About one unified description of the first‐ and second‐order phase transitions in the phase‐field crystal model

Abstract

Modeling of crystal micro-structures and their dynamics during fast phase transitions can be performed by the phase-field crystal (PFC) model in the hyperbolic formulation (Modified Phase Field Crystal [MPFC] model). This method is suitable for a continual modeling of the atomic density field at diffusion time intervals (slow diffusion dynamics) and short intervals of atomic flux relaxation (fast structural relaxation). Since the PFC model describes transitions of the first and second order, we present a description of both transitions in a unified manner. We show how phase transitions of each order can be treated using specific analytical transformations. To justify the unified approach to description of the first-and second-order transformations, we provide results of numerical simulation of phase changes between homogeneous structure and Body Centered Cubic (BCC) crystal lattice. The set of benchmarks for different domains shows coincidence of instantaneous atomic distributions and free energies in both forms.

Structure diagram and dynamics of formation of hexagonal boron nitride in phase-field crystal model

“…Now, after substitution of equations (2.3) to (2.2) and expanding equation (2.2) around the reference density nfalse(bold-italicrfalse)=0 one can derive the local free energy as [23,33] Fid≈kBTρ0(12n2−a6n3+v12n4). Here the linear and higher-order expansion terms are neglected. This free energy contribution as discussed earlier in works [8,33,34] could be reduced to the Swift–Hohenberg form of free energy.…”

“…Here the linear and higher-order expansion terms are neglected. This free energy contribution as discussed earlier in works [8,33,34] could be reduced to the Swift-Hohenberg form of free energy. Expansion of the excess part of the free energy equation (2.1) made around the density field ρ(r) = ρ(r) − ρ 0 ≈ 0 could be derived as [4,35,36]…”

“…The dynamical PFC equation is based on the partial differential equation of parabolic type. The classical PFC model can carry out simulations on diffusive time scales predicting a structure evolution during the first- and second-order phase transitions [8] at longer timescales in comparison to accessible times provided by the molecular dynamics simulations [9,10]. For a wide class of processes and materials evolving under high driving forces of transformations [11], the PFC model is advanced to describe either slow diffusion dynamics or fast relaxation of atomic structure by the partial differential equation of hyperbolic type, which results in the hyperbolic or modified form of the PFC model (MPFC model) [10,12].…”

Approximation of correlation functions in phase‐field crystal model by machine learning approach

“…This free energy contribution as discussed earlier in works 23,29,30 could be reduced to the Swift-Hohenberg free energy form Equation (1).…”

Structural phase-field crystal model for Lennard–Jones pair interaction potential

“…Moreover such free energy introduces the first-and second-order phase transitions [28,39]. The pair correlation function could be approximated in a reciprocal space (space of k-vectors) in a manner proposed by Ramakrishnan and Yusoff (RY) [20,21] as:…”