which does not alter the energy of the cell, when maintained at a constant temperature. The work done by external pressures is evidently nothing, and that done by gravity is (or may be) nothing. Yet an electrical current is produced. The work done (or which may be done) by the current outside of the cell is the equivalent of the work (or of a part of the work) which might be gained by allowing the gases to mix in other ways. This is equal, as has been shown by ​Lord Rayleigh, to the work which may be gained by allowing each gas separately to expand at constant temperature from its initial volume to the volume occupied by the two gases together. The same work is equal, as appears from equations (278), (279) on page 156 (see also page 159), to the increase of the entropy of the system multiplied by the temperature.
It is possible to vary the construction of the cell in such a way that nitrogen or other neutral gas will not be necessary. Let the cell consist of a U-shaped tube of sufficient height, and have pure hydrogen at each pole under very unequal pressures (as of one and two atmospheres respectively) which are maintained constant by properly weighted pistons, sliding in the arms of the tube. The difference of the pressures in the gas-masses at the two electrodes must of course be balanced by the difference in the height of the two columns of acidulated water. It will hardly be doubted that such an apparatus would have an electromotive force acting in the direction of a current which would carry the hydrogen from the denser to the rarer mass. Certainly the gas could not be carried in the opposite direction by an external electromotive force without the expenditure of as much (electromotive) work as is equal to the mechanical work necessary to pump the gas from the one arm of the tube to the other. And if by any modification of the metallic electrodes (which remain unchanged by the passage of electricity) we could reduce the passive resistances to zero, so that the hydrogen could be carried reversibly from one mass to the other without finite variation of the electromotive force, the only possible value of the electromotive force would be represented by the expression $t{\frac {d\eta }{de}}$, as a very close approximation. It will be observed that although gravity plays an essential part in a cell of this kind by maintaining the difference of pressure in the masses of hydrogen, the electromotive force cannot possibly be ascribed to gravity, since the work done by gravity, when hydrogen passes from the denser to the rarer mass, is negative.
Again, it is entirely improbable that the electrical currents caused by differences in the concentration of solutions of salts (as in a cell containing sulphate of zinc between zinc electrodes, or sulphate of copper between copper electrodes, the solution of the salt being of unequal strength at the two electrodes), which have recently been investigated theoretically and experimentally by MM. Helmholtz and Moser, are confined to cases in which the mixture of solutions of different degrees of concentration will produce heat. Yet in cases in which the mixture of more and less concentrated solutions is not ​attended with evolution or absorption of heat, the electromotive force must vanish in a cell of the kind considered, if it is determined simply by the diminution of energy in the cell. And when the mixture produces cold, the same rule would make any electromotive force impossible except in the direction which would tend to increase the difference of concentration. Such conclusions would be quite irreconcilable with the theory of the phenomena given by Professor Helmholtz.
A more striking example of the necessity of taking account of the variations of entropy in the cell in a priori determinations of electromotive force is afforded by electrodes of zinc and mercury in a solution of sulphate of zinc. Since heat is absorbed when zinc is dissolved in mercury, the energy of the cell is increased by a transfer of zinc to the mercury, when the temperature is maintained constant. Yet in this combination, the electromotive force acts in the direction of the current producing such a transfer. The couple presents certain anomalies when a considerable quantity of zinc is united with the mercury. The electromotive force changes its direction, so that this case is usually cited as an illustration of the principle that the electromotive force is in the direction of the current which diminishes the energy of the cell, i.e., which produces or allows those changes which are accompanied by evolution of heat when they take place directly. But whatever may be the cause of the electromotive force which has been observed acting in the direction from the amalgam through the electrolyte to the zinc (a force which according to the determinations of M. Gaugain is only one twenty-fifth part of that which acts in the reverse direction when pure mercury takes the place of the amalgam), these anomalies can hardly affect the general conclusions with which alone we are here concerned. If the electrodes of a cell are pure zinc and an amalgam containing zinc not in excess of the amount which the mercury will dissolve at the temperature of the experiment without losing its fluidity, and if the only change (other than thermal) accompanying a current is a transfer of zinc from one electrode to the other,—conditions which may not have been satisfied in all the experiments recorded, but which it is allowable to suppose in a theoretical discussion, and which certainly will not be regarded as inconsistent with the fact that heat is absorbed when zinc is dissolved in mercury,—it is impossible that the electromotive force should be in the direction of a current transferring zinc from the amalgam to the electrode of pure zinc. For, since the zinc eliminated from the amalgam by the electrolytic process might be re-dissolved directly, ​such a direction of the electromotive force would involve the possibility of obtaining an indefinite amount of electromotive work, and therefore of mechanical work, without other expenditure than that of heat at the constant temperature of the cell.
None of the cases which we have been considering involve combinations by definite proportions, and, except in the case of the cell with electrodes of mercury and zinc, the electromotive forces are quite small. It may perhaps be thought that with respect to those cells in which combinations take place by definite proportions the electromotive force may be calculated with substantial accuracy from the diminution of the energy, without regarding, the variation of entropy. But the phenomena of chemical combination do not in general seem to indicate any possibility of obtaining from the combination of substances by any process whatever an amount of mechanical work which is equivalent to the heat produced by the direct union of the substances.
A kilogramme of hydrogen, for example, combining by combustion under the pressure of the atmosphere with eight kilogrammes of oxygen to form liquid water, yields an amount of heat which may be represented in round numbers by 34000 calories. We may suppose that the gases are taken at the temperature of 0° C., and that the water is reduced to the same temperature. But this heat cannot be obtained at any temperature desired. A very high temperature has the effect of preventing to a greater or less extent, the combination of the elements. Thus, according to M. Sainte-Claire Deville, the temperature obtained by the combustion of hydrogen and oxygen cannot much if at all exceed 2500° C., which implies that less than one-half of the hydrogen and oxygen present combine at that temperature. This relates to combustion under the pressure of the atmosphere. According to the determinations of Professor Bunsen in regard to combustion in a confined space, only one-third of a mixture of hydrogen and oxygen will form a chemical compound at the temperature of 2850° C. and a pressure of ten atmospheres, and only a little more than one-half when the temperature is reduced by the addition of nitrogen to 2024° C., and the pressure to about three atmospheres exclusive of the part due to the nitrogen.
