Perturbation Solution of the Cahn Hilliard Equations

EasyChair Preprint 14282

Abstract

This paper presents a perturbation solution for the one-dimensional Cahn-Hilliard (CH) equation with a small perturbation parameter, epsilon. The CH equation, which describes phase separation processes in binary mixtures, is given by:

\[ \frac{\partial \phi(x,t)}{\partial t} = \Delta \left( \phi(x,t) \left( \phi(x,t)^2 - 1 \right) \right) - \frac{\epsilon^2}{2} \Delta \phi(x,t), \quad x \in (0, L), \, t > 0, \]

where \(\phi(x,t)\) represents the order parameter. To tackle this nonlinear partial differential equation, we employ a small perturbation expansion in terms of \(\epsilon\), assuming a series solution of the form:

\[ \phi(x,t) = \phi_0(x,t) + \epsilon \phi_1(x,t) + \epsilon^2 \phi_2(x,t) + \cdots \]

We derive the governing equations for each order of \(\epsilon\) by substituting the series expansion into the original CH equation and equating terms of the same order. The leading-order equation is a nonlinear diffusion equation, while the first-order and second-order corrections result in linear equations with coefficients dependent on the lower-order solutions. The solutions are obtained iteratively, starting from the leading order, and incorporating boundary conditions to ensure physical consistency.

This perturbation approach provides an approximate analytical framework for understanding the dynamics of the CH equation under small perturbations, offering insights into the behavior of phase separation and pattern formation in binary mixtures. The method demonstrates the power of perturbative techniques in handling complex nonlinear systems, making it a valuable tool for theoretical and applied studies in materials science and related fields.

Keyphrases: MGA alloys, Recycling, non-linear, perturbation, phase transformtion

@booklet{EasyChair:14282,
  author    = {Rahul Basu},
  title     = {Perturbation Solution of the Cahn Hilliard Equations},
  howpublished = {EasyChair Preprint 14282},
  year      = {EasyChair, 2024}}