Angle-adjustable density field formulation for the modeling of crystalline microstructure

Abstract

A continuum density-field formulation with particle-scale resolution is constructed to simultaneously incorporate the orientation dependence of interparticle interactions and the rotational invariance of the system, a fundamental but challenging issue in modeling structure and dynamics of a broad range of material systems across variable scales. This generalized phase field crystal type approach is based upon the complete expansion of particle direct correlation functions and the concept of isotropic tensors. Through applications to the modeling of various two-and three-dimensional crystalline structures, our study demonstrates the capability of bond angle control in this continuum field theory and its effects on the emergence of ordered phases, and provides a systematic way of tunable angle analysis for crystalline microstructures.

Control of phase ordering and elastic properties in phase field crystals through three-point direct correlation

“…As detailed in Ref. [35], a more general PFC-type density-field formulation can be developed to incorporate any orders of multi-point direct correlations that are rotationally invariant. Here we give a brief description to show how it leads to the free energy functional used in this study integrating the effect of three-point correlation.…”

“…components of a rank-K tensor T (K) . Both C (m) and Ĉ(m) are rotationally invariant, and thus T (K) should be a 2D or 3D isotropic Cartesian tensor which can be written as linear combinations of products of q i • q j and (q k × q l ) • q p , leading to [35] Ĉ(m) (q 1 , q 2 , ...,…”

“…The phase field crystal (PFC) method, an effective density-field approach for modeling atomistic details of material systems on diffusive time scales [1][2][3][4][5], has been widely applied to the study of various structural and dynamical phenomena in a broad range of areas, such as solidification [3,[6][7][8], elastic and plastic deformation of materials [9][10][11][12][13][14], crystal growth [4,[15][16][17], dislocation dynamics [18][19][20][21][22][23], grain boundary structures and dynamics [24][25][26][27][28], ferromagnetics and ferroelectrics [29], quasicrystals [30,31], heterostructures and stacked multilayers of two-dimensional (2D) materials [32,33], among many others. The PFC models can be connected to or derived from classical density functional theory (cDFT) through the expansion of direct correlation functions [3,6,34,35]. The original PFC model contains only two-point direct correlation where crystal structures and ordered patterns are controlled by a single microscopic lattice length scale.…”

“…More importantly, the incorporation of multi-point correlations can enable the study of broader material systems and enrich the properties of the PFC modeling. Limited attempts have been made to explore the influence of multipoint correlations in PFC [35,[39][40][41][42]. For example, Seymour and Provatas [39] approximated the three-point direct correlation function as the product of two-point direct correlations to model structures with a specific bond angle, such as triangular, honeycomb, or square, and Kocher and Provatas [40] used both three-and four-point direct correlations to model vapor-liquid-solid transitions.…”

“…Alternatively, Alster et al [41] expanded the three-point correlation function in terms of Legen-dre polynomials in Fourier space and constructed various crystalline phases (including the ABX 3 perovskite structure) from their PFC model. Recently, we developed a general PFC formulation to incorporate any multipoint direct correlations satisfying the condition of rotational invariance [35], from which effects of bond-angle dependency and adjustment can be achieved through the four-point correlation, and a variety of 2D and 3D crystal structures, such as 3D diamond cubic phase and 2D rhombic or 3D simple monoclinic structure with tunable bond angles, can be stabilized. Based on this approach, an efficient PFC modeling for 2D vapor-liquid-solid coexistence and transitions was subsequently developed by considering the three-point direct correlation [42].…”