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NBSIR 82-2524

NAT'L INST. OF STAND & TECH

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Effects of Water and Other

Dielectrics on Crack Growth

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Materials Science

Fracture and Deformation Division

Washington, DC 20234

May 1982

Interim Report

Prepared for

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Office of Naval Research

department of the Navy

Arlington, VA

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EFFECTS OF WATER AND OTHER (jcio.

DIELECTRICS ON CRACK GROWTH

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C. ;

S. M. Wiederhorn, S. W. Freiman, E. R. Fuller, Jr. and C. J. Simmons

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Materials Science

Fracture and Deformation Division

Washington, DC 20234

May 1982

Interim Report

Prepared for Office of Naval Research

Department of the Navy

Arlington, VA

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director
f
Effects of Water and other Dielectrics on Crack Growth

S. M. Wiederhorn, S. W. Freiman, E. R. Fuller, Jr.

and C. J. Simmons

National Bureau of Standards

Washington, D.C. 20234

ABSTRACT

The effect of water and a variety of organic liquids on the crack

growth rate in soda lime silica glass was investigated. When water is

present in organic liquids, it is usually the principal agent that

promotes subcritical crack growth in glass. In region I, subcritical

crack growth is controlled primarily by the chemical potential of the

water in the liquid; whereas in region II, crack growth is controlled

by the concentration of water and the viscosity of the solution formed

by the water and the organic liquid. In region III, where water does

not affect crack growth, the slope of the crack growth ciirves can be

correlated with the dielectric constant of the liquid. It is suggested

that these latter results can be explained by electrostatic interactions

between the environment and charges that foimi during the rupture of Si-0

bonds

.

Effects of Water and other Dielectric Fluids on Crack Growth

S. M. Wiederhorn, S. W. Freiman, E. R. Fuller, Jr.

and C. J. Simmons

National Bureau of Standards

Washington, D.C. 20234

1. Introduction:

Subcritical crack growth in glasses is believed to result primarily

from a stress-enhanced chemical reaction between a chemical environment,

usually water, and the Sl-0 bonds of the glass. The reaction occurs

preferentially at crack tips where stresses are high. Prior investigations of subcritical crack growth in glass have indicated a rather complex

dependence of crack velocity on the applied stress Intensity factor and

on the amount of water in the environment. If crack velocity is plotted

as either an exponential, or power function of applied stress intensity

factor, a trimodal curve is obtained. Figure 1; each of the three regions

of the curve results from a different mechanism of crack growth.^

Region I crack growth behavior is characterized by a dependence of

crack velocity on stress intensity factor, K^, and on the partial

pressure of water in the environment.^ If either of these two quantities

are increased, the crack velocity is observed to increase. Crack growth

studies conducted in either nitrogen gas^ or in alcohols^ suggest that

1
the crack velocity is reaction rate limited in region I, and therefore,

is controlled by the chemical potential of water in the environment at

any level of applied stress intensity. Hence, identical crack growth

behavior should be obtained in environments in which the chemical

potential of water is the same.

Region II crack growth behavior is also characterized by a

dependence of crack velocity on the amount of water in the environment.

In contrast to region I behavior, however, the crack velocity in region

II does not depend strongly on the applied stress intensity factor.

Theoretical interpretations of region II crack growth behavior lead to

the conclusion that the crack velocity is controlled by the rate of

transport of water from the environment to the crack tip.^’^’^ Although

it is recognized that crack growth in region II is limited by factors that

affect transport processes, experimental data to quantify this relation

have not yet been obtained.

Region III crack growth behavior is characterized by a strong

dependence of crack growth rate on stress intensity factor, and by a

complete absence of any dependence of crack growth rate on water in the

environment. Of the three regions of crack growth, region III behavior

is the least understood. Region III crack growth has been observed in

vacuum for a variety of commercial silicate glasses, but not for glasses

that have a high concentration (>90 percent) of silica.^ Because of

these observations it has been suggested that region III crack growth is

controlled by fracture mechanisms that do not depend on environment.

2
Studies of region III crack growth in normal alcohols, however, conflict

with this view.^’^ In normal alcohols, it has heen shown hy Freiman^

that the slope of the crack growth curves is not as steep as that

obtained in vacuum, or in dry nitrogen. Furthermore Freiman^ and

Richter‘S have demonstrated that the positions of the crack growth curves

depend on the chain length of the alcohol. Hence, in some cases an

environmental dependence of region III crack growth is apparent.

This paper provides additional experimental data to clarify our

understanding of subcritical crack growth in glass. Crack growth data

obtained in alcohols confirm the proposed role of chemical potential in

controlling the crack growth rate in region I. In region II, however,

water concentration, rather than chemical potential, will be shown to

control the rate of crack growth. A model for diffusion controlled crack

growth is developed to show that fluid viscosity is also a parameter of

importance. Finally, data is presented to confirm the fact that crack

growth behavior in region III is affected by the chemical composition of

the test media. Crack growth in region III is influenced by an electrostatic interaction between the environment and charges that develop at

the crack tip during the fracture process.

2. Experimental Procedure

Crack propagation studies were conducted on soda lime silica

glass in normal alcohols, heptane, formamide, trichloroethylene and

acetonitrile. The alcohols were selected as test media because the

3
solubility of water in alcohols decreases with increasing alcohol chain

length, so that it is possible to produce solutions of water and alcohol

that have equal chemical potentials even though they contain different

concentrations of water. These solutions were used to elucidate the

effect of water concentration on region II crack growth behavior. Where

possible, the partial pressure of water in the solutions was determined

by direct analysis with an immersible hygrometer. The amount of water

in solution, (i.e. the concentration) was determined by the Kar 1-Fisher

technique. This procedure permitted us to make independent comparisons

of effects of chemical composition and chemical potential on crack growth

in region II. A set of ethyl and butyl alcohol solutions, used primarily

for the study of region I crack growth, was made by adding predetermined

amounts of water to the "pure" alcohols. The chemical potential of

these solutions were then determined by reference to phase equilibrium

7*8 diagrams for these solutions. The viscosities of solutions used

for region II crack growth studies were determined by direct analysis

using a Cannon-Fenske viscosimeter (Table 1)

.

Crack growth studies in region I, II and III were conducted on sodalime-silica glass microscope slides using both the double cantilever beam

technique and the double torsion technique. 9’ 10 Region I studies were

conducted using double cantilever beam specimens, 7 x 25 x 2 mm, with a

0.5 mm deep groove along the midplane on both sides of the specimen to

control the direction of crack growth. Double torsion specimens,

(ungrooved) 75 x 25 x 1 mm, were used for crack growth measurements in

regions I, II and III. In regions I and II the load relaxation method

4
was used to collect crack growth data, whereas in region III, both load

relaxation and constant displacement techniques were used.

Studies in region II were made in saturated alcohols prepared by

mixing excess water with the alcohols and agitating the liquids several

times a week for six months. A chemical analysis of the water in

solution and a measirrement of the partial pressure of the water in the

alcohol are presented in Table 1. With the exception of the octyl

alcohol, all of the alcohols were found to be saturated with water. We

were somewhat surprised by the observation that the octyl alcohol used

in our study was not fully saturated, and have no explanation for this

result. However, crack growth studies (region II ) to be discussed

below are consistent with this observation.

The double torsion technique was also used to investigate crack

growth behavior in region III. To determine if the position of the

crack velocity curve in alcohol differed significantly from that in dry

nitrogen, crack propagation data were obtained by the relaxation method

on the same specimen, first in dry nitrogen, then in alcohol. Three

crack velocity curves were first obtained in nitrogen gas. Without

disturbing the specimen, either methyl, amyl, or decyl alcohol were

added to the test chamber and three more crack velocity curves were

obtained. This procedure enabled us to reduce systematic variations

and thus obtain crack propagation curves that accurately reflect the

positions of the crack growth curves in alcohol, relative to those in

dry nitrogen gas.

