Question

Discuss the analogy between Cahn-Hillard time evolution equation and Fick's second law of diffusion. What is he limitation or applicability of Fick diffusion equation to phase dissolution(homogenization) versus phase segregation in a colloidal system

Answer 1

The Cahn-Hillard equation describes microstructural evolution in in cases where the microstructure is completely described by the composition field variable:

--------(1) where the composition c(x,t) is a function of position x and the time t, is the chemical potential and M is the mobility.

The classical equation that describes microstructural evolution through diffusion is the Fick’s second law, which has the following form:

--------(2)

where D is the diffusion coefficient.

This strong resemblance between Fick’s second law (Eq. 2) and Cahn-Hillard equation (Eq. 1) is not accidental; it can indeed be considered as a generalization of Fick’s second law. in reality, CH is the result of a study of spinodal decomposition — a phenomenon which also brought out the limitations of Fick’s second law.

For phase dissolution, in materials undergoing glass transition, the flow becomes non-Fickian. Also, it is applicable when the mixture is extremely dilute and each species is interacting with the bulk only.

For colloidal phase segregation, Stoke-Einstein theory is applicable and it is non-Fickian in nature