A New Structural Phase Field Crystal Approach for Modelling Graphene

This paper introduces a new structural phase field crystal (PFC) type model that expands the PFC methodology to a wider class of structurally complex crystal structures than previously possible. Specifically, our new approach allows for stabilization of graphene, as well as its coexistence with a disordered phase. It also preserves the ability to model the usual triangular and square lattices previously reported in 2D PFC studies. Our approach is guided by the formalism of the classical field theory, wherein the the free energy functional is expanded to third order in PFC density correlations. It differs from previous PFC approaches in two main features. First, it utilizes a hard-sphere repulsion to describe two-point correlations. Second, and more important, is that it uses a rotationally invariant three-point correlation function that provides a unified way to control the formation of crystalline structures that can be described by a specific bond angle, such as graphene, triangular or square symmetries. Our new approach retains much of the computational simplicity of previous PFC models and allows for efficient simulation of nucleation and growth of polycrystalline 2D materials. In preparation for future applications, this paper details the mathematical derivation of the model and its equilibrium properties, and uses dynamical simulations to demonstrate defect structures produced by the model.Comment: Updated figures, included comparison to experimental paper by Huang et a

New density functional approach for solid-liquid-vapor transitions in pure materials

A new phase field crystal (PFC) type theory is presented, which accounts for the full spectrum of solid-liquid-vapor phase transitions within the framework of a single density order parameter. Its equilibrium properties show the most quantitative features to date in PFC modelling of pure substances, and full consistency with thermodynamics in pressure-volume-temperature space is demonstrated. A method to control either the volume or the pressure of the system is also introduced. Non-equilibrium simulations show that 2 and 3-phase growth of solid, vapor and liquid can be achieved, while our formalism also allows for a full range of pressure-induced transformations. This model opens up a new window for the study of pressure driven interactions of condensed phases with vapor, an experimentally relevant paradigm previously missing from phase field crystal theories.Comment: 5 pages, 4 figures, peer reviewed versio

Quantitative phase field modeling of solute trapping and continuous growth kinetics in rapid solidification

Solute trapping is an important phenomenon in rapid solidification of alloys, for which the continuous growth model (CGM) is a popular sharp interface theory. Using matched asymptotic analysis, we show how to quantitatively map the sharp interface behavior of a binary alloy phase field model onto the CGM kinetics of Aziz et al. [1], with a controllable partition coefficient k(V ). We demonstrate the parameterizations that allow the phase field model to map onto the corresponding CGM or classical sharp interface models. We also demonstrate that the mapping is convergent for different interface widths. Finally we present the effect that solute trapping can have on cellular growth in a directional solidification simulation. The treatment presented for solute trapping can be easily implemented in different phase field models, and is expected to be an important feature in future studies of quantitative phase field modeling in rapid solidification regimes, such as those relevant to additive manufacturing

Thermo-Density Coupling in PFC Type Models for the Study of Rapid Crystallization

We self-consistently derive a formalism that couples a Phase Field Crystal (PFC) density field to thermal transport. It yields a theory for non-uniform transient temperature and density evolution, and includes local latent heat release during atomic rearrangements of the PFC density field. The basic formalism is applied to the basic PFC model, demonstrating the approach's capacity to capture solidification and recalescence. With an aim towards linking physical temperature to PFC temperature, a new classical density field theory for solid/liquid/vapor systems is then derived. It presents a different approach to those traditionally used in the PFC literature while still retaining the major advantages that have become the hallmark of PFC modelling. The new model is based entirely on physical density and temperature scales. We end the paper by applying the thermal-density coupling formalism to this new multi-phase density functional theory/PFC model.Comment: 14 pages, 10 figure

Beyond the Ginzburg-Landau theory of freezing: Anisotropy of the interfacial free energy in the Phase-Field Crystal model