5
In a second study of region III crack growth, the constant displacement rate technique was used to obtain crack growth data in dry nitrogen

gas, methyl alcohol, decyl alcohol, acetonitrile and heptane. At least

10 experimental runs were made at each displacement rate in order to

reduce systematic error that resulted in variations in measured K^. The

mean value and standard deviation of were obtained for each displacement rate measurement, and at least three displacement rates were used

to determine the position and slope of each crack velocity curve.

Displacement rates used in these studies were selected so that the

velocities were in region ill. The velocities had to be low enough,

however, to assure that cavitation of the fluid at the crack tip did not

occur. Some experimental results in region III were also obtained on

vitreous silica (C7940) in octyl alcohol using the applied moment,

double cantilever beam technique. Since silica glass normally does

not exhibit subcritical crack growth in region III, the demonstration

of subcritical crack growth in octyl alcohol is a clear indication of

an interaction between glass and alcohol.

3. Results

The crack propagation data for soda-lime-silica glass in ethyl and

butyl alcohols are shown in Figures 2,3. The shapes and positions of

these curves are similar to those obtained earlier on this glass tested

in nitrogen gas^ and in other alcohols^ In regions I and II, crack

velocity is observed to decrease as the amount of water in the alcohols

is decreased. In region III, the data for the butyl alcohol appears to

6
approach a common curve, as it does for nitrogen gas^. Hence the

mechanisms for fractiare of soda lime silica glass in ethyl and butyl

alcohols are probably Identical to those observed in nitrogen gas and in

long chain alcohols studied earlier by Freiman^ and Richter^

.

The crack growth data from region I of the present paper (Figures

2 and 3) can be compared with earlier data collected in nitrogen gas^

by plotting the logarithm of the crack velocity as a function of the

logarithm of the relative humidity of the water in the alcohol. To do

this, crack velocities were taken from Figures 2 and 3 for an applied

stress intensity factor of 0.563 MPa-m , which is the same value of

* used in the earlier study. The relative humidities were determined from

the concentrations of water in the alcohols, using experimental phase

diagrams^* Q . The resultant plot. Figure 4, indicates a relationship

between crack velocity and relative humidity that is roughly linear

over the range of relative humidities studied. Furthermore, the alcohol

data are found to lie relatively close to the earlier data collected in

nitrogen gas. Since the scatter of crack growth curves obtained by the

double cantilever beam technique can be as much as a factor of v2 along

the velocity axis, the difference between the three sets of data can be

attributed to experimental scatter. A comparison of the

* This procedure necessitated a linear extrapolation of the region I v-K^

data for the butyl alcohol (Figure 3) slightly beyond the range of the

data in region I.

7
ethyl and butyl alcohol data with crack growth data collected by

Freiman^ on long chain alcohols was also made; agreement between the

two sets of data was similar to that shown in Figure 4. These results

confirm earlier conclusion that crack growth in region I is controlled

by the relative humidity (i.e. partial pressure) of the environment in

contact with the crack tip.^’^ As will be discussed in section 4.1, such

behavior is consistent with reaction rate theory, in which the effect of

environment on the crack growth rate depends on the chemical potential

or activity of the water in the alcohol.

In contrast to the results presented above, crack growth in region

II does not appear to be controlled directly by the relative humidity

of the water in the alcohol. This conclusion is supported by plotting

the region II crack growth data presented in Figures 2 and 3 as a

function of relative humidity of water in the alcohol. Data plotted in

this manner are compared in Figure 5 with similar data obtained by

Freiman from studies of crack growth in octyl alcohol^. To assure

ourselves that the two sets of data were consistent, new crack growth

data were collected in nominally saturated octyl alcohol. The crack

velocity in region II for the octyl alcohol was found to be in line with

the earlier data by Freiman. A comparison of the data collected in ethyl

alcohol and butyl alcohol with the data collected in octyl alcohol shows

that the ethyl and butyl alcohol data lie far from the curve obtained

for the octyl alcohol. Since the separation of the data is much greater

than can be explained by experimental scatter, we conclude that factors

other than the relative humidity of water control crack growth in

region II.

8
The conclusion stated above is supported by the double torsion

studies (relaxation method) conducted in saturated alcohols, Figure 6.

With the exception of the octyl alcohol, which was not fully saturated,

the crack velocity in region II is observed to decrease as the chain

length of the alcohol is increased. This decrease in velocity correlates

directly with the concentration of water in these alcohols, and inversely

with the viscosity of the alcohols. Since the alcohols used to obtain

Figure 6 were fully saturated (with the exception of the octyl alcohol)

,

they had the same relative humidity of water as the water with which

they were in contact. The change of crack velocity of approximately 1.5

orders of magnitude in region II, therefore, cannot be explained by a

variation in the relative humidity of the water in the alcohol solutions.

A model will be presented in Section 4.3 of this paper, which explains

region II crack growth behavior in terms of water concentrations and

viscosity of the water-alcohol solution.

The data obtained in region III by the double torsion technique

(relaxation method) is presented in Figure 7, for methyl, amyl and decyl

alcohol. In agreement with other studies^ *^, our data shows a small,

but systematic shift of the crack propagation curve to higher values of

Kj as the length of the alcohol chain is increased. Furthermore,

regardless of alcohol chain length, the curves obtained in alcohol had

slopes that were less than those obtained in dry nitrogen gas.

The most surprising aspect of the data shown in Figure 7 has to do

with the position of the crack growth curve obtained in decyl alcohol.

9
Three separate runs using this alcohol resulted in data that were

similar to that shown in Figure 7. In all cases, the crack growth curve

in decyl alcohol crossed the curve obtained in nitrogen gas, suggesting

that at higher crack growth rates, decyl alcohol was retarding the crack

growth. This finding, and the observation of a systematic shift and a

lower slope of the crack growth data in the alcohols suggests that the

alcohols are involved in some intimate way with the fracture process in

region III. To further check the possibility of an intimate interaction

between organic fluids and the silica network of the glass, experimental

data were collected on silica glass in octyl alcohol. The results of

this study are shown in Figure 8. If silica were tested in nitrogen

gas, it would not have been possible to obtain crack growth data in

region III since fracture occurs abruptly in vitreous silica once the

crack has passed from region II. ^ The fact that region III crack growth

was obtained in octyl alcohol supports our contention of a direct effect

of the alcohol on the fracture process of glass.

To substantiate the data shown in Figure 7, data were obtained

using the double torsion technique, constant displacement rate method.

The data obtained in this way (Figure 9) support the data shown in

Figure 7; the slope of the curve obtained in dry nitrogen, or in decyl

alcohol is greater than that obtained in methyl alcohol. Furthermore,

the data for decyl alcohol and nitrogen gas fall in a significantly

higher range of than the data for methyl alcohol. It is interesting

to note that the data for decyl alcohol also falls in a higher range of

Kj than that for nitrogen gas, a finding that is consistent with the

10
data in Figure 7. However, because of the experimental scatter in both

the nitrogen gas and decyl alcohol curves this difference in location

cannot be construed as being significant from this figure alone.

Finally, data obtained in acetonitrile also had a slope that was

significantly lower than that obtained in nitrogen gas, whereas the

slope of the curve obtained in heptane was similar to that of nitrogen

gas.

4.0 Discussion of Results

4.1 Reaction Rate Theory

The results presented in this paper can be understood in terms of

chemical reaction rate theory, which has been used recently by

Wiederhorn, Fuller and Thomson^ ^ to derive a model for crack growth in

ceramics and glasses. To ease our discussion of the present set of

data, a brief review of the model is presented in this section, giving

the underlying theory and the equations that are pertinent to the

results. In Section 4.4, the model is extended to account for electrostatic interactions that are believed to occur between the environment

and charges that form during rupture of Si-0 bonds at the crack tip.

