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The Specific Heat ∆C

. Table 2.5 shows the calculated values of ∆C

for some of the metals. The maximum estimated errors in ∆C

and ∆C

are

10 and 20%, respectively. The significance of ∆C

in diffusion studies

arises from the relationship:

∆C

 [∂(∆H

)∂T]

. (52)

Thus, the parameter ∆C

governs the variation of ∆H

with respect to tem-

perature, which is responsible for curvature in the diffusivity plot [n(D)

vs. 1T plot]. The change in ∆H

in going from temperature T

to T

given by Eq. (52) as:

∆H

 ∆H

 (T

 T

)∆C

. (53)

∆C

is assumed to be constant in Eq. (53). As shown by Eq. (52), ∆C

indeed a function of temperature. However, justification for assuming the

invariance of ∆C

with respect to temperature is provided by the data in

Table 2.6, where the maximum variation in ∆C

over a range of more than

500 K is less than 0.3 R. This is significantly smaller than the inherent

error in the measurement of ∆H

. Therefore, in the temperature range of

to T

, an inherent curvature in the diffusivity plot will be observed only

when the magnitude of the right-hand side of Eq. (53) is significantly

larger than the sum of systematic errors in the determination of ∆H

Furthermore, it may be indicative of a change in the mechanism of diffu-

sion only when the deviation in ∆H

exceeds the limit given by Eq. (53)

over the temperature range under consideration.

Several earlier attempts to estimate ∆C

have been reported in the

literature. Using the Debye theory of specific heat and thermodynamic

relationships between ∆C

and ∆C

, Levinson and Nabarro

[71]

derived the

following expression for the temperature variation of the entropy for

vacancy formation in noble metals:

(∂∆S

∂T)

  {(V

B)(∆V

V)}  21  10

4

J mol

1

. (54)

Equation (54) was then used to examine the possibility of curvature in

the Arrhenius plot for vacancy formation in metals. Over a temperature

range of 400 K, the maximum variation expected in the vacancy forma-

tion entropy is only about 2 J mol

1

. Levinson and Nabarro concluded

that at this magnitude, no departures from linearity are expected in the

Arrhenius plots of vacancy concentration.

Nowick and Dienes

[72]

first write ∆C

as:

∆C

 (∆C

 ∆C

)  ∆C

. (55)

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SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL. 103

For temperatures higher than Debye temperature, Nowick and Dienes

argue that ∆C

is very nearly zero and, therefore, the contribution to ∆C

comes only from the first term on the right-hand side of Eq. (55). Further,

using Eqs. (48) and (51), they arrive at the following equation for ∆C

∆C

 0.5 R[(∆bb

)  (∆VV

)]. (56)

Here, b

and V

are the compressibility and specific volume of the

matrix, and ∆b and ∆V are the changes in these parameters per mole of

defects, respectively. Nowick and Dienes further observe that both the

quantities within the bracket are positive, and the bracketed term will

closely approximate unity. Therefore,

∆C

≈ 0.5R. (57)

The value of ∆C

given by Eq. (57) is small, and the observed devia-

tions in the diffusivity plots at low as well as high temperatures cannot be

accounted for by the inherent temperature variation of ∆H

. Nowick and

Dienes concluded that curvatures in the diffusivity plots might be mani-

festations of the multiplicity of diffusion mechanisms.

Gilder and Lazarus

[73]

used Grüneisen Eq. (48) and the Debye theory

of specific heat for estimation of ∆C

. In their analysis, the contributions

to ∆C

arising from the change in the phonon spectrum during the forma-

tion of an activated vacancy complex were considered responsible for the

diffusion process. An additional contribution to ∆C

arises from the gen-

eration of new vibrational modes around the vacancy. According to Gilder

and Lazarus, ∆C

is given by:

∆C

 ∆C

 a

T∆V

{(2a

a

)  (b

b

)}b. (58)

Here, a

represents the volume thermal expansion coefficient of the matrix.

and a

are the compressibility and volume thermal expansion coeffi-

cient, respectively, for activated vacancy. According to Gilder and Lazarus,

∆C

for metals can be as large as 15R and can account, in principle, for the

curvature in their diffusivity plots. However, in the absence of precise data

on the high-temperature values of a

, a

, b

and b

, Eq. (58) cannot be

applied directly. They further state that to a good approximation:

∆C

 m

T, (59)

and

 2(a

∆V

∆C

V. (60)

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Here, g

is the Grüneisen constant and ∆C

is the specific heat of the

matrix at constant volume. Gilder and Lazarus determined m

through the

curve fitting of the diffusion data by:

T ln D  (∆H

R)  (ln D

)T  12(m

R)T

. (61)

Tables 2.5 and 2.6 compare the values of ∆C

for some cases given by

Eqs. (50), (51), and (59). The two sets of values differ by 31 to 82% for

different metals; the difference narrows to 1% for copper at 1200 K.

