Evaluation of the Methods for Calculating the

Evaluation of the Methods for Calculating the ConcentrationDependent Diffusivity in Binary Systems SRIDHAR K. KAILASAM, JEFFREY C. LACOMBE, and MARTIN E. GLICKSMAN Boltzmann and Matano developed a procedure for the solution to Fick’s second law when the diffusivity is a function of concentration. The procedure requires the determination of the so-called Matano interface. The accuracy of the resulting solution depends heavily on the precise location of the Matano interface, the determination of which is laborious and often inaccurate. Three alternative procedures by Sauer and Freise, Wagner, and den Broeder, modifications to the original Boltzmann–Matano (B–M) method, were developed such that the diffusivity can be calculated without having to determine the location of the Matano interface. However, none of these derivations quantifies the extent to which the modified methods arrive at the same result as that obtained from the standard B–M analysis. This article serves to apply the B–M method, and the modification suggested by den Broeder, to various analytical concentration profiles containing different degrees of noise and to compare the results quantitatively. In addition to these analytical functions, concentration profiles obtained from interdiffusion experiments were studied. The two methods are shown to be equivalent in terms of the accuracy of the result. The advantages or disadvantages of one method over the other are illustrated with examples.

I.

INTRODUCTION

LINEAR solutions to Fick’s second law require constant

diffusivity, D. However, in reality, the diffusion coefficient may vary due to a range of parameters such as temperature, pressure, composition, and crystal orientation. The variation of diffusivity with concentration becomes particularly important when the concentration difference across a diffusion couple is large. In order to address the problem of a concentration-dependent diffusivity, Boltzmann[1] developed the theory and Matano,[2] the experimental procedure, now known collectively as the Boltzmann–Matano (B–M) method. From their analysis, one does not obtain a diffusion solution for concentration as a function of time. Instead, one calculates the concentration-dependent diffusivity, D(C ), at any specified time. The result, D(C ), is given by the integrodifferential expression

1 2Z e xdC

1 dx D(C ) 5 2 2t dC

C8

[1]

C8 C2

The distances, x, are measured from the so-called Matano interface (x 5 0). The Matano interface defines a plane in the diffusion zone about which there is a balance of mass. In other words, the amount of material lost by diffusion on one side of the Matano interface is equal to the amount of material gained by diffusion on the other side of the Matano interface for the component in consideration. Graphically, this can be represented by the solid line in the inset in Figure 1 that makes the two hatched areas equal (i.e., A1 5 A2). Mathematically, this condition can be written as

SRIDHAR K. KAILASAM, Graduate Student, JEFFREY C. LACOMBE, Postdoctoral Research Associate, and MARTIN E. GLICKSMAN, Professor, are with the Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590. Manuscript submitted December 4, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A

C8

C1

C2

C8

e xdC 5 e xdC

[2]

where C2 and C+ represent the left- and the right-hand-side concentrations of a component at the ends of the diffusion couple. Once this plane is located, it is designated as x 5 0, and all distances on the penetration profiles are recalculated with respect to the Matano interface. It has been pointed out that the procedure of finding the location of the Matano interface can be tedious and inaccurate. Inasmuch as distances are measured with respect to the Matano interface, errors in its determination result in errors in all subsequent calculations. The other drawback of this method is that large errors arise in the calculation of diffusivity near the ends of the penetration curves. Toward the ends of the penetration curve, the integral in Eq. [1] becomes very small. Also, the concentration gradient, dC/ dx, vanishes, and its inverse, dx/dC, becomes unbounded. Because the numerical evaluation of the integral and the inverse of the slope toward the ends of the penetration profile are difficult to perform accurately, large uncontrolled errors are introduced in the determination of the diffusivity. Another commonly used assumption is that the Matano interface, determined for any one component in the diffusion couple, is identical for all the remaining interdiffusing components. Although this assumption may be reasonable for binary diffusion couples, it may not be valid for multicomponent systems, wherein there are three or more interdiffusing components, or in systems undergoing volume change during diffusion. However, we will restrict discussions here to the case of a binary alloy. den Broeder’s Approach In order to simplify the standard analysis, den Broeder introduced the concept of “relative concentration.”[3] This variable, c, is given by VOLUME 30A, OCTOBER 1999—2605