Now 10 calories at 2500° C. are to be regarded as reversibly convertible into one calorie at 4° C. together with the mechanical work representing the energy of 9 calories. If, therefore, all the 34000 calories obtainable from the union of hydrogen and oxygen under atmospheric pressure could be obtained at the temperature of ​2500° C., and no higher, we should estimate the electromotive work performed in a perfect electro-chemical apparatus in which these elements are combined or separated at ordinary temperatures and under atmospheric pressure as representing nine-tenths of the 34000 calories, and the heat evolved or absorbed in the apparatus as representing one -tenth of the 34000 calories. This, of course, would give an electromotive force exactly nine-tenths as great as is obtained on the supposition that all the 34000 calories are convertible into electromotive or mechanical work. But, according to all indications, the estimate 2500° C. (for the temperature at which we may regard all the heat of combustion as obtainable) is far too high, and we must regard the theoretical value of the electromotive force necessary to electrolyze water as considerably less than nine-tenths of the value obtained on the supposition that it is necessary for the electromotive agent to supply all the energy necessary for the process.
The case is essentially the same with respect to the electrolysis of hydrochloric acid, which is probably a more typical example of the process than the electrolysis of water. The phenomenon of dissociation is equally marked, and occurs at a much lower temperature, more than half of the gas being dissociated at 1400° C. And the heat which is obtained by the combination of hydrochloric acid gas with water, especially with water which already contains a considerable quantity of the acid, is probably only to be obtained at temperatures comparatively low. This indicates that the theoretical value of the electromotive force necessary to electrolyze this acid (i.e., the electromotive force which would be necessary in a reversible electro-chemical apparatus) must be very much less than that which could perform in electromotive work the equivalent of all the heat evolved in the combination of hydrogen, chlorine and water to form the liquid submitted to electrolysis. This presumption, based upon ​the phenomena exhibited in the direct combination of the substances, is corroborated by the experiments of M. Favre, who has observed an absorption of heat in the cell in which this acid was electrolyzed. The electromotive work expended must therefore have been less than the increase of energy in the cell.
In both cases of composition in definite proportions which we have considered, the compound has more entropy than its elements, and the difference is by no means inconsiderable. This appears to be the rule rather than the exception with respect to compounds which have less energy than their elements. Yet it would be rash to assert that it is an invariable rule. And when one substance is substituted for another in a compound, we may expect great diversity in the relations of energy and entropy.
In some cases there is a striking correspondence between the electromotive force of a cell and the rate of diminution of its energy per unit of electricity transmitted, the temperature remaining constant. A Daniell's cell is a notable example of this correspondence. It may perhaps be regarded as a very significant case, since of all cells in common use, it has the most constant electromotive force, and most nearly approaches the condition of reversibility. If we apply our previous notation (compare (691)) with the substitution of finite for infinitesimal differences to the determinations of M. Favre, estimating energy in calories, we have for each equivalent (32.6 kilogrammes) of zinc dissolved
* $(V''-V')\Delta e={\text{24327}}^{\text{cal}},$$\Delta \epsilon =-{\text{25394}}^{\text{cal}},$$\Delta Q=-{\text{1067}}^{\text{cal}}.$
$\Delta Q$
$-\Delta Q$
$(V''-V')\Delta e$
$-\Delta Q$
$-\Delta \epsilon$
It appears that under certain conditions neither heat nor cold is produced in a Grove's cell. For M. Favre has found that with different degrees of concentration of the nitric acid sometimes heat ​and sometimes cold is produced. When neither is produced, of course the electromotive force of the cell is exactly equal to its diminution of energy per unit of electricity transmitted. But such a coincidence is far less significant than the fact that an absorption of heat has been observed. With acid containing about seven equivalents of water $({\text{HNO}}_{6}+{\text{7HO}})[{\text{HNO}}_{3}+3{\tfrac {1}{2}}{\text{H}}_{2}{\text{O}}]$, M. Favre has found
* $(V''-V')\Delta e=46781^{\text{cal}},$$\Delta \epsilon =-41824^{\text{cal}},$$\Delta Q=4957^{\text{cal}};$
* $({\text{HNO}}_{6}+{\text{7HO}})[{\text{HNO}}_{3}+{\tfrac {1}{2}}{\text{H}}_{2}{\text{O}}],$ $(V''-V')\Delta e=49847^{\text{cal}},$$\Delta \epsilon =-52714^{\text{cal}},$$\Delta Q=-2867^{\text{cal}}.$
This absorption of heat in the cell he has observed in other cases, in which the chemical processes are much more simple.
For electrodes of cadmium and platinum in hydrochloric acid his experiments give
* ${\begin{aligned}(V''-V')\Delta e&=9256^{\text{cal}},&\Delta \epsilon &=-8258^{\text{cal}},\\\Delta W_{\text{P}}&=-290^{\text{cal}},\qquad &\Delta Q&=1288^{\text{cal}}.\end{aligned}}$
For electrodes of zinc and platinum in the same acid one series of experiments gives
* ${\begin{aligned}(V''-V')\Delta e&=16950^{\text{cal}},&\Delta \epsilon &=-16189^{\text{cal}},\\\Delta W_{\text{P}}&=-290^{\text{cal}},\qquad &\Delta Q&=1051^{\text{cal}};\end{aligned}}$
* ${\begin{aligned}(V''-V')\Delta e&=16738^{\text{cal}},&\Delta \epsilon &=-17702^{\text{cal}},\\\Delta W_{\text{P}}&=-290^{\text{cal}},\qquad &\Delta Q&=-674^{\text{cal}}.\end{aligned}}$
* $(V'-V'')\Delta e=34825^{\text{cal}}$$\Delta Q=2113^{\text{cal}},$
​whence
* $\Delta \epsilon -\Delta W_{\text{P}}=36938.$
$\Delta W_{\text{P}}$
$-\Delta W_{\text{P}}$
The great difference in the results of the two series of experiments relating to electrodes of zinc and platinum in hydrochloric acid is most naturally explained by supposing some difference in the conditions of the experiment, as in the concentration of the acid, or in the extent to which the substitution of zinc for hydrogen took place. That which it is important for us to observe in all these cases is that there are conditions under which heat is absorbed in a galvanic or electrolytic cell, so that the galvanic cell has a greater electromotive force than can be accounted for by the diminution of its energy, and the operation of electrolysis requires a less electromotive force than would be calculated from the increase of energy in the cell,—especially when the work done against the pressure of the atmosphere is taken into account.
It should be noticed that in all these experiments the quantity represented by $\Delta Q$ (which is the critical quantity with respect to the point at issue) was determined by direct measurement of the heat absorbed or evolved by the cell when placed alone in a calorimeter. The resistance of the circuit was made so great by a rheostat placed outside of the calorimeter that the resistance of the cell was regarded as insignificant in comparison, and no correction appears to have been made in any case for this resistance. With exception of the error due to this circumstance, which would in all cases diminish the heat absorbed in the cell (or increase the heat evolved), the probable error of $\Delta Q$ must be very small in comparison with that of $(V'-V'')\Delta e$, or with that of $\Delta \epsilon$, which were in general determined by the comparison of different calorimetrical measurements, involving very much greater quantities of heat.
