Abstract

Understanding the intricate relationship between processing, microstructure, and properties in materials science is paramount. Experimental approaches offer glimpses into these domains, yet limitations exist in visualizing real-time microstructure evolution. To bridge this gap, the utilization of materials modeling and simulation emerges as a powerful tool.

This readme introduces the fundamental principles of phase-field modeling, a widely adopted simulation technique in materials science. The practical aims to provide an entry-level understanding of the phase-field method by delving into its foundational equations, numerical solutions employing the finite volume method, and practical applications.

Phase-Field Model
The core of the phase-field method lies in its representation of microstructures using order parameters. These parameters serve as variables that delineate boundaries between phases or structures within a material system. The intricacies of defining these order parameters are explored, elucidating their conservative or non-conservative nature based on the evolution of the system over time.

Thermodynamics and Energy Formulation
Central to the phase-field approach is its thermodynamic underpinning. Total energy formulations encapsulate bulk free energy and gradient energy, pivotal in characterizing system behavior. The bulk free energy, modeled through Landau energy, elucidates the relative energetically-favorable states, while gradient energy compensates for diffusive interfaces. Additional energy terms, depending on specific material systems, further refine the model's accuracy.

Finite Volume Method
The readme progresses to elucidate the finite volume method, a robust approach for solving partial differential equations. This method discretizes domains into control volumes, enabling the interpretation and solution of differential equations in integral form. Emphasis is placed on understanding conservation equations, their physical interpretations, discretization, and boundary conditions to derive solutions effectively.

By comprehensively detailing the theoretical foundations and numerical methodologies of the phase-field model, this readme aims to equip materials scientists with an introductory toolkit to simulate and comprehend microstructure evolution—a critical facet in advancing materials science research and development.