computation of reequilibration data in solids

Computation of Reequilibration Data in Solids

Frangois Morin Applications de l'Hydrogkne et de l'fhectrochimie, Znstitut de Recherche &Hydro-Qukbec, Varennes, Que'bec, JOL 2P0, Canada

Received 8 January 1985; accepted 25 March 1985

Reequilibration processes are often encountered within solids and they have long been described mathe- matically. However, computation of reequilibration data over entire processes is often difficult. An algorithm has been developed specifically for this purpose. It allows a fast and efficient computation of reequilibration parameters. Anisotropic diffusion can, moreover, be taken into account.

INTRODUCTION

A reequilibration process within a solid is encountered when boundary conditions are suddenly modified and there is a tendency for a new equilibrium state to be reached. Such a definition encompasses a rather great variety of phenomena: drying of porous and moisture absorption and desorption of com- posite material^,^,^ temperature reequilibration within solid^,^ or chemical gradient relaxation.6-8 All these physical processes can be described by analogous mathematical solutions. This is particularly true for simple physical and boundary conditions for which the usual analytical solution is fully applicable.2,5,6,9-10 In practice, this analytical solution is often applied in an approximated form,4,S, 11-20 although there is a loss of physical information in doing SOY Actual de- parture from the true analytical solution may originate from various factors like a slow surface reaction," an eventual variation of the chemical diffusion c~efficient,'~ or, for a some- what comparable case, the absolute applica- bility of an analytical solution to potential relaxation in batteries.20*21 On the other hand, a precise point by point computation of the analytical solution for the entire reequilibration process is rather cumbersome and it is seldom used.

This article describes an algorithm which leads to a simple and accurate computation of the reequilibration parameters over the whole process. This algorithm can equally be applied to uniaxial and multidimensional processes and, also, to anisotropic diffusion and to widely different sample dimensions. A

brief account of the original analytical solution and some of the approximations usually derived from it, are given first.

GENERAL ANALYTICAL FORMULATION

A diffusional process where the surface chemical equilibrium of a solid with its sur- roundings is suddenly changed to a distinct value, can be represented as follows:

( a c l a t ) = B(a2c /ax2 + a2c/ay2 + d2C/dZ2)

(1) Equation (1) is the three-dimensional ex-

pression for Fick's second law. 15 is the chemical diffusion coefficient, c the concentration of the migrating species and x , y , and z, the dis- tances along each of the three diffusion axis. Boundary conditions are then given by

c(x,y,z,O) = co at t = 0 ( 2 4 and

C(O,Y , z, t ) = Cf (2b) c(x , O,z, t ) = cf at t > 0 (24

C b , Y , 0, t ) = Cf ( 2 4 The extension of eq. (1) to a three-

dimensional diffusion process and its integra- tion for a brick-shaped sample with overall dimensions 2a1, 2a2, and 2a3 would lead to a general solution of the following type:

(3) where

cr = full) ' f(T2) * f(T3)

m

f(T2) = (2/n2) c (n + 1/2)y n=O - exp[-Ti - ( n + 1/2)'. n2] (4)

Journal of Computational Chemistry, Vol. 6, No. 6, 514-519 (1985) 0 1985 by John Wiley & Sons, Inc. CCC 0192-8651/85/060514-06$04.00

Reequilibration Data in Solids 515

c r = (cf - C ) / ( C ~ - cO) (5)

T , = b t / a : (6) c is the average defect concentration

within the sample. c, is a dimensionless pa- rameter which can be easily replaced by a corresponding weight or electrical conduc- tivity variation. Equation (4) is cumbersome to handle, specifically at the beginning of the relaxation process where a large value of n (n + 1 being the number of terms in the exponential series) is required and where the data may be scarce due to fast initial reequilibration rates. Depending on the actual availability of experimental data, the full analytical solution represented by eqs. (3146) is usually replaced by either of two types of approximations: one for short time durations and one for long reequilibration times. In the first case, diffusion in a semiinfinite medium is considered. Hence, it can be shown that c, is related to time t through the following parabolic law:22

(7) where A is the sample surface normal to the diffusion axis and V is the volume of the sample. At the opposite end of the relaxation process, i.e., for large values of T , all but the first exponential term in eq. (4) may be neglected. A simple semilogarithmic expression can thus be obtained from eqs. (3)-(6)

