Thermodynamic consistency and fast dynamics in phase-field crystal modeling

Abstract

A general formulation is presented to derive the equation of motion and demonstrate thermodynamic consistency for several classes of phase-field (PF) and PF crystal (PFC) models. It can be applied to models with a conserved and non-conserved phase-field variable, describing either locally uniform or periodic stable states, and containing slow as well as fast thermodynamic variables. The approach is based on an entropy functional formalism previously developed in the context of PF models for locally uniform states [P. Galenko and D. Jou, Phys. Rev. E 71 (2005) p.046125] and thus allows to extend several properties of the latter to PF models for periodic states, i.e., PFC models.Phase-field (PF) modeling has become a versatile tool for studying the dynamics of systems out of equilibrium and is used in numerous applications of materials science [1][2][3][4]. For instance, considering a material that is disordered at high temperature and has two stable phases at low temperature. Upon quenching the material from high to low temperature, grains with different stable phases will develop, grow, and compete with each other. PF modeling is able to describe the time evolution of such a process. In a PF model for this example, a continuous function of space and time (x, t) is introduced -the PF variable -that assumes a different constant value for both stable phases. Near an interface between two grains, the value of changes rapidly. The PF variable in this example can be interpreted as an order parameter to represent the relative mass fraction of both phases. One advantage of PF modeling is that the PF contains information on the location of all interfaces in the system, without the need for explicit interface tracking. Another advantage is that it focuses on general features that are common to the dynamics of classes of systems, and thus helps to identify generic features in newly investigated systems. System-specific details are incorporated by the interpretation given to the PF variable, and by the values given to the phenomenological parameters in the model equations. Systems that are similar according to appropriate criteria, are therefore described by the same PF model. Important classes of such models developed for slow and rapid phase

Gibbs–Thomson condition for the rapidly moving interface in a binary system

“…For fast transformations and transitive processes, equations of a hyperbolic type have been suggested to take deviations from local equilibrium conditions into account [13][14][15][16]. In this sense, equations of a hyperbolic type were naturally derived within a spectrum of tasks arising in systems evolving far from thermodynamic equilibrium [17][18][19][20][21]. The validity of hyperbolic type models has been verified in molecular dynamics simulations of solute trapping effect by the rapidly moving fronts [22] and by coarse graining derivations of phase field equations [23].…”

Correlated noise effect on the structure formation in the phase‐field crystal model

“…8,[15][16][17][18][19][20][21][22] The MPFC model can be considered as a singularly perturbed PFC model, which takes into account the large deviations from the thermodynamic equilibrium. 23,24 In addition to our recent results on the fast dynamics of the PFCs, 25,26 the present work is devoted to the further advancement in the description of the atomic formation using the MPFC model. Namely, we numerically investigate the behavior of the MPFC model formulated in a form of hyperbolic stochastic equation, where the noise source is defined as a correlated in space and time variable, which makes the noise as colored one.…”

Classical dynamical density functional theory: from fundamentals to applications

“…Note that all these approaches are based on the local validity of the laws of thermodynamics, which do not hold for systems far from equilibrium. Finally, Hütter and Brader [522] employed (in nonequilibrium thermodynamics) an entropy density s( r, t) that is a functional of mass and energy density and obtained a dynamic equation for s. Entropy production has also been discussed in the context of PFC models [587].…”