Consider a chemical reaction of the general type:

aA + 7?B+...^X^^ZL + wM+... (1)

11
If the reverse reaction is negligible, the rate of reaction is given by:

Rate = [A]^ [B]^ . . , (2)

where [A], [B], . . . are the concentrations of the reactants. The

order of the reaction with regard to each component is given by the

exponents in equation 2 (i.e. the order with regard to A is a), and the

overall order of the reaction is given by the s-um a + 2? + . . . . The

reaction rate constant, k^, can be determined by transition rate theory

and is given by the following relation: 1^*15

k

r

K (kT/h) exp (-Ag'^/RT) (f/ f/ . . ./f"^)

O R D

(3)

where k is the transmission coefficient and is usually assumed equal

to 1; AG^ is the Gibbs free energy of activation; and f , f ,

. . . , o A a

f are the activity coefficients for the reactants and for the activated

state. The Gibbs free energy of activation is the standard Gibbs free

energy change associated with the activation process, and can be

related to the standard chemical potentials of the reactants and the

activated complex.

(M

By combining equations 2 and 3, and by using the definition of

chemical activity (a = [A] f . )

,

the rate of chemical reaction can be

-M. ^

12
expressed in terms of the chemical activity of the reactants:

^

"I ^

Rate = (K/f ) (kT/h) a^ . . . exp (-AG^/RT) (5)

where a., a„, . . . are the chemical activities of reactants A, B, etc.

A B

Equation 5 can also be expressed in terms of the chemical potential.

Pa* , of the reactants:

Rate = (K/f ) (kX/h) exp (ayAd+ />y + . . . - y^/RT (6) O

where the chemical potentials of the reactants, y., y„, etc., are given A B

by the following type of relation: y. = d. + RT In a . The free energy

riO ri. change in the exponent of equation 6 includes effects of chemical

composition on reaction rate. By contrast, compositional effects on

reaction rate are included in the activities of equation 5.

For purposes of this paper, the crack growth rate is assumed to be

proportional to the rate of chemical reaction of the crack tip, and the

reaction is assumed to be first order with respect to the water in

k

solution. For these conditions, equations 5 or 6 can be written as:

The assumption of a first order reaction may not be valid for very low

activities of water. In an earlier study^, it was noted that the slight

curvature of the data for nitrogen shown in Figure 4 suggested a change

in the order of the chemical reaction from first order at high relative

humidities to one-half order at low relative humidities. This observation was later substantiated for long chain alcohols by Freiman.^ At

the high humidities used in the present paper, the data for ethyl and

butyl alcohols can be represented by a first order reaction of the

water in solution with the glass.

13
V = Vosa exp (y + y - )/RT ( 7 ) so g o

where a is the activity of the water in solution, and where y , y

and y^ refer respectively to the water in solution, the reactive

species in the glass (presumably Si-0 bonds) and the activated complex.

The standard state for the water is taken as pure water in the liquid

state at 1 atmosphere pressure and a temperature of 25° C, a convention

that follows the practice used in references 7 and 8. The frequency

factor, kT/h, the transmission coefficient, k, the activity coefficient,

f^, and geometric factors relating the number of bonds broken to the

crack velocity are all incorporated into v^ of equation 7. It is worth

noting that the dependence of crack velocity on the amount of water in

the environment is all contained in the activity a^, whereas the

dependence of crack velocity on stress is all contained within the

exponent of equation 7.

Although the free energy of activation in equation 7 contains three

terms (AG^ = y^ - y - y ), these terms are not all equivalent with

o g so regard to water enhanced crack growth. The chemical potentials of both

the glass, y^, and the activated state, y^, depend on the state of

stress at the crack tip, and hence, are the prime determinants of the

slope of the v-K^ curves shown in Figures 1-3. As the stress applied

to the crack tip is increased, y^ - y must decrease to account for

o g an increase in the rate of crack growth. The slope of the v-K^. curve

depends on the sensitivity of y^ - y to the applied stress, the greater

o g

the sensitivity, the steeper the slope of the v-K^ plot. By contrast

14
the chemical potential of the water does not depend on the state of stress

at the crack tip, and hence, has no effect on the slope of the v-K^ curve.

It is for this reason that all of the curves in region I of Figures 1-3

have approximately the same slope.

±

A quantitative treatment of the dependence of applied stress on AG^

has been developed by modelling the crack tip as an elliptical slit in a

continuum. The chemical reaction is assumed to occur at the tip of

the slit and the individual bonds at the crack tip are assumed to be

acted upon by stresses calculated from the Inglis relation. The

following expression is obtained for the free energy of activation. ^

AG^ = - TAS^+ AE ^-[2 K^/(7ip)'^] AV^.- yV)/p (8)

where AS^, AE^and AV^ are, respectively, the activation entropy, the

activation energy and the activation volume. The parameter p is the

radius of curvature of the crack tip, and is assumed to be of the order

•k of the lattice, or network spacing in the solid: 0.5 nm for glass.

The term containing on the right hand side of the equation

is the stress to which the reacting species at the crack tip are

subjected. This portion of the activation free energy determines

the stress dependence of crack growth rate. More specifically,

if crack growth data are fitted to the following empirical relation:

A curved crack tip is assumed primarily to estimate the stress level

at the crack tip. The nature and structure of real crack tips in

glass are, at present, not fully understood.

15
V = exp (-E* + bK^)/RT, then the slope of the v-K^ data, b, can be

used to determine the activation volume for crack growth. Equating b to

the coefficient of in equation 8, the following relation is obtained

for Av"^:

= (b/2) (irp)^ (9)

This expression will be used later in the paper to relate the

crack growth data (i.e. the slope b) to a variable calculated from an

electrostatic model of environmental interactions at crack tips.

The last term on the right hand side of equation 8 is included to

account for the fact that surface curvature can modify the chemical

potential of the reacting species. The parameters y and V are the

surface tension and molar volume of the reacting species (the strained

Si-0 bonds of glass for example) before reaction, while y ^ and V^are

these same quantities for the activated complex on surface during

reaction. The inclusion of this surface energy term follows the approach

first taken by Charles and Hillig^® in their dissolution model of static

fatigue.

4.2 Region I Crack Growth

The crack growth data collected in the present paper can best be

compared with equation 7 of the theory presented in Section 4.1. Since

V and the chemical potentials y , y and y^ are constants for a given

o so g o

° temperature and state of stress, the equation implies that at a constant

applied stress intensity factor, the crack velocity will depend only on

16
the activity, a^, of water in the alcohol solution. If the standard

state of the chemical potential of the water in the alcohol is taken as

p\ire water, then the chemical activity of the fully saturated alcohols

used in the present studies is '^^l, at satioration, (i.e. the chemical

potentials of the water are almost the same for all of the alcohols).

F\irthemore, if water vapor at room temperature is assumed to behave as

a perfect gas, then the activity of the water is given by the ratio of

the partial pressure, p^, of water over the solution to that, 5 over

pure water: a = p /p . Since p /p is defined as the relative

s s so s so humidity, the crack velocity is predicted to be proportional to the

relative humidity of the water, as is observed experimentally (Figure U).

Hence we conclude that the chemical activity, or what is equivalent, the

chemical potential of water in solution is the prime driving force for

crack growth in region I.

The above observation has rather broad implications with regard to

the effect of water on the lifetime of structural ceramics. As long as

the chemical potential of water is the same in two solutions, the response

of cracks in structural ceramics to stress will be Identical with regard

to region I crack growth. Since the lifetime of ceramics is controlled

mainly by region I (i.e. slow) crack growth, the chemical potential of

water in solution is the prime environmental determinant of structural

reliability. Although several investigations of the effect of relative

humidity on component lifetime are consistent with this prediction, ^

this subject has not been studied systematically and certainly warrants

further investigation.

