We present and discuss the development of an unconditionally stable algorithm used to solve the evolution equations of the phase field crystal (PFC) model. This algorithm allows for an arbitrarily large algorithmic time step. As the basis for our analysis of the accuracy of this algorithm, we determine an effective time step in Fourier space. We then compare our calculations with a set of representative numerical results, and demonstrate that this algorithm is an effective approach for the study of the PFC models, yielding a time step effectively 180 times larger than the Euler algorithm for a representative set of material parameters. As the PFC model is just a simple example of a wide class of density functional theories, we expect this method will have wide applicability to modeling systems of considerable interest to the materials modeling communities.
We present maximally-fast numerical algorithms for conserved coarsening systems that are stable and accurate with a growing natural time-step ∆t = At 2/3 s . For non-conserved systems, only effectively finite timesteps are accessible for similar unconditionally stable algorithms. We compare the scaling structure obtained from our maximally-fast conserved systems directly against the standard fixed-timestep Euler algorithm, and find that the error scales as √ A -so arbitrary accuracy can be achieved.PACS numbers: 02.60.Cb, 64.75.+g Phase-ordering kinetics studies the evolution of structure after a quench from a disordered phase into an ordered phase. The later stages of most phase-ordering processes in simple systems show universal scaling behavior described by a single growing length scale which increases as a power law in time, L(t) ∼ t α , where 0 < α ≤ 1 [1]. For the scalar order-parameter systems considered in this paper, α = 1/2 and 1/3 for non-conserved and conserved dynamics, respectively [1]. While these growth exponents and their universality can be understood in terms of interfacial motion leading to domain coarsening [2], the time-independent scaled structure that result is less well understood.Computer simulation is an effective technique to systematically study these non-linear non-equilibrium coarsening systems. To maintain accuracy, the discretized dynamics must move interfaces at most a small fraction of the interfacial width, ξ, in a single timestep ∆t. This determines a maximal or natural timestep of coarsening systems, ∆t nat ∼ ξ/(dL/dt) ∼ t 1−α , that grows in time. Unfortunately common time-discretizations are unstable for timesteps above a fixed threshold determined by the lattice spacing ∆x [3]. Any such fixed timestep algorithm is increasingly inefficient at late times compared to the natural timestep. Various algorithms have been proposed to make simulations more efficient, including the cell-dynamical-scheme [4] and Fourier spectral methods [5]. However, these approaches still require a fixed time-step for numerical stability.There is a newly developed class of unconditionally stable semi-implicit algorithms [6,7] that impose no stability constraints on the timestep ∆t, which is then determined by accuracy considerations. Since we generally expect larger ∆t to lead to larger errors, there is a tradeoff between speed and accuracy. This tradeoff is best resolved by picking growth rates for ∆t that induce an error in the correlations that is approximately constant in magnitude throughout the scaling regime, where the magnitude can be chosen to be less than other systematic sources of error such as initial transients or finite-size effects. While errors of single growing timesteps can be small [7], this begs the question of how much those singlestep errors accumulate in correlations at late times after the quench. For example, we might expect that some types of single-step errors would be benign, given the irrelevance of small amounts of random thermal noise to the scaled structure [1]....
Given an unconditionally stable algorithm for solving the Cahn-Hilliard equation, we present a general calculation for an analytic time step Deltatau in terms of an algorithmic time step Deltat. By studying the accumulative multistep error in Fourier space and controlling the error with arbitrary accuracy, we determine an improved driving scheme Deltat=At(2/3) and confirm the numerical results observed in a previous study [Cheng and Rutenberg, Phys. Rev. E 72, 055701(R) (2005)].
We contribute to the more detailed understanding of the phase-field crystal model recently developed by Elder et al (2002 Phys. Rev. Lett. 88 245701), by focusing on its noise term and examining its impact on the nucleation rate in a homogeneously solidifying system as well as on successively developing grain size distributions. In this context we show that principally the grain size decreases with increasing noise amplitude, resulting in both a smaller average grain size and a decreased maximum grain size. Despite this general tendency, which we interpret based on Panfilis and Filiponi (2000 J. Appl. Phys. 88 562), we can identify two different regimes in which nucleation and successive initial growth are governed by quite different mechanisms.
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
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