An efficient algorithm for solving the phase field crystal model

Cheng, Mowei; Warren, James A.

doi:10.1016/j.jcp.2008.03.012

Cited by 141 publications

(108 citation statements)

References 24 publications

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“…The quench depth takes the value of γ = −0.025. Similar experiments were reported in, e.g., [5], [17].…”

Section: A Homogeneous Crystallization In a Supercooled Liquidsupporting

confidence: 90%

“…It is therefore of great importance to study scalable parallel algorithms for the PFC equation. Although numerical methods for the PFC equation have been investigated in a number of publications, e.g., [5], [14], [17], [26], [28], works dedicated to parallel algorithms are not yet to be seen. There are some successful studies on scalable parallel algorithms for some other phase-field problems such as the Cahn-Hilliard equation [29], [31] and the coupled Allen-Cahn/Cahn-Hilliard equations [24], [27], [30].…”

Section: Introductionmentioning

confidence: 99%

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A Scalable Implicit Solver for Phase Field Crystal Simulations

Yang

Cai

2013

2013 IEEE International Symposium on Parallel &Amp; Distributed Processing, Workshops and PHD Forum

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Abstract-The phase field crystal equation has become a popular model for simulating micro-structures in materials science but is very computationally expensive to solve. A highly scalable solver for phase field crystal modeling is presented in this paper. The equation is discretized with a stabilized implicit finite difference method and the time step size is adaptively controlled to obtain physically meaningful solutions. The nonlinear system arising at each time step is solved by using a parallel Newton-Krylov-Schwarz algorithm. In order to achieve good performance, low-order homogeneous boundary conditions are imposed on subdomain interfaces in the Schwarz preconditioner. Experiments are carried out to exploit optimal choices of the preconditioner type, the subdomain solver and the overlap size. Numerical results are provided to show that the solver is scalable to thousands of processor cores.

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“…The quench depth takes the value of γ = −0.025. Similar experiments were reported in, e.g., [5], [17].…”

Section: A Homogeneous Crystallization In a Supercooled Liquidsupporting

confidence: 90%

Section: Introductionmentioning

confidence: 99%

A Scalable Implicit Solver for Phase Field Crystal Simulations

Yang

Cai

2013

2013 IEEE International Symposium on Parallel &Amp; Distributed Processing, Workshops and PHD Forum

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“…To verify our analysis we solved the PFC dynamic equation (2) with N = 3 via a pseudo-spectral algorithm [31,32], using periodic boundary conditions in systems of sizes ranging from 256 2 to 1024 2 . We restricted our parameter space to ψ 0 = −0.2, r = −0.15, λ = 0.02, and τ = 0 for simplicity.…”

Section: Arxiv:13066357v1 [Cond-matmtrl-sci] 26 Jun 2013mentioning

confidence: 99%

Exploring the Complex World of Two-Dimensional Ordering with Three Modes

2013

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The world of two-dimensional crystals is of great significance for the design and study of structural and functional materials with novel properties. Here we examine the mechanisms governing the formation and dynamics of these crystalline or polycrystalline states and their elastic and plastic properties by constructing a generic multi-mode phase field crystal model. Our results demonstrate that a system with three competing length scales can order into all five Bravais lattices, and other more complex structures including honeycomb, kagome and other hybrid phases. In addition, nonequilibrium phase transitions are examined to illustrate the complex phase behavior described by the model. This model provides a systematic path to predict the influence of lattice symmetry on both structure and dynamics of crystalline and defected systems. . On larger length scales much progress has been made in the self-assembly of 2D crystals using particles of nano or micron size that are easier to tailor for specific functionalities and to observe. Colloidal crystals, for example, play a vital role in the study of structural properties of crystalline systems and the development of engineered, functional materials [7][8][9]. In addition, another novel technique for artificial lattice ordering is built on the trapping of ultracold atoms (e.g., 87 Rb) in optical superlattices produced by overlaying laser beams [10,11], as utilized for the study of many-body quantum physics. [7,11]. Thus it is of fundamental importance to identify the universal mechanisms underlying these distinct modes of crystallization, based on the general principle of symmetry [12]. It is also important to understand the nature of topological defects which occur frequently in such systems and are known to determine the electronic and mechanical properties of the sample [2]. Unfortunately it is very difficult to model and predict the nature of such defected states, due to multiple length and time scales involved in the non-equilibrium crystallization processes.In this work we develop a dynamic model that can be applied to the study of crystallization with a variety of ordered and defected structures. We adopt the phase field crystal (PFC) formalism [13][14][15][16], in the spirit of the Alexander-McTague analysis of crystallization based on Landau theory [12]. The advantage of this PFC approach is that one can study polycrystal formation in terms of the atomic number density on diffusive time scales that are many orders of magnitude larger than that of classical microscopic models such as molecular dynamics. One can also apply renormalization techniques [17][18][19] on the PFC equation to study problems that involve both micro and meso scales such as epitaxial growth [20] and surface patterning in ultra-thin films [21].Recently a great deal of progress has been made on generalizing the PFC formulation to include more crystal symmetries [22][23][24][25], although in 2D current PFC studies are restricted to triangular and square states. The basic idea is to incorp...

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“…A wide range of phenomena has been studied via this PFC method for both pure and binary material systems, including solidification [25,32,33], grain nucleation and growth [26,30,34,35], phase segregation [25,32], quantum dot growth during epitaxy [3,32,36,37], surface energy anisotropy [38,39], formation and melting of dislocations and grain boundaries [40][41][42][43], commensurate/incommensurate transitions [44,45], sliding friction [46,47], and glass formation [48]. In addition, recent work has been conducted to extend the modeling to incorporate faster time scales associated with mechanical relaxation [49,50], and to develop new efficient computational methods [28,[51][52][53].…”

Section: Introductionmentioning

confidence: 99%

Phase-field-crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures

2010

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The dynamics of phase field crystal (PFC) modeling is derived from dynamical density functional theory (DDFT), for both single-component and binary systems. The derivation is based on a truncation up to the three-point direct correlation functions in DDFT, and the lowest order approximation using scale analysis. The complete amplitude equation formalism for binary PFC is developed to describe the coupled dynamics of slowly varying complex amplitudes of structural profile, zerothmode average atomic density, and system concentration field. Effects of noise (corresponding to stochastic amplitude equations) and species-dependent atomic mobilities are also incorporated in this formalism. Results of a sample application to the study of surface segregation and interface intermixing in alloy heterostructures and strained layer growth are presented, showing the effects of different atomic sizes and mobilities of alloy components. A phenomenon of composition overshooting at the interface is found, which can be connected to the surface segregation and enrichment of one of the atomic components observed in recent experiments of alloying heterostructures.

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An efficient algorithm for solving the phase field crystal model

Cited by 141 publications

References 24 publications

A Scalable Implicit Solver for Phase Field Crystal Simulations

A Scalable Implicit Solver for Phase Field Crystal Simulations

Exploring the Complex World of Two-Dimensional Ordering with Three Modes

Phase-field-crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures

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