Solutions of the Cahn-Hilliard Equations

Solutions of the Cahn–Hilliard Equations

Second Revised and Expanded Edition

Rahul Basu

The Cahn–Hilliard equation is one of the central mathematical models of modern materials science. It describes phase separation, spinodal decomposition, and the evolution of diffuse interfaces in binary alloys, polymers, batteries, thin films, and many other systems. This book provides a unified, practical treatment of the equation from first principles through to working computer codes.

Starting from the free-energy functional and the derivation of the classical fourth-order PDE, the text develops matrix formulations, linear stability analysis, and a range of numerical methods: finite differences, sparse iterative solvers (including conjugate gradient), semi-implicit and energy-stable time integration, Fourier spectral methods, and mixed finite elements. Complete, runnable implementations in MATLAB, Python (NumPy/SciPy), and the Wolfram Language are included, together with verification benchmarks (mass conservation, energy dissipation, and community spinodal-decomposition tests).

Worked examples cover one- and two-dimensional spinodal decomposition, coarsening kinetics, and an application to Fe–45Cr at 750 K. Later chapters survey advanced topics—thin-film dewetting, grain growth, solidification, lithium-ion electrodes, multiphase flow, and emerging directions such as physics-informed neural networks and high-performance computing.

Written for postgraduate students, researchers, and engineers in computational materials science, applied mathematics, and numerical analysis, the book emphasises methods that are both mathematically sound and immediately usable. All major algorithms are accompanied by sparse-matrix formulations, stability criteria, and source listings that can be adapted to new free energies, mobility laws, and spatial dimensions.