PIIYSlOAELSEVIER

Physica A 219 (1995) 253-289

Viscoelasticity of vesicle dispersions
Kazuhiko Seki a,1, Shigeyuki Komura b,2
a Department of Energy Sciences, Faculty of Science and Engineering, Tokyo Institute of Technology,
Meguro-ku, Tokyo 152, Japanb Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan
Received 30 June 1994; revised 27 January 1995

Abstract
Linear viscoelastic properties of dispersions of vesicles and microemulsions are investigated indetail. Each droplet is considered to be coated by a film of amphiphific molecules (fluid membrane) whose shape is mainly governed by the bending rigidity rather than the surface tension. In
addition to this elastic property, we also take into account the membrane compressibility whichplays an important role in the macroscopic viscoelastic behavior of the system. For dilute emulsions, we calculated the effective complex viscosity by generalizing Taylor's old calculation. The
mechanical force balance condition at the membrane can be expressed by using the generalized
Laplace's formula derived for compressible fluid membranes. On the basis of the cell model, we
also calculated the effective complex viscosity for nondilute emulsions. Comparing various characteristic time scales both for vesicles and microemulsions, we discuss the detectable relaxation
modes in the viscoelastic experiments.

1. Introduction
Properties of flexible fluid membranes made o f amphiphilic molecules have attracted
great interest in connection with various fields such as the statistical physics, quantum
field theory, physical chemistry or biophysics [ 1 ]. Such membranes are typically realizedin spherically closed membranes of lipid bilayers called "vesicles" which are exemplified
by red blood cells. This type of thin-walled fluid vesicles have received great attention asmodels of cell membranes. Shape transformation among various conformations can be
I Address after 1 April 1994: Theoretical Chemistry Lab., National Institute of Materials and Chemical
Research, Tsukuba 305, Japan. e-mail address: ck938©nime, go. j p
2 e-mail address: komura@ton, scphyn . k y o t o - u . ac. j p
0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved
SSDI 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 1 7 1-9

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caused by changing, e.g., the osmotic conditions, the temperature or the composition of
the lipids [2]. These properties of vesicles might be closely related to the physiological
functions of biomembranes. Similar properties are also realized in a microemulsion
system being homogeneous mixtures of oil, water and surfactants [3]. In both cases,
amphiphilic molecules orient their polar heads towards water and their aliphatic tails
away from it, decreasing the surface tension drastically to the level of practically zero.
Instead of the surface tension, it is widely recognized that the deformation of themembrane is mainly governed by the elastic bending energy [4]. Since the associatedbending rigidity is known to be of the order of kBT, membranes can easily fluctuate dueto thermal agitations. Although microemulsion droplets differ from vesicles by several
decades in length scale (see Section 6), the ruling physics behind it is expected to be
similar.
Such sheets of membranes have two important features. First, a typical size of a vesicle
formed by bilayers is in the/zm range while its thickness is in the nm range so that thebilayer can be regarded as a two-dimensional system embedded in a three-dimensional
space. Second, an ordinary membrane is in a fluid state at room temperature and it
is supposed to have no in-plane shear resistivity. The curvature model was originallyproposed for such a fluid membrane according to a phenomenological consideration [4].
In order to make our understanding of the bending energy and associated hydrodynamics deeper, not only static measurements but also dynamical measurements ofthe membrane system are quite important. In fact, there have been several dynamicalmeasurements of vesicles (fluorescence microscopy [ 5], optical video-microscopy [ 612], reflection interference contrast microscopy [ 13-15] ) and microemulsions (neutron
scattering [ 16-18], dynamical Kerr effect [ 19] ). By knowing the relaxation time ofa small deformation of a vesicle or a microemulsion droplet, one can quantitatively
determine the bending rigidity. Along with these experiments, several people calculated
the time correlation function of the out-of-plane displacement for a spherically closed
fluid membrane. The first work by Shneider, Jenkins and Webb [5] was generalized byMilner and Safran to the case of non-zero spontaneous curvature [20]. The important
assumption in their calculations is that the total area is a conserved quantity as well as
the total volume both for vesicles and microemulsion droplets. In order to incorporate
these two constraints simultaneously, they introduced the idea of "constant excess area"
together with the unknown Lagrange multiplier. On the other hand, Van der Linden,
Bedeaux and Borkovec insisted that only the area should be kept constant for vesicles,
whereas only the volume constraint is necessary for microemulsion droplets since thesupply and the loss of surfactants from the bulk phase would take place in a shortenough time scale compared to the deformation of droplets [21]. Van der Linden et al.
or Komura and Seki [22] obtained different dynamical correlation functions from theprevious result by assuming that only the total volume should be kept constant. Somepeople calculated the relaxation time upon taking into account the compressibility of themembrane [23-27].
Recently, Smeulders, Blom and Mellema performed a viscoelastic measurement onan emulsion of relatively small vesicles, r0 --~ 40-100 nm [28-30]. A distinct feature

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of their observation is that there exist two relaxation processes; one related to thetranslational ordering of the vesicles that are subjected to a shear flow, and the other,appearing at higher frequencies, has been attributed to the membrane deformation. Fromthe characteristic time of the latter relaxation process, they determined several elastic
constants and viscosities of the membrane as well as the bending rigidity by changing
the size of the droplets. Their experimental data has been analyzed according to thetheoretical prediction by Oldroyd with a modified relaxation time including the bending
rigidity [31,32]. One of their interesting findings is the deformation mode dependent
on the shear modulus, although Helfrich assumed vanishing of it in his original model[4]. Recently, viscosity of the L3-phase in amphiphilic systems has been also reported[33,34].
Up to now there has been no complete calculation of the effective viscosity for an
emulsion of fluid membranes governed by the bending energy except for some special
cases. In the present article, we consider the viscoelasticity of an emulsion of vesicles
or a microemulsion system. Since the viscosity can be regarded as an example of a
response function, its calculation is also interesting from the point of view of statistical mechanics. We take into account the membrane compressibility which plays animportant role in the viscoelastic behavior. In order to grasp the physical properties ofthe membrane, we calculated the effective complex viscosity by generalizing Taylor's
old calculation [35,36]. The hydrodynamical boundary condition at the membrane represents the balance of force at the membrane [22], and we employ the generalized
Laplace formula derived for compressible fluid membranes [22,37,38]. The spirit of
our calculation is similar to the recent work by Onuki [26]. In addition to the dilutecase, we also derived the complex viscosity for nondilute emulsions on the basis of thecell model.
In fact, the calculation of the effective viscosity of dispersed systems has a very
long history. The first hydrodynamic calculation was given by Einstein for suspensionsof solid spheres in the steady shear flow [39]. Taylor extended Einstein's approachto the liquid droplets in the continuous aqueous phase [35]. Both of the results werelimited to the dilute case. Later Fr61ich and Sack [40] and Oldroyd [31] argued theviscoelasticity of suspensions and emulsions, respectively. For nondilute dispersions,extensive calculations have been performed by several people by using the cell model[41,42].
The outline of this article is as follows. In Section 2, after some mathematical definitions, we give a description of the compressible fluid membrane and the mechanicalboundary condition at the interface. In Section 3, the constitutive equation and formal
expression of the effective viscosity are presented. The viscoelasticity is calculated for
dilute and nondilute dispersions in Sections 4 and 5, respectively. In the last section,comparing various characteristic time scales for vesicles and microemuisions, we discuss
the detectable relaxation modes in the viscoelastic experiments.

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2. Closed fluid membranes
2.1. PreparationsFirst, we will collect some formulas from differential geometry. One can, in general,parameterize a two-dimensional membrane in a three-dimensional space by two real
inner coordinates s = (s 1, s2). The shape of the membrane is then described by a threedimensional vector r = r ( s ) . At each point on the membrane, there are two tangent
vectors ri = Or/Os i with i = 1,2. The outward unit normal vector n is perpendicular tothese tangent vectors, i.e., n = (rl x r 2 ) / I r l x rzl.
All properties related to the intrinsic geometry of the membrane are expressed interms of the metric tensor defined by the inner product of the tangential vectors(2.1)

gij = ri . rj.

Two important quantities are the determinant and the inverse of the metric tensor which
will be denoted byg = det(gij)andg/J = ( g i j ) - l .

(2.2)

In addition, one has to consider the (extrinsic) curvature tensor given by(2.3)

hij = n • rij = - n i " r j ,where rij = 02r/c~siOsJ and ni = O n / O s i. The third expression follows from the partial
derivative of n . rj = 0. Similar to Eq. (2.2), the determinant and the inverse of thecurvature tensor are denoted by
h=det(hij)andh i J = ( h i j ) -1.

(2.4)

The mean curvature H and the Gaussian curvature K are calculated according to
H = lgiJhij = - ½ ( c J + c2),

(2.5)

K = h / g = CLC2,

(2.6)

and

respectively, where cl and c2 are two principle curvatures.
The covariant derivative D j of f i and f i are
Djf i = Ojf i + F~jf kandO j f i = Ojfi - F ~ f k ,(2.7)

respectively, with the Christoffel symbols F~,
I "k = gklrl • rij,

(2.8)

and Oi =-- O/Os i.
The metric and the curvature tensors are determined by the vector r ( s ) and consist of
six independent functions since both tensors are symmetric. In order to solve the inverse

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257

problem, namely, to determine the function r(s) from the fundamental tensors, one has
to solve the following equations:ni = -hi Jrj,

(2.9)

rij = F~.rk + hun,

(2.10)

andwhich are called Weingarten equation and Gaussian equation, respectively.

