3D pattern formation from coupled Cahn-Hilliard and Swift-Hohenberg equations: Morphological phases transitions of polymers, bock and diblock copolymers

Abstract

This work proposes a new mathematical model of a ternary mixture composed of coupled Cahn-Hilliard and Swift-Hohenberg equations solved in three dimensions (3D) through a pseudo-spectral implicit method (a numerical method of the fast Fourier transform). The 3D numerical solutions dynamic is analyzed through its structure factor and growth law to study its behavior and some observed phase transitions. These numerical solutions evolve into glassy and crystalline phases in the form of 3D patterns. The new model has a free energy functional that considers small monomers of polymer (or copolymeric) chains taken into the Edwards free energy, as well as solvent-monomers and solvent-stretch free energies whose dynamic is given by the Cahn-Hilliard and Allen-Cahn equations. Isotropic and anisotropic phases from polymers, block, and diblock copolymers blend such that these phases have a morphological diversity of classical and complex structures. As an application of the dynamics of the new ternary model, it has a diversity of morphologies which allows obtaining porous polymeric materials manufactured by a novel 3D printing technique: the mathematical design process and 3D printing assisted manufacturing (MDP-3DPAM). This method allows the proposal of a new technique for the design and creation of polymeric materials with controlled pore size distribution starting with computational modeling.