-

(1 - c,)' = 4(A/V)' * 5 > t / ~

log, c, = -.ir-'((a,' + a i 2 + a,')Bt/4 + log,(512/.ir6) (8)

Analogous expressions also hold for single and two-dimensional processes. Equations (7) and (8) lend themselves readily to simple graphical curve fitting computation of the chemical diffusion coefficient. Despite this usefulness, these approximations should be used with caution:17 short time diffusion data are usually not reliable due to errors re- sulting from slow surface reactions; moreover, only a small portion of the relaxation data is thus used. Confusion may also arise from the fact that a linear plot of log, c, vs. time is not limited to eq. (8), but that it is also expected from some rate limiting surface reactions.l* Moreover, the range of validity of the semilogarithmic approximation, in eq. (81, becomes severely restricted as the relative dimensions of the sample are varied.17

One of the major aims of reequilibration studies is to verify the degree of correspon-

dence between the actual physical process and its mathematical counterpart, basically represented by eqs. (1) and (2). The whole reequilibration process then needs to be fully analyzed. This can be done, for example, by calculating 6 vs. c,, the degree of reequilibration of the process, with c, varying over its full range from unity down to zero. The calculated value of 6, in Figure 1, is then called BaPp. * Bapp should theoretically be a perfect horizontal line. For real experiments, however, large departures from this trivial behavior may be expected, reflecting either usual experimental uncertainties, more de- fective experimental conditions, o r some unexpected physical phenomena. l7 One very common effect is that of a partly limiting surface reaction typified by curve b. Uncer- tainties are usually amplified at both ends of experimental curves and may then overlap with additional physical phenomena. The type of plot shown in Figure 1 is, nonetheless, a very good means for both verifying the ap- plicability of eqs. (1) and (2) and obtaining a better estimate on the actual value of the true chemical diffusion coefficient 6.

FLOWCHART FOR THE COMPUTATION OF 6

Computing 5> by means of the implixit sys- tem represented by eqs. (3146) requires the use of a numerical method. A simple path, making use of the Newton-Raphson method, is shown in Figure 2 for a unidimensional process. Only the most illustrative steps are shown in this figure. First, limit values of T are found in accordance with the experimental value of c,. Using these limit values of

- Dapp

(lQm cm?s )

1.0 0.0 1-0 t- co

t i m q

Figure 1. Graphical representation of reequilibration data. Exponent m is generally comprised between 4 and 7. Shaded areas indicate the regions of maximum uncertainty.

516

a. Searching for limits of T

To=b t b > highest T

value

COIWUTATION OF (Cr) talc

I I

LOWERING T: TI = T1-l x R, R < 1

Morin

b. Solving for the implicit equation

r - - -L - - -1 ibid .

steps 2 and 3 I

- - -+-- -

,

PRINT

Figure 2. Flowchart showing the computation of 5 for reequilibration along a single axis.

T, the implicit equation can then be solved with the desired accuracy for T and then for fi. Both numerical steps involved the same type of calculation for c,. Either a very low or a very large value of T may be initially cho- sen to start the iteration process. Fewer terms in the exponential series are required as T increases, and this can be taken into account to shorten the calculations by weighing the value of n accordingly. At the opposite end of the reequilibration process, n increases dra- matically as c, gets closer to unity, in order to maintain the same degree of accuracy on T. This may be a major time consuming opera- tion as shown in Table I. This point becomes

increasingly important in the cases of two- and three-dimensional shapes. This is more evident in Figure 3 where the flowchart of Figure 2 is extended to a two-dimensional case. Since c, here is equal to the product of f(T,) - f(T2), a larger value of n + 1, the number of terms in the exponential series, is required for each function f ( T J to preserve the same accuracy for c,. This tendency of f (Ti) is strongly emphasized when a2 is not equal but larger than a,. These limitations are even more important for the three-dimensional case, and they also explain why the loga- rithmic approximation, represented by eq. (8), is less applicable to two- and three-

517 Reequilibration Data in Solids

Table I. Accuracy achieved in the computation of T by means of the usual approximations.

