This lecture series introduces the phase field method using FiPy as a practical computational framework. The course is structured to guide readers from foundational numerical concepts to advanced phase field applications, including solidification and multi-physics generalizations.
The material is organized into six sections, each designed to span approximately one to two weeks of study. Every section consists of multiple lectures originally supported by complementary learning materials such as presentations and interactive examples. The overall progression is intentionally linear, allowing readers with no prior experience in Python or phase field modeling to gradually develop both theoretical understanding and practical implementation skills.
Introduction and Learning Expectations
The introductory section establishes the essential numerical foundations required for phase field simulations. Core principles such as finite volume discretization, numerical stability, and the distinction between implicit and explicit solution schemes are reviewed. In addition, basic concepts related to non-linear solvers are discussed to prepare readers for more advanced modeling tasks.
Readers are guided through a step-by-step Python workflow that introduces elementary programming constructs, including array creation, loop control, and terminal output. This foundational scripting experience ensures that participants can comfortably follow and modify FiPy-based simulation scripts later in the course.
The section also introduces the simplest FiPy features, including mesh generation, variable definition and initialization, boundary condition specification, equation construction, time stepping, and solver iteration. These topics are presented with the explicit goal of connecting theoretical numerical concepts to their computational realization.
Finite Volumes, Stability, and Non-Linear Systems
Subsequent lectures in the introductory phase explore finite volume methods in greater detail, emphasizing their advantages for conservation laws and multi-physics problems. Stability considerations for both finite volume and finite difference systems are examined, highlighting common numerical pitfalls and best practices.
The treatment of non-linear systems provides the mathematical context required for understanding phase field equations. Readers learn how non-linearity arises in material models and how iterative solvers are used to obtain stable numerical solutions.
A Detailed FiPy Simulation Example
This section presents a more in-depth exploration of the FiPy library and its available computational tools. While not intended as a complete reference, the lectures introduce different grid types in one, two, and three dimensions, along with equation terms, boundary condition formulations, viewers, and solver iterators.
Practical exercises are used to illustrate the differences between implicit and explicit numerical schemes. By comparing alternative implementations of diffusion problems, readers gain an intuitive understanding of accuracy, stability, and computational cost.
Spinodal Decomposition and the Phase Field Method
The phase field method is formally introduced through the example of spinodal decomposition. The theoretical background is presented first, followed by a numerical implementation that translates the governing equations into a working FiPy simulation.
Central to this discussion is the Cahn–Hilliard equation, which provides a mathematical framework for modeling conserved order parameters. A concise Python script demonstrates how these theoretical ideas are implemented computationally.
Ferroelectrics and Non-Conserved Order Parameters
Building on the treatment of conserved systems, the lecture series extends phase field modeling to non-conserved order parameters. Ferroelectric materials serve as an enabling application, allowing the introduction of order–disorder transformations in two-dimensional systems.
The Allen–Cahn equation is presented as the governing model for non-conserved dynamics, and its numerical behavior is explored through targeted computational experiments.
Solidification Phase Field Modeling
Once both conserved and non-conserved order parameters have been established, the lectures introduce the phase field formulation for liquid–solid phase transformations. The phase field variable is used to link thermodynamic equilibrium information between parent and product phases.
The thermal diffusion equation is revisited and modified to incorporate the phase field variable. Physical parameters such as interface thickness and surface tension are used to calibrate the energetic terms of the model, ensuring consistency with experimental observations.
The discussion then expands from single-component to binary systems using concepts from solution thermodynamics. Regular solution models are introduced as a starting point, with clear pathways for extending the framework to more complex systems.
Advanced Concepts and Generalization
Advanced lectures introduce orientational phase fields to capture crystallographic effects during grain growth and impingement. Surface tension anisotropy is incorporated to accurately model dendritic growth, while wetting phenomena at walls, edges, and corners are also addressed.
In the concluding lecture, phase field models are derived from thermodynamic principles. Applications include wave propagation, electromigration in metallic and semiconducting systems, and charge transport in ionic materials. The advantages and limitations of the phase field approach are critically examined.
Credits
The lecture series was developed by Edwin García, Assistant Professor of Materials Engineering at Purdue University, with a focus on computational materials science and phase field modeling.
Acknowledgment
This work was made possible through financial support from the Center for Theoretical and Computational Materials Science at the National Institute of Standards and Technology.