simulations were also undertaken where Gaussian noise was added to the ideal concentration profiles. Also, published interdiffusion profiles from experiments were studied. The two methods, B–M and dB, are implemented either numerically or analytically using various concentration profiles as input, and the results are compared. Finally, the advantages and disadvantages in using these methods are discussed, which can aid a user in selecting the best procedure for determining the concentration-dependent diffusivity. II. RESULTS

Fig. 1—Concentration profiles obtained using Eqs. [5] through [8]. Note that all the profiles are sigmoidal and have identical concentration-distance range. Inset: Sketch identifying the Matano interface from the penetration curve.

C(x) 2 C C+ 2 C2

2

c5

[3]

Concentration profiles can be analyzed in one of three different ways to obtain the concentration-dependent diffusivity. Numerical techniques can be applied to the concentration-distance data to implement the two methods. Alternatively, one could fit a function to the raw data and execute the two B–M and dB methods analytically. Finally, the data could be fit to spline functions and such functions could be used in the numerical solutions of the two methods. In this article, the results from implementation of the two methods via numerical and analytical techniques on the raw data are discussed. A. Comparison of the Two Methods via Numerical Solution The following four functions were selected to yield “S”shaped one-dimensional concentration profiles. The functions are C(x) 5 0.25 ? [tanh (x) 1 2]

After a series of steps, it was shown by den Broeder that Eq. [1] can be rewritten as D(C ) 5

1 2Z

1 dx 2t dC

C8

H

x*

(1 2 c*)

2`

1`

1 c*

e (C

+

e

J

[4]

2 C(x)) dx

x*

The use of a relative concentrationvariable in solving inverse diffusion problems was first introduced by Sauer and Freise.[4] Using this variable, Sauer and Freise derived an expression for D(C ) that was later termed the “modified B–M method.” However, the broader significance of that result was not emphasized by Sauer and Freise. Later, Wagner pointed out that the salient feature of the modified B–M method, as derived by Sauer and Freise, is that it avoids the need to determine the position of the Matano interface.[5] Though den Broeder acknowledges the precedence of these two earlier articles, he states that his derivation leading to Eq. [4] is considerably simpler than the earlier approaches. However, den Broeder never compared his method quantitatively with the B–M method, but merely implied that the two methods are equivalent. The purpose of this article is to provide a careful comparison between the original B–M method and the den Broeder (dB) method. For comparison, several analytical functions were chosen to generate sets of concentration-distance data. However, such profiles are considered “ideal,” as they are noise-free and lack the scatter that would always be present in real experimental data. To address realistic situations, 2606—VOLUME 30A, OCTOBER 1999

C(x) 5 0.75 2 0.5 ? exp [(20.3 ? x) ]

[6]

C(x) 5 0.5 1 [3.49 3 10

[7]

23

(C(x) 2 C2 ) dx

[5] 3

and

F

C(x) 5 0.75 2 0.25 ? erfc

tan

21

(x)]

14Dt2G x

[8]

Each of the preceding functions was used to generate concentration profiles over a range from 23 to 13 (arbitrary distance units) in steps of 0.01. Equations [5] through [8] were constructed so that in all cases, the ordinate, representing the concentration of one of the components in the binary alloy, ranged from 0.25 to 0.75 (atomic fraction). The penetration profiles obtained from implementing Eqs.[5] through [8] are shown in Figure 1. In using Eq.[6], values for x must be positive in order to obtain the desired range of the concentration. Equation [6] was computed for values of x ranging from 0 to 16, and the data were adjusted to obtain the desired profile. Because the assumed concentration data are antisymmetric about C 5 0.5, x 5 0 is the Matano interface. This is true for all the analytical functions selected. The first and second derivatives of the composition profiles were computed to ensure that Fick’s first and second laws were not violated; their continuity ensures a proper solution. The time for interdiffusion was chosen to be t 5 0.5 (arbitrary units) while implementing either method on all of the analytical cases. This was done so that the denominator in. Eq. [8] became unity when the diffusivity was also set to 0.5, thereby simplifying the data generation. Because the complementary error function is consistent only with a METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 4—Variation of diffusivity with concentration when C 5 f(arctan (x)).

