Boltzmann–Matano analysis

The Boltzmann-Matano method is used to convert the partial differential equation resulting from Fick's law of diffusion into a more easily solved ordinary differential equation, which can then be applied to calculate the diffusion coefficient as a function of concentration.

Ludwig Boltzmann worked on Fick's second law to convert it into an ordinary differential equation, whereas Chujiro Matano performed experiments with diffusion couples and calculated the diffusion coefficients as a function of concentration in metal alloys. Specifically, Matano proved that the diffusion rate of A atoms into a B atom crystal lattice is a function of the amount of A atoms already in the B lattice.

The importance of the classic Boltzmann-Matano method consists in the ability to extract diffusivities from concentration-distance data. These methods, also known as inverse methods, have both proven to be reliable, convenient and accurate with the assistance of modern computational techniques.

Boltzmann’s Transformation

Boltzmann’s Transformation converts Fick's second law into an easily solvable ordinary differential equation. Assuming a diffusion coefficient D that is in general a function of concentration c, Fick's second law is:

${\frac {\partial c}{\partial t}}={\frac {\partial }{\partial x}}\overbrace {\left[D(c){\frac {\partial c}{\partial x}}\right]} ^{\text{Flux}}$

where t is time and x is distance.

Boltzmann's transformation consists in introducing a variable ξ, defined as a combination of t and x:

$\xi ={\frac {x}{2{\sqrt {t}}}}$

The partial derivatives of ξ are:

${\frac {\partial \xi }{\partial t}}=-{\frac {x}{4t^{3/2}}}=-{\frac {\xi }{2t}}$

${\frac {\partial \xi }{\partial x}}={\frac {1}{2{\sqrt {t}}}}$

To introduce ξ into Fick's law, we express its partial derivatives in terms of ξ, using the chain rule:

${\frac {\partial c}{\partial t}}={\frac {\partial c}{\partial \xi }}{\frac {\partial \xi }{\partial t}}=-{\frac {\xi }{2t}}{\frac {\partial c}{\partial \xi }}$

${\frac {\partial c}{\partial x}}={\frac {\partial c}{\partial \xi }}{\frac {\partial \xi }{\partial x}}={\frac {1}{2{\sqrt {t}}}}{\frac {\partial c}{\partial \xi }}$

Inserting these expressions into Fick's law produces the following modified form:

$-{\frac {\xi }{2t}}{\frac {\partial c}{\partial \xi }}={\frac {1}{2{\sqrt {t}}}}{\frac {\partial }{\partial x}}\left[D(c){\frac {\partial c}{\partial \xi }}\right]$

Note how the time variable in the right-hand side could be taken outside of the partial derivative, since the latter regards only variable x.

It is now possible to remove the last reference to x by using again the same chain rule used above to obtain ∂ξ/∂x:

$-{\frac {\xi }{2t}}{\frac {\partial c}{\partial \xi }}={\frac {1}{4t}}{\frac {\partial }{\partial \xi }}\left[D(c){\frac {\partial c}{\partial \xi }}\right]$

Because of the appropriate choice in the definition of ξ, the time variable t can now also be eliminated, leaving ξ as the only variable in the equation, which is now an ordinary differential equation:

$-2\xi {\frac {\mathrm {d} c}{\mathrm {d} \xi }}={\frac {\mathrm {d} }{\mathrm {d} \xi }}\left[D(c){\frac {\mathrm {d} c}{\mathrm {d} \xi }}\right]$

This form is significantly easier to solve numerically, and one only needs to perform a back-substitution of t or x into the definition of ξ to find the value of the other variable.

The Parabolic Law

Observing the previous equation, a trivial solution is found for the case dc/dξ=0, that is when concentration is constant over ξ. This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time ($x\propto {\sqrt {t}}$), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the distance ($t\propto x^{2}$); the square term gives the name parabolic law.

Matano’s Geometry

The experimental configuration and geometric requirements needed to apply the desired transformed equation in the previous section was suggested by C. Matano and hence the name, Boltzmann-Matano analysis. Matano described the linear flow geometry in which to apply the Boltzmann-transformed diffusion equation, for the express purpose of extracting diffusivity data from an observed concentration field. The following figure defines the Matano’s diffusion geometry. Given the initial conditions, we can begin the process to determine an expression for the diffusivity as a function of known concentration data.

$C=C_{L},\left(x<0,t=0\right)$ $C=C_{R},\left(x>0,t=0\right)$

The first integral with the given boundary conditions is shown by the following.

$-2\int \limits _{C_{R}}^{C'}\xi dC=\int \limits _{C_{R}}^{C'}d\left(D(C)\left({\frac {dC}{d\xi }}\right)\right)$

Carrying out the integration indicated on the right-hand side of the previous equation, we get the following.

