Phase-field simulation of void evolution

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Phase-field simulation of void evolution


Author: Xiao, Zhi-hua
Title: Phase-field simulation of void evolution
Degree: Ph.D.
Year: 2013
Subject: Metals -- Fatigue.
Materials science -- Simulation methods.
Hong Kong Polytechnic University -- Dissertations
Department: Dept. of Mechanical Engineering
Pages: xxiii, 217 leaves : ill. (some col.) ; 30 cm.
Language: English
InnoPac Record:
Abstract: Void in metal is an empty space encased by a sharp metallic surface. The void formation and growth process in irradiated metal is a complicated process involving multiple spatial (from 10-1 to 102 nanometers) and temporal (from nanoseconds to several days, months or even years) scales, as well as interactions with other types of defects, such as point defects (vacancies and self-interstitials), line defects (dislocations), plane defects (grain boundary) and volume defects (precipitates). Since metal with the supersaturated point defects is in a meta-stable state, local random fluctuations of vacancy concentration may result into the nucleation of void embryos. After the nucleation, the void embryo will grow or shrink depending on whether the net vacancies or interstitials flow in. Action of the void surface tension causes a vacancy emission from the void. Due to the dependence of the vacancy emission rate on the void surface curvature, void embryos with sizes larger than some critical one will continuously grow from the supersaturated solution of vacancies, while the smaller embryos will be re-dissolved. Since it is difficult to simulate the complex sharp interface structure using a numerical method for the cases of void ensemble, which involves a complex topological change of surface of multiple voids in a system, the phase-field method, using the concept of a diffuse interface, is a good alternative choice for the simulation of void evolution. The phase-field method is a powerful numerical simulation tool for studying microstructure evolution during phase transformation. In order to quantitatively simulate void evolution in metals using the phase-field method, the free energy functional of the system is first developed. In this functional the vacancy concentration is the only order parameter, which evolution is governed by the Cahn-Hilliard equation. The vacancy concentration is unity in the void, close to zero in the matrix, and between one and zero within the diffuse interface region. Thus, in the phase-field approach voids are treated as a kind of precipitates of vacancies. In this thesis, a single void dynamics after the nucleation, when only vacancies are present in the metal matrix, is quantitatively studied under various conditions by using a phase-field method. The results obtained with sharp boundary approach of void evolution of classical thermodynamics are used as the benchmark for the results obtained with the phase-field method. Since the realistic void-metal interface is very sharp, in order to effectively model the void evolution through using a diffuse interface to mimic the sharp interface, the phase-field model should be built properly with the physical mechanisms of void evolution maintained. In order to be consistent with the classical thermodynamics, the phase-field model should be able to reproduce the classical thermodynamics of void evolution in conditions under the sharp interface limit. The void-metal diffuse interface is customarily modeled by a Ginzburg-type gradient energy term with a coefficient which is parameterized from surface tension. The interfacial energy in the diffuse interface approach consists of two parts: the gradient energy due to the variation of vacancy concentration across the interface, and the local free energy due to the vacancies in non-equilibrium state within the diffuse interface. The competition between these two parts determines the thickness of the void-metal interface. The larger the local free energy due to the non-equilibrium vacancies, the narrower the interface will be; and the larger the gradient energy, the wider the interface will be. Within the interface region, the local free energy due to the non-equilibrium vacancies is equal to the gradient energy for flat interface case in equilibrium state because the chemical potential is constant zero across the interface in equilibrium state. For the curved interface case, the relationship between these two kinds of energy is more complicated because the chemical potential is spatially non-zero constant across the interface in equilibrium state. The chemical potential in the curved interface case is inversely proportional to the void radius in equilibrium state.
In the present work, following the results obtained by A. A. Semenov and C. H. Woo, the gradient energy coefficient is treated as a constant independent of void size. Realistic concentrations of single vacancies, which correspond to the real experimental conditions, are used in the simulations. The real, rather than the reduced, time is used as well. This allows us to make a direct comparison between the results obtained by the phase-field model and those derived from the sharp boundary approach. The simulations are performed by using the material parameters of molybdenum and copper in three-dimensional space. The vacancy concentration varies across many orders of magnitude across the interface region. In order to maintain the stability of numerical scheme, tiny time steps and spatial grid sizes are used. For the high supersaturation of vacancy concentration, the developed phase-field model reproduces very well the results of the sharp boundary approach on the behavior of single void evolution within the classical thermodynamics framework. Around the critical point for void evolution, due to the sensitivity of void growth behavior to the parameters of system conditions, the results obtained with phase-field method deviate slightly from those obtained with sharp boundary approach. The ultrafine spatial scales of the void-metal diffuse interface and the fourth-order parabolic non-linear partial differential equation of the Cahn-Hilliard equation, both of which require using a very tiny time step and spatial grid size, present a challenge to numerically efficient modeling of the evolution of a void ensemble under irradiation conditions because this tiny time step and spatial grid size result in enormous calculations for numerical simulations in three-dimensional Cartesian coordinates of a system of large domain.

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