Nonlinear relaxation patterns in the Cahn-Hilliard equation: an exact solution

Area of Science:

• Materials Science
• Mathematical Physics
• Chemical Engineering

Background:

• The Cahn-Hilliard equation models phase separation in materials.
• Understanding the dynamics of evolving structures is crucial for materials design.

Purpose of the Study:

• To derive exact solutions for the 1-D Cahn-Hilliard equation.
• To analyze the behavior of these solutions and their implications for pattern formation.

Main Methods:

• Introduction of a modified order parameter g, where g''=v'.
• Separation of variables technique applied to the new equation.
• Expression of solutions using generalized hypergeometric functions.

Main Results:

• Exact solutions for the order parameter v were obtained.
• Solutions exhibit infinite gradients at zeros and zero derivatives at extrema.
• Pattern amplitude decays as the inverse square root of time.

Conclusions:

• The derived solutions provide a theoretical framework for understanding pattern evolution.
• Relaxation patterns observed in these solutions may explain compartmentalization in Cahn-Hilliard type models.