We propose a computationally-efficient approach to multiscale simulation of polycrystalline materials, based on the phase field crystal (PFC) 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.PACS numbers: 81.16. Rf, 05.10.Cc, 61.72.Cc, 81.15.Aa Why is it so hard to predict the properties of real materials? Unlike simple crystalline solids, real materials, produced by a wide range of processing conditions, contain defects and multiple grains that strongly impact mechanical, thermal, electrical response, and give rise to such important phenomena as plasticity, hysteresis, work hardening and glassy relaxation. Moreover, it is frequently the case that a faithful description of materials processing requires simultaneous treatment of dynamics at scales ranging from the nanoscale up to the macroscopic. For example, dendritic growth, the generic mode of solidification of most metals and alloys, involves the capillary length at the nanoscale, the emergent pattern dimensions on the scale of microns, the thermal or particle diffusion length on the scale of 10 −4 m, in addition to the grain and sample size.Despite these obstacles, progress in rational material design requires a fundamental understanding of the way in which useful properties emerge as the mesoscale is approached. Questions that must be addressed include: What is the collective behavior of assemblies of nanoscale objects? How best to achieve target mesoscale properties from nanoscale constituents? And how can the properties at nano-, meso-and intermediate scales simultaneously be captured quantitatively and predictively?A number of computational approaches to handle the range of length scales have been proposed recently [1, 2], including quasi-continuum methods [3,4,5,6], the heterogeneous multiscale method [7,8], multi-scale molecular dynamics [9,10,11,12], multigrid variants [13] and extensions of the phase field model [14]. These techniques strive to provide a unified description of the many scales being resolved, but in some cases require non-systematic ways to link the disparate scales to enable treatment of sufficiently large mesoscale systems. This can introduce spurious modes and excitations, and difficulties associated with the transition between scales [2,8]. Most of this work is limited to crystalline materials with a few isolated defects [15].In this Letter, we propose a novel theoretical approach to these difficulties, by combining the phase field crystal (PFC) formalism [16,17] with renormalization group (RG) [18,19] and related methods (see, e.g.[20]), developed for the analysis of hydrodynamic instabilities in spatially-extended dynamical systems [21,22,23,24,25,26,27,28]. We present effective equations at the mesoscale, from which the atomic density can readily be reconstructed...

We derive a set of rotationally covariant amplitude equations for use in multiscale simulation of the two dimensional phase field crystal (PFC) 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 scales 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.

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.

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We review the use of phase field methods in solidification modeling, describing their fundamental connection to the physics of phase transformations. The inherent challenges associated with simulating phase field models across multiple length and time scales are discussed, as well as how these challenges have been addressed in recent years. Specifically, we discuss new asymptotic analysis methods that enable phase field equations to emulate the sharp interface limit even in the case of quite diffuse phase-field interfaces, an aspect that greatly reduces computation times. We then review recent dynamic adaptive mesh refinement algorithms that have enabled a dramatic increase in the scale of microstructures that can be simulated using phase-field models, at significantly reduced simulation times. Combined with new methods of asymptotic analysis, the adaptive mesh approach provides a truly multi-scale capability for simulating solidification microstructures from nanometers up to centimeters. Finally, we present recent results on 2D and 3D dendritic growth and dendritic spacing selection, which have been made using phase-field models solved with adaptive mesh refinement.