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The ability to use external magnetic fields to influence the microstructure in polycrystalline materials has potential applications in microstructural engineering. To explore this potential and to understand the complex interactions between…

Materials Science · Physics 2019-04-03 R. Backofen , K. R. Elder , A. Voigt

The phase field crystal (PFC) model captures the elastic and topological properties of crystals with a single scalar field at small undercooling. At large undercooling, new foam-like behavior emerges. We characterize this foam phase of the…

Pattern Formation and Solitons · Physics 2015-05-18 Nicholas Guttenberg , Nigel Goldenfeld , Jonathan Dantzig

In a recent class of phase field crystal (PFC) models, the density order parameter is coupled to powers of its mean field. This effectively introduces a phenomenology of higher-order direct correlation functions acting on long wavelengths,…

Materials Science · Physics 2024-08-21 Daniel L. Coelho , Duncan Burns , Emily Wilson , Nikolas Provatas

Phase-field methods offer a versatile computational framework for simulating large-scale morphological evolution. However, the applicability and predictability of phase-field models are inherently limited by their ad hoc nature, and there…

Materials Science · Physics 2025-10-30 Jaehyeok Jin , David R. Reichman

Material properties controlled by evolving defect structures, such as mechanical response, often involve processes spanning many length and time scales which cannot be modeled using a single approach. We present a variety of new results…

Materials Science · Physics 2015-06-19 Joel Berry , Nikolas Provatas , Jörg Rottler , Chad W. Sinclair

We review how phase-field models contributed to the understanding of various aspects of crystal nucleation including homogeneous and heterogeneous processes, and their role in microstructure evolution. We recall results obtained both by the…

A nonlocal phase-field crystal (NPFC) model is presented as a nonlocal counterpart of the local phase-field crystal (LPFC) model and a special case of the structural PFC (XPFC) derived from classical field theory for crystal growth and…

Numerical Analysis · Mathematics 2026-05-20 Qiang Du , Kai Wang , Jiang Yang

We present a phase-field crystal (PFC) model for solidification that accounts for thermal transport and a temperature-dependent lattice parameter. Elasticity effects are characterized through the continuous elastic field computed from the…

Materials Science · Physics 2023-01-26 Maik Punke , Steven M. Wise , Axel Voigt , Marco Salvalaglio

We investigate bulk thermodynamic and microscopic structural properties of amorphous solids in the framework of the phase-field crystal (PFC) model. These are metastable states with a non-uniform density distribution having no long-range…

Soft Condensed Matter · Physics 2024-09-11 Shaho Abdalla , Andrew J. Archer , László Gránásy , Gyula I. Tóth

Rapid solidification experiments on thin film aluminum samples reveal the presence of lattice orientation gradients within crystallizing grains. To study this phenomenon, a single-component phase-field crystal (PFC) model that captures the…

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…

Computational Physics · Physics 2009-11-13 Mowei Cheng , James A. Warren

The study of polycrystalline materials requires theoretical and computational techniques enabling multiscale investigations. The amplitude expansion of the phase field crystal model (APFC) allows for describing crystal lattice properties on…

Computational Physics · Physics 2019-04-25 Simon Praetorius , Marco Salvalaglio , Axel Voigt

We introduce a Phase Field Crystal (PFC) model for particles with n-fold rotational symmetry in two dimensions. Our approach is based on a free energy functional that depends on the reduced one-particle density, the strength of the…

Soft Condensed Matter · Physics 2023-02-02 Robert F. B. Weigel , Michael Schmiedeberg

The phase field crystal (PFC) approach extends the notion of phase field models by describing the topology of the microscopic structure of a crystalline material. One of the consequences is that local variation of the interatomic distance…

Materials Science · Physics 2016-05-18 V. Heinonen , C. V. Achim , T. Ala-Nissila

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…

Materials Science · Physics 2018-05-09 Paul Jreidini , Gabriel Kocher , Nikolas Provatas

Phase field crystal (PFC) theory, extensively used for modelling the structure of solids, can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Standard derivations neglect a term of form…

Soft Condensed Matter · Physics 2019-09-04 Andrew J. Archer , Daniel J. Ratliff , Alastair M. Rucklidge , Priya Subramanian

We investigate the equilibrium properties of bcc-liquid interfaces modeled with a continuum phase-field crystal (PFC) approach [K. R. Elder and M. Grant, Phys. Rev. E 70, 051605 (2004)]. A multiscale analysis of the PFC model is carried out…

Materials Science · Physics 2009-11-13 Kuo-An Wu , Alain Karma

The amplitude expansion for a magnetic phase-field-crystal (magnetic APFC) model enables a convenient coarse-grained description of crystalline structures under the influence of magnetic fields. Considering higher-order magnetic coupling…

Materials Science · Physics 2022-11-01 Rainer Backofen , Marco Salvalaglio , Axel Voigt

We review our recent modeling of crystal nucleation and polycrystalline growth using a phase field theory. First, we consider the applicability of phase field theory for describing crystal nucleation in a model hard sphere fluid. It is…

Soft Condensed Matter · Physics 2007-05-23 L. Granasy , T. Pusztai , T. Borzsonyi

The phase-field crystal equation, a parabolic, sixth-order and nonlinear partial differential equation, has generated considerable interest as a possible solution to problems arising in molecular dynamics. This is because the phase-field…

Mathematical Physics · Physics 2015-07-29 Philippe Vignal , Lisandro Dalcin , Donald L. Brown , Nathan Collier , Victor M. Calo