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