This paper re-visits the weakly fourth order anisotropic Ginzburg-Landau (GL) theory of freezing. First we determine the anisotropy of the interfacial free energy in the Phase-Field Crystal (PFC) model analytically, and prove that it remains finite at the critical point as a direct consequence of the one-mode dominance of the model. Next, we derive the leading order PFC amplitude model and show the formal analogy to traditional weakly 4th order anisotropic GL theories. We conclude that the material-independent anisotropy appearing in emergent GL theory coincides with the remnant anisotropy of the generating PFC model. As a result, we show that the reduced temperature {\epsilon} does not enter into the interfacial free energy anisotropy for metallic materials in both the Phase-Field Crystal model and the emerging Ginzburg-Landau theories. Finally, we investigate the possible pathways of calibrating anisotropic Ginzburg-Landau theories.Comment: 13 pages, 2 figure

Classical nucleation theory in the phase-field crystal model

A full understanding of polycrystalline materials requires studying the process of nucleation, a thermally activated phase transition that typically occurs at atomistic scales. The numerical modeling of this process is problematic for traditional numerical techniques: commonly used phase-field methods' resolution does not extend to the atomic scales at which nucleation takes places, while atomistic methods such as molecular dynamics are incapable of scaling to the mesoscale regime where late-stage growth and structure formation takes place following earlier nucleation. Consequently, it is of interest to examine nucleation in the more recently proposed phase-field crystal (PFC) model, which attempts to bridge the atomic and mesoscale regimes in microstructure simulations. In this work, we numerically calculate homogeneous liquid-to-solid nucleation rates and incubation times in the simplest version of the PFC model, for various parameter choices. We show that the model naturally exhibits qualitative agreement with the predictions of classical nucleation theory (CNT) despite a lack of some explicit atomistic features presumed in CNT. We also examine the early appearance of lattice structure in nucleating grains, finding disagreement with some basic assumptions of CNT. We then argue that a quantitatively correct nucleation theory for the PFC model would require extending CNT to a multi-variable theory

Spacing Characterization in Al-Cu Alloys Directionally Solidified Under Transient Growth Conditions

We study spacing selection in directional solidification of Al-Cu alloys under transient growth conditions. New experimental results are presented which reveal that dendritic spacing versus solidification rate evolves in an almost step-wise fashion, consistent with previous theoretical predictions of Langer and co-workers. Phase field simulations of directional solidification with dynamical growth conditions approximating those in the experiments confirm this behavior. Changes in dendrite arm spacing is shown to be consistent with dendrite instabilities confined, initially, to sub-domains, rather than the entire system. This is due to the rapid variation in growth conditions, which prevent the system from adapting as a whole but, rather, in a succession of quasi-isolated domains.Comment: 10 pages, 17 figures, submitted to Acta Materiali

A Phase Field Crystal Study of Solute Trapping

In this study we have incorporated two time scales into the phase field crystal model of a binary alloy to explore different solute trapping properties as a function of crystal-melt interface velocity. With only diffusive dynamics, we demonstrate that the segregation coefficient, K as a function of velocity for a binary alloy is consistent with the model of Kaplan and Aziz where K approaches unity in the limit of infinite velocity. However, with the introduction of wave like dynamics in both the density and concentration fields, the trapping follows the kinetics proposed by S. Sobolev[Phys. Rev. A. 199:383386, 1995.], where complete trapping occurs at a finite velocity.Comment: 22 pages, 8 figure

Solidification in a channel

We simulate solidification in a narrow channel through the use of a phase-field model with an adaptive grid. In different regimes, we find that the solid can grow in fingerlike steady-state shapes, or become unstable, exhibiting unsteady growth. At low melt undercoolings, we find good agreement between our results, theoretical predictions, and experiment. For high undercoolings, we report evidence for a new stable steady-state finger shape which exists in experimentally accessible ranges for typical materials.Comment: 4 pages, 4 figure

Microscopic Treatment of Solute Trapping and Drag

The long wavelength limit of a recent microscopic phase field crystal (PFC) theory of a binary alloy mix- ture is used to derive an analytical approximation for the segregation coefficient as a function of the interface velocity, and relate it to the two-point correlation function of the liquid and the thermodynamic properties of solid and liquid phases. Our results offer the first analytic derivation of solute segregation and solute drag de- rived from a microscopic model, and analytically support recent molecular dynamics and fully numerical PFC simulations. Our analytical result also provides an independent framework, motivated from classical density functional theory, from which to elucidate the fundamental nature of solute drag, which is still highly contested in the literature