Number of (n + 1) terms in Eq. (4) for a 0.1%

Error on AT) by means of

AT) T accuracy on AT) eq. (10) 0.99 7.8563-5 72 <0.05% 0.95 1.9643-3 17 <0.05% 0.90 7.8553-3 8 <0.05% 0.80 3.1423-2 4 <0.05% 0.70 7.0693-2 3 <0.05% 0.60 1.2573-1 3 <0.05% 0.55 1.5923-1 2 0.05%

0.40 2.8643-1 1 1.3% 0.50 1.9673-1 2 0.2%

f i r s t

OF n ACCORDING

a x i s

I I

I

I t

-1 I I I I I I I I I I

- J

I

I I I I I I I I I

OF n ACCORDING

VALUE TJ TO THE CURRENT

SOLVING FOR

I

I I I I

I I I I I

t t I

Figure 3. Modifications to be added to the flowchart, in Figure 2, when two rectangular axis are involved in the reequilibration process.

dimensional problems. At this point, a very useful observation on eq. (7) can be made. For a unidimensional process, this equation is equivalent to writing:

(1 - c,)~ = (4/n) * T (gal

c, = 1 - 2 W T (9b) f ( T ) being equal to c, is also equal to the

right-hand term within the validity range of the parabolic approximation. Each specific function can thus also be written as:

(10) within the same validity range, described by a maximum value of Ti. Table I shows that the accuracy on Ti computed from eq. (10) de- creases rather slowly in the range where a large number of exponential terms would alternatively be required for eq. (4). Opti- mization of both computing time and accuracy would be achieved by making a more extensive use of the parabolic function stated in eq. (10). Thus, f (Ti) in eq. (3) could be replaced alternatively by its most suitable expression according to the value of Ti, either from eq. (4) or (10). Moreover, the parabolic approximation for the initial two-dimensional reequilibration process then becomes:

or

f (Ti) = 1 - 2VT&

c, = 1 - 2(a;l + a , ' ) / G * fi + (4/na,az) Dt (11)

The calculation of fi in eq. (11) is straight- forward since the appropriate root for f i t can be algebraically calculated. These modifications are now included in the flowchart of Figure 4. At c, > 0.57, eq. (11) is the sole expression used. Below this value, a smooth transition goes on between the parabolic ex-

518 Morin

pression, represented by eq. (lo), and the exponential series limited to a few terms. The preceding algorithm also can be quite easily extended to a three-dimensional reequilibration process: it would be analogously dealt as the product of f(T,) * f(T,) - f ( T 3 ) with f ( T J either coming from eq. (4) or (10). Im- provement on computation efficiency is even greater for this last case.

ADDITIONAL REMARKS

A thorough comparison of computation times would require that all corresponding parameters be properly defined. Nonetheless, it can be safely stated that an acceleration factor of an order of magnitude has been ob- served while going from the flowchart of Fig- ure 3 to that of Figure 4. Another natural extension of the present algorithm is the calculation of reequilibration parameters in anisotropic materials. Little mathematical difficulty is encountered in considering dis-

a . Searching for l i m i t s of T (T2(T,l

I

PRINT J-!J I

I

I I

A

Figure 4. Flowchart of an improved algorithm for a two-dimensional process.

tinct diffusion coefficients for all axis in eq. (1). This would lead to modified but still constant ratios between TI, TP, and T,. For example, reordering between T I and T , accordingly can be introduced in the flowchart from Figure 4. Furthermore, an unknown value of an anisotropic diffusion ratio eventu- ally may be estimated from experimental data if the process theoretically represented by eqs. (1) and (2) reasonably holds: this ratio can thus be estimated by iterative computation of the reequilibration data with pro- gressively varying anisotropic ratios until a horizontal line for fi is obtained.

CONCLUSION

An efficient algorithm has been suggested for the computation of reequilibration data. It leads to a fast and accurate computation of experimental data with a smooth transition from a parabolic to an exponential rate law. A major improvement is obtained mainly for two- and three-dimensional processes. The application of the present algorithm can be readily extended to anisotropic rate proper- ties in solids.

References

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Requilibration Data in Solids 5 19

17. F. Morin, J. Electrochem. SOC., 128, 2439 (1981). 18. F. Morin, Surface Reaction Kinetics of Rutile in

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computation of reequilibration data in solids

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