$-2\int \limits _{C_{R}}^{C'}\xi dC=\left[D(C)\left({\frac {dC}{d\xi }}\right)\right]$

Implementing the boundary condition where the solute distribution in Matano’s geometry provides a gradient, dC/dξ, which vanishes C→CR, the equation further simplifies.

$-2\int \limits _{C_{R}}^{C'}\xi dC=D(C'){\frac {dC}{d\xi }}$

Rearranging the equation in terms of D(C’), the equation can be written by the following.

$D(C')=-2\left({\frac {d\xi }{dC}}\right)\int \limits _{C_{R}}^{C'}\xi dC$

Finally, transforming back into ordinary (x,t) space-time coordinates by substituting ξ, we define the Boltzmann-Matano solution.

$D(C')=-{\frac {1}{2t}}\left({\frac {dx}{dC}}\right)\int \limits _{C_{R}}^{C'}(x-X_{M})dC$

Matano’s Interface

After determining the Boltzmann-Matano solution, one unknown still remains unsolved, the Matano interface. The Matano interface can be defined as a plane within the diffusion couple, across which equal amounts of mass have diffused to the left and to the right. The Matano interface may be determined through a mass balance condition, where the solute loss on the left side is equal to the solute gained on the right side.

$\int \limits _{-\infty }^{X_{M}}\left[C_{L}-C(x)\right]dx=\int \limits _{X_{M}}^{\infty }\left[C(x)-C_{R}\right]dx$

Integrating by parts, the equation transforms to equivalent integrals with C as the running variable instead of x.

$[C_{L}-C(x)]x-\int \limits _{C_{L}}^{C_{M}}xdC=[C(x)-C_{R}]-\int \limits _{C_{M}}^{C_{R}}xdC$

Now evaluating the integrals by applying the Matano boundary conditions (defined in the previous section), the equation simplifies.

$(C_{L}-C_{M})X_{M}-\int \limits _{C_{L}}^{C_{M}}xdC+(C_{M}-C_{R})X_{M}-\int \limits _{C_{M}}^{C_{R}}xdC=0$

And simplifying the equation even more, we get the following.

$(C_{L}-C_{R})X_{M}-\int \limits _{C_{L}}^{C_{M}}xdC-\int \limits _{C_{M}}^{C_{R}}xdC=0$

We can further simplify the equation by setting the origin at X_M (X_M=0).

$\int \limits _{C_{L}}^{C_{M}}xdC+\int \limits _{C_{M}}^{C_{R}}xdC=0$

Since diffusion time is fixed, the following substitution can be made.

$\int \limits _{C_{R}}^{C_{L}}xdC=\int \limits _{C_{M}}^{C_{L}}xdC=0$

where

$\xi ={\frac {x-X_{M}}{2{\sqrt {t}}}}$

Do note that the location of the Matano interface can also be defined from the concentration-distance data by balancing the area denoted A5 against the summation of the areas A1+A2+A3.

Application of Boltzmann-Matano Method

Take for example two pairs of different alloys (A-B alloys) that are joined to form two classical diffusion couples. On the left, the alloy has a mole fraction of 0.25 of component B. And on the right, the alloy has a mole fraction of 0.75 of component B. We are given the following concentration-distance data.

We are assuming that one pair of alloys, exhibits a constant diffusivity, and upon diffusion yields a concentration field described by the usual linear Grube-Jedele diffusion couple solution.

$C(x,t)=0.75-0.25erfc\left({\frac {x}{4Dt}}\right)$

And then for the second pair of alloys, we assume it has a concentration-dependent diffusivity. For this example, we choose t=0.5.

$C\left(x,0.5\right)=0.25\left[tanh(x)+2\right]$

These equations are both ideal solutions for each of the pairs of alloys. Identical numerical procedures were applied to the two concentration fields, to implement the Boltzmann-Matano procedure for this example. Steps were taken for this particular example to determine the Boltzmann-Matano solution as well as the mass balance for the Matano’s interface. The following plot is the result for the calculation for the diffusivity for each set of alloys.

This example shows that when implementing the Boltzmann-Matano method, one can inversely solve for the diffusivity from given concentration-distance data. However, large errors occur at the end point data and could propose problems in any complex concentration profiles. Note as a reminder, this approach is for the classic diffusion couple. And when implemented in complex data, the errors seen in this ideal example may be significantly large and unacceptable in actual experimental data.

Sources

• M. E. Glicksman, Diffusion in Solids: Field Theory, Solid-State Principles, and Applications, Wiley, New York, 2000.
• Matano, Chujiro. "On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System)". Japanese Journal of Physics. Jan. 16, 1933.