In considering the numbers which have been cited, we should remember that when hydrogen is evolved as gas the process is in general very far from reversible. In a perfect electrochemical ​apparatus, the same changes in the cell would yield a much greater amount of electromotive work, or absorb a much less amount. In either case, the value of $\Delta Q$ would be much greater than in the imperfect apparatus, the difference being measured perhaps by thousands of calories.
It often occurs in a galvanic or electrolytic cell that an ion which is set free at one of the electrodes appears in part as gas, and is in part absorbed by the electrolytic fluid, and in part absorbed by the electrode. In such cases, a slight variation in the circumstances, which would not sensibly affect the electromotive force, would cause all of the ion to be disposed of in one of the three ways mentioned, if the current were sufficiently weak. This would make a considerable difference in the variation of energy in the cell, and the electromotive force cannot certainly be calculated from the variation of energy alone in all these cases. The correction due to the work performed against the pressure of the atmosphere when the ion is set free as gas will not help us in reconciling these differences. It will appear on consideration that this correction will in general increase the discordance in the values of the electromotive force. Nor does it distinctly appear which of these cases is to be regarded ​as normal and which are to be rejected as involving secondary processes.
If in any case secondary processes are excluded, we should expect it to be when the ion is identical in substance with the electrode upon which it is deposited, or from which it passes into the electrolyte. But even in this case we do not escape the difficulty of the different forms in which the substance may appear. If the temperature of the experiment is at the melting point of a metal which forms the ion and the electrode, a slight variation of temperature will cause the ion to be deposited in the solid or in the liquid state, or, if the current is in the opposite direction, to be taken up from a solid or from a liquid body. Since this will make a considerable difference in the variation of energy, we obtain different values for the electromotive force above and below the melting point of the metal, unless we also take account of the variations of entropy. Experiment does not indicate the existence of any such difference, and when we take account of variations of entropy, as in equation (694), it is apparent that there ought not to be any, the terms ${\frac {d\epsilon }{de}}$ and $t{\frac {d\eta }{de}}$ being both affected by the same difference, viz., the heat of fusion of an electrochemical equivalent of the metal. In fact, if such a difference existed, it would be easy to devise arrangements by which the heat yielded by a metal in passing from the liquid to the solid state could be transformed into electromotive work (and therefore into mechanical work) without other expenditure.
The foregoing examples will be sufficient, it is believed, to show the necessity of regarding other considerations in determining the electromotive force of a galvanic or electrolytic cell than the variation of its energy alone (when its temperature is supposed to remain constant), or corrected only for the work which may be done by external ​pressures or by gravity. But the relations expressed by (693), (694), and (696) may be put in a briefer form.
If we set, as on page 89,
* $\psi =\epsilon -t\eta ,$
* $d\psi =d\epsilon -td\eta ;$
| $V''-V'=-{\frac {d\psi }{de}}+{\frac {dW_{\text{G}}}{de}}+{\frac {dW_{\text{P}}}{de}};$ | (697) |
| $(V''-V')de\leqq -d\psi +dW_{\text{G}}+dW_{\text{P}}.$ | (698) |
* $\zeta =\epsilon -t\eta +pv,$
$p$
$v$
* $d\zeta =d\epsilon -td\eta +pdv,$
| $V''-V'=-{\frac {d\zeta }{de}},$ | (699) |
| $(V''-V')de\leqq -d\zeta .$ | (700) |
* ↑ [This paper was originally printed in two parts, divided at this point. For dates see heading, p. 55.]
* ↑ The effect of distorting stresses in a solid on the phenomena of crystallization and liquefaction, as well as the effect of change of hydrostatic pressure common to the solid and liquid, was first described by Professor James Thomson. See Trans. R. S. Edin., vol. xvi, p. 575; and Proc. Roy. Soc., vol. xi, p. 473, or Phil. Mag., ser. 4, vol. xxiv, p. 395.
* ↑ A suffixed $m$ stands here, as elsewhere in this paper, for all the symbols $m_{1},m_{2}$, etc., except such as may occur in the differential coefficient.
* ↑ For an example of this method, see Thomson and Tait's Natural Philosophy, vol. i, p. 705. With regard to the general theory of elastic solids, compare also Thomson's Memoir "On the Thermo-elastic and Thermo-magnetic Properties of Matter" in the Quarterly Journal oj Mathematics, vol. i, p. 57 (1855), and Green's memoirs on the propagation, reflection, and refraction of light in the Transactions of the Cambridge Philosophical Society, vol. vii.
* ↑ The values of $F$ and $G$ given in equations (434) and (438), which are here deduced at length, may be derived from inspection of equation (430) by means of the usual theorems relating to the multiplication of determinants. See Salmon's Lessons Introductory to the Modern Higher Algebra, 2d ed., Lesson III; or Baltzer's Theorie und Anwendung der Determinanten, §5.
* ↑ See Thomson and Tait's Natural Philosophy, vol. i, p. 711.
* ↑ To illustrate the difference between variations which are reversible, and those which are not, we may conceive of two entirely different substances meeting in equilibrium at a mathematical surface without being at all mixed. We may also conceive of them as mixed in a thin film about the surface where they meet, and then the amount of mixture is capable of variation both by increase and by diminution. But when they are absolutely unmixed, the amount of mixture can be increased, but is incapable of diminution, and it is then consistent with equilibrium that the value of $\delta \epsilon$ (for a variation of the system in which the substances commence to mix) should be greater than the second member of (477). It is not necessary to determine whether precisely such cases actually occur; but it would not be legitimate to overlook the possible occurrence of cases in which variations may be possible while the opposite variations are not. It will be observed that the sense in which the term reversible is here used is entirely different from that in which it is frequently used in treatises on thermodynamics, where a process by which a system is brought from a state A to a state B is called reversible, to signify that the system may also be brought from the state B to the state A through the same series of intermediate states taken in the reverse order by means of external agencies of the opposite character. The variation of a system from a state A to a state B (supposed to differ infinitely little from the first) is here called reversible when the system is capable of another state B' which bears the same relation to the state A that A bears to B.
* ↑ It will be understood that the $^{S}$ here used is not an algebraic exponent, but is only intended as a distinguishing mark. The Roman letter $S$ has not been used to denote any quantity.
* ↑ It is here supposed that the thermodynamic properties of the homogeneous masses have already been investigated, and that the fundamental equations of these masses may be regarded as known.
* ↑ The suffixed $\mu$ is used to denote that all the potentials except that occurring in the denominator of the differential coefficient are to be regarded as constant.