Fig. 2—Variation of diffusivity with concentration when C 5 f (tanh (x)). The solid line represents the analytical solution, D(x), obtained from the dB’s method.

Fig. 5—Variation of diffusivity with concentration when C 5 f(erfc (x)).

Fig. 3—Variation of diffusivity with concentration when C 5 f(exp (x)).

constant diffusivity, the B–M and dB analysis methods should both yield a constant value of 0.5 for the diffusivity, thereby providing verification of the proper implementation of the two methods. Where the concentration-distance data were extracted from actual experiments, the appropriate interdiffusion time was noted from the published information. The concentration-dependent diffusivity was calculated for each of the analytical functions using the B–M and dB methods. The results are shown in Figures 2 through 5. Because the time and penetration distances have arbitrary units, the calculated diffusivities are also in arbitrary units. METALLURGICAL AND MATERIALS TRANSACTIONS A

In Figures 3 through 5, the displayed concentration range has been truncated to 0.3 to 0.7, instead of 0.25 to 0.75, because the diffusivity, as explained earlier, close to the ends of the diffusion couple cannot be evaluated accurately using either the B–M or dB method. In the case of the complementary error function, the diffusivity is not a function of the composition and is equal to 0.5 (Figure 5). This indicates that the two methods yield consistent results. More importantly, the graphs show that both methods yield nearly the same constant value of diffusivity in each case, proving that the dB method provides an alternative and convenient approach compared to the B–M method without compromising the result. An interesting result is that the variation of diffusivity with concentrationis notablydifferent for each concentration VOLUME 30A, OCTOBER 1999—2607

profile. Even though all the concentration profiles have an identical range for the abscissa and the ordinate, and differ slightly in their behavior, the analysis returns diffusivity variations that are remarkably different from one case to another. Unlike the cases analyzed here, the functional form of the concentration-distance data, obtained experimentally, is never known in advance. Experimentally obtained concentration-distance data could be regressed to some appropriate function. The choice of the fitting function is not straightforward, and this issue is discussed subsequently in Section IV. B. Analytical Solution of Concentration-Dependent Diffusivity In an effort to make additional comparisons, an analytical approach was adopted to solve for the concentration-dependent diffusivity. Equation [5] was selected arbitrarily to calculate the concentration-dependent diffusivity using dB’s formula Eq. [4]. The dB formula contains integrals of the type * Cdx, whereas the original B–M method contains integrals of the type * xdC. To evaluate integrals of the type * xdC, one would have to invert Eq. [5] to express position as a function of concentration. This, at best, could result in an awkward expression and, at worst, may not be possible. Hence, the B–M formulation was not solved analytically. Using Eq. [5] and expressing all the terms that appear in the dB formulation as a function of x, one obtains Eqs. [9] through [12];

c5

1 1 tanh (x) 2

[9a]

12c5

1 2 tanh (x) 2

[9b]

dC(x) 5 0.25 ? sech2 (x) dx

[10]

x

e (C(x) 2 C(2`)) dx

2`

[11]

1 5 {2x 1 log [cosh (x)] 1 log 2} 4 and 1`

e (C(`) 2 C(x)) dx 5 14{2x 1 log [cosh (x)] x

[12]

1 log 2} Substituting the expressions given by Eqs. [9] through [12] into the dB result, the analytical expression for the concentration-dependent diffusivity as a function of x becomes D(x) 5 cosh2 (x) ? [log(2 ? cosh (x)) 2 x ? tanh (x)]

[13]