2.2. Compressible fluid membranes
Using the notations in the previous subsection, we consider an emulsion system where
each dispersed droplet is considered to be coated by a fluid membrane consisting ofamphiphilic molecules such as lipids or surfactants. The equilibrium shape of the fluid
membrane is determined by the minimization of the following shape energy [4,37]:

Hf = f ~r(ps)dA + ½Kf (cl + c2 - co)2dA.
A0(2.11)

A0

In the above, dA = v/-~dslds 2 is the surface element while A0 is the surface of aparticular closed membrane over which the integrals are performed. (We use f for thesurface integral in order to distinguish from the volume integral f . ) Furthermore, o'(ps)is the bare surface tension dependent on the areal density Ps of amphiphilic moleculeson the membrane [43], K the bending rigidity and co the spontaneous curvature.
It is important to mention that due to the incompressible assumption for the ambient
fluids (see later Eq. (3.2)), the shape energy Eq. (2.11) should be minimized under
the constraint of constant enclosed volume. This constraint is only a mathematicalrequirement in the present paper and one can also argue the case of compressible
surrounding fluids [25]. Appropriate constraint for the real systems is a matter ofdiscussion as has been stated in the introduction. In principle, one can also consider thesituation in which K and/or co depend also on Ps. However this effect is expected tobe negligibly small compared to the bare surface tension [23] and we shall put thesequantities as constants.
Exchange of molecules between the membrane and the ambient fluid takes placethrough desorption and adsorption. Desorption is a thermally activated process. The
characteristic time for a molecule to remain in the membrane is proportional to exp(z~E/
kBT), where AE is the energy barrier per molecule associated with the desorption
process. For a phospholipid in the bilayer membrane, the estimated typical sticking time(at room temperature) ranges from several hours to a couple of days, i.e., "~ 105 [s],which provides a measure of the time scale within which a new chemical equilibrium
can be attained. This estimation implies that such a relaxation process takes a rather
long time in the case of bilayers made of lipids such as red blood cells. Hence one canassume that the number of molecules in the membrane remains almost constant during

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the time scale of the experiments which is much shorter than the sticking time [44,45].
Keeping these facts in mind, we assume throughout this paper that the total number ofamphiphilic molecules N is a conserved quantity. Here N is given by the integral of ps
over the surface of the membrane, i.e.,
N =/psdA.

(2.12)A0

By further assuming that the density fluctuation is induced due to the local areal changeof the membrane, the variation of Eq. (2.12) yields

0 = ~psdA + ps6dA,(2.13)

which will be used in the following discussion.
Some comments are necessary for the above assumption. Typical sticking time scale
for a single chain surfactant in micelles is much shorter compared to the lipids and isin the order of 10-5-10 -3 [s]. Hence for a supramolecular structure of surfactants, Eq.
(2.13) holds only for a short time scale. One should note that there exists also an open
system, such as black films, where the lipid bilayer exchanges molecules with the outersystem. In this case, N is no longer a constant while Ps remains unchanged with respect
to the small change in the total area [44].
By calculating the first variation of the shape energy Eq. (2.11) with respect to an
infinitesimal displacement normal to the membrane under the constraint of fixed N, we
obtain the following normal restoring force due to the membrane [5]:FII = 2 2 ( p s ) H - K(2H + co) (2H 2 - 2K - coil) - 2x~7~BH,(2.14)

where E ( P s ) is the apparent surface tension defined by [43]ao(ps)~Y(ps) - o ( p s ) - p s - - ,Ops(2.15)

and 272B is the Laplace-Beltrami operator on the surface given by

V ~8 = -~gOi( giJv~cgj ) .

(2.16)In order to make this paper self-contained, we give a concise derivation of Eq. (2.14)
in Appendix A. The second term on the r.h.s, of Eq. (2.15) represents the reduction of
the surface tension.
It is important to notice that in the shape energy Eq. (2.11 ), the compressibility of thefluid membrane has been implicitly taken into account through the change of Ps. Thiscan be seen as follows. Due to the fluctuation of the local density ~ps of amphiphilicmolecules around the equilibrium (or mean) value of Ps denoted as p~0, the apparent
surface tension Eq. (2.15) can be different from point to point on the surface being
expressed as ~(Ps) = ~(pso) + ~3~(ps), where [23,43,46,47]

6X(ps) = -B-B-~3ps,psowith B = P~o ( o2~r(ps).
ps=pso(2.17)

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In the above, B is called the two-dimensional (in-plane) compression modulus and isput hereafter as a constant. The former relation in Eq. (2.17) can be also regarded as
an equation of state of the membrane as a two-dimensional fluid [ 27 ].
It is known that the compressible nature of the membrane leads to a considerable effecton the hydrodynamical properties of the interface [52]. The change in the membrane
shape due to the motion of the surrounding fluid is coupled to the change in ps which
in turn affects the surface tension as seen by Eq. (2.17). If ps varies over the surface,
the apparent surface tension Eq. (2.15) is not constant as well and this fact gives riseto the tangential restoring force due to the membrane such that [55]F ± , = grad±~ 2 ( p s ) ,(2.18)

where grad±,, is the a-component of the gradient operator on the surface given by
grad±`" = cgx,n`'nl~ a x # '(2.19)

while n`" are components of the normal vector (which should not be confused withni = an/Osi). (We shall use Greek indices for the range 1, 2, 3.)

In order to construct the boundary condition that must be satisfied at the interface
between two viscous fluids in motion, we follow the discussion by Landau and Lifshitz[52]. According to Eqs. (2.14) and (2.18), the balance of force per unit membraneleads to(11`'l~ - 17'~)n~ -t- Fiin`" -+- F±`" = 0,(2.20)or more explicitly(//,,/~ --//~/3) nfl + grad±`'2?(p.~)+ [ 2 2 ( p s ) H - K(2H + c0) ( 2 H 2 - 2K - c o i l ) - 2KV2BH]n`" = 0.

(2.21)

where, as will be given later in Section 3.1,//`'t~ is the fluid stress tensor outside of themembrane and //'¢~ is that inside of the membrane. Here and below we shall use theprime in order to distinguish the quantities of the fluid inside from the corresponding
quantities of the fluid outside. For the direction normal to the membrane, Eq. (2.21)
reduces to the well-known Laplace's formula when K = 0 and ~(Ps) represents a
constant surface tension [52]. A more general case has been discussed by Onuki and
Kawasaki [ 38,43 ].
In addition to the equation of motion for the ambient fluids with the boundary condition Eq. (2.21 ), we need another equation since the variable Ps has been introducedin our problem. This is an equation of continuity of amphiphilic molecules, expressing the conservation of the local number of molecules (Eq. (2.13)). According to thehydrodynamics of two-dimensional fluids [27,38,53], this can be written in general as
Ops + d i v ± ( p s V s ) + ps 0g = 0.
cgt
2g Ot

(2.22)

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See Eq. (2.2) for g and Vs is the velocity of amphiphilic molecules composing themembrane and will be put equal to those of ambient fluids. (The allowance of theslippage between the membrane and the fluids was considered by Oldroyd [32] andrecently by Onuki [ 26,38].) The precise mathematical definition of the term divL (psVs)
in the language of differential geometry is given in Appendix B. The last term in Eq.
(2.22) simply expresses the density fluctuation due to the local areal change.
We finally comment that another way of incorporating the membrane compressibility
is to add a compressional energy term such as(~Ps~ ~
Hc = I B "f_)o\~--~ / dA,(2.23)Ao

to Eq. (2.11 ) if the surface tension tr(ps) in the first term is put as a constant (zero)[48-51]. However, the compressibility is more generally taken into account in Eq.
(2.11) than by Eq. (2.23).
2.3. Spherically closed membranes

For our later purpose, we summarize here several expressions related to a sphericallyclosed fluid membrane of radius r0 at zero temperature. By specifying the internal
coordinates as (s 1, s 2) = (0, ~o), we introduce the following three unit vectors as a local
basis:

sin0coser = [ sin0sin~o ] ,\cos0 /

e0 =(cos0cos ( sancho,
cos0sinq~ ] ,-sinO /e¢ =cos~o ~ .

(2.24)

With these notations, the undeformed reference state is described by(2.25)

R = roer.

Now we consider the membrane slightly distorted from the reference state. Any
deformed state of the membrane without any overhangs can then be parameterized byusing the normal vector N in the reference state, i.e., N = (Rl x R2)/IRI x R21, in thefollowing way:r = R + g(O,~o,t)N = [ro + ~(O,~o,t)]er.

(2.26)

Here the variable g(O, q~, t) represents the transverse (out-of-plane) displacement field
which can generally depend on time t. (Here we did not include the in-plane displacements which are irrelevant in the bending energy up to the second order in terms ofdisplacement fields. In-plane displacements should be taken into account in describing
polymerized elastic membranes.) A straightforward calculation up to first order in termsof the out-of-plane displacement g yields the following expression for the normal vector:n ~ er10g

ro ~-~e0

1

~g

e~,
r0 sin 0 a~o(2.27)

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261

and for twice the mean curvature H and the Gaussian curvature K:
2
1
2H ~ - - - +(2 + V 2) g(O, ~o, t),r0(2.28)

70

and

I

1

K ~ r--~- ~--~(2 + V 2 ) g(O, q~, t),(2.29)

respectively, whereV~_-sinO00

sinO

1 32
+ - -sin- - 2 0 3q~2"(2.30)The linearized equation of continuity can be written from Eq. (2.22) as [27]3776p, ~ -P~o div±vscq

2pso 3g
ro 3t '

(2.31 )

whered i v ± v s - -rosin0-1[ ~o( s i n O v s o ) + ff-~Cv',~
3
1(2.32)

is the usual two-dimensional divergence of v~ in the spherical coordinate.
For our later calculations, it is convenient to expand the function g(O, q~, t) in termsof the spherical harmonics Ynm(O, ~o),Ig ( O, ~p, t) = Z

g"" ( t ) Yn,, ( O, q~) .