* ↑ If liquid mercury meets the mixed vapors of water and mercury in a plane surface, and we use $\mu _{1}$ and $\mu _{2}$ to denote the potentials of mercury and water respectively, and place the dividing surface so that $\Gamma _{1}=0$, i.e., so that the total quantity of mercury is the same as if the liquid mercury reached this surface on one side and the mercury vapor on the other without change of density on either side, then $\Gamma _{2(1)}$ will represent the amount of water in the vicinity of this surface, per unit of surface, above that which there would be, if the water- vapor just reached the surface without change of density, and this quantity (which we may call the quantity of water condensed upon the surface of the mercury) will be determined by the equation $\Gamma _{2(1)}=-{\frac {d\sigma }{d\mu _{2}}}\cdot$ (In this differential coefficient as well as the following, the temperature is supposed to remain constant and the surface of discontinuity plane. Practically, the latter condition may be regarded as fulfilled in the case of any ordinary curvatures.) If the pressure in the mixed vapors conforms to the law of Dalton (see pp. 155, 157), we shall have for constant temperature $dp_{2}=\gamma _{2}d\mu _{2},$ where $p_{2}$ denotes the part of the pressure in the vapor due to the water-vapor, and $\gamma _{2}$ the density of the water-vapor. Hence we obtain $\Gamma _{2(1)}=-\gamma _{2}{\frac {d\sigma }{dp_{2}}}\cdot$ For temperatures below 100° centigrade, this will certainly be accurate, since the pressure due to the vapor of mercury may be neglected. The value of $\sigma$ for $p_{2}=0$ and the temperature of 20° centigrade must be nearly the same as the superficial tension of mercury in contact with air, or 55.03 grammes per linear meter according to Quincke (Pogg. Ann., Bd. 139, p. 27). The value of $\sigma$ at the same temperature, when the condensed water begins to have the properties of water in mass, will be equal to the sum of the superficial tensions of mercury in contact with water and of water in contact with its own vapor. This will be, according to the same authority, 42.58 + 8.25, or 50.83 grammes per meter, if we neglect the difference of the tensions of water with its vapor and water with air. As $p_{2}$, therefore, increases from zero to 236400 grammes per square meter (when water begins to be condensed in mass), $\sigma$ diminishes from about 55.03 to about 50.83 grammes per linear meter. If the general course of the values of $\sigma$ for intermediate values of $p_{2}$ were determined by experiment, we could easily form an approximate estimate of the values of the superficial density $\Gamma _{2(1)}$ for different pressures less than that of saturated vapor. It will be observed that the determination of the superficial density does not by any means depend upon inappreciable differences of superficial tension. The greatest difficulty in the determination would doubtless be that of distinguishing between the diminution of superficial tension due to the water and that due to other substances which might accidentally be present. Such determinations are of considerable practical importance on account of the use of mercury in measurements of the specific gravity of vapors.
* ↑ With respect to questions which concern only the form of surfaces of discontinuity, such precision as we have employed in regard to the position of the dividing surface is evidently quite unnecessary. This precision has not been used for the sake of the mechanical part of the problem, which does not require the surface to be defined with greater nicety than we can employ in our observations, but in order to give determinate values to the superficial densities of energy, entropy, and the component substances, which quantities, as has been seen, play an important part in the relations between the tension of a surface of discontinuity, and the composition of the masses which it separates. The product $\sigma s$ of the superficial tension and the area of the surface, may be regarded as the available energy due to the surface in a system in which the temperature and the potentials $\mu _{1},\mu _{2}$, etc.—or the differences of these potentials and the gravitational potential (see page 148) when the system is subject to gravity—are maintained sensibly constant. The value of $\sigma$, as well as that of $s$, is sensibly independent of the precise position which we may assign to the dividing surface (so long as this is sensibly coincident with the surface of discontinuity), but $\epsilon _{S}$, the superficial density of energy, as the term is used in this paper, like the superficial densities of entropy and of the component substances, requires a more precise localization of the dividing surface.
* ↑ In the case of infinitesimal changes in the nature of the film, the sign $\Delta$ must be interpreted, as elsewhere in this paper, without neglect of infinitesimals of the higher orders. Otherwise, by equation (501), the above expression would have the value zero.
* ↑ With respect to the sense in which this term is used, compare page 79.
* ↑ See page 104, where the term stable is used (as indicated on page 103) in a less strict sense than in the discussion which here follows.
* ↑ For example, in applying our formulae to a microscopic globule of water in steam, by the density or pressure of the interior mass we should understand, not the actual density or pressure at the center of the globule, but the density of liquid water (in large quantities) which has the temperature and potential of the steam.
* ↑ To make the physical significance of the above more clear, we may suppose the two processes to be performed separately in the following manner. We may suppose a large mass of the same phase as that which has the volume $v'$ to exist initially in the interior of the other. Of course, it must be surrounded by a resisting envelop, on account of the difference of the pressures. We may, however, suppose this envelop permeable to all the component substances, although not of such properties that a mass can form on the exterior like that within. We may allow the envelop to yield to the internal pressure until its contents are increased by $v'$ without materially affecting its superficial area. If this be done sufficiently slowly, the phase of the mass within will remain constant. (See page 84.) A homogeneous mass of the volume $v'$ and of the desired phase has thus been produced, and the work gained is evidently $(p'-p'')v'$. Let us suppose that a small aperture is now opened and closed in the envelop so as to let out exactly the volume $v'$ of the mass within, the envelop being pressed, inwards in another place so as to diminish its contents by this amount. During the extrusion of the drop and until the orifice is entirely closed, the surface of the drop must adhere to the edge of the orifice, but not elsewhere to the outside surface of the envelop. The work done in forming the surface of the drop will evidently be $\sigma s$ or ${\tfrac {3}{2}}(p'-p'')v'.$. Of this work, the amount $(p'-p'')v'$ will be expended in pressing the envelop inward, and the rest in opening and closing the orifice. Both the opening and the closing will be resisted by the capillary tension. If the orifice is circular, it must have, when widest open, the radius determined by equation (550).
* ↑ It is true that such a case as we are now considering is formally excluded in the discussion referred to, which relates to a plane surface, and in which the system is supposed thoroughly stable with respect to the possible formation of any different homogeneous masses. Yet the reader will easily convince himself that the criterion (520) is perfectly valid in this case with respect to the possible formation of a thin sheet of the phase $C$, which, as we have seen, may be treated simply as a different kind of surface of discontinuity.
* ↑ If we represent phases by the position of points in such a manner that coexistent phases (in the sense in which the term is used on page 96) are represented by the same point, and allow ourselves, for brevity, to speak of the phases as having the positions of the points by which they are represented, we may say that three coexistent phases are situated where three series of pairs of coexistent phases meet or intersect. If the three phases are all fluid, or when the effects of solidity may be disregarded, two cases are to be distinguished. Either the three series of coexistent phases all intersect,—this is when each of the three surface tensions is less than the sum of the two others,—or one of the series terminates where the two others intersect,—this is where one surface tension is equal to the sum of the others. The series of coexistent phases will be represented by lines or surfaces, according as the phases have one or two independently variable components. Similar relations exist when the number of components is greater, except that they are not capable of geometrical representation without some limitation, as that of constant temperature or pressure or certain constant potentials.