With the exception of Na, K, b-Zr, and b-Ti, the experimentally

observed nDvs. 1T Arrhenius plots for other metals do not exhibit a

continuous curvature. Instead, they may be broken, indicating contribu-

tions of more than one Arrhenius dependence and diffusion process.

Therefore, in the case where the nDversus 1T plot is only broken, its

representation by a continuous curve can compromise the accuracy of m

and the corresponding ∆C

values. In contrast, the expression for ∆C

derived here depends on the relationships among the various diffusion

parameters. It is validated by the appropriate values for ∆S

, u, and ∆V

obtained from one basic equation, which is Eq. (43).

2.3.5 Reliability of Estimated Parameters

The reliability of the calculated values of various parameters can be

judged by two considerations. (1) Their magnitude should be close to the

normally acceptable values. We consider this important because this kind

of independent estimate of the diffusion parameters has not been carried

out before. (2) The errors associated with the calculated values should lie

within reasonable limits. The evaluation of errors in the calculated values

of , ∆S

, u, ∆C

, and ∆C

has been carried out using standard procedures;

[74]

the details are given in Appendix 2B.

The estimated values are indeed close to the normally accepted

values, and the errors in their estimation arising from those in the

experimental data are also within reasonable limits. An error of not

more than 10% is expected in the evaluation of ∆S

, about 90% in the

estimation of u, a maximum of 10% in ∆C

, and a maximum of 20%

in ∆C

. Therefore, with respect to the two considerations mentioned

earlier, the method of calculation proposed here has proved to be cor-

rect and yields reliable values of the diffusion parameters. However,

improvement in the precision of the concerned parameter can lower

the error limits.

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SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL. 105

2.4 Summary

Two different approaches to estimating diffusion parameters from the

bulk properties have been discussed in this chapter. First, empirical as

well as semi-empirical correlations between diffusion and the bulk prop-

erties are discussed. These include the correlation of activation energy for

self-diffusion with the cohesive energy, melting point, latent heat of

fusion, bulk modulus, and the Debye temperature. The Zener’s hypothe-

sis, which paved the way for development of relationships between the

diffusion parameters and the elastic modulus, is also discussed. Finally, a

correlation between the activation energy for self-diffusion in metals with

the valence bond parameter and that for solvent diffusion in dilute alloys

with electron-to-atom ratio is presented.

Secondly, the empirical treatment of diffusion data is followed by the

derivation of a thermodynamic relationship between the activation vol-

ume and activation entropy for self-diffusion. These two diffusion param-

eters are shown to be related to each other through the compressibility and

the volume expansion coefficient. Because this equation does not contain

any arbitrary constant, it is used to obtain the values of other diffusion

parameters. Satisfactory agreement is obtained between the calculated and

experimental values for all the parameters.

The correlations between the bulk properties and the diffusion param-

eters discussed here can be used to rationalize new data and estimate the

diffusion rates in the absence of experimental values.

Acknowledgment

The authors thank Dr. R. Nakamura for help in preparing the figures.

Appendix 2A. Taylor Series Expansion of ∆G

The general form of the Taylor series for f(x) about x  h is:

f(x)  f(h)  f(h)(x  h)  f(h)(x  h)

2!

 f(h)(x  h)

3!  ...........

(A1)

The Taylor series permits us to obtain an algebraic expression for any

parameter in terms of a desired variable. The condition necessary is that

the derivatives of f(x) should exist at f(x)  f(h). Equation (36) is

obtained from Eq. (A1) when f(x)  ∆G

, x  V, and h  V

. In order to

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justify the procedure to derive Eq. (36), we try the expansion of ∆G

as a

function of T and P. Like V, both T and P are thermodynamic state prop-

erties of the matrix; therefore, the expansion of ∆G

as a function of these

two variables is valid.

We first expand ∆G

as a function of temperature to obtain:

∆G

 ∆G

 [∂(∆G

)∂T](∆T)  [∂

(∆G

)∂T

](∆T)

2!  ......... (A2)

Here, ∆G

is the value of ∆G

at T  0.

From the basic thermodynamic equation:

∆G

 ∆H

 T∆S

, (A3)

we obtain:

∆G

 ∆H

(A4)

for T  0.

Further, by definition:

[∂(∆G

)∂T)]

∆S

, (A5)

and

∆T  T. (A6)

Substituting from Eqs. (A4), (A5), and (A6) into (A2), after ignoring

second and higher order terms, we obtain:

∆G

 ∆H

 T∆S

which is the same as Eq. (A3).