(2.33)

11,111As usual, we have g~m(t) = ( - 1 ) m g , , _ m ( t ) in order to ensure that the displacement
field is real (the asterisk denotes the complex conjugate value) and the summation
with a prime runs over n = 1,2 . . . . and [m I < n. The term for ( n , m ) = (0,0) is
excluded here due to the constraint of constant enclosed volume. (In our later notation,
the summation without any prime includes the ( n , m ) = (0,0) term.) Hereafter, thewell-known relation
V~,,(0,~)= - n ( n + l) ~ , , ( 0 , ~ )(2.34)

will be used frequently.

3. Effective viscosity of dispersions
3.1. Constitutive equation
Dispersions of small spherical droplets can be regarded as a homogeneous fluid when
we concern the phenomena that occur in a much larger length scale than the average

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262

size of the dispersed droplets. One example is the rheological behavior of dispersions
where one asks the stress needed to cause a given bulk motion. On measuring therheological property, a dispersion can be considered as a homogeneous fluid with aneffective viscosity. Here we have addressed the term homogeneous in a statistical sense,
since the exact position and motion of droplets may differ for different realizationsof experiments even if the macroscopic conditions such as the boundary conditions areprepared in the same manner. Hence we may observe, in principle, the ensemble averageof the fluid velocity (v) instead of its exact value v. However, the ensemble average can
not be calculated directly and we assume the ergodicity property of the system, namely,
the equality of the ensemble average and the volume average. Study of the effective
viscosity of dispersions has a fairly long history and its theoretical basis has been wellestablished. In this section, we shall briefly outline the basic theory by Batchelor who
provided a general constitutive equation of suspensions [54] and extend it for emulsions.
First we consider an emulsion of fluid droplets being coated by a fluid membrane(discontinuous phase) and immersed in another fluid (continuous phase). As in theprevious section, quantities with prime refer to those on the inside of the droplets while
notations without prime correspond to those on the outside. Both fluids are assumed
to be Newtonian in its stress behavior. Denoting the fluid velocity by v (';, the stresstensors are given by

H('),#3 = - P ( ' ) 6,,/3 + °'~'),~ = - P)6,~t~+ r/() \ Ox/3 + Ox,, el ,(3.1)

where primes with parentheses imply that the above equation holds both for inside andoutside the membrane. In the last equation, r/~') are the dynamic viscosities and wehave also assumed that both fluids are incompressible, i.e.,
divv (') = 0.

(3.2)

Notice that this condition leads to the constraint of constant enclosed volume.
The average of the stress tensor over the volume V large enough to contain many
droplets is(Ha~) = 1 / H,~/~dV.

(3.3)

V

This integral can be decomposed into the sum of the integrals over the volume of atypical droplet having the volume V0 and the remaining part of the bulk V. Then wehave:0"o 0".I +c/.;.,..
v

v?

where c is the number of droplets per unit volume. Notice that V = V(1 - cVo) holds.
Since we have assumed that V is completely occupied by the continuous phase (without

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263

prime), V0 must include the interfacial layer. In order to stress that this boundary lieson the outer surface of the membrane, we use the notations V0> and A~, which standfor the volume including the membrane and the membrane outer surface, respectively.
It is convenient to express Eq. (3.4) in terms of surface integrals. For this purpose
we employ the relations( Ova ~

1=-

OvadV = 1Ox:Vava dV+c / van:dA,

(3.5)

Ag

7

and

/ n2~dV= f n , , x : , d A -f @x:dV,(3.6)

where the Gauss theorem has been applied in Eq. (3.5). Then Eq. (3.4) can be expressedas(Ha#) = --~+c

P6azdV + rI ~xB + aXa /[Ha~:x:n~,- rl (Van: + V:na)] dA - cAg~x~, X:aV.

(3.7)

v~

In general, the stress tensor can be divided into an isotropic part and an anisotropic
part. The isotropic part is determined by the external conditions and is irrelevant to theargument of the constitutive equation. Hence we shall drop the first term on the r.h.s, ofEq. (3.7) and neglect hereafter the difference in the isotropic part of the stress tensor.
Moreover the last term also vanishes since OH~/Ox~ = 0 within the droplet as longas the inertial forces can be neglected. The derived constitutive equation is, therefore,[52,54]:o.o o.,; / [Ha~,x:n~,- rl(van E + vBna)] dA.

(Ha:} = r / \ Ox: + axa / + c

(3.8)

AX
Here the second term on the r.h.s, represents the extra stress due to the presence of thedispersed particles.
Although Eq. (3.8) has been first derived for a suspension of rigid particles dispersed
in a Newtonian fluid [52,54], it is also applicable for emulsions where fluid droplets are
immersed in another Newtonian fluid. To see this more clearly, we derive an alternative
form of Eq. (3.8) [55]. If we denote the inner surface of the membrane by A~, wehave/ lI:,x:n,dA = / r/ ( v'~n: + v'~na)dA,
AS

AS

(3.9)

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264

where the isotropic pressure term has been dropped out. Using this equation and Eq.
(3.8), we obtain(n.e} = n \ axe + axo I + c [a( IIarxl~nrdA - Af~ Illarx~nrdA+ v~n,~)da - 77'/A~(u~nfl-~-v~na)da].

(3.10)Taking the limit of the membrane thickness going to infinitely small and employing the"stick" boundary condition (continuity of the velocity) at the interface, we have [55](//~e) = r / \ a-x--~#+ axe~ +c(Har-ll'~r) xenrdA
Aa~ C ( ~Irl') i ( v~nl~+ v#n~)dA.

(3.11 )

Aa!

In the above, the stress difference (H~r - II'~r) xl~nr can be written in terms of thesurface tension and the bending rigidity [43,56].
In case there is no force associated with the interface, i.e. //,,r = / / ' r ' Eq. (3.11)
reduces to(v,,nt~+ vl~n~)dA.

(//~,#) = r / \ 3x# + 3x,, / - c(~7 - r/')(3.12)A>

Here the change in the viscosity due to the presence of droplets is proportional to thedifference of the viscosities between the inside and outside of the droplet. Detailed
discussions are left in Sections 4 and 5.

3.2. Formal calculation
In this subsection, we shall give a formal calculation of the effective viscosity on
the basis of Eq. (3.8) under an imposed shearing motion. The flow field around aspherical droplet is assumed to be the "creeping flow" which satisfies the stationary
Stokes equation [5,20,21,58] :r/(')V2v(') = grad P('),(3.13)and simultaneously the incompressibility condition Eq.
)° The solution to Eqs.
(3.2) and (3.13) in spherical coordinates is expressed in terms of three scalar functions:¢(r, t), x(r, t) and P(r, t) where ~p and X give solutions to the homogeneous equation

K. Seki, S. Komura/Physica A 219 (1995) 253-289

265

V2U = 0. Since all of these functions satisfy the Laplace equation (obviously ~72P = 0),
they can be expanded in terms of solid spherical harmonics:

O'(r,t) = y~O~.,(t)Yn,,(O,~o),(3.14)

Yn.,(O,q~),(3.15)(r)-~o ° Ynm(0, ~P),

(3.16)~(r,t) = ~-'~d/nm(t) (r~°r)n+ly.m(O,~o),(3.17)

n,D|,x~(r't) = ~-~X~"'(t)

-~o

rl,nl

P'(r't)=~'P~m(t)
n,ltl

and
!

?l,rllI

X(r,t) = ~-'~Xnm(t) (~)n+lYnm(O,~o),(3.18)n,ill

P(r, t) = ~ Pnm(t)n~nl(3)"+' Y,m(O, q~),

(3.19)

describing the inside and outside of the fluid, respectively. For the pressure fields P~
and P, we included (n,m) = (0,0) mode (without prime in the summation) becauseof the existence of the finite non-fluctuating hydrostatic pressure which satisfies(Poo - P~o)
~ ro3 + 2~(Pso)r(~ + Kcoro(coro -- 2) = O.

