Unconventional Phase Field Simulations of Transforming Materials with Evolving Microstructures

Abstract

Abstract Transforming materials are a class of multi-functional materials that couples multiple physical couplings, such as, mechanical and thermal in shape memory alloy (SMA), mechanical and electrical in ferroelectrics, mechanical and magnetic in ferromagnetic shape memory (FSMA), mechanical and chemical with electrical as in ion battery. These materials exhibit their characteristic properties because of their underlying microstructures. In this thesis, we would like to address to the fundamental question: Why do materials form microstructures, and how do microstructures evolve? The thesis starts with an introduction to phase field simulation for modeling and simulating microstructures in material systems with multiple phases or variants, as a literature review. In order to simulate Austenite-Marteniste interfaces in shape memory alloys that appears in different physical length scales and temperature range, a two-level phase field approach is developed, which enables us to further simulate special two dimensional structures such as tunnels and tents, as well as possible interfaces in shape memory alloys. The new approach also allows us to study thermal hysteresis and to numerically verify the critical dependence of the hysteresis length on the crystal symmetry of the participating Austenite and Martensitic phases. Rigorous analysis in one dimension is also carried out to study domain formation and switching, which serves as foundation for the methodology of phase field simulation and the choices of parameters. Conventional phase field simulation assumes strong periodicity in all physical dimensions, which makes it inconvenient to study the simulations related to experiments such as functional materials being probed in various atomic force microcopy (AFM). To extend the possibilities for simulations, we release the periodicity, first along the out-of-plane direction and then along all directions. Numerical schemes for solving Maxwell equations (for electrical and magnetic orderings), elasticity equations (for mechanical ordering) and phase field equations (as a diffusion-type PDE) are formulated in these geometric configuration, utilizing eigen-expansion, finite difference and Chebyshev spectral method. These approaches are then used to simulate electromechanical responses in piezoelectric and ferroelectric materials, as well as domain formations in multi-functional materials, for instance, shape memory alloys and ferroelectrics. As a remark, Chebyshev spectral method for general geometry without any periodicity also works for inhomogeneous materials and arbitrary boundary conditions, which becomes an ideal candidate for investigating problems in the most general contexts and configurations. Modeling, numerical solution and quantitative analysis in lithium ion electrode are also considered. A simplified model (as differential equation) for the dynamics of lithium ions due to externally applied voltage is set up and solved. Method of line (MOL) is adopted to solve the solution numerically to maintain conservative system, resulting in simulation of ion dynamics and corresponding deformation as deflection as detected by Electrochemical Strain Microscopy (ESM). Harmonic analysis is carried out semi-analytically to the system subjected to sinusoidal voltage (AC). Ion dynamics and deformation due to combined DC-AC bias are simulated and further compared with existing experimental data to estimate diffusivity and local concentration of ions in the electrode.