IT
4.3 Region II Crack Growth

The mechanism described in Section 4.1 will control crack growth as

long as the number of water molecules reaching the crack tip maintains

the activity of water at the crack tip constant. When the velocity is

fast enough, the water will be used up at a rate faster than can be

supplied by diffusion from the environment. When this happens, the rate

of transport will control the rate of crack motion.

Using a one dimensional model of diffusive transport of water to a

crack tip it has been shown that the rate of crack growth in region II

can given by:^

V = 0.0275 D X /n5 (10)

o

where D is the diffusivity of water in the alcohol, 6 is the

diffusive boundary layer thickness in front of the crack tip, is the

molar concentration of the water in the solution and n is the order of

* the chemical reaction. For relative humidities >0.1, the parameter n

has been shown to be 1 for water reactions in nitrogen gas and will be

assumed to be equal to 1 for the alcohols.

The constant 0.0275 is appropriate when all variables in equation 10

are expressed in SI units.

18
The diffusivity , D, of water in alcohols has been measured in only a

21 few of the shorter chain length alcohols. Hence in order to compare

the data with equation U, it is necessary to use a theoretical expression

for the diffusion coefficient. For this purpose we use the Stokesp p Einstein relation:

D = (kT)/ (6irrn) ( 11 )

where p is the fluid viscosity, r is the radius of the diffusing

molecule and k is the Boltzman constant. Substituting equation 11 into

equation 10, the following equation is obtained for the crack velocity

in region II:

V = (0.0275 kT/(6iir6 ) ) (X /n)

o

( 12 )

If the form of equation 12 is correct, then the crack velocity in

region II should be proportional to X^/p (i.e. the ratio of the molar

concentration of water in the alcohol to the viscosity of the alcohol).

The data plotted in this manner (Figure 10 ) scatters about a line of

slope 1, as would be expected from the theory. Much of the scatter in

Figure 10 can be attributed to the uncertainty in selection of the

characteristic velocity of region II. As can be seen from examination

of Figure 6, the error in estimating the crack velocity in region II can

be as much as a factor of 2. Fiirthermore , the fact that crack velocity

in region II exhibits a small but significant dependence on applied

stress intensity factor contributes to the uncertainty in selecting

19
characteristic velocities. Considering these sources of scatter in

the experimental data, the correlation between the variables shown in

Figure 10 lends credence to our model of diffusion controlled crack

growth.

The data shown in Figure 10 can be used to obtain an estimate of the

diffusive boundary layer, 6, at the crack tip. The intercept on Figure

10 is approximately 2.5 x 10 m/s; substituting this value of crack

velocity into equation 12 and assuming T = 300 K and r (the molecular

radius of H2O) = 0.3 nm a value of 6 = 0.8 ym is obtained. This value

is approximately one-third that obtained in earlier studies of crack

growth in nitrogen gas. Since the collision distances in liquids

(^^>.5 nm) are less than the collision distances in gases (^^^50 nm at

standard temperature and pressure), the boundary layer, 6 , in a liquid

is expected to be less than that in a gas. The thickness of the boundary

is likely to vary from one type of liquid to another, depending on the

nature of the interactions that occur between the liquid and the water

that is dissolved in it.

4.4 Region III Crack Growth

The data presented in this paper and in the papers by Freiman^ and

Richter^ suggest that alcohols and other organic liquids affect crack

growth in Region III. The influence of these environments is indicated

by small shifts in the position of the crack propagation curves and by

small decreases in the slopes of these curves when compared to the

20
position and the slope of the curve obtained in nitrogen gas. Tvo

possible mechanisms were considered to explain our experimental

observations. The first involves viscous drag of the liquid on the

freshly formed fracture surfaces, which would retard crack growth,

causing small shifts in the crack velocity curves. If viscous drag

were an important mechanism, then the crack growth curve would be

expected to shift to higher values of as the viscosity of the liquid

increased. This type of shift in the position of the crack growth

curve is in fact observed for the alcohols. The possibility of a

viscous drag contribution to the crack, growth behavior was, however,

rejected for two reasons: (l) the shape of the crack growth curves

obtained experimentally differs from that predicted from fluid dynamic

equations; (2) viscous drag cannot explain the experimental crack growth

behavior at low velocities, since all of the crack growth curves would

be expected to join up with the nitrogen curve as the viscous drag

effects vanished. To support these conclusions, detailed discussion of

viscous drag is given in Appendix A.

The second mechanism considered to explain our experimental findings

involves electrostrictive interactions between the glass and the fluid at

the crack tip. If one accepts the chemical interaction model of

subcritical crack growth in ceramic materials, then the slope of the

crack propagation curves in regions I or III is determined by the

activation volume, AV , of the chemical reaction that occurs at the

crack tip.^^ When fracture is caused by the reaction of water with

t- glass, AV represents the difference in molar volume between the

21
reactants and the chemical complex in the activated state. If fracture

occurs in the absence of water, AV^ then represents the change in volume

that occurs as the strained Si-0 bonds move into an activated state.

Referred to as a fission reaction in the chemical literature, this is

the type of reaction of primary interest in fracture processes.

When a fission reaction occurs in a liquid environment, it has

been shown that the activation volume normally consists of two parts:

one part relates to the stretching of the bond that is being broken;

the second part relates to volume changes in the fluid surrounding the

reacting molecule.^ The second part is usually attributable to

electric charge fonnation or destruction during the chemical reaction,

which results in a localized electrostriction, and hence a change in

volume in the fluid surrounding the reacting species. This change in

volume must be included as part of the activation volume because the

volume change is concomitant with the fission reaction. For normal

chemical reactions, charge formation or destruction results in rather

large contributions to the activation volume. In fact, the electrostatic

portion of the activation volume is usually 3 to 10 times that due to

changes in bond length alone. Hence, if charges are either formed or

destroyed during the rupture of a silicon-oxygen bond in glass, electrostatic effects in the glass during crack growth will have to be

considered.

22
In order to model the effect of environment on a fission reaction

in glass, a procedure similar to that used by Wiederhorn, Fuller and

Thomson^ ^ is employed. The procedure models the crack tip as an

ellipse and uses the stress at the major axis of the ellipse, calculated

from the Inglis relation, as the driving force for the chemical

reaction. The crack tip stress then replaces pressure as the thermodynamic parameter usually used to discuss chemical reactions. The crack

tip stress is of course a negative pressure. The effect of charge

formation is then treated by elementary solvation theory: the

charged atoms or molecules formed during the reaction being approximated by

charged conducting spheres; the medium surrounding the reacting species

being treated as a continuum with a dielectric constant, e. For the

fission of a silicon-oxygen bond during crack growth in glass, positive

and negative charges are assumed to form on the silicon and oxygen

atoms respectively as they are separated. This assumption that a full

unit of charge develops during the activated state is based on the work

of Akmed et al.,^^ who reported charge fomation on silicon and oxygen

atoms during the crushing of silica glass. Suggestions of charge

formation during fracture were also made in earlier glass literature.

The procedure used in the present paper to estimate the electrostatic component of the activation volume differs somewhat from that

used by Wiederhorn, Fuller and Thomson, in that the reacting species

(the oxygen and the silicon) are assumed to lie at a stressed surface

of the glass. This assumption is necessary in order to evaluate the

effect of environment on the reacting species. Furthermore, we assume

23
that the electrostatic component of the activation volume (or activation

free energy) is entirely the result of charging the oxygen and silicon

atoms during the fission process. Finally, we assume that the reaction

is sufficiently slow that the static component of the dielectric

constants can be used to estimate the electrostrictive processes at the

crack tip, and that both the glass and the environment at the crack tip

can be treated as a continuum. These last two assumptions are usually

made to treat the kinetics of chemical reactions in liquids .