(3.20)

This relation is called the "capillarity condition" [21,22], and comes from the requirement that the undeformed reference sphere Eq. (2.25) is always a solution of theequilibrium shape equation [ 37 ] :P - P ' = ~1"(3.21)

According to the general solution of the Stokes equation given by Lamb [57], thevelocity field on the inside of the droplet is given by [58]v'(r,t) =

~--~(

(n+3)

~P~nn'(t)grad+X~'m(t)r°tr+ 2r/'(n + 1)(2n + 3) P~nm(t) r2gradn,nl

77'(n + 1 ) (2n + 3) P~nm(t)rY..,(O, q~),while the corresponding solution for outside the droplet is(3.22)

K. SekL S. Kornura/Physica A 219 (1995) 253-289

266

Iv(r,t) -v°~(r,t) = Z( ~g"'(t)grad+ X"m(t)r°tr

2r/n(2n(n-2)-1) Pnm(t)r2grad

n,m

(n+l)

) (~)(n+l,-~ ~Tn(2n - 1) Pnm(t)rYnm(O,q~),(3.23)

where v ~ ( r , t ) is the unperturbed flow given by the boundary condition at infinite
distance from the droplet given later in Eq. (3.24). In Eqs. (3.22) and (3.23), both thegradient and rotation operators act on the solid spherical harmonics outside the large
bracket as well. Lamb showed that the radial component of the velocity involves ~p andP while r • rotv is only a function of X [57,58] and the terms involving X always
separate out during the calculation. Hence, as far as our present purpose is concerned,
we can ignore X ~'~ without loss of generality and this simplifies the problem to someextent [5,20,21,58].
At large distances from the droplet, the velocity field is described by a linear velocity
distributionv~(r, t) = p~#(t)x~,(3.24)

where p~a(t) is a second-order symmetric tensor with p,~,, = 0 since the fluids areincompressible. Although p ~ ( t ) can vary with time arbitrarily, we shall consider only
an oscillatory motion in the form of e i°~t with which all the time dependence will betaken into account hereafter. Following the symmetry argument on p,~# by Oldroyd [ 31 ],it suffices to consider only a pure elongational flow represented by( 00)

p~B( t) =

0
0

--F 0
0 2F

ei~°t,(3.25)in Cartesian coordinates. In spherical coordinates, Eq. (3.24) becomesv ~ (r, t) = Fgrad [r2Y2o(O,q~)] e iwt,

(3.26)a10
1 agrad = er~r + eOr-~ + e¢ r sin 0 0~o"(3.27)

with

The unknown coefficients in Eqs. (3.22) and (3.23) are determined according to theboundary conditions at r = r0 and it turns out that only/'2o contributes to the calculationof the effective viscosity. To the lowest order (in the absence of droplets), the first termin Eq. (3.8) is given by [59]rl\Ox a + ax,~/ =rl \~xa + #x,~ ]°"~t~"(3.28)

By using the following abbreviation for the surface integral in Eq. (3.8):= c~( [H,~rxt~nz, - rl(v,~n~ + v#n,~)] dA,
A~(3.29)

K. Seki, S. Komura/Physica A 219 (1995) 253-289

267

we show in Appendix C that [42,59]j~ =P 2 0 . o0
- 4r/F q~o',~,(3.30)

where ¢ = (4rr/3)cr3o is the volume fraction of the total droplets. Notice that J,~
vanishes in the absence of the droplets.
Since the effective viscosity r/* is defined as\ ax~ + ax, / '( 3.31 )it can be formally expressed asr/*=r/

1 - ~.]'(3.32)

from Eqs. (3.28) and (3.30). We see that the knowledge of P2o/F is sufficient todetermine the effective viscosity.
In the present paper, we restrict ourselves to the case that the deformation of themembrane is small enough. A qualitative picture of the deformation process can beeasily obtained. When the droplet of radius r0 is deformed according to a shear flow ofmagnitude F, stress tending to elongate the droplet is of order r/F. In the case of thevanishing surface tension, opposing r/F is a stress of magnitude K/r 3, which tends tomaintain the droplet in a spherical shape. Thus, the shape of the droplet in a steady shear
flow is governed by a non-dimensional parameter, rlFr3o/K, with which the condition ofsmall deformation can be expressed as rlFr3/K << 1. The above dimensionless quantity
is analogous to the capillary number ~TFro/X of the system where the surface tension
plays an important roll.

4. D i l u t e caseIn this section, we calculate the effective viscosity of a dilute dispersion. To achievethis, we have to determine /)20 in Eq. (3.32) subjected to the appropriate boundary
conditions. For such, we employ the "stick" boundary condition as has been argued
in the foregoing sections and was used by several authors before [21,22,27]. Thiscondition requires that the velocity of the molecules composing the membrane and thefluid velocity on both sides of the membrane are equal. Hence from Eq. (2.26), thecondition is written asOrv = v ' =Vs = -~-,

(4.1)

at the surface. Since the velocity field is linear in the fluctuating amplitude L one can
impose the boundary conditions at r = r0. The first set of boundary conditions comesfrom the continuity of the velocity. In the er-direction, this is written as

K. Seki, S. Komura/Physica A 219 (1995) 253-289

268

Vr(r = ro) = 1ro

~-~(

,
n
2pt'~
~o)eiWtn~b'n,, + 2rf(2n + 3) rO nm) Ynm(O,

rl,rB

=

---

n +

r0

( n + 1) r2pnm) ynm(O,~o)ei~Ot_+_v~(r = ro )2r/(2n - 1)1)~Pn,,

n,m

I

=~

itOgnmYnm( O, ~o) e iwt,

(4.2)

II,11!

where the last equation has been obtained by taking the time derivative of Eq. (2.33).
Likewise, the continuity of the velocity in the e0-direction leads tovo(r = ro) = ro2 tm) OYnm itot(n + 3)roP~
2rl'(n + 1 ) ( 2 n + 3)
~e

m
ll,nl

'±(o

"~ --

ro n fl'n(n - 2) r2pnm"~ OYnm ion,
2r/~-2n~-l)) --~-e + v ~ ° ( r = r o ) .

nm

(4.3)

The continuity condition in the e¢-direction results in the equivalent condition as Eq.
(4.3).
An additional set of boundary conditions follows from the balance of force on themembrane presented by Eq. (2.21). By noticing that the Laplace-Beltrami operator is
now the usual Laplacian operator on the sphere
V~B = V~r02'(4.4)

the force balance equations in the er and eo-directions up to first order in terms of g are
given by(P-/
OVr _ 2r/,Ov'r'~ 2 ~ Z ( p s )P') - ~2r/-&-rO r / + ro

1 [Z(p~o)r~_K(V2+2coror4_ $c0r0)] 1 2 2 (2 + V~)g(O,~o,t) = O,

(4.5)

and
-r/k.~0~- +

+

+

-

(ps) =0,(4.6)

at r = ro, respectively. In the above, we used the capillarity condition Eq. (3.20). Noticealso that nr ~ 1, while no --~ - ( l / r 0 ) ( 8 g / 0 0 ) , which is proportional to g to the lowest
order. In order to eliminate 8Z(ps) from the above equations, we obtain the following
relation from Eqs. (2.17), (2.31) and (4.1):

3
2B Og_~6Z(ps) = B div±v' + - - - Ot
ro 3t

K. Seki, S. Komura/PhysicaA 219 (1995) 253-289

269

I

rg

n ( n + l)~b~,,+

n,t/I

2r/l(n +n+;n+3)P[mrg)Y,m(O,~o)
1)

eiwt

I

+iw2B Z gnmYnn,(O, q~)ei'',ro n,Ol

(4.71)

at r = ro. Substitution of Eq. (4.7) into the time derivative of Eqs. (4.5) and (4.6)
yieldsiw P,Im _._____z,_ 2rln(n _ 1 )F + - 2n- + 3 P~m
2n'n(n - 1) rg[(~nm--iw P . m - 2 r / ( n + l ) ( n + 2 ) r ° 2

1-iw~(nron(n+l))]
2n--1 P.m

1 2 21- 1)(n + 2) {2;(pso)ro2 + K [(n + 1) - 2coro + gCoro]} g..,+-25-3'o2Bn( n + 1) ( O.m' + 2r/(n+n+31)(2n+3) P~.,ro' 2)-iw~~"m = O,

(4.81)

n+3' °2)-iw--~og,m=O,+rBon(n+ 1) ( ~b,m+ 2rl,(n+l)(2n+3)P~mr
2B

(4.9)

and
iw

i
2 ~ l ( n - 1) O'rg"---Z'- 2 r / ( n -n(n + 2)p, )I ) F + ( n + 1)(2n+3) .m+iw(2rl(n+2)~byo, _ ( n 1- 1) ~) (2nt+~ i)_ ~ Pnm)respectively, for n _> 1.
As far as we consider the shearing motion given by Eq. (3.25), it is sufficient to takeonly (n, m) = (2, 0) mode in the above conditions. Introducing the abbreviation__
1 2 2\K (6 - 2coro + gCoro),
S - X(p~o) + r~

(4.10)and omitting the suffixes 20 of all the functions, Eqs. (4.2), (4.3), (4.8), (4.9) reduceto the following equations to be solved (we have also eliminated g2o):
lp,
2 ,
1
3 0
7rf + rT~p = ~--~P- r°2 +2/',
5 p,+314r/----S

70
15

,

(4.11)

3 0= r 2 + 3r,

1
4 (2S
7r/,ro2 (4roS - 6roB - io)~7'r~) P' + r--~o -~o=i~°rl(-3 P + ~O + 4F)(4.12)

B + i w l ) O/ro(4.13)