We now expand ∆G

as a function of pressure:

∆G

 ∆G

 [∂(∆G

)∂P](∆P)  [∂

(∆G

)∂P

](∆P)

2! ........ (A7)

Here, ∆G

is the value of ∆G

at P  0.

When pressure is a variable, the standard equation for ∆G

is:

∆G

 ∆H

 P∆V

 T∆S

, (A8)

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SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL. 107

where ∆V

is the activation volume. Therefore, at P  0:

∆G

 ∆H

 T∆S

. (A9)

By definition,

[∂(∆G

)∂P)]

 ∆V

, (A10)

and

∆P  P. (A11)

Substituting from Eqs. (A9), (A10), and (A11) into (A7), after ignoring

second and higher order terms, we obtain:

∆G

 ∆H

 P∆V

 T∆S

which is the same as Eq. (A8).

The results obtained from the expansion of ∆G

as a function of T and

P allow us to draw the following conclusions:

1. We retrieve the basic equations for ∆G

in both cases.

Therefore, the procedure used is justified.

2. The truncation of the Taylor series after first term, and

ignoring the second and higher order terms, are justified

because the results obtained are true and correct.

3. The results obtained by Taylor series expansion of ∆G

as a

function of T and P are mathematically trivial because they

do not yield any new result. As shown in this appendix, only

the expansion of ∆G

as a function of V yields a new result.

Appendix 2B. Evaluation of Errors in Estimated

Parameters

2B.1 ∆S

Using Eq. (47), the error in the estimation of ∆S

is given by:

∆(∆S

)∆S

 ∆(ab)(ab)  ∆(∆V

V)(∆V

V). (B1)

Equation (B1) shows that the error in estimating ∆S

originates from the

errors in ab and the experimental determination of activation volume

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∆V

V. In contrast, the error in V

can be neglected, because this is usu-

ally determined to an accuracy of better than 1 in 1000.

According to Hanneman and Gatos,

[75]

the ab values are approxi-

mately pressure- and temperature-independent at room temperature and

above. We can therefore conclude that the error in ∆S

is limited to the

error in the experimental measurement of ∆V

V. Asurvey of the experi-

mental data for a number of metals shows that the error in measurement

of ∆V

V varies from 1 or 2%, as in the case of Zn and Cd,

[65, 66]

to ∼7%,

as in case of tin.

[76]

Consequently, the maximum possible error in the ∆S

value estimated from Eq. (47) does not exceed 10%.

2B.2 u

From Eq. (46), the expression for estimation of u is given as:

u  D

[fa

exp(∆S

R)], (B2)

and the corresponding equation for estimation of error is:

∆uu  (∆D

D

)  (∆ff)  2(∆aa)

 [exp(∆S

R)exp(∆S

R)].

(Β3)

The error in u is dependent on the errors in the determination of fre-

quency factor D

, lattice parameter a, and activation entropy ∆S

. Because

the correlation factor f is a pure number for self-diffusion in metals, it does

not contribute to the error.

In a conventional measurement of self-diffusion in metals through

radioactive tracer technique, the absolute error in the magnitude of D

can

vary from 50 to 80%.

[77, 78]

The error in the determination of the lattice

parameter rarely exceeds 5%.

[79]

With regard to the error in ∆S

, the pre-

ceding paragraph shows that the maximum possible error in its magnitude

may amount to 10%. A summation of all these errors shows that use of

Eq. (B3) to estimate u could be in error by a maximum of about 95%.

2B.3 ∆C

Because g is a constant, use of Eq. (50) gives the error in estimation

of ∆C

as:

∆(∆C

)∆C

 [∆(∆S

)∆S

]  [∆(1  aT)(1  aT). (B4)

Since aT is usually of the order of 10

3

, contribution by the last term to

the error in ∆C

can be neglected. Thus, the error in ∆C

can be put at a

maximum of 10%, the same as the error in ∆S

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SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL. 109

2B.4 ∆C

From Eq. (51), the error in ∆C

is given by:

∆(∆C

)∆C

 [∆(∆C

)∆C

]  [∆(aT)aT]  [∆(ab)(ab)]

 [∆(∆V

)∆V

(B5)

It has already been stated that aT is usually of the order of 10

3

and ab

is practically temperature-independent. Therefore, the error in the estima-

tion of ∆C

is the sum of the errors in ∆C

and ∆V

, and can be put at a

maximum of 20%.

The preceding analysis shows that the error in the estimated diffusion

parameters comes from the error inherent in measurement of ∆V

and D

Sometimes the diffusion parameters are measured with a precision of

∼5%. We have indicated here the maximum possible error limits for all the

parameters. These could be significantly lower if the diffusion data are

measured with greater precision. In other words, the accuracy of the val-

ues calculated by the present method basically depends on the precision

of diffusion parameters.

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