* ↑ If we set $V=-{\frac {\Gamma _{'}}{\gamma '}}-{\frac {\Gamma _{''}}{\gamma ''}},$ (a) $H_{S}=\eta _{S}-{\frac {\Gamma _{'}}{\gamma '}}\eta _{V'}-{\frac {\Gamma _{''}}{\gamma ''}}\eta _{V''},$ (b) and in like manner $E_{S}=\epsilon _{S}-{\frac {\Gamma _{'}}{\gamma '}}\epsilon _{V'}-{\frac {\Gamma _{''}}{\gamma ''}}\epsilon _{V''},$ (c) we may easily obtain, by means of equations (93) and (507), $E_{S}=tH_{S}+\sigma -pV.$ (d) Now equation (580) may be written $d\sigma =-H_{S}dt+Vdp.$ (e) Differentiating (d), and comparing the result with (e), we obtain $dE_{S}=tdH_{S}-pdV.$ (f) The quantities $E_{S}$ and $H_{S}$ might be called the superficial densities of energy and entropy quite as properly as those which we denote by $\epsilon _{S}$ and $\eta _{S}$. In fact, when the composition of both of the homogeneous masses is invariable, the quantities $E_{S}$ and $H_{S}$ are much more simple in their definition than $\epsilon _{S}$ and $\eta _{S}$, and would probably be more naturally suggested by the terms superficial density of energy and of entropy. It would also be natural in this case to regard the quantities of the homogeneous masses as determined by the total quantities of matter, and not by the surface of tension or any other dividing surface. But such a nomenclature and method could not readily be extended so as to treat cases of more than two components with entire generality. In the treatment of surfaces of discontinuity in this paper, the definitions and nomenclature which have been adopted will be strictly adhered to. The object of this note is to suggest to the reader how a different method might be used in some cases with advantage, and to show the precise relations between the quantities which are used in this paper and others which might be confounded with them, and which may be made more prominent when the subject is treated differently.
* ↑ See Proc. Roy. Soc., vol. ix, p. 255 (June, 1858); or Phil. Mag., ser. 4, vol. xvii, p. 61.
* ↑ When, however, homogeneous masses which have not coexistent phases are brought into contact, the superficial tension may increase with the course of time. The superficial tension of a drop of alcohol and water placed in a large room will increase as the potential for alcohol is equalized throughout the room, and is diminished in the vicinity of the surface of discontinuity.
* ↑ It is here supposed that we have chosen for components such substances as are incapable of resolution into other components which are independently variable in the homogeneous masses. In a mixture of alcohol and water, for example, the components must be pure alcohol and pure water.
* ↑ From the experiments of M. E. Duclaux (Annales de Chimie et de Physique, ser. 4, vol. xxi, p. 383), it appears that one per cent. of alcohol in water will diminish the superficial tension to .933, the value for pure water being unity. The experiments do not extend to pure alcohol, but the difference of the tensions for mixtures of alcohol and water containing 10 and 20 per cent. water is comparatively small, the tensions being .322 and .336 respectively. According to the same authority (page 427 of the volume cited), one 3200th part of Castile soap will reduce the superficial tension of water by one-fourth; one 800th part of soap by one-half. These determinations, as well as those relating to alcohol and water, are made by the method of drops, the weight of the drops of different liquids from the same pipette) being regarded as proportional to their superficial tensions. M. Athanase Dupré has determined the superficial tensions of solutions of soap by different methods. A statical method gives for one part of common soap in 5000 of water a superficial tension about one-half as great as for pure water, but if the tension be measured on a jet close to the orifice, the value (for the same solution) is sensibly identical with that of pure water. He explains these different values of the superficial tension of the same solution as well as the great effect on the superficial tension which a very small quantity of soap or other trifling impurity may produce, by the tendency of the soap or other substance to form a film on the surface of the liquid. (See Annales de Chimie et de Physique, ser. 4, vol. vii, p. 409, and vol. ix, p. 379.)
* ↑ We have sometimes given a physical expression to a supposition of this kind, problems in which the peculiar condition of matter in the vicinity of surfaces discontinuity was to be neglected, by regarding the system as surrounded by a rigid and impermeable envelop. But the more exact treatment which we are now to give the problem of equilibrium would require us to take account of the influence of the envelop on the immediately adjacent matter. Since this involves the consideration of surfaces of discontinuity between solids and fluids, and we wish to limit ourselves at present to the consideration of the equilibrium of fluid masses, we shall give up the conception of an impermeable envelop, and regard the system as bounded simply by an imaginary surface, which is not a surface of discontinuity. The variations of the system must be such as do not deform the surface, nor affect the matter external to it.
* ↑ The term actual component has been defined for homogeneous masses on page 64, and the definition may be extended to surfaces of discontinuity. It will be observed that if a substance is an actual component of either of the masses separated by a surface of discontinuity, it must be regarded as an actual component for that surface, as well as when it occurs at the surface but not in either of the contiguous masses.
* ↑ Concerning another kind of conditions of chemical equilibrium, which relate to the molecular arrangement of the components, and not to their sensible distribution in space, see pages 138–144.
* ↑ Compare page 146, where a similar problem is treated without regard to the influence of the surfaces of discontinuity.
* ↑ If the fluid system is divided into separate masses by solid diaphragms which are permeable to all the components of the fluids independently, the conditions of equilibrium of the fluids relating to temperature and the potentials will not be affected. (Compare page 84.) The propositions which follow in the above paragraph may be extended to this case.
* ↑ We may here remark that a nearer approximation in the theory of equilibrium and stability might be attained by taking special account, in our general equations, of the lines in which surfaces of discontinuity meet. These lines might be treated in a manner entirely analogous to that in which we have treated surfaces of discontinuity. We might recognize linear densities of energy, of entropy, and of the several substances which occur about the line, also a certain linear tension. With respect to these quantities and the temperature and potentials, relations would hold analogous to those which have been demonstrated for surfaces of discontinuity. (See pp. 229–231.) If the sum of the tensions of the lines $L'$ and $L''$, mentioned above, is greater than the tension of the line $L$, this line will be in strictness stable (although practically unstable) with respect to the formation of a surface between $A$ and $C$, when the tension of such a surface is a little less than that represented by the diagonal $\alpha \gamma$. The different use of the term practically unstable in different parts of this paper need not create confusion, since the general meaning of the term is in all cases the same. A system is called practically unstable when a very small (not necessarily indefinitely small) disturbance or variation in its condition will produce a considerable change. In the former part of this paper, in which the influence of surfaces of discontinuity was neglected, a system was regarded as practically unstable when such a result would be produced by a disturbance of the same order of magnitude as the quantities relating to surfaces of discontinuity which were neglected. But where surfaces of discontinuity are considered, a system is not regarded as practically unstable, unless the disturbance which will produce such a result is very small compared with the quantities relating to surfaces of discontinuity of any appreciable magnitude.
