DDFT calibration and investigation of an anisotropic phase-field crystal model

Choudhary, Muhammad Ajmal; Li, Daming; Emmerich, Heike; Löwen, Hartmut

doi:10.1088/0953-8984/23/26/265005

Cited by 13 publications

(12 citation statements)

References 40 publications

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“…We consider four interaction kernels J 1 , J 2 , J 3 and J 4 , which are defined as: Remark 4.1. The interaction kernel J 4 has six-fold anisotropic shape, and the nucleation and growth simulation in this setting is very similar to the simulation of the anisotropic PFC equation presented in [21].…”

Section: Nucleation and Growth With Anisotropic Interaction Kernelsmentioning

confidence: 63%

Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations

Guan

Lowengrub

Wang

et al. 2014

Journal of Computational Physics

151

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Section: Nucleation and Growth With Anisotropic Interaction Kernelsmentioning

confidence: 63%

Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations

Guan

Lowengrub

Wang

et al. 2014

Journal of Computational Physics

151

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“…Here, a ij is a symmetric matrix and b ijkl is a tensor of rank 4 with the symmetry of an elastic tensor: i ↔ j, k ↔ l, (i, j) ↔ (k, l) [137]. Choudhary et al [138,139] proved that based on a functional of the form (59) further crystal lattices can be Taylor & Francis and I. T. Consultant assessed as hexagonal, bcc, and corresponding sheared structures, for which they have presented the elastic parameters and identified the stationary states.…”

Section: The Original Pfc Model and Its Generalisationsmentioning

confidence: 99%

Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview

Emmerich

Löwen

Wittkowski

et al. 2012

Advances in Physics

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346

417

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Here, we review the basic concepts and applications of the phase-field-crystal (PFC) method, which is one of the latest simulation methodologies in materials science for problems, where atomic-and microscales are tightly coupled. The PFC method operates on atomic length and diffusive time scales, and thus constitutes a computationally efficient alternative to molecular simulation methods. Its intense development in materials science started fairly recently following the work by Elder et al. [Phys. Rev. Lett. 88 (2002), p. 245701]. Since these initial studies, dynamical density functional theory and thermodynamic concepts have been linked to the PFC approach to serve as further theoretical fundaments for the latter. In this review, we summarize these methodological development steps as well as the most important applications of the PFC method with a special focus on the interaction of development steps taken in hard and soft matter physics, respectively. Doing so, we hope to present today's state of the art in PFC modelling as well as the potential, which might still arise from this method in physics and materials science in the nearby future.Keywords: phase-field-crystal (PFC) models, static and dynamical density functional theory (DFT and DDFT), condensed matter dynamics of liquid crystals and polymers, nucleation and pattern formation, simulations in materials science, colloidal crystal growth and growth anisotropy * Corresponding authors. Emails: heike.emmerich@uni-bayreuth. de, hlowen@thphy.uni-duesseldorf.de, and grana@szfki.hu

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“…Further, it extends the concept of thermodynamic consistency, previously discussed predominantly in the context of PF models [3][4][5][6], to PFC models. Thus, in addition to the concepts of periodicity inherent to the amplitude approach of SH [7] and the calibration via classical density functional theory [9], thermodynamic consistency, as it follows from the unified framework we present in this letter, poses a third theoretical foundation for PFC modeling. Introducing this concept to the field of PFC modeling, we hope to contribute a new formalism that can help to derive further generalized PFC models, which extend the applicability of PFC models in materials science.…”

Section: By Usingrmentioning

confidence: 99%

“…The main goal of this letter is to present a unified description of these classes and extend the concept of thermodynamic consistency, which has previously predominantly been discussed in the context of PF models for locally uniform states [3][4][5][6], to PF crystal (PFC) models for periodic states. Therefore, we introduce a third theoretical foundation for PFC models besides the previously discussed concepts of amplitude equations [7,8] and the calibration via classical density functional theory (for a most recent application, see, e.g., [9] and references therein) with the aim thus to facilitate further generalization and applicability of PFC models in materials science.…”

mentioning

confidence: 99%

Thermodynamic consistency and fast dynamics in phase-field crystal modeling

Cheng

Kundin

et al. 2012

Philosophical Magazine Letters

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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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DDFT calibration and investigation of an anisotropic phase-field crystal model

Cited by 13 publications

References 40 publications

Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations

Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations

Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview

Thermodynamic consistency and fast dynamics in phase-field crystal modeling

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