Project Overview
This project models the phase separation process in binary alloy systems using the Cahn–Hilliard equation, a nonlinear fourth-order partial differential equation that describes how concentration evolves due to diffusion and interfacial energy effects.
The simulation captures how initially mixed compositions evolve into distinct regions ("domains") over time due to thermodynamic instabilities, governed by spinodal decomposition dynamics. We implement a semi-implicit numerical scheme to study the coarsening behavior of concentration patterns.
🔬 Key Physics Concepts
Phase Separation: The spontaneous formation of distinct regions with different compositions in an initially mixed binary system.
Cahn–Hilliard Equation: A fourth-order PDE that models the temporal evolution of concentration in phase-separating systems.
Spinodal Decomposition: A mechanism of phase separation where small fluctuations grow exponentially due to thermodynamic instability.
Coarsening: The process where small domains merge to form larger ones, reducing interfacial energy over time.
🎯 Project Objectives
- Understand the Cahn–Hilliard equation and phase separation
- Implement semi-implicit numerical schemes for PDEs
- Study how initial fluctuations grow into patterns
- Observe and quantify coarsening behavior over time
- Analyze Fourier spectra and dominant wavelengths
🔧 Numerical Methods
- Spatial discretization using finite differences
- Periodic boundary conditions in 1D
- Semi-implicit backward Euler time integration
- Fourier analysis for pattern characterization
- Biharmonic solver for fourth-order equations
📊 Key Results
- Concentration field evolution from random perturbations
- Domain growth and coarsening dynamics
- Power spectrum analysis showing wavelength scaling
- Validation of theoretical coarsening laws
- Visualization of spinodal decomposition process