* ↑ See note on page 288. We may here add that the linear tension there mentioned may have a negative value. This would be the case with respect to a line in which three surfaces of discontinuity are regarded as meeting, but where nevertheless there really exists in stable equilibrium a filament of different phase from the three surrounding masses. The value of the linear tension for the supposed line, would be nearly equal to the value of $W_{S}-W_{V}$ for the actually existing filament. (For the exact value of the linear tension it would be necessary to add the sum of the linear tensions of the three edges of the filament.) We may regard two soap-bubbles adhering together as an example of this case. The reader will easily convince himself that in an exact treatment of the equilibrium of such a double bubble we must recognize a certain negative tension in the line of intersection of the three surfaces of discontinuity.
* ↑ Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires, vol. i, p. 294.
* ↑ Sitzungsberichte der Wiener Akademie (mathemat.-naturwiss. Classe), B. xl, S. 607. The calculation of formula (656) and that of the factor (${\tfrac {2}{3}}$) applied to the formula of Poiesuille, to adapt it to a current between plane surfaces, have been made by means of the general equations of the motion of a viscous liquid as given in the memoir referred to.
* ↑ Ibid., p. 653; or Mémoires des Savants Étrangers, vol. ix, p. 532.
* ↑ The experiments of M. Plateau (chapter VII of the work already cited) show that this is the case to a very remarkable degree with respect to a solution of saponine. With respect to soap-water, however, they do not indicate any greater superficial viscosity than belongs to pure water. But the resistance to an internal current, such as we are considering, is not necessarily measured by the resistance to such motions as those of the experiments referred to.
* ↑ These spherical air-films are easily formed in soap-water. They are distinguishable from ordinary air-bubbles by their general behavior and by their appearance. The two concentric spherical surfaces are distinctly seen, the diameter of one appearing to be about three-quarters as large as that of the other. This is of course an optical illusion, depending upon the index of refraction of the liquid.
* ↑ This will appear more distinctly if we consider a particular case. Let us consider a thin plane sheet of a crystal in a vacuum (which may be regarded as a limiting case of a very attenuated fluid), and let us suppose that the two surfaces of the sheet are alike. By applying the proper forces to the edges of the sheet, we can make all stress vanish in its interior. The tensions of the two surfaces are in equilibrium with these forces, and are measured by them. But the tensions of the surfaces, thus determined, may evidently have different values in different directions, and are entirely different from the quantity which we denote by $\sigma$, which represents the work required to form a unit of the surface by any reversible process, and is not connected with any idea of direction. In certain cases, however, it appears probable that the values of a and of the superficial tension will not greatly differ. This is especially true of the numerous bodies which, although generally (and for many purposes properly) regarded as solids, are really very viscous fluids. Even when a body exhibits no fluid properties at its actual temperature, if its surface has been formed at a higher temperature, at which the body was fluid, and the change from the fluid to the solid state has been by insensible gradations, we may suppose that the value of $\sigma$ coincided with the superficial tension until the body was decidedly solid, and that they will only differ so far as they may be differently affected by subsequent variations of temperature and of the stresses applied to the solid. Moreover, when an amorphous solid is in a state of equilibrium with a solvent, although it may have no fluid properties in its interior, it seems not improbable that the particles at its surface, which have a greater degree of mobility, may so arrange themselves that the value of <r will coincide with the superficial tension, as in the case of fluids.
* ↑ The differential coefficients of $\sigma$ with respect to the direction-cosines of the surface appear to be discontinuous functions of the latter quantities.
* ↑ The potential $\mu _{1}''$ is marked by double accents in order to indicate that its value is to be determined in the fluid mass, and to distinguish it from the potential $\mu _{1}'$ relating to the solid mass (when this is in a state of isotropic stress), which, as we shall see, may not always have the same value. The other potentials $\mu _{2}$, etc., have the same values as in (659), and consist of two classes, one of which relates to substances which are components of the fluid mass (these might be marked by the double accents), and the other relates to substances found only at the surface of discontinuity. The expressions to be multiplied by the potentials of this latter class all have the value zero.
* ↑ In equation (387), the density of the solid is denoted by $\Gamma$, which is therefore equivalent to $\gamma _{1}'$ in (661).
* ↑ The possibility that the new solid matter might differ in composition from the original solid is here left out of account. This point has been discussed on pages 79–82, but without reference to the state of strain of the solid or the influence of the curvature of the surface of discontinuity. The statement made above may be generalized so as to hold true of the formation of new solid matter of any kind on the surface as follows:—that new solid matter of any kind will not be formed upon the surface (with more than insensible thickness), if the second member of (661) calculated for such new matter is greater than the potential in the fluid for such matter.
* ↑ It seems probable that a tendency of this kind plays an important part in some of the phenomena which have been observed with respect to the freezing together of pieces of ice. (See especially Professor Faraday's "Note on Regelation" in the Proceedings of the Royal Society, vol. x, p. 440; or in the Philosophical Magazine, 4th ser., vol. xxi, p. 146.) Although this is a body of crystalline structure, and the action which takes place is doubtless influenced to a certain extent by the directions of the axes of crystallization, yet since the phenomena have not been observed to depend upon the orientation of the pieces of ice we may conclude that the effect, so far as its general character is concerned, is such as might take place with an isotropic body. In other words, for the purposes of a general explanation of the phenomena we may neglect the differences in the values of $\sigma _{IW}$ (the suffixes are used to indicate that the symbol relates to the surface between ice and water) for different orientations of the axes of crystallization, and also neglect the influence of the surface of discontinuity with respect to crystalline structure, which must be formed by the freezing together of the two masses of ice when the axes of crystallization in the two masses are not similarly directed. In reality, this surface—or the necessity of the formation of such a surface if the pieces of ice freeze together—must exert an influence adverse to their union, measured by a quantity $\sigma _{II}$, which is determined for this surface by the same principles as when one of two contiguous masses is fluid, and varies with the orientations of the two systems of crystallographic axes relatively to each other and to the surface. But under the circumstances of the experiment, since we may neglect the possibility of the two systems of axes having precisely the same directions, this influence is probably of a tolerably constant character, and is evidently not sufficient to alter the general nature of the result. In order wholly to prevent the tendency of pieces of ice to freeze together, when meeting in water with curved surfaces and without pressure, it would be necessary that $\sigma _{II}\geqq 2\sigma _{IW}$, except so far as the case is modified by passive resistances to change, and by the inequality in the values of $\sigma _{II}$ and $\sigma _{IW}$ for different directions of the axes of crystallization. It will be observed that this view of the phenomena is in harmony with the opinion of Professor Faraday. With respect to the union of pieces of ice as an indirect consequence of pressure, see page 198 of volume xi of the Proceedings of the Royal Society; or the Philosophical Magazine, 4th ser., vol. xxiii, p. 407.
