Unveiling the Free-Energy Landscape

Abstract

Phase separation plays a crucial role in the development of microstructures in multiphase materials, influencing their functional properties. This thesis addresses the inverse problem of inferring the free-energy function directly from microstructures developed by diffusion-induced phase separation. The Cahn–Hilliard equation is employed to generate a dataset of two-dimensional phase-field simulations, with the free-energy function parameterized using 5 B-spline control points. A diverse set of microstructural descriptors is implemented, including statistical descriptor (two-point correlation function), topological descriptors (Betti numbers), geometric metrics (edge length, curvature, pore size), and black-box features extracted from a pre-trained ResNet50V2. Neural network models are trained to map these descriptors to the underlying free-energy parameters. Quantitative evaluation shows that the combined descriptor model achieved accurate predictions, while SHAP analysis confirms the physical relevance of the learned feature–to-parameter relationships. Qualitative comparisons of predicted free-energy functions and re-constructed microstructures demonstrates that the re-constructed morphologies closely reproduce the characteristic patterns of the target morphologies. This study demonstrates the feasibility of descriptor-based machine learning frameworks for free-energy inference, providing a foundation for future applications in data-driven materials design and the analysis of experimental microscopic images.