Figure 2 shows the comparison between the results obtained from the two methods implemented numerically and the result from the analytical approach. The solid line represents the analytical solution for diffusivity. Eq. [13], cross-plotted against C(x), Eq. [5]. The solution for diffusivity from the analytical approach differs from the numerical solutions. 2608—VOLUME 30A, OCTOBER 1999

The differences arise from the fact that the expressions given by the dB method involve infinite integrals in x, the computation of which is feasible by analytical approaches only. This is of practical significance because real diffusion data are always limited to some finite interval in x. With experimental data, it is noted that the analytical solution is ascertainable only if the functional form describing the concentration profile is determined via regression. C. Concentration Data with Noise In all of the aforementioned cases, the concentration was uniquely defined by an analytical function for every position in the diffusion couple. In reality, the concentration data would be obtained by experimental methods with some noise or scatter occurring in the data. In this section, the application of the two methods to data with scatter in the concentration values is explored to determine if the conclusions about the usefulness of the two methods still hold under more realistic conditions. In order to simulate data with scatter in concentration, the following procedure was adopted. The composition-distance relationship defined by Eq. [7] was modified by adding Gaussian noise to the analytically derived concentration values. The noise level applied here is representative of the typical scatter observed in concentration measurements using an electron microprobe. The factors contributing to the scatter in concentration data during an electron microprobe analysis are the atomic number of the element, the concentration of the element, availability of standards, X-ray counting statistics, time of counting, instrumentation error, operator error, etc. With these factors, a reasonable estimate of the scatter at any given position (for a well-characterized element) is #1 pct. Two different levels of noise were implemented in this study, characterized by their standard deviations of 0.333 and 1 pct. Random numbers based on these standard deviations were generated and then added to the concentration profile given by Eq. [7]. Because the magnitude of the scatter is small, it is hard to visually discern the presence of scatter in the modified concentration-distance data. Using a computer program, the concentrationdependent diffusivity for 1000 such simulated data sets was evaluated using the two methods with the appropriate statistics (mean and standard deviation) recorded. The result for D(C ) for the case of 0.333 pct noise is plotted in Figure 6. The case of no noise was included to confirm the validity of the numerical application of the two methods. It was found that the results, though not plotted in Figure 6, are identical to the numerical solutions plotted in Figure 4. Figure 6 refers to the case of 0.333 pct noise added to the ideal concentration data. It is clear that the B–M and dB methods, when applied to data containing scatter, yield identical values for the concentration-dependent diffusivity. Though the mean diffusivities derived from the two methods are identical, and nearly the same as the values obtained from the case where no noise was added (dashed line in Figure 6), there now exists a statistical spread in the diffusivity measurements characterized by the standard deviation. The spread is more significant toward the ends of the diffusion couple. Not shown here is the case of 1 pct noise added to the scatter-free concentration data. Though the trend is similar to that shown in Figure 6, the magnitude of the spread is much larger, particularly near the ends of METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 6—Variation of diffusivity with concentration with 0.333 pct “noise” in the concentration data. The dashed line represents the B–M solution for concentration data that were noise-free.

the couple. This confirms the expectation that the uncertainty in the calculated diffusivity correlates with the scatter in the original concentration data. The following caveats apply regarding the numerical techniques. From the simulations of the discrete concentration profile data containing scatter, it was found that the concentration values do not necessarily increase monotonicallyover the range of distance. If the separation between adjacent data points during the simulations was too small, then the local up and down swings in the concentration data due to scatter suggested that the local gradient at a point on the penetration curve becomes negative. The physics of diffusion shows that the chosen concentration-distance functions cannot have negative gradients. By increasing the distance between data points, this problem can be mitigated. However, increasing the measurement spacing implies fewer concentration data for a given concentrationrange. If the number of data points is reduced too much, the numerical integration steps will incur larger errors. In this work, 50 data points were used to represent the penetration profile, which mimics the number of points sampled in a typical microprobe experiment. With this number of data points, the local slope at any point remained positive, and the elements remained small enough to be handled effectively by the numerical integration scheme. Thus, in situations where the scatter in the concentration data is 1 pct or more, a discrete numerical approach (point by point) to calculate D(C ) is not recommended. Instead, a regression scheme using a smooth mathematical function or a spline should be applied to the discrete data. Thereafter, the analytical function or spline functions could be used to evaluate the concentration-dependent diffusivity. D. Verification Using Experimental Data Concentration-distancedata from the published interdiffusion profiles in a Cu-Ni diffusion couple annealed at 1054 8C for 312 hours in N2 atmosphere were used.[6] After data extraction, the two methods to obtain the concentrationdependent diffusivity were applied to the data and the results METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 7—Variation of diffusivity with concentration from interdiffusion data in a Cu-Ni diffusion couple.