* ↑ A suffixed $m$ is used to represent all the symbols $m_{1},m_{2}$, etc., except such as may occur in the differential coefficient.
* ↑ See O. Lehmann, "Ueber das Wachsthum der Krystalle," Zeitschrift fur Krystallographie und Mineralogie, Bd. i, S. 453; or the review of the paper in Wiedemann's Beiblätter, Bd. ii, S. 1.
* ↑ That it is necessary that certain relations shall be precisely satisfied in order that equilibrium may subsist between a liquid and gas with respect to evaporation, is explained (see Clausius, "Ueber die Art der Bewegung, welche wir Wärme nennen," Pogg. Ann., Bd. o, S. 353; or Abhand. über die mechan. Wärmetheorie, XIV) by supposing that a passage of individual molecules from the one mass to the other is continually taking place, so that the slightest circumstance may give the preponderance to the passage of matter in either direction. The same supposition may be applied, at least in many cases, to the equilibrium between amorphous solids and fluids. Also in the case of crystals in equilibrium with fluids, there may be a passage of individual molecules from one mass to the other, so as to cause insensible fluctuations in the mass of the solid. If these fluctuations are such as to cause the occasional deposit or removal of a whole layer of particles, the least cause would be sufficient to make the probability of one kind of change prevail over that of the other, and it would be necessary for equilibrium that the theoretical conditions deduced above should be precisely satisfied. But this supposition seems quite improbable, except with respect to a very small side. The following view of the molecular state of a crystal when in equilibrium with respect to growth or dissolution appears as probable as any. Since the molecules at the corners and edges of a perfect crystal would be less firmly held in their places than those in the middle of a side, we may suppose that when the condition of theoretical equilibrium (665) is satisfied several of the outermost layers of molecules on each side of the crystal are incomplete toward the edges. The boundaries of these imperfect layers probably fluctuate, as individual molecules attach themselves to the crystal or detach themselves, but not so that a layer is entirely removed (on any side of considerable size), to be restored again simply by the irregularities of the motions of the individual molecules. Single molecules or small groups of molecules may indeed attach themselves to the side of the crystal but they will speedily be dislodged, and if any molecules are thrown out from the middle of a surface, these deficiencies will also soon be made good; nor will the frequency of these occurrences be such as greatly to affect the general smoothness of the surfaces, except near the edges where the surfaces fall off somewhat, as before described. Now a continued growth on any side of a crystal is impossible unless new layers can be formed. This will require a value of $\mu _{1}''$ which may exceed that given by equation (665) by a finite quantity. Since the difficulty in the formation of a new layer is at or near the commencement of the formation, the necessary value of $\mu _{1}''$ may be independent of the area of the side, except when the side is very small. The value of $\mu _{1}''$ which is necessary for the growth of the crystal will however be different for different kinds of surfaces, and probably will generally be greatest for the surfaces for which $\sigma$ is least. On the whole, it seems not improbable that the form of very minute crystals in equilibrium with solvents is principally determined by equation (665), (i.e., by the condition that $\textstyle \sum \displaystyle (\sigma s)$ shall be a minimum for the volume of the crystal except so far as the case is modified by gravity or the contact of other bodies), but as they grow larger (in a solvent no more supersaturated than is necessary to make them grow at all), the deposition of new matter on the different surfaces will be determined more by the nature (orientation) of the surfaces and less by their size and relations to the surrounding surfaces. As a final result, a large crystal, thus formed, will generally be bounded by those surfaces alone on which the deposit of new matter takes place least readily, with small, perhaps insensible truncations. If one kind of surfaces satisfying this condition cannot form a closed figure, the crystal will be bounded by two or three kinds of surfaces determined by the same condition. The kinds of surface thus determined will probably generally be those for which $\sigma$ has the least values. But the relative development of the different kinds of sides, even if unmodified by gravity or the contact of other bodies, will not be such as to make $\textstyle \sum \displaystyle (\sigma s)$ a minimum. The growth of the crystal will finally be confined to sides of a single kind. It does not appear that any part of the operation of removing a layer of molecules presents any especial difficulty so marked as that of commencing a new layer; yet the values of $\mu _{1}''$ which will just allow the different stages of the process to go on must be slightly different, and therefore, for the continued dissolving of the crystal the value of $\mu _{1}''$ must be less (by a finite quantity) than that given by equation (665). It seems probable that this would be especially true of those sides for which $\sigma$ has the least values. The effect of dissolving a crystal (even when it is done as slowly as possible) is therefore to produce a form which probably differs from that of theoretical equilibrium in a direction opposite to that of a growing crystal.
* ↑ The freezing together of wool and ice may be mentioned here. The fact that a fiber of wool which remains in contact with a block of ice under water will become attached to it seems to be strictly analogous to the fact that if a solid body be brought into such a position that it just touches the free surface of water, the water will generally rise up about the point of contact so as to touch the solid over a surface of some extent. The condition of the latter phenomenon is $\sigma _{SA}+\sigma _{WA}>\sigma _{SW},$ where the suffixes $_{S},_{A},$, and $_{W}$ refer to the solid, to air, and to water, respectively. In like manner, the condition for the freezing of the ice to the wool, if we neglect the æolotropic properties of the ice, is $\sigma _{SW}+\sigma _{IW}>\sigma _{SI},$ where the suffixes $_{S},_{A},$, and $_{I}$ relate to wool, to water, and to ice, respectively. See Proc. Roy. Soc., vol. x, p. 447; or Phil. Mag., 4th ser., vol. xxi, p. 151.
* ↑ It is here supposed that the gravitational potential may be regarded as constant for each electrode. When this is not the case the expression may be applied to small parts of the electrodes taken separately.
* ↑ If the electrolytic fluid dissolved the mercury as well as the zinc, equilibrium could only subsist when the electromotive force is zero, and the composition of the electrodes identical. For when the electrodes are formed of the two metals in different proportions, that which has the greater potential for zinc will have the less potential for mercury. (See equation (98).) This is inconsistent with equilibrium, according to the principle mentioned above, if both metals can act as cations.
* ↑ See his memoir: "Relations entre les phenomenes electriques et capillaires," Annales de Chimie et de Physique, 5° série, t. v, p. 494.