A schematic diagram of the model used in the present paper to

account for the electrostatic component of the activation volume is

given in Figure 11. Charges are assumed to form at a flat interface

during the reaction. Each Si and 0 atom in the activated state is

assumed to react independently with the glass and the surrounding

environment during the fission reaction. This assumption is often

made in evaluating the electrostatic component of the activation

volume and tends to neglect dipole or quadrupole effects on reaction

rate. The configuration shown in Figure 11 was chosen because

electrostatic effects of the dielectric fluid on the rupture process

are easily estimated from this model. The model does not take into

account effects that result from curvature of the crack near the

crack tip or stearic hinderance due to limits in the dimensions of

the crack near its tip. It is assumed that the change in free energy,

AG^^, when a charge forms at the interface can be estimated by the energy

required to charge conducting spheres of radius, r_ . and r , equal to the

D X L/ radius of the silicon and oxygen atoms (0.04 and 0.14 nm) respectively.

Each atom is assumed to develop a full unit of charge as rupture occurs,

24
a positive charge for the silicon atom and a negative charge for the

oxygen atom. The free energy of formation of each charged atom,

* is given by:

AG = (N e^/r)/(e +e„) (l3)

GS M

per mole of charge formed for each ion. N is Avogadro's constant, and

and are the dielectric constants of the glass and the liquid

respectively.

The electrostatic component of the activation volume for the reaction is determined from equation 13, using the relation, = 8AG^^/9P,

where P is the press'ure that constrains the chemical reaction. When the

reaction occirrs at the surface of the glass, this press'ure is equal to

the negative of the crack tip stress. Hence for a chemical reaction

constrained by a pressure, P, the activation volume is given by:

AV^s = (N^ e^/r) 31n r/3P - ( e^+e2)"^3£i/9P] (l^)

for each ion. The derivation of equation (lU) is based on the assumption

that the pressure is applied only to the glass, hence the partial

derivative is taken only with regaurd to e^. Equation 1^+ can be -written

in terms of the bulk modulus B:

AVes = (N^ e^/r)[(3B)”^(e^+£2)"^ - ( e^+£2)"^3e^/9P] ( 15 )

*This equation is derived in Appendix B.

25
From equation 15, we note that the electrostatic contribution to the

activation volume depends on four parameters: the size of the

electrostatic charge formed during the reaction; the value of the bulk

modulus; the value of the dielectric constants ( and e^) and the

partial derivative of the dielectric constant with regard to the applied

pressure. As. already mentioned, equation 15 neglects any dipole, or

quadrupole contributions to The partial derivative of with

regard to P is usually negative for solids; however, some positive values

have been measured.^^ » 30 yhen 9e^/3P is negative, then the activation

volume is positive. However when 9e^/3P is positive, the activation

volume, AV , can be positive or negative depending on the size of the

6 S two terms in the square brackets. Finally, the dependence of AV on

0 s environment depends primarily on the dielectric constant, e^, of the

environment

.

Crack growth data can be compared with the electrostrictive model

for crack growth by plotting AV , the total activation volume (equation

9), as a function of AV , the electrostatic portion of the activation

es

-t volume. It is worth noting that AV determined from equation 9 contains

both the stretching and the electrostatic components of the activation

volume. For crack growth studies conducted on soda-lime-silica glass in

different environments, is expected to depend primarily on the

dielectric constant of the fluid at the crack tip. For fluids that have

a high dielectric constant (i.e. water, formamide) AV is expected to

0 s be small, and as a consequence, the value obtained for AV^will be due

primarily to stretching of the Si-0 bonds by the applied forces. For

26
environments that have low dielectric constants (i.e. alkanes, vacuum)

AV will he large and AV will contain components due both to electroes static interactions and to stretching. Fig\ire 12 presents a plot of AV

* as a fixnction of AV . Despite the experimental scatter, a correlation

6 S between AV and AV is apparent from the figure. The figure

0 s demonstrates a definite dependence of crack growth data on the

dielectric constant of the test media, and as such lends support to

the idea that electrostatic interactions between the environment and

the highly strained bonds at the crack tip influence crack growth. The

scatter of the data in Figure 12, illustrates the difficulty of

obtaining accurate values of the slope of the v-K^ data in region III,

especially when the slopes are high, as they are in alkanes, nitrogen

gas, and vacuum. Further confirmation of the theory will require the

development of more accurate methods of determining the slope of crack

growth curves. These techniques should then be applied to crack growth

The authors coiild find no values of 8e^/9p in the literature for

soda-lime-silica glass. The value of 3e^/3P (-9.T2 x 10 Pa

used for Figure 12 was obtained by Colwell^® on an alkali containing

glass made at the National Bureau of Standards. The fluid dielectric

constants and the slopes, b, of the crack growth cirrves used for

Figure 12 are given in Table 2

.

27
studies in low dielectric constant liquids, <8, where electrostatic

effects are expected to be greatest. Furthennore , a variety of glasses

and single phase ceramics should be studied to evaluate the effect of

9e^/3P on crack growth data.

5 . Summary

This paper presents the results of a series of studies to determine

the effect of water and a variety of organic liquids on crack growth in

soda-lime-silica glass. Throughout the paper the crack growth process

is viewed as a chemical reaction and the data are interpreted in terms

of chemical reaction rate theory. In the presence of a chemically

active environment, rupture of the Si-0 bonds of the glass network is

viewed as a fission reaction in which Si-0 bonds, strained by the crack

tip stresses, are cleaved by the chemical environment. In the absence

of a chemical reaction between the glass and the environment the fission

reaction occurs as a result of a stress induced cleavage of the Si-0

bonds of the glass network structure. The three regions of crack growth,

observed in crack growth studies are interpreted in terms of reaction

rate theory.

In region I, crack growth in organic liquids is the result of a

chemical reaction between the active component in the environment

(usually H2 O) and the glass; the rate of growth is controlled by the

chemical potential of the active component. This finding confirms those

of earlier studies and is consistent with the chemical reaction rate

28
theory of crack growth. In region II the rate of crack growth is

determined by diffusion of the reactant to the crack tip. We show

that crack growth in region II is not controlled by the chemical potential

of the reactant, but by its concentration. Using the Stokes-Einstein

relation for the diffusivity, a linear relation is obtained between

the crack velocity and the ratio of the concentration of water to

the viscosity of the organic fluid. Finally in our discussion of

crack growth in region III, we Introduce the idea that electrostrictive

effects at crack tips can influence the crack growth behavior of solids.

The concept of an electrostatic component to the activation volume is

commonly used by physical chemists to explain the effects of solvents

on rates of chemical reactions. In this paper, a model was developed

to quantify the effect of dielectric media on crack growth rate in glass.

The model follows classical chemical lines and has the virtue of providing

a relation between the slope of v-K^. data and the dielectric constants of

the glass and the test medium. A plot of the total activation volume

obtained from the slope of crack growth data versus the electrostatic

component of the activation volume calculated from theory, lends support

to the model and suggests that electrostrictive effects influence the

growth of cracks in glass. Despite this agreement, the approximate

nature of the model should be recognized. Further studies may show

that some variables neglected in the model are in fact important to our

understanding of region III crack growth. Clearly, additional data is

needed to fully evaluate the theory presented in this paper.

29
Acknowledgment: This work was supported by the Office of Naval Research,

Metallurgy and Ceramics Program. The authors are grateful for the

technical assistance of A.C. Gonzalez, D. Kravitz, and G.S. White.

