Coarse-Graining of Atomistic Description at Finite Temperature

Citation

Kulkarni, Yashashree (2007) Coarse-Graining of Atomistic Description at Finite Temperature. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/W9M0-HX47. https://resolver.caltech.edu/CaltechETD:etd-11102006-152125

Abstract

This thesis presents a computational method for seamlessly bridging the atomistic and the continuum realms at finite temperature. The theoretical formulation is based on the static theory of the quasicontinuum and extends it to model non-equilibrium finite temperature material response.

At non-zero temperature, the problem of coarse-graining is compounded by the presence of multiple time scales in addition to multiple spatial scales. We address this problem by first averaging over the thermal motion of atoms to obtain an effective temperature-dependent energy on the macroscopic scale. Two methods are proposed to this end. The first method is developed as a variational mean field approximation which yields local thermodynamic potentials such as the internal energy, the free energy, and the entropy as phase averages of appropriate phase functions. The chief advantage of this theory is that it accounts for the anharmonicity of the interaction potentials, albeit numerically, unlike many methods based on statistical mechanics which require the quasi-harmonic approximation for computational feasibility. Furthermore, the theory reduces to the classical canonical ensemble approach of Gibbs under the quasi-harmonic approximation for perfect, isotropic, infinite crystals subjected to uniform temperature. In the second method, based on perturbation analysis, the internal energy is derived as an effective Hamiltonian of the atomistic system by treating the thermal fluctuations as perturbations about an equilibrium configuration.

These energy functionals are then introduced into the quasicontinuum theory, which facilitates spatial coarse-graining of the atomistic description. Finally, a variational formulation for simulating rate problems, such as heat conduction, using the quasicontinuum method is developed. This is achieved by constructing a joint incremental energy functional whose Euler-Lagrange equations yield the equilibrium equations as well as the time-discretized heat equation.

We conclude by presenting the results for numerical validation tests for the thermal expansion coefficient and the specific heat for some materials and compare them with classical theory, molecular dynamics results, and experimental data. Some illustrative examples of thermo-mechanical coupled problems such as heat conduction in a deformable solid, adiabatic tension test, and finite temperature nanoindentation are also presented which show qualitative agreement with expected behavior and demonstrate the applicability of the method.

Thesis Files