K. Seki, S. Komura/PhysicaA 219 (1995)253-289

270,

7r/'r~ (9r°B + 8iwrf r2)p' + 7r
=itor/(-~r/P-~o~24-~o + itor/') ~p'+6r)(4.14)The set of these four linear homogeneous equations in the five unknowns P~0, ¢60,
P20, ~020 and F provide sufficient information to calculate P2o/F from which we cancalculate the effective viscosity according to Eq. (3.32). After some calculation, thefinal expression is obtained asr/*/r/=1, 24BS+ [(23E- 16)B+4(5E+2)S] f+ (E- 1)(19E+ 16)f 2 ,3 48BS + [2(23E + 32)B + 4 0 ( E + 1)S] f + (2E + 3) ( 19E + 16)f2 <P,(4.15)
where E = r/t/r/, f = iwror/.
The equation which determines the overdamped mode of the membrane deformation
can be obtained by putting the denominator of Eq. (4.15) equal to zero. When neither
B nor S is zero (B 4 : 0 and S v~ 0), it exhibits, in general, two characteristic times.
On the other hand, in the case of B = S = 0, we find a stationary r/* given by
r/*/r/= l +5(E- 1~--~3)~b,(4.16)

which was first derived by Taylor [36]. In such a case there is no force due to themembrane and the fluid stress tensors balance at the interface. The previous argument
leading to Eq. (3.12) corresponds to this case and we see that the viscosity change is
proportional to r / - r/' as it should be [55].
When either one of B or S is zero or tends to infinity, only a single characteristic
time shows up. This case can be represented in general byGr

r/* = r/~ + 1 + iw-------~'(4.17)

which coincides with the outcome of the "Maxwell model" in the phenomenologicalrheology. In the above, r is the relaxation time, G the relaxation strength and r/~ isthe constant viscosity when w + c~. r represents the time scale dividing the shorttime Hookian regime and the long time Newtonian regime. As we show in Appendix
D, r corresponds to the longest relaxation time of the membrane deformation for each
limiting case. We shall collect below the results of r/M, r, G and r/~ = lira 77* for severaloJ--*0cases depending on the relation between B and S.
(i) B > > S > 0
This limit corresponds to the case where the membrane is incompressible. Taking
the limit of B/S ~ c~ in Eq. (4.15), we find5(23E - 16) ~b,
r/a/r/= l + 2(23ET3-~(4.18)

271

K. Seki, S. Komura/Physica A 219 (1995) 253-289

r -~r023E + 32,
S
24

(4.19)

S
2880
G - ro ( 2 3 E + 3 2 ) 2~'(4.20)

r/~/r/= 1 + -~b.

(4.21)

andEq. (4.21) does not depend on the fluid viscosity and coincides with that of dilute
suspensions of rigid particles [39].
(ii) S > > B > 0

In the limit of S/B ~ ec, we have
5E+2
r/a/r/= 1+ --~b,2 ( E + 1)
r/ro 5(E + 1)
7"--

B

6

B

9

'

G=--

r0 5 ( E + 1)

2~,

(4.22)(4.23)(4.24)

and r/(~ is identical to Eq. (4.21).
(iii) S > B = 0
In this limit the membrane is fully compressible. By putting B = 0, we have
5 ( E - 1)
r/~o/r/= 1 + - - ~ b ,2E+3(4.25)

r/ro (2E + 3) ( 19E + 16)
r = -,
S
4 0 ( E + 1)

(4.26)S20
r0 (2E + 3) 2~b'G=

(4.27)

andr/~/'q=

5E+2
1 + 2 ( E + 1)~b.

(4.28)

Eq. (4.28) was first derived by Taylor for the dispersions of almost undeformableliquid droplets with a large surface tension [35]. The case of E = 1 in Eq. (4.28)
was also discussed by Schwartz and Edwards [60,61].
(iv) B > S = 0
In this case we have
5 ( E - l)~b '2E T 3n/n = 1 +Tm,~r/ro ( 2 E + 3 ) ( 1 9 E + 16)
B2(23E + 32)--(4.29)(4.30)

272

K. Seki, S. Komura/Physica A 219 (1995) 253-289

G-

B

5

~b,

(4.31)

5 ( 2 3 E - 16)
4).
2(23E + 32)(4.32)

r0 ( 2 E + 3) 2

andr/~/r/= 1 +In Appendix D, we have listed the sequence of the relaxation time of the membrane
deformation for general n in accordance with the cases considered here. Notice that all
the relaxation times given above are recovered by putting n = 2 in the corresponding
expressions.
Another limit can be taken in the small frequency limit by neglecting the f2 terms
in Eq. (4.15). In this case, the effective viscosity can be also expressed in terms of theMaxwell model, Eq. (4.17), withr/~/r/= 1 +

5 [ ( 2 3 E - 16)B + 4 ( 5 E + 2)S]~b,
2 [ ( 2 3 E + 3 2 ) B + 2 0 ( E + 1)S]rlro ( 2 3 E + 3 2 ) B + 2 0 ( E + 1)S

r = --

BS
BS

G=

24

(4.33),(4.34)l)S]2~b,(4.35)

720(4B + S)

r0 [ ( 2 3 E + 32)B + 2 0 ( E +and r/~ is identical to Eq. (4.21).
In all these cases, both r/~ and r/~ approach to Einstein's result by taking the limitof r/' ~ c~. It should be emphasized that as long as the membrane is characterized
by the non-zero B and S, the steady state effective viscosity coincides with that of asuspension of solid particles [52,62]. Moreover, neither B nor S appears in the steadystate viscosity, while the relaxation time and the relaxation strength depend explicitly on
B and S. Hence an appropriate dynamical experiment should be performed in order tograsp the elastic properties of the membrane. Although these statements might appear
obvious in the early calculations by Oldroyd [31,32], it should be stressed that he did
not take into account the effect of the bending rigidity K. We also comment that the
recent work by Onuki agrees with our calculation [26].

5. N o n d i l u t e caseIn the previous section, we have analyzed the dilute case which is valid only when thedroplet concentration is sufficiently low and the effect of droplets on the macroscopic
properties scales linearly with the volume fraction. As the volume fraction increases,
such a dilute picture breaks down and the presence of the droplets affects the flow fieldaround the other droplets.
As far as the stationary effective viscosity is concerned, studies of nondilute suspensions of rigid particles are reviewed in the book by Happel and Brenner [58], andK. Seki. S. Komura/Physica A 219 (1995) 253-289

273

Fig. 1. Schematicrepresentationof the cell model.
notable among them is what is called the "cell models" used by Simha [63] and Happel[64]. In this model the hydrodynamic interactions between particles are taken into account to some extent. The result obtained by using Simha's model reduces to that given
by Einstein's theory for a very dilute system [65]. However, Happel's prediction islarger than both Einstein's and Simha's results even for small concentrations. This trend
is reversed at large concentrations when the theoretical viscosity is compared with theexperimental data. In the case of nondilute emulsions, Choi and Schowalter proceeded
the analysis by means of Simha's cell model [41 ]. In this section, we shall follow theirscheme to calculate the effective viscosity of a nondilute dispersion.
In the cell model, we have extra boundary conditions at r = rl > r0, where the velocity
is equal to the velocity of a homogeneous system (see Fig. 1). rl can be regarded as
the hydrodynamic interaction radius. The results of the dilute case are recovered bytaking the limit of rl --~ oc. The velocity field inside of the droplet, r < r0, is given
by Eq. (3.22) and the fluid at r > rl is described by the flow Eq. (3.26). In-between(r0 _< r _< rl ) the velocity field is expressed as a superposition of Eqs. (3.22) and(3.23):v ( r , t)

=

±[(
n,m

"
(n + 3)P~mr grad
On,,,grad + 2r/( n + 1 ) (2n + 3 )r/(n + 1 ) (2n + 3) P~"~mr+Onmgrad(n - 2)
Pnmr2grad
2 r / n ( 2 n - 1)

(n ---~ 1) n|( F 0 " ) ' ~+ r l n ( 2 n _ l)t-'nmrj \ r /(n+l)]

Ynm(O,q~) ei°)t

•

(5.1)

The first set of boundary conditions comes from the continuity of the velocity field at

K. Seki, S. Komura/Physica A 219 (1995) 253-289

274

r = ro. In the er-direction, this is written as(n0,m + 2~/'(2nn + 3) r2p,'~ynm(O.~o)ei,.,to ,m,]

Vr(r = ro) = ro1 ~
n,m

1±(
~

--

_

__

_na/j

r0

n

.1!
m

2r/(2n + 3) r°rnm

1) r2pnm~ y,m(O,~)ei,O,zTqtzn - l)/+(n + 1)Ohm-., (n+!

= Z ieOgnmYnm(0, ~0)e i°n,

(5.2)

.,m

where the last expression has been obtained by taking the time derivative of Eq. (2.33).
Likewise, the continuity of the velocity in the eo-direction yields

,,(
,'(

+ 3)
roPnm
O~m + 2r/'(n+(n 1)(2n+3)
] a0

+On,,

(n - 2) r~P,m) c~Y,m io,,2r}n(2n- 1)/ --~--e .

vo(r=ro)=-~o ~2 t ~ OYnmeiwt(n + 3)2~t!
2r/(n + 1) (2n + 3) rornm(5.3)

The other set of boundary conditions follows from the balance of forces on themembrane given by Eq. (2.21). Similar to the previous section, we haveiw

(P~m' ,,.
{Y'O~mP~,,,) - 2 n ( n - 1) ~ r~,, n(n+)]riO~,n)1)
r2 J-~--n~3 ('°Inn,- pint.,,[(
Onmn(n+l))]--ioJ P.,.- 2 r / ( n + l ) ( n + 2 ) r2
2n--1 P.m
1 (n - 1) (n + 2) {.,~(Ps0)r2-+-K [(n+ 1) - 2c0ro + ~cor0
1 2 21}~nm-'tW--r4
j
2B

(

n+3

,2)

4B

+--~o n( n + l) 0.'m+ 2r/'(n+ 1) ( 2n + 3 ) P~'nr° - ieo--~-g.mro =0,

(5.4)

and

ioo [2 ( n - 1) t"7'__:'m
\ r°2
0..,
+iw 2r/(n+2) r2

r/0~'., ~n(n + 2)
(Pn'm - Pn"m)]] + ( n + 1)(2n+3)( n - l ) ( n + 1)p.m)

n+ 3

, 2)

+ B n ( n + 1) (0n'm + 2r/t(n q- 1) (2n + 3) Pnmro roforn_> 1.

ia'-2B-~oe"m= O,

(5.5)

275

K. Seki, S. Komura/Physica A 219 (1995) 253-289

In Simha's cell model, the velocity is required to be continuous at r = rl. Hence theconditions are for er and e0-direction as
2 ntt "~
Vr(r=rl): I l~-~'(r-r~O)n-I (n~'t''nm 2r/(2nrl + 3) rlr.mJ

I ~-~(ro~"+' (
--

(n +

rl n,m \ rl /(n+ l) r~Pnm']] YnmCO,~p)e~°'r1)Ohm 2r/(2n
- 1)
]1

I

= ~ nr~- ' FY,,, ( O, ~o) e i°~',

(5.6)

II,m

and
rl
\ro]

vo(r = rl) = ~0

~P~'m+

1 ~-'~ (r~°) n+' (~On m

-~ --

rl

n,m

(n + 3)2~/,"~2r/(n + 1) (2n + 3) r°Gm)( n - 2 ) rZPnm)] OYnmei~O,
2r/n(2n- 1)--~!