Helpful discussions with R. M. Thomson, J. W. Cahn, B. R. Lawn and

H. Richter are also gratefully acknowledged.

r

30
REFERENCES

1. S. M. Wiederhorn, "Influence of Water Vapor on Crack Propagation in

Soda-Lime Glass", J. Am. Ceram. Soc. , 407-414 (1967).

2. K. Schonert, H. Umhauer and W. Klemm, "The Influence of Temperature

and Environment on the Slow Crack Propagation in Glass", pp 474-482

in Proc. 2nd Int. Conf. on Fracture, Biyhten, 1969, Chapman and

Hall, London (1970).

3. S. W. Freiman, "Effect of Alcohols on Crack Propagation in Glass,"

J. Am. Ceram. Soc. 51_, 350-353 (1974).

4. H. Richter, "The Effect of Different Liquids on the Transition from

Slow to Fast Crack Propagation in Soda-Lime-Glass," pp 618-624 in

The Physics of Non-Crystalline Solids, G. H. Frischat, ed. , Trans.

Tech. Publications, CH-4711 Aedermannsdorf , Switzerland (1977).

5. C. L. Quackenbush and V. D. Frechette, "Crack-Front Curvature and

Glass Slow Fracture," J. Am. Ceram. Soc. 402-406 (1978).

6. S. M. Wiederhorn, H. Johnson, A. M. Diness and A. H. Heuer, "Fracture

of Glass in Vacuum," J. Am. Ceram. Soc., 336-341 (1974).

7. M. Randall and H. P. Weber, "The Activity of the Constituents in

Mixtures of n-butyl Alcohol and Water at 30 C," J. Phys. Chem. 44 , 917-920 (1940).

8. H. R. Null, Phase Equilibrium in Process Design, Wiley-Interscience,

New York (1970)

.

9. S. M. Wiederhorn, "Subcritical Crack Growth," pp 613-646 in Fracture

Mechanics of Ceramics, 2, Microstructure Materials and Applications,"

R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, eds. , Plenum Press,

New York (1974).

10. B. J. Pletka, E. R. Fuller, Jr., and B. G. Koepke, "An Evaluation

of Double-Torsion Testing-Experimental," pp 19-37 in Fracture

Mechanics Applied to Brittle Materials, ASTM STP 678, S. W. Freiman,

ed. , American Society for Testing and Materials (1979).

11. T. A. Michalske and V. D. Frechette, "Dynamic Effects of Liquids on Crack Growth Leading to Catastrophic Failure in Glass," J. Am. Ceram.

Soc., 603-609 (1980).

12. S. W. Freiman, D. R. Mulville, and P. W. Mast, "Crack Propagation

Studies in Brittle Materials", J. Mat. Sci. 1527-33 (1973).

13. S. M. Wiederhorn, E. R. Fuller, Jr., and R. Thomson, "Micromechanisms

of Crack Growth in Ceramics and Glasses in Corrosive Environments,"

Metal Science 1^ , 450-458 (1980)

.

31
14. S. Glasstone, K. J. Laidler, and H. Eyring, Theory of Rate Processes,

McGraw-Hill, New York, 1941.

15. G. Kohnstam, "The Kinetic Effects of Pressure," pp 355-408 in

Progress in Reaction Kinetics, G. Porter, ed., Pergamon Press,

London (1970).

16. W. J. Moore, Physical Chemistry, Prentice-Hall Inc., Englewood

Cliffs, N.J. (1972).

17. C. E. Inglis, "Stresses in a Plate due to the Presence of Cracks

and Sharp Corners," Trans. Inst. Naval Archit. 219- (1913).

18. R. J. Charles and W. B. Hillig, pp 511-27 in Symposium on Mechanical

Strength of Glass and Ways of Improving It. Florence, Italy,

September 25-29, 1961. Union Scientifique Continentale du Verre,

Charleroi, Belgium, 1962.

19. R. E. Mould, "Strength and Static Fatigue of Abraded Glass under

Controlled Ambient Conditions: IV. Effect of Surrounding Medium,"

J. Am. Ceram. Soc. 481-91 (1961).

20. J. E. Ritter, Jr., and C. L. Sherburne, "Dynamic and Static

Fatigue of Silicate Glasses," J. Am. Ceram. Soc. 601-605 (1971).

21. P. A. Johnson and A. L. Babb, "Liquid Diffusion of Non-Electrolytes,"

Chem. Rev. 387-453 (1956).

22. J. 0*M. Bockris and A. K. N. Reddy, Modern Electrochemistry,

vol. 1, Plenum Press, New York (1970).

23. S. D. Hamann, "Chemical Kinetics," pp 163-207 in High Pressure

Physics and Chemistry, vol. 2, R. S. Bradley, ed.. Academic Press,

New York (1963)

.

24. E. Whalley, "Use of Volumes of Activation for Determining Reaction

Mechanisms," Advan. Phys. Org. Chem. 93-162 (1964).

25. C. A. Eckert, "High Pressure Kinetics in Solution," Ann. Rev. Phys.

Chem. 239-264 (1972).

26. K. A. Akhmed-zade, V. V. Baptizmanskii, V. A. Zakrevskii, and

E. E. Tomashevskii, "Paramagnetic Centers Produced by Mechanical

Destruction of Silica," Soviet Physics-Solid State, 14, 351-354

(1972).

27. R. E. Benson and J. E. Castle, "Reactions of Freshly Formed

Surfaces of Silica," J. Phys. Chem. 840-843 (1958).

28. W. A. Weyl, "Crystal Chemical Considerations of the Surface

Chemistry of Silica," Research _3, 230-235 (1950).

32
29 . R. S. Bradley, "Other Miscellaneous Effects of Pressure," pp 325-337

in High Pressure Physics and Chemistry, vol. 2, R. S. Bradley, ed.

,

Academic Press, New York (1963).

30. J. H. Colwell, "Stable Pressure Transducer," National Bureau of

Standards Report, NBSIR 76-1116, July 1976.

31. S. M. Wiederhorn and L. H. Bolz, "Stress Corrosion and Static

Fatigue of Glass," J. Am. Ceram. Soc. 543-548 (1970).

32. S. W. Freiman, "Effect of Straight-Chain Alkanes on Crack

Propagation," J. Am. Ceram. Soc. 339-340 (1975).

33. S. W. Freiman, "Temperature Dependence of Crack Propagation in

Glass in Alcohols," J. Am. Ceram. Soc. 340-341 (1975).

34. P. C. Paris and G. C. Sih, "Stress Analysis of Cracks" in Fracture

Toughness Testing and Its Applications , ASTM STP 381 (1965),

pp 80-81.

35. G. I. Barenblatt, "The Mathematical Theory of Equilibrium Cracks

in Brittle Fracture," Adv. Appl. Mech. 1_^ 55-126 (1962).

36. G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge

University Press, Cambridge (1967).

37. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and

Products (Academic Press, New York, 1980), p. 527 (formula 4.224.11).

38. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc.

New York, 1962)

.

33

APPENDIX A

The Effect of Viscous Flow on Crack Motion

In an earlier section of this paper, the idea that the position of

the v-K^ curves in alcohol could he explained hy viscosity effects was

rejected. In this section we develop equations that relate the crack

motion to the viscosity of the fluid at the crack tip and show why the

idea was rejected.

Assume that the crack can he represented hy a parabola that

intersects an external surface. Figure lA. The parabola is selected

because the crack has the shape of a parabola near its crack tip in

the linear elastic approximation. The crack length is equal to c, the

y-coordinate is perpendicular to the plane of the crack and the x

coordinate lies in the plane of the crack. As the crack moves along,

it sucks fluid into the crack opening, which is exposed to an infinite

reservoir of fluid at an ambient pressure, P . In order for the fluid

to flow to the crack tip, the fluid pressure decreases from the crack

opening to the crack tip. Therefore the fluid within the crack is at a

negative pressiire relative to the ambient pressiire, P . The negative

Qj pressure loads the surface of the crack in such a way as to tend close

the crack. Thus, the closing force on the crack surface produces a

negative stress intensity factor, K which opposes the positive, applied

P

stress intensity factor, K , that is driving the crack to propagate.

