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

Abstract

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...

Phase-field-crystal models and mechanical equilibrium

“…This procedure is described in detail in Refs. [21][22][23][24][25][26]. The reason it is interesting to consider the complex amplitudes as opposed to n itself is that the magnitude and phase of η j describe different physical features and more importantly naturally separate out the elastic part, as will be explained in the next section.…”

“…Amplitude formulation bridges the gap between the conventional PFC model and more macroscopic phase field models and was introduced by Goldenfeld and collaborators [21][22][23] for the two-dimensional triangular phase of the PFC model and has been extended to three-dimensional bcc and fcc crystals and binary alloys [24,25] and to include the miscibility gap in the density field [26]. This approach considers variations of the amplitudes of a periodic density field.…”

Phase-field-crystal study of grain boundary premelting and shearing in bcc iron

“…(56) to convert from µ to T . We obtain these probability distributions by conducting simulations with noise for various undercoolings and plotting the resulting histograms, as seen in Fig.…”

“…More generally, decreasing this parameter makes the freezing transition more weakly first order and increases the width of the solid-liquid interface in units of lattice spacing. To gain additional insights into the role of this parameter, we also compute the disjoining potential using amplitude equations [56][57][58][59][60][61][62][63] . These equations can be formally derived from the PFC model in the small ǫ limit using similar multiscale expansions introduced previously to analyze continuum models of pattern formation [64][65][66][67][68][69][70][71] .…”

Exploring the Complex World of Two-Dimensional Ordering with Three Modes

“…The advantage of this PFC approach is that one can study polycrystal formation in terms of the atomic number density on diffusive time scales that are many orders of magnitude larger than that of classical microscopic models such as molecular dynamics. One can also apply renormalization techniques [17][18][19] on the PFC equation to study problems that involve both micro and meso scales such as epitaxial growth [20] and surface patterning in ultra-thin films [21].Recently a great deal of progress has been made on generalizing the PFC formulation to include more crystal symmetries [22][23][24][25], although in 2D current PFC studies are restricted to triangular and square states. The basic idea is to incorporate interparticle interactions through a two-point direct correlation function that (i) has N peaks in Fourier space (corresponding to N different characteristic length scales) and (ii) is isotropic.…”