=Zr"t-IFOY~+e i','t,

(57"

n,m

respectively.
As before, it is sufficient to take only (n,m) = (2,0) mode in the above conditions.
Using the abbreviation of Eq. (4.10) and omitting the suffixes 20 of all the functions,Eqs. (5.2), (5.3), (5.4), (5.5), (5.6) and (5.7) are reduced to the following equations
to be solved:
1

,,

2

1P' + ~O'= ff-V~P + ~
7rf
5 p,
1~-,,

3 ,

,,

1p
+2r/

_ 3

(5.8)

r~b'

5__~_p,, 3 .

+ 70 ¢' = 14,7

+ rq ¢' +

(5.9)

¢"

1
4(2S
7r/,r02(4roS - 6roB - ioJrfr2)p ' + -r~- ~oBro + io~ r/'

4 ~ , \) + iwrl ( 3- e +-~o2 4~) ,
=iwrI ( 7- - ~ P" +-~o

3.%+ (9ro8 + 8iom'ro~)e ' + r-~ 70 + i,o,',)+,

6 , \) +i¢or/ ( 3~p=iwr/ (7-~ P"+~0~9-24~ ) ,)~b'(5.10)(5.11)

K. Seki, S. Komura/PhysicaA 219 (1995) 253-289

276
1( r--I "]2pt, -k- 2r °2 ptt +

\ro/
5
rl
lnr/ \ ro /P" +1 ( r- o- ~ 3 p~ \rl/

O + ---~o3 ( r- 0- ~ 3 0 = 2 F 'r~ \ r l /--• = 3/'.

(5.12)(5.13)

r21 -~lThe set of these six linear homogeneous equations for the seven unknowns P~0, ¢~0,P2o, ¢2o, P~, ¢ ~ and /" are sufficient to calculate P2o/Y from which we calculate
the effective viscosity Eq. (3.32) as before. After a tedious calculation, the effective
viscosity is expressed by introducing the notations f = io~ro~7, E = rl'/r ! and y = ro/rl
as

C4 + C s f -q- C6f 2 .
r/*/r/= 1 q- 10CI ._bC2f+C3f2cp,(5.14)

where
C1 = 24BS(4 - 25y 3 + 42y 5 - 25y 7 + 4yl°),C 2 = 4 [ ( 2 3 E + 32)B + 2 0 ( E +I)S]- 2 5 [ ( 2 3 E - 16)B + 4 ( 5 E + 2 ) S ]y3+42 [(23E - 8)B + 20ES] y5- 2 5 [ ( 2 3 E + 4)B + 4 ( 5 E - 2)S] y7+ 4 ( E - 1 ) (23B + 20S)y l°,(73 = 2 ( 2 E + 3) (19E + 6) - 25(E - 1) ( 19E + 16)y 3+49(E-1 ) ( 1 9 E + 16)y 5 - 2 5 ( E - 1 ) ( 1 9 E + 18)y 7+76(E-

1)2y 1°,C4 = 24BS( 1 - y7),
C5 = (23E - 16) B + 4 ( 5 E + 2) S - ( E - 1 ) (23B + 20S)y 7,
C6= ( E -I ) ( 1 9 E + 16) - 1 9 ( E - 1)2y7.

(5.15)

Notice that y = ro/rj introduces the effect of interaction among droplets. In the limitof y ~ 0, the above results reduce to Eq. (4.15) of the dilute case. Since the interactions
between the droplets are effectively accounted for by the introduction of the radius rl,
y can be an explicit function of the volume fraction ~b, such as [42,63]
y3 = g~b,(5.16)

where g is a constant number. The Oldroyd's result is recovered when g = (2/5) 2[31]. Choi and Schowalter, on the other hand, chose g = 1 [41] and Thomas proposed
g = 0.656 for ~b < 0.1 [66].
As regards the characteristic relaxation times, they satisfy the equations obtained byputting the numerator and denominator of Eq. (5.14) equal to zero. When neither B

K. Seki, S. Komura/Physica A 219 (1995) 253-289

277

nor S is zero (B ~ 0 and S ~ 0), each equation determines two characteristic times.
In this case, we can write the effective viscosity in the form

.1 + iwr2-

O.)2/~2~7* = 71o1 + io~r] - w2vl ,(5.17)where1+

10( 1 - yT)
4 - 25y 3 + 42y 5 - 25y 7 + 4y lo~b,~ 1 + 54)(1 + -~gq~) ,(5.18)

rlro ((23E+32) B + 2 0 ( E + l ) S

rl ~ ~

24

rIro((23E+32) B + 2 0 ( E + l ) S

r2 ~ ~
121

v2~

24

+

25(4B+S)
)
8
g~b ,

+

5 (4B + S ) ( 5 g 8(5.19)

2) ~b)(r/r0) 2 (2E + 3 ) ( 1 9 E + 16) /'1 +
125
"~
B~
48
~,
4 (2E + 3) g~b),07r°)2(2E+3)(19E+16) ( 1+25BS
48
4(2E+3)(5.20)'(5.21))(5g-2)~b.

(5.22)

Eq. (5.18) coincides with the steady state viscosity of a suspension of rigid spheres
derived by Simha [63]. In the absence of the bending rigidity, K = 0, this result canbe included as a special case of that given by Oosterbroek and Mellema, where more
complex mechanical properties such as the viscosity and shear elasticity of the film wereincorporated [42]. On the other hand, in the case of B = S = 0 we find a stationary r/*
given by

,7+/,7 = 1 + t0c~,/,~1+5(E-I)2E+3 4) ( 1 + 22(2E
5 ( E -+~ ) ) g~b) .

(5.23)

When either one of B or S is zero or tends to infinite, the effective viscosity can be
written formally in the following form

.1 + iwr2
rl* = 71o1 + iwr~'(5.24)

where rj, r2 are the characteristic times. We give below the explicit expressions of r/~,rl and "/'2 according to the relation between B and S.
(i) B > > S > 0
r/~ is identical to Eq. (5.18),
r/ro(23E+32
rl~ T
24

~
+

)
g~b ,

(5.25)

K. Seki, S. Komura/Physica A 219 (1995) 253-289

278

r/r0 ( 23E + 32

7"2~ -g- \ ~5'~+ ~ (5g- 2) 4,,/.

(5.26)(ii) S > > B > 0
r/j is identical to Eq. (5.18),
r/r°(5(E61)rt ~--ff- - -rlro(5(E61)- -7"2~--ff-

+

-~

)
g~b ,

5

(5.27)+g(5g-2)~b) .

(5.28)(iii) S > B = 0,7;/n = i+

2 ( 5 E + 2) - 1 0 ( E - 1)y 7
4 (E + 1) - 5 ( 5 E + 2) y3 + 42Ey5 _ 5 (5E - 2) y7 + 4(E -1)y 16(b5E+2 ( 5(se+2)g4, ~..~l+2(E+I~ ~b 1+ 4 - ~J'

7"1 "~

7"2 , ~

(5.29)(i+5 19E+16 3)g,k)
4 (E+ 1)(2E+'
nro(2E+3)(19E+16) (1+1
19E+16(5.30)~ro(2E+3)(19E+I6)S

4 0 ( E + 1)

S

40(E+l)4(E+l)(2E+3))(5g-2)~b.
(5.31)(iv) B > S = 0
r/Ur / = 1 + [10 (23E - 16) - 230(E - l)y 7]/ [4 (23E + 32) - 25 (23E - 16) y3 --F42 (23E - 8) y5- 25 (23E + 4) y7 + 92(E - 1)yl°] q~5(23E-16),~l+2(23E+32)~b

7" I

7"2 ~ ,(

25(23E-16))
1 + 4 ( 2 3 E + 3 2 ) g~b ,~Tro(2E + 3) (19E + 16)
B2(23E + 32)

25

19E + 16

(5.32))

1 + --~ (23E+.32)(2E+3)gck , (5.33)

519E+ 16
r/ro ( 2 E + 3 ) ( 1 9 E + 16) (I+4(23E+32)(2E+3)
B
2(23E + 32))(5g-2)~b .
(5.34)

Eq. (5.29) was derived by several authors by means of various methods [41,67,681.

K. Seki, S. Komura/PhysicaA 219 (1995) 253-289

279

Before closing this section, we point out that the relaxation time 7- and the relaxation
strength G in the Maxwell model Eq. (4.17) can be related to the quantities discussed
in this section. In fact, ~- corresponds ~-1 while G can be written asG=-(5.35)

which characterizes the elastic modulus for t << 7- [69]. Thus ~'1 - 7-2 is proportional tothe elastic part of the stress tensor [40].