A-l
To calculate the decrease in pressure from the crack opening to the

crack tip, and to estimate the negative stress intensity factor, we

assume that the coordinate system is fixed to the crack tip. Then the

crack surface appears to recede from the crack tip at a velocity v^,

which is also the velocity of the crack. The half width of the crack,

y^, is determined hy the standard (plane strain) fracture mechanics

relation for the displacement in the vicinity of a crack that is subject

to a stress intensity factor (see, for example. Ref. 34)

= [2K^(1-v)/G]^(x/2tt) (1A)

where G is the shear modulus of the solid and v is Poisson's ratio.

The stress intensity factor, K^, therefore describes the stresses and

displacements at the crack tip and has to be equal to the sum of and

K : K = K + K . This approach to the crack tip stresses is similar

p I a p

to the approach used by Barenblatt^^ in his discussion of crack tip

forces.

At any distance, x, from the crack tip, the pressure drop across a

section, dx, of the crack is given by the Poiseille flow equation for a

parallel sided channel;

V = (l/3ri) (dP/dx) y

^ (2A)

o o

where ri is the viscosity of the fluid in the crack. As noted by

Batchelor, this approximation is valid provided the slope of the

A-2
walls of the channel change slowly with distance along the channel.

Substituting equation lA into equation 2A and integrating the pressure

drop from a point within the crack to the crack mouth, the following

equation is obtained for the pressiure, P, at any point within the

channel

P = P + 6irv n [G/2K (l-v)]^ In (x/c) (3A) a o 1

Note that as the position x approaches the crack tip (i.e. x -> o), large

negative press\rres occur within the fluid.

The negative stress int.ensity factor, K , that resiolts from the

P

reduction of pressure within the crack can be calculated from a standard

equation for a two dimensional crack within a solid that is subject to

an applied stress a(x):

K = 2 (c/7r)^/2 f a(x’) dx'/[c^-(x’ (4a) ^

*b

where x' = c-x measures the distance from the crack mouth.

The stress a(x*) at any point on the crack surface is determined

from the difference between the pressure within the crack and the

external pressure: o(x’

)

= P - P . Note from 3A that a(x'

)

is a

3. negative quantity, as it should be for a closing force on the crack tip.

Substituting the equation for a(x') into equation hA and integrating,^^

the following equation is obtained for K :
K = -1.5 (TTln2 + kc) V T] [G/K^(l-v]^

p o 1

= -8.76 V^n [G/K^(l-v)]^ (5A)

where C = 0.915966 is Catalan's constant.

For a constant crack length, c, K depends only on the crack

P

velocity, v^, and the crack tip stress intensity factor, K^; since

is a slowly varying function of v^ for ceramic materials, depends

primarily on v^.

Normally when a v-K curve is determined experimentally, it is K

-L £i that is measured and presented as a plotting parameter. As long as

is small, the v-K curve and the v-K curve will be indistinguishable.

X di However when K becomes large relative to K , the experimental curve

P 3. will diverge from the "true" v-K^ ciorve. This occurs when part of the

applied load is transferred from the specimen to the fluid within the

crack. The range of velocities over which this occurs can be calculated

by evaluating equation 5A and comparing it with crack tip stress

intensity factor. As shown in Table lA, where P-P and K are calculated

a p as a function of velocity for water, the transition range over which

these viscous effects are important occur over a relatively narrow range

of crack velocities. The data in Table lA are plotted in Figure 2A to

give a predicted v-K curve. We note that the curve bends over rather

Si

A-h
sharply, over a velocity range of about two orders of magnitude.

Behavior of this type has been observed for water by Schonert et al.^

and by Michalske and Frechette for water and for methyl alcohol.

Since the shape of the curves observed in the present paper are not

similar to those obtained for the alcohols in region III, we conclude

that viscous effects of the crack tip fluid played little if any role

in the present experiments, thus justifying o\ir statements in section

U.3.

A-5
' 'iJ 'v;/: ; .s
APPENDIX B

Electrostatic Energy of a Charged Sphere on a Dielectric Interface

The model used in section ^+.3 to describe the region III crack growth

behavior relates the change in free energy during the crack-tip fission

reaction to the electrostatic energy required to charge a conducting sphere

embedded on a dielectric interface (see Figure 12). In this appendix we

derive an expression for this electrostatic energy.

In a linear dielectric medium, the electrostatic energy of a system

of charges is related to the electric field E that they produce by the

relation:^®

where e is the dielectric constant and the volume integral is over all

space. Accordingly, the problem reduces to finding the electric field E,

or its scalar potential <J), since then E = - V$, and integrating the results

according to Eq. (IB).

The electrostatic potential $ is determined as a solution of Poisson's

equation:

W

es (IB)

V^<J) = - (2B)

B-1
vhere x is the charge density. The boundary conditions for the present

problem require: (l) that the surface of the conducting sphere is at a

constant potential; (2) that the displacement field, eE, normal to the

sphere is equal to Utt times the surface charge density; (3) that the

surface charge density integrated over the surface of the sphere is Q;

and (4) that the tangential components of E is continuous across all

boundaries. Rather than solve Eq. (2B) directly, we note that a solution

of Poisson's equation is unique when the potential is specified on all

closed conducting surfaces (Dirchlet boundary conditions), as in the

present problem. Thus, if we obtain a solution by any means, we have

the unique solution.

A solution to the present problem is obtained from the electrostatic

potential of a point charge Q embedded on a planar interface between to

• ^ ft two semi-infinite dielectrics. This solution:

$(R) 2Q

(^2 ^ 1

^^ (3B)

has spherical equipotential surfaces and hence gives the proper form of

the solution for distance greater than the radius r of the conducting

sphere (i.e., R>r). Inside the sphere the potential is constant:

2Q

( £2

for R"^ r (4b)

B-2
The electric field vector is given by the negative of the gradient of the

potential, and only has a radial component since the potential is

spherically symmetric:

1

0 , for R<r

2Q

( , for r<R (5B)

It is easy to verify that this solution satisfies the remaining boundary

conditions

.

The electrostatic energy required to charge the conducting sphere is

obtained by integrating Eq. (5B) in Eq. (IB):

W = -^ / E^ d^R +

^

^ dielectric 2

'2 ^ ' 8tt /dielectric 1

"2 (6b)

In section U.3 the charge Q is taken to be a full unit of electronic

charge, Q = e.

B-3

Table 1: Viscosity, Water Concentration and Relative Humidity of Alcohols

used in Region II Crack Growth Study.

Alcohol Viscosity Water Concentration Percent Relation

( Pa • s

)

(mol/liter) Humidity

Butyl 3.13 X

10"^ 9.2 100

Amyl 3.^6 X 10 ^ U.08 100

Hexyl 3.95 X 10"^ 3.0U 100

Heptyl 5.16 X

10"^ 2.52 100

Octyl 6.31 X

10"^ 1.13 50

Decyl 10.99 X 10”^ 1.68 100

Ethyl (ifO

-3*

1.1 X 10 0.555 7

Butyl (0.2^)

-3*

2.95 X 10 0.111 5

Butyl il%)

-3*

2.95 X 10 0.555 21

Butyl

-3*

2.95 X 10 ^ 2.78 70

The viscosities of the partially saturated ethyl and butyl alcohols

were assiuned to be equal to the pure alcohols at 20 C.
Table 2: A comparison of the slopes, b, of crack growth data with the

dielectric constant, e^, of the test medium. All crack growth

measurements were made on soda-lime-silica glass; slopes were

determined from an empirical fit to the equation: v = v exp

bKj/RT.

