Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model.

We implement an adaptive mesh algorithm for calculating the space and time dependence of the atomic density field in microscopic material processes. Our numerical approach uses the systematic renormalization-group formulation of a phase-field crystal model of a pure material to provide the underlying equations for the complex amplitude of the atomic density field--a quantity that is spatially uniform except near topological defects, grain boundaries, and other lattice imperfections. Our algorithm employs a hybrid formulation of the amplitude equations, combining Cartesian and polar decompositions of the complex amplitude. We show that this approach leads to an acceleration by three orders of magnitude in model calculations of polycrystalline grain growth in two dimensions.

Renormalization-group theory for the phase-field crystal equation.

We derive a set of rotationally covariant amplitude equations for use in multiscale simulation of the two-dimensional phase-field crystal model by a variety of renormalization-group (RG) methods. We show that the presence of a conservation law introduces an ambiguity in operator ordering in the RG procedure, which we show how to resolve. We compare our analysis with standard multiple-scale techniques, where identical results can be obtained with greater labor, by going to sixth order in perturbation theory, and by assuming the correct scaling of space and time.

Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model.

We propose a computationally efficient approach to multiscale simulation of polycrystalline materials, based on the phase field crystal model. The order parameter describing the density profile at the nanoscale is reconstructed from its slowly varying amplitude and phase, which satisfy rotationally covariant equations derivable from the renormalization group. We validate the approach using the example of two-dimensional grain nucleation and growth.

Numerical modeling of long bone adaptation due to mechanical loading: correlation with experiments.

The process of external bone adaptation in cortical bone is modeled mathematically using finite element (FE) stress analysis coupled with an evolution model, in which adaptation response is triggered by mechanical stimulus represented by strain energy density. The model is applied to experiments in which a rat ulna is subjected to cyclic loading, and the results demonstrate the ability of the model to predict the bone adaptation response. The FE mesh is generated from micro-computed tomography (microCT) images of the rat ulna, and the stress analysis is carried out using boundary and loading conditions on the rat ulna obtained from the experiments [Robling, A. G., F. M. Hinant, D. B. Burr, and C. H. Turner. J. Bone Miner. Res. 17:1545-1554, 2002]. The external adaptation process is implemented in the model by moving the surface nodes of the FE mesh based on an evolution law characterized by two parameters: one that captures the rate of the adaptation process (referred to as gain); and the other characterizing the threshold value of the mechanical stimulus required for adaptation (referred to as threshold-sensitivity). A parametric study is carried out to evaluate the effect of these two parameters on the adaptation response. We show, following comparison of results from the simulations to the experimental observations of Robling et al. (J. Bone Miner. Res. 17:1545-1554, 2002), that splitting the loading cycles into different number of bouts affects the threshold-sensitivity but not the rate of adaptation. We also show that the threshold-sensitivity parameter can quantify the mechanosensitivity of the osteocytes.

A quantitative analysis on latent heat of an aqueous binary mixture.

The latent heat during phase change of water-NaCl binary mixture was measured using a differential scanning calorimeter, and the magnitude for two distinct phase change events, water/ice and eutectic phase change, were analyzed considering the phase change characteristics of a binary mixture. During the analysis, the latent heat associated with each event was calculated by normalizing the amount of each endothermic peak with only the amount of sample participating in each event estimated from the lever rule for the phase diagram. The resulting latent heat of each phase change measured is 303.7 +/- 2.5 J/g for water/ice phase change, and 233.0 +/- 1.6 J/g for eutectic phase change, respectively regardless of the initial concentration of mixture. Although the latent heats of water/ice phase change in water-NaCl mixtures are closely correlated, further study is warranted to investigate the reason for smaller latent heat of water/ice phase change than that in pure water (335 J/g). The analysis using the lever rule was extended to estimate the latent heat of dihydrate as 115 J/g with the measured eutectic and water/ice latent heat values. This new analysis based on the lever rule will be useful to estimate the latent heat of water-NaCl mixtures at various concentrations, and may become a framework for more general analysis of latent heat of various biological solutions.