6. Discussion
In this paper, we have examined the effective dynamical viscosity of dispersions of
fluid droplets coated by amphiphilic molecules. The shape of the fluid membrane isconsidered to be mainly governed by the compression modulus B and the bending
rigidity K. We obtained its general expression in terms of B and S (Eq. (4.10)) bothfor dilute and nondilute cases. Several limiting cases have been also derived. The timescale of the viscoelastic behavior is characterized by the longest relaxation time of themembrane deformation when either one of B or S is zero or infinity. According to
the analysis of the dilute limit, it is found that when both B and S are non-zero, thestationary effective viscosity ~7~ reduces to Einstein's result. In case either one of B or
S is zero, r/~ depends on the viscosity of the inside fluid r/' while Taylor's result can be
reproduced when B = 0 and S 4: 0. A cell model for the nondilute case yields Simha's
result for ~Tt~when neither B nor S is zero.
In order to estimate the value of the bending rigidity K from the observed characteristic time, we have to eliminate X (Pso) from S (see Eq. (4.10)). (For the present
discussion, we ignore the spontaneous curvature co.) This can be done by an independent observation of X(p~o) according to the "capillarity condition" Eq. (3.20) [911.
In practice, however, it is difficult to perform such a measurement in a sufficient accuracy and many people assumed that ,? (p~0) ~ 0. In such a case, it is possible toestimate K from the measured relaxation time of the membrane deformation [5-1211.
Such a simplification can be justified provided X (P~o) << K/r g is satisfied. The quantitative estimation of this condition should be done separately for bilayer vesicles and for
microemulsion droplets since they differ by several decades in length scale.
For a lipid vesicle of size r0 ~ 10 -5 [m], various people observed that the bending
rigidity is typically K ~-- 1 x 10 -19 [J] [5-12] and hence we have K/r20 ~ 1 x 10 - 9[ N / m ] . As for the surface tension, some people claim that it is less than 10 -5 [ N / m ][70] or 10 -8 I N / m ] [9] while others insist , ~ 1 × 10 -8 [ N / m ] [10]. For a microemulsion droplet of size r0 ~-- 5 × 10 - 9 [ m ] , the measured bending rigidities are K -~1 × 10 -z° [J] by SANS technique [ 16-18] or K ~_ 1 × 10 -21 [J] by dynamic Kerr effectmeasurement [19] yielding K/r~ -~ 4 × 10 -4 or 10 -5 [ N / m ] , respectively. (See Table
1.) These values can be compared with the relatively small two-dimensional pressure

K. SekL S. Komura/Physica A 219 (1995) 253-289

280

Table 1
Orders of various hydrodynamical time scales. We used vc "" 103 [ m / s ] , p ~' 103 [ k g / m 3] and r/_~ 10 - 3

INs/m21ro [mlB IN/m]K [J]VesicleMicroemulsion droplet

10 - 510 - l10 -1910 - 8
10 - 2 *10 -20K/r 2 [N/ml

10-9

10-4

Wc [ l / s ]

108
104
106
107

1011
1010
1010
109

1/% ll/s]o9B [ I / s ]I/~'B ll/sl
Ws [ l/sl

10 6

10 9

oJK [ I / s }102

1/~-~ [l/sl

10-l

109
107

* See the text.

7 x 10 -5 [ N / m ] [ 17], although they claim that it is different from the macroscopic
surface tension.
In either case, it is somewhat delicate to ignore the apparent tension £ (P,o) in
Eq. (4.10). For the simplicity of the present argument, however, we assume here thetensionless case, i.e., 2; (P~0) ~ 0. In order to extract the property due to the bending
rigidity in an emphasized way, it is recommended to prepare droplets of small size.
As we have done in Sections 4 and 5, several limiting cases of the effective viscosity
have been considered according to the relation between B and S ~ K/r 2. The question
is to which case the real system corresponds. By a mechanical experiment, Evans andcoworkers found B "~ 0.14 [Nm -1 ] for egg lecithin bilayers [70-72]. Therefore fora vesicle, we have typically B / S ~ Br2/K ~-- 1.4 x 108 and the membrane can beconsidered to be almost incompressible. It should be noticed that even S ~ K/r~ isnegligibly small compared to B, this does not mean S = 0 which corresponds to thedifferent limiting case.
Another possible argument may be given as follows. Let us regard the membrane as ahomogeneous elastic shell with a thickness a. According to the shell theory [29,69,73],
two-dimensional compression modulus B and the bending rigidity K are given byB-Ya
2(1-o-)

andYa 3
K = 12(1 --0"2) '(6.1)

respectively, where Y and o- are Young's modulus and Poisson ratio for an isotropic
material in three dimensions. Then one can see that B / S is approximately determinedby B / S ,~ ( r o / a ) 2 [ 11 ]. For a bilayer membrane, Evans took into account the fact that
membrane is not a continuum in the thickness direction and obtained different thickness
dependence of the bending rigidity as [72,74,75]
K = Ba2/2.

(6.2)

K. Seki, S. Komura/Physica A 219 (1995) 253-289

281

In this case we also have the same scaling behavior as B/S ~ (ro/a) 2. For a lipid
vesicle of size r0 -~ 10 -5 [m] and a _~ 4 × 10 -9 [m] [ 11,29], one finds B/S _~ 6 × 106
corresponding to the incompressible case B >> S 4= 0 as before. For a microemulsion
droplet, however, since the size and thickness are typically r0 ~ 5 x 10 -9 [m] anda ~- 2 × 10 - 9 [ m ] , B and S take rather comparable values. Thus we cannot simply
employ the incompressible condition for the interface of microemulsion droplets and it
might be better to take both B and i< into account in the consideration of the effective
viscosity.
There are several time scales involved in the present problem; each of them reflecting
the corresponding mode coupled to the ambient fluids. We consider now whether thesetime scales can be well separated from each other. As mentioned above, we ignore
the time scales which come from the surface tension ~. In Section 3.1, we used thestationary Stokes equation Eq. (3.13) to describe the creeping motion of the surrounding
fluids. This can be justified as long as the Reynolds number of the fluids R = pvro/~7
is sufficiently small ( p is the fluid mass density). For a given frequency to, Reynolds
number can be given in terms of the viscous diffusion rate l / r , 7 ~ rl/pr 2 as R ~ cot o.
Hence the above condition sets the upper boundary value of the allowed frequency range
such that to << 1/7~7. On the other hand, the incompressible condition of the fluids Eq.
(3.2) requires to << toc ~ Vc/ro where v,: is the speed of sound.
For the modes related to the compression modulus B, we have the oscillating mode
toB ~ (B/pr3) ]/2 and the overdamped decaying mode (or Lucassen mode [76-78])
1/r8 ~ B/~Tro. Moreover there exists an additional surface mode connected with theredistribution of ps for compressible membranes. This mode is known to have an unusual
dispersion law [23,24,38,46,47]:
to,. -

x/~_2_--i(B2 )l/3
~

(6.3)

For the bending modes, on the other hand, we have the oscillating mode to,,(K/prS) I/2 and the overdamped mode l / r , , ~ K/r/r~. Notice that Eq. (6.3) is derived in the limit of to >> 1/r, 7, to,, and may be irrelevant to the viscosity measurement.
Order estimations of these time scales both for vesicles and microemulsion droplets arelisted in Table 1. (Since there is no published experimental value of B for microemulsion
droplets, as far as we know, it is roughly estimated by B ,,~ K/a 2 with a ~- 10 - 9 [ m ] .)For the estimation of these quantities, we used vc ~- 103 [ m / s ] , r/ _~ 10 -3 [ N s / m 2]
and p ~_ 103 [kg/m3].
For vesicles, bending modes are observable since 1/7-,, << 1/7-n, while incompressible
condition for the membrane is regarded to be reasonable since coB, 1/7-B,to~. >> 1/7,7.
It can be also seen that the overdamped bending mode 1/7-,, is well separated fromoscillating bending mode to,,. In the microemulsion droplet case, although the allowed
frequency range is rather high, to << 1/7-,~ -~ 101° [ l / s ] , several modes are in thesame order close to 1/7-o and might be difficult to separate from each other. Moreoverthese frequencies lie near the upper limit frequency which is measurable by mechanicalK. SekL S. Komura/Physica A 219 (1995) 253-289

282

experiments. The longest relaxation time l / r , _~ 107 [ I / s ] turns out be quite large
compared to the vesicles and becomes rather comparable to 1/zn "~ 109 [ l / s ] . Henceboth of the modes should be taken into account in this case as mentioned before.

Acknowledgements
We would like to thank Prof. K. Kitahara and Prof. A. Onuki for their interest anduseful comments. Fruitful interactions with Dr. Y. Fujitani, Dr. H. Kodama and Miss T.
Ayako are also acknowledged.

Appendix A. Derivation of the membrane force
In this appendix, we briefly give the derivation of the restoring force due to themembrane Eq. (2.14) following mainly the calculation by Zhong-can and Helfrich[37]. In the present case, o ' ( p D in Eq. (2.11) is no more a Lagrange multiplier anddepends on Ps. We introduce a virtual displacement denoted by e ( s I , s 2) which is thedifference between a point on the actual surface and some point on the neighboring
varied surface ~,= r + en.

(A.1)

Notice that r describes not only the undeformed surface but also the deformed shapeand e can be different from the actual displacement g introduced in Eq. (2.26). Let ususe g0, h,7, g and /~ for the metric tensor, curvature tensor, determinant of the metrictensor and mean curvature of the varied surface, respectively. The tangent vector of thevaried surface isri = ri + ~i~ + ~ni,

(A.2)

with which we findg0 ~ gu - 2ehu-

(A.3)

Hence
~ g( 1 - 4ell).