are shown in Figure 7. The B–M and dB methods provide nearly identical results, except near the ends of the couple. The results from the current analysis, though not identical, are in reasonable agreement with the published values. One likely reason for the source of error could be attributed to the process of data extraction using a digitized image of the published graphs and specialized data extraction software. Given that the comparison between the published data and the calculated values for the Cu-Ni system resulted in minor differences, further efforts were undertaken to repeat the comparison using the data from a different system. Data were extracted once again from digitized interdiffusion profiles from a Ta-W system annealed at 2100 8C for 16 hours in Ar atmosphere.[7] Specifically, the penetration profile for Ta diffusion was used. For our own B–M analysis, the location of the Matano interface was assumed to be the same as that specified in the journal article. One notable difference between the originally published work and this article is that whereas the earlier article predicted a constant diffusivity over a range of 20 to 80 at. pct Ta, the present work suggest nearly constant diffusivity only over the range of 20 to 60 at. pct Ta. Otherwise, good agreement in the results from the two methods is noted. Although slight variations occurred in the diffusivity data calculated here from those published, again, the differences could be attributed to the process of extracting the data from the published concentrationdistance profiles. III.

COMPARING THE B–M AND dB METHODS

The dB method offers a simpler procedure than the B–M method, as the user does not have to determine the Matano interface. Although den Broeder did not point out any additional advantages, the B–M method becomes inapplicable if extrema are present in the concentration profiles. Such behavior is often encountered in the interdiffusion of multicomponent alloys. In multicomponent diffusion, the flux of each component is influenced by the concentration gradient of all of the components in the alloy.[8] As a result of the VOLUME 30A, OCTOBER 1999—2609

interactions among the various interdiffusing components in the alloy, it is possible that unique “multicomponent”effects, such as uphill diffusion, extrema in the concentration profiles, zero flux planes, etc., occur in the interdiffusion zone of a couple.[9] The appearance of extrema in the concentration profiles makes the implementation of the B–M method via Eq. [1] infeasible, because the position, x, is no longer a single-valued function of concentration, C. Hence, integrals of the type * xdC cannot be evaluated uniquely. However, one can still calculate D(C ) using the dB method. The dB method uses integrals of the type * Cdx that can be calculated precisely. Another advantage of the dB method lies in its applicability to another feature of multicomponent diffusion. In the case of multicomponent diffusion, it cannot be assumed that the Matano interface is the same for all components. One would have to determine the Matano interface for each of the components prior to determining the diffusivity using the B–M method. Clearly, this is a time-consuming and laborious process compared to the evaluation of the diffusivity as suggested by the dB method. Despite these advantages, the dB method cannot be applied universally. For instance, interdiffusion experiments in ternary and other multicomponent alloys are designed such that the concentration of one of the components is identicalin the terminal alloys at the start of the interdiffusion experiment. Such couples, referred to as “Darken-like” couples, are employed to study the interactions between the interdiffusing components.[8] In such couples, the relative concentration variable for the component with identical concentration in the terminal alloys is indeterminate, and the dB method cannot be implemented. On the other hand, it could be argued that in such couples, the variation in concentration is small, and so the diffusivity would vary only slightly. IV.