°

Test Dielectr;

Medium Constant

Formamide 109

Water 78

Acetonitrile 39

Methanol 33

Hexanol 13.3

Heptanol 12.2

Octanol 10.3

Decanol 8.0

Trichloroethylene 3.4

Slope, b Test

MKS units Method

0.168 D.C.B.

0.111 D.C.B. , ref. 31

0.232 Double Torsion

0.182 . Double Torsion

0.169 Double Torsion

0.223 D.C.B. , ref. 33

0.283 D.C.B. , ref. 33

0.228 D.C.B. , ref. 33

0.240 D.C.B. , ref. 33

0.252 Double Torsion

0.391 Double Torsion

0.463 D.C.B.

Alkanes 2

Heptane 2

Nitrogen 1

1

0.488 D.C.B., ref. 32

0.361 Double Torsion

0.772 Double Torsion

0.724 D.C.B. , ref. 32

0.636 Double Torsion

0.542 Double Torsion

0.823 D.C.B., ref. 1

Vacuum 0.88 D.C.B. , ref. 6
Table lA. Effect of viscous forces on crack motion calculated from

equations 3A and 5 A. The pressure is calculated for a 1 mm

crack at a distance of 1 ym from the crack tip. (G = 28 GPa,

V = 0 . 2h and n = 1 mPa»s)

Crack Velocity

(m/s) (mPa*m

P - P

a (mPa)

K

P 1/

(mPa*m^)

1 11— oO rH o

11— 0.570 -13.6 -0.205

5.72 X 10"^ 0.560 - 8.1 -0.122

2.63 X 10"^ 0.543 - 3.9 -0.059

1.20 X 10"^ 0.530 - 1.9 -0.028

5.25 X 10"^ 0.505 - 0.91 -o.olU

2 .k 0 X 10"^ O.U90 - 0.4I1 -0.007

1oo oo X rH o

11— O.U70 -0.20 -0.003

T T

Figure 1 Effect of relative humidity on crack growth in soda-limesilica glass (ref. l).
Figure 2

STRESS INTENSITY FACTOR, Kj, MPa-m'^2

Crack propagation in ethyl alcohol containing vater. The

concentration of vater is expressed in volume percent.
Figure 3 Crack propagation in butyl alcohol containing water . The

concentration of water is expressed in volume percent.
Figure U . Crack grovth rate in region I as a function of relative

humidity. Data are taken from figures 2 and 3j^at an applied stress intensity factor of 0.563 MPa-m .
PERCENT RELATIVE HUMIDITY

Figure 5. Crack growth rate in region II as a function of relative

humidity. The data was taken from figures 2 and 3 using

the plateau values of the velocity and relative humidities

calculated from the concentration using thermodynamic data

reported in references 7 and 8.
o

o

0) >

o

<0 o

J I I I I

I I 50 .55 .60 .65 .70 .75 .80

Stress intensity Factor, Kj, MPa-m''^*

Figure 6. Crack growth data on soda-lime-silica glass collected "by

the double torsion technique (relaxation method) in

alcohols. With the exception of the octyl alcohol, all

alcohols were fiilly saturated. The octyl alcohol was '''50 percent saturated.
;SE

ra Q.S

(AM

O

S/IU ‘A ‘<H30|3A

E

<Q Q.s

o> S)

s/LU ‘A ‘AlpoiaA >13BJ0

"e

(0 a.S

M

M

0)w

S)

o3 o -—'• •H ^ , fH o

•H

CO

1 0) 0) e

•H c

rH o

I •H

cd

Cd o X

CO CD rH a a; o u

0) 0) P

o

0) •rH •—

1

c

rHo o

o

HJ

aJ C

cd O

nd •rlCO

-P o

?0

U (U bO rH

o o

cd 'd o a;

1—

1

•p M

>> c

0 10 •H CO M1 cd 0) rH K bO

0>

bD

•H

Pin

s/uj ‘A ‘AjpoiaA M^bjo

in

dry

nitrogen

gas,

and

in

methyl,

amyl

and

decyl

alcohols.

Note

the

shift

in

the

curve

to

higher

values

of

and

the

increase

in

the

slope

of

the

crack

growth

curve

as

the

chain

length

of

the

alcohol

is

increased.
Stress intensity Factor,

K|, MPa-m’^^

Figure 8. Region III crack growth data collected on silica glass

(CT9^0) by the double cantilever beam technique (constant

moment method )

.
Crack

Velocity,

V,

m/s

Crack

Velocity,

V,

m/s

Kj, MPa-m^'^^

(0E

>

o

o

0) >

o

(0

k. o

Stress Intensity Factor,

K|, MPa-m^^^

0.7 0.8 0.7 0.8

Stress Intensity Factor,

K|, MPa-m'/2

Stress Intensity Factor,

K|, MPa-m^/2

Figure 9. Region III crack grovth data collected on soda-lime-silica

glass ty the double torsion technique (plateau method) in

dry nitrogen gas and in methyl alcohol, decyl alcohol and

heptane

.
Concentration/Viscosity

3 (Kg-moles/m /Pa*s)

Concentration/Viscosity

(g-moles/l/Poise)

Figure 10. Plot of the crack velocity in region II as a functionof

the ratio of the molar concentration to the viscosity.

Crack growth data on the alcohols taken from figures 2, 3

and 6.
a

o

Dielectric Glass

o

a

Figure 11. Schematic of model used to determine the electrostatic

component of the activation volume.
Electrostatic Component of Activation Volume,

AVgg, cm^/mol

Figure 12. Plot of the total activation volume (.eq. 9) as a function

of the electrostatic component of the activation volume

(eq. 15). The value of 3e-/3P (-9.72 x lO” ^ Pa~M was taken

from data by Colwell^® collected on an alkali containing glass

made at NBS. Double cantilever beam data are indicated by

circles, double torsion data by triangles. Data taken from

the literature are identified by reference numbers on the

figure. ACN is an abreviation for acetonitrile; FMD is an abbreviation for formamide; TCE is an abbreviation for

trichlorethylene. The other chemical symbols are standard

chemical designations.
Pa

Figure lA: Schematic diagram of parabolic shaped surface crack.
Stress Intensity Factor, MPa-m^'^^

Figure 2A: Calculated effect of fluid viscosity on v-K^ curve: sodalime-silica glass in water. The portion of the curve is

linear extrapolation of data collected at velocities less

than 10 m/s.
NBS-114A (REV. 2 >ac) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 82-2524

2. Performini Organ. Report No. 3. Publication Date

June 1982

4. TITLE AND SUBTITLE

Effects of Water and other Dielectrics on Crack Growth

5. AUTHOR(S) Wiederhorn, S. W. Freiman, E. R. Fuller, Jr. and C. J. Simmons

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL,BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

S. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

10.

SUPPLEMENTARY NOTES

[ ] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200‘word or less factual surhmary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

The effect of water and a variety of organic liquids on the crack growth rate in

soda lime silica glass was investigated. When water is present in organic liquids, it

is usually the principal agent that promotes subcritical crack growth in glass. In

region I, subcritical crack growth is controlled primarily by the chemical potential of

the water in the liquid; whereas in region II, crack growth is controlled by the

concentration of water and the viscosity of the solution formed by the water and the

organic liquid. In region III, where water does not affect crack growth, the slope of

the crack growth curves can be correlated with the dielectric constant of the liquid.

It is suggested that these latter results can be explained by electrostatic interactions

between the environment and charges that form during the rupture of Si-0 bonds.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Glass; fracture; strength; static fatigue; subcritical crack growth; cracks

13. AVAILABILITY

[X~| Uni imited

( I For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

60

15. Price

S9.00

USCOMM-DC 8043-PS0