(A.4)

The local areal change is obviously
V ~ ~ v ~ ( 1 - 28H).

(A.5)

In addition, we have~ij ~ gjj .~_ 2 e ( 2 H g i j _ K h i J ) ,(A.6)

h(i ,~ hij -t- F,ij "~ e( K g 0 - 2 H h i j ) - ~k F k ,(A.7)

K. Seki, S. Komura/Physica A 219 (1995) 253-289

283

where the Christoffel symbols F~ are defined in Eq. (2.8). In the above, we have used
the Weingarten equation Eq. (2.9) and the relation
hi.igjk hkt = 2Hhit - Kgil.

(A.8)

From Eqs. (2.5), (A.6) and (A.7), one can obtain the mean curvature of the varied
surface as
1 _ i j 1"~
f-1 ,,~ H + e ( 2 H 2 - K) + 2s
L~ie'j"(A.9)

Next the first variation of H f (Eq. (2.11 )) is given by
6Hf = 8Hb + ¢ 8o"(Ps) da + 1 0 " (ps) 6dA,
J

t /

Ao

A0

(A.10)

where
6 H b = 1K6

f(2H+ co)2da

Ao½K¢ [ (2H + co) 2 6dA + 4(2H + co) (6H)dA].

(A.l 1 )J

AoAccording to Eq. (2.13), the second term on the r.h.s, of Eq. (A.10) becomesf6o" (Ps) dA =Aoi 00-ps_,-~/ o0-(ps)6' dA'ops

Ao

Ao

~

( Tps) dA = -Ps(A.I2)

Using that the first variation of A is

6fdA=-feeHda,
A0(A.13)

Aoand after the integration of eij and ek by parts, we obtain
6Hf = ~ e[ -2.Z (Ps) H + K(2H + co) (2H 2 - 2K - coil) + 2K~7~sH] dA,
ao(A.14)

where ,~ (Ps) and V2LBare defined by Eqs. (2.15) and (2.16), respectively. Finally the
restoring force due to the membrane is calculated by
Ell=

6Hf

8~'

which simply leads to Eq. (2.14).

(A.15)

284

K. Seki, S. Komura/Physica A 219 (1995) 253-289

Appendix B. Definition of the divergence operator
In this appendix, we give the mathematical definition of the divergence operator
appearing in Eq. (2.22) in the general geometry [53]. First we raise the indices by(B.1)

r i = giJrj,

with which we define(B.2)

Ois = Vs" r i.

Then the two-dimensional divergence stands for
diVL (p~Vs) = I¢~i( x/~psv~).
x/g

(B.3)

Appendix C. Derivation of (3.30)
In this appendix, we show the derivation of Eq. (3.30) by following the paper by
Schowalter, Chaffey and Brenner [59]. Using Eqs. (3.28) and (3.29) the stress tensor
Eq. (3.8) can be written as(1L~) =n \axil + axaJ +Jail.

(C.l)

We can obtain 7/* defined in Eq. (3.31) by evaluating Jail. By substituting Eq. (3.26)
into Eq. (3.28), we haveo "~ = 2~IF

Or~o

2Y2o(O,~O)erer -+- ----~erea)"~

e ia't.

(C.2)

The viscous stress tensor of the unperturbed flow is expressed in terms of the sphericalharmonics ~0(0, ~o). We shall find J~t~ having this symmetry. Since the integral in Eq.
(3.29) can be taken over a large closed surface A lying entirely in the ambient fluid
enclosing the droplet but no other ones, only the terms of order 1/r 2 are relevant in thevelocity field Eq. (3.23). Thus it is sufficient to consider only
(C.3)

v ( r , t) - v ~ ( r , t) = ~--~Qr,

where
Q = P20

Y20(0, ~o)e iwt.

Then we can readily see thatnail - ~°Q= ~ (xa
\
Ox~

(C.4)

oQ)

.

(C.5)

K. Seki. S. Komura/Physica

Substituting

Eqs. (C.5)(I

Jap =cand (C.3)

aQ
zxpr3dRa;into Eq. (3.29),

+ i

(

(C.7)

-;

to

+i

x,xpQrdfl

JQ
rxpr3df2

JJ

To evaluate Eq. (C.8)
gradeof area on a unit sphere. With the aid of Euler’spolynomials,

can be simplified

Jap=c

Jx...xpQt-da),
CC.61

x,xax,,crdfl““,aQ&$=rr=-3Q,nEq. (C.6)285

one obtains

J

where d.0 = sin BdedqP is an element
theorem for homogeneous

A 219 (I 995) 253-289

a

>

.

((33)

we note that

= -Pze? 3; [YU(&sD) +y3,-I(U)]

e,

(
+;

[Y3l(U)e,. + 3Y30(0, (p) e,- Y3,-l(&cp)]

e’O’,

(C.9)

>
and

e, =

(C.10)

Owing to the orthogonality
Eq. (C.8), hence
5
J = --cPpy$
2(Iof the sphericalYdf9, rp)e,e,dfl

only the first term remains

erwt.

in

(C.11)

1’

From the explicit expression
appears in Eq. (C.11)

harmonics,

y2a = (3 cos2 0 - 1) /2, it can be seen that the variable

only through e,.. By integrating

f&de,
Jsin3

rp

over p, one finds

?7

5lr

J = - -cP2ori
2co)de[e,e,+ eye,10n+2

sin 0 cos2 ~9Y2,(0, gP)dBe, e,(C.12)

eiot .

J
0

Evaluating

the integrals

as
0

?T

J

sin3 eu20(e,p)de

0

= -

J

sinecos2eY20(e,~)de

0

= -$,

(C.13)

286

K. Seki, S. Komura/PhysicaA 219 (1995) 253-289

we finally obtain
1
(
aY20
"~
J = -~P20~b ,2Y2°(0'~°)erer -l- - ~ - e r e o ; e i°'t,(C.14)

where ~b = (47r/3)cr3o . Comparing Eq. (C.14) with Eq. (C.2), we obtain Eq. (3.30).

Appendix D. Relaxation times
In this appendix, we list the sequence of the relaxation time Tnm associated withthe membrane deformation in accordance with the various limiting cases discussed
in the text. Given ~'.m, the time correlation function of the out-of-plane displacement(g.m (t)g.m (0)*) is found to be proportional to exp(--t/Tnm). In order to calculate rnm,
we let v °~ (r, t) = 0, i.e., no external flow. In such a situation, the time dependent flow
fields inside and outside of the droplet are purely generated by the shape fluctuationof the membrane. Combining Eqs. (4.2), (4.3), (4.8) and (4.9) with F = 0, one has
5 homogeneous equations for 5 unknowns P~m, Cn~m' P.m, ~P.m and gnm. The nontrivial
solutions can be found when ~'.m = -ioJ satisfies the following quadratic equation foreach set of ( n , m ) [27]:A.m (l/r.m) 2 - B.m (1/r.m) + Grim =0,

(D.1)

where
Anm

~

[2(n-1)(n+l)E+2n2+l][(2n2+4n+3)E+2n(n+2)],(D.2)

nnm = 3 n m ( n - 1)n(n + 1 ) ( n + 2)(2n + 1 ) ( E + 1)

~r0
B+n(n+l)[(n-1)(2n2+5n+5)E+(n+2)(2n2-n+2)],
~ro
BSnm

Cn m wr/Zr ~ ( n - 1)n2(n + 1)2(n + 2 ) ,

(D.3)
(D.4)

with
__

1

223

K [n(n + 1) - 2coro + ~coroJ .
S,m = X(ps0) + r2

(D.5)

For n = 2, Eq. (D.1) reduces to the equation which is obtained by putting the denominator of Eq. (4.15) equal to zero. We list below the limiting expressions of rnm according
to the relation between B and Snm.
(i) B )> Snm ) 0
1

7"nm( n - 1)n(n + 1)(n + 2)
Snmrlro (n - l) (2n 2 + 5 n + 5 ) E + ( n + 2 ) (2n 2 - n + 2 ) "(D.6)

K. Seki, S. Komura/Physica A 219 (1995) 253-289

287(ii) S,,, >> B > 0

1

rnm

B
n(n + 1)rlro ( 2 n + 1 ) ( E +

1)

(D.7)

(iii) S.., > B = 0l
TnmS,,,,,
7~ro(n-- l)n(n+l)(n+2)(2n+l)(E+l)

x
[2(n-

1 ) ( n + 1 ) E + 2n 2 + 1] [(2n 2 + 4 n + 3) E + 2n(n +2)]"(D.8)(iv) B>S.m=O

1

B

Trim

T/ro
x

n(n+l)[(n-1) (2n 2 + 5 n + 5 )[2(n-1)(n+l)E+2n2+l]

E+(n+2)(2n 2 - n + 2 ) ][(2nZ + 4n + 3) E + 2n(n + 2) ] "(D.9)

For n = 2, Eqs. ( D , 6 ) , ( D . 7 ) , ( D . 8 ) and ( D . 9 ) give Eqs. ( 4 . 1 9 ) , ( 4 . 2 3 ) , ( 4 . 2 6 ) and( 4 . 3 0 ) , respectively. Eq. ( D . 6 ) was first derived in Refs. [5,20] for E = 1 and a moregeneral case was o b t a i n e d recently in Refs. [26,27] while Eq. ( D . 8 ) was calculated inRefs. [ 2 2 , 2 4 ] .

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