CONCLUSIONS

It can be concluded that the methods for measuring concentration-dependent diffusivity developed by Sauer and Freise, Wagner, and den Broeder provide useful modifications to the B–M method. The expression for diffusivity as a function of concentrationobtained from each of the modified methods is identical. The salient and common feature of these modifications is the use of the relative concentration variable. It has been shown here that for most situations the modified methods are easier and quicker to implement. However, cases exist where the modified methods would not be applicable. Over the years, the formulation that is most commonly referred to is the one developed by den Broeder.[10,11] However, the concept upon which the dB method is based was formulated originally by Sauer and Freise. For the sake of propriety, fairness, and precise reference to the early works in this field, the modification involving the use of a relative concentration variable should be referred to as the Sauer, Freise, and den Broeder method. Upon comparison of the results from the application of the two methods on generated concentration profiles, it can be concluded that the two methods essentially yield identical results. Comparing the results from the numerical analysis and the analytical solution to dB’s method reveals that the

2610—VOLUME 30A, OCTOBER 1999

analytical approach, unlike the numerical approach to solving for D(C ), is accurate over the entire composition range as long as the functional form of the concentration profile can be established. Finally, applying the two methods to previously published concentration-distancedata from interdiffusion in the Cu-Ni and Ta-W systems provided results that are in good agreement with the published variation of diffusivity with concentration. Results from the application of the methods to concentration profiles simulated to contain noise/scatter in the data indicate that the uncertainty in the diffusivity values increases as the scatter in the concentration gets larger. Furthermore when a numerical approach is adopted, special attention needs to be paid to the calculation of the gradients and integrals. Otherwise, there exists a possibility of introducing large errors in diffusion analysis. When the scatter in concentration becomes large, a better methodology in evaluating the concentration-dependent diffusivity is to fit the raw data to an appropriate function and then to use this function for subsequent analytical analysis. Because the choice of the fitting function affects the magnitude of the diffusivity, the choice must be made judiciously. It would not be sensible to fit a complementary error function equation to a sigmoidal profile, where the concentration differences in the composition of the terminal alloys are large. The complementary error function is consistent only with a constant diffusivity, a condition that is generally invalid when the concentration difference across a diffusion couple exceeds 5 to 10 at. pct. Issues such as this warn the alert user that prior to choosing a fitting function, one needs to know more than just the appropriate numerical techniques. Other factors to consider critically are the magnitude of the concentration difference, the boundary conditions for the diffusion problem, thermodynamic data, etc. An alternative approach that is used successfully is to fit the raw data to appropriate spline functions and to determine the diffusivity via numerical techniques. ACKNOWLEDGMENTS The authors are grateful to Dr. Y. Liang (Assistant Professor, Department of Geology, Brown University) for providing valuable suggestions and Dr. D. Wark (Research Associate Professor, Department of Geology, RPI) for advice on microprobe-related information. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

L. Boltzmann: Wiedemanns Ann. Phys., 1894, vol. 53, p. 959. C. Matano: Jpn. J. Phys., 1933, vol. 8, p. 109. F.J.A. den Broeder: Scripta Metall., 1969, vol. 3 (5), pp. 321-25. F. Sauer and V. Freise: Z. Elektrochem., 1962, vol. 66, p. 353-63. C. Wagner: Acta Metall., 1969, vol. 17, pp. 99-107. L.C.C. da Silva and R.F. Rhines: Trans. Am. Inst. Min. Metall. Eng., 1951, vol. 191, pp. 155-73. A.D. Romig, Jr. and M.J. Cieslak: J. Appl. Phys., 1985, vol. 58 (9), pp. 3425-29. L.S. Darken: Trans. Am. Inst. Min. Metall. Eng., 1948, vol. 180, pp. 430-37. M.A. Dayananda and C.W. Kim: Metall. Trans. A, 1979, vol. 10A, pp. 1333-39. G.W. Roper and D.P. Whittle: Scripta Metall., 1974, vol. 8, pp. 1357-62. I. Thibon, D. Ansel, M. Boliveau, and J. Debuigne: Z. Metallkd., 1998, vol. 98 (3), pp. 187-91.

METALLURGICAL AND MATERIALS TRANSACTIONS A