* ↑ Philosophical Magazine, vol. xlix, p. 311.
* ↑ Annalen der Physik und Chemie, Neue Folge, Band iii, February, 1878.
* ↑ J. Regnauld, Comptes Rendus, t. li, p. 778.
* ↑ Gaugain, Comptes Rendus, t. xlii, p. 430.
* ↑ See Rühlmann's Handbuch der mechanischen Wärmetheorie, Bd. ii, p. 290.
* ↑ Comptes Rendus, t. lvi, p. 199; and t. lxiv, 67.
* ↑ Pogg. Ann., Bd. cxxxi (1867), p. 161.
* ↑ These numbers are not subject to correction for the pressure of the atmosphere, since the 34000 calories relate to combustion under the same pressure.
* ↑ Unless the received ideas concerning the behavior of gases at high temperatures are quite erroneous, it is possible to indicate the general character of a process (involving at most only such difficulties as are neglected in theoretical discussions) by which water may be converted into separate masses of hydrogen and oxygen without other expenditure than that of an amount of heat equal to the difference of energy of the matter in the two states and supplied at a temperature far below 2500° C. The essential parts of the process would be (1) vaporizing the water and heating it to a temperature at which a considerable part will be dissociated, (2) the partial separation of the hydrogen and oxygen by filtration, and (3) the cooling of both gaseous masses until the vapor they contain is condensed. A little calculation will show that in a continuous process all the heat obtained in the operation of cooling the products of filtration could be utilized in heating fresh water.
* ↑ Sainte-Claire Deville, Comptes Rendus, t. lxiv, p. 67.
* ↑ See Mémoires des Savants Étrangers, sér. 2, t. xxv, no. 1, p. 142; or Comptes Rendus, t. lxxiii, p. 973. The figures obtained by M. Favre will be given hereafter, in connection with others of the same nature.
* ↑ See Mém. Savants Étrang., loc. cit., p. 90; or Comptes Rendus, t. lxix, p. 35, where the numbers are slightly different.
* ↑ A comparison of the experiments of different physicists has in some cases given a much closer correspondence. See Wiedemann's Galvanismus, etc., 2$^{te}$ Auflage, Bd. ii, §§ 1117, 1118.
* ↑ Mém. Savants Étrang., loc. cit., p. 93; or Comptes Rendus, t. lxix, p. 37, and t. lxxiii, p. 893.
* ↑ Comptes Rendus, t. lxviii, p. 1305. The total heat obtained in the whole circuit (including the cell) when all the electromotive work is turned into heat, was ascertained by direct experiment. This quantity, 7968 calories, is evidently represented by $(V''-V')\Delta e-\Delta Q$, also by $-\Delta e+\Delta W_{\text{P}}$. (See (691 ).) The value of $(V''-V')\Delta e$ is obtained by adding $\Delta Q$, and that of $-\Delta \epsilon$ by adding $\Delta W_{\text{P}}$, which is easity estimated, being determined by the evolution of one kilogramme of hydrogen.
* ↑ Ibid.
* ↑ Mém. Savants Étrang., loc. cit., p. 145.
* ↑ Ibid., p. 142.
* ↑ It should perhaps be stated that in his extended memoir published in 1877 in the Mémoires des Savants Étrangers, in which he has presumably collected those results of his experiments which he regards as most important and most accurate, M. Favre does not mention the absorption of heat in a cell of this kind, or in the similar cell in which cadmium takes the place of zinc. This may be taken to indicate a decided preference for the later experiments which showed an evolution of heat. Whatever the ground of this preference may have been, it can hardly destroy the significance of the absorption of heat, which was a matter of direct observation in repeated experiments. See Comptes Rendus, t. lxviii, p. 1305.
* ↑ Except in the case of the Grove's cell, in which the reactions are quite complicated, the absorption of heat is most marked in the electrolysis of hydrochloric acid. The latter case is interesting, since the experiments confirm the presumption afforded by the behavior of the substances in other circumstances. (See page 343.) In addition to the circumstances mentioned above tending to diminish the observed absorption of heat, the following, which are peculiar to this case, should be noticed. The electrolysis was performed in a cell with a porous partition, in order to prevent the chlorine and hydrogen dissolved in the liquid from coming in contact with each other. It had appeared in a previous series of experiments (Mém. Savants Étrang., loc. cit., p. 131; or Comptes Rendus, t. lxvi, p. 1231), that a very considerable amount of heat might be produced by the chemical union of the gases in solution. In a cell without partition, instead of an absorption, an evolution of heat took place, which sometimes exceeded 5000 calories. If, therefore, the partition did not perfectly perform its office, this could only cause a diminution in the value of $\Delta Q$. A large part at least of the chlorine appears to have been absorbed by the electrolytic fluid. It is probable that a slight difference in the circumstances of the experiment—a diminution of pressure, for example,—might have caused the greater part of the chlorine to be evolved as gas, without essentially affecting the electromotive force. The solution of chlorine in water presents some anomalies, and may be attended with complex reactions, but it appears to be always attended with a very considerable evolution of heat. (See Berthelot, Comptes Rendus, t. lxxvi, p. 1514.) If we regard the evolution of the chlorine in the form of gas as the normal process, we may suppose that the absorption of heat in the cell was greatly diminished by the retention of the chlorine in solution. Under certain circumstances, oxygen is evolved in the electrolysis of dilute hydrochloric acid. It does not appear that this took place to any considerable extent in the experiments which we are considering. But so far as it may have occurred, we may regard it as a case of the electrolysis of water. The significance of the fact of the absorption of heat is not thereby affected.
* ↑ It will be observed that in using the formulæ (694) and (696) we do not have to make any distinction between primary and secondary processes. The only limitation to the generality of these formulae depends upon the reversibility of the processes, and this limitation does not apply to (696).
* ↑ M. Raoult has experimented with a galvanic element having an electrode of bismuth in contact with phosphoric acid containing phosphate of bismuth in solution. (See Comptes Rendus, t. lxviii, p. 643.) Since this metal absorbs in melting 12.64 calories per kilogramme or 885 calories per equivalent (70$^{kil}$), while a Daniell's cell yields about 24000 calories of electromotive work per equivalent of metal, the solid or liquid state of the bismuth ought to make a difference of electromotive force represented by .037 of a Daniell's cell, if the electromotive force depended simply upon the energy of the cell. But in M. Raoult's experiments no sudden change of electromotive force was manifested at the moment when the bismuth changed its state of aggregation. In fact, a change of temperature in the electrode from about fifteen degrees above to about fifteen degrees below the temperature of fusion only occasioned a variation of electromotive force equal to .002 of a Daniell's cell. Experiments upon lead and tin gave similar results.