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Original Articles

Nucleation and growth by a phase field crystal (PFC) model

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Pages 813-820 | Received 08 Mar 2007, Accepted 31 May 2007, Published online: 09 Jan 2009

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Michael te Vrugt, Hartmut Löwen & Raphael Wittkowski. (2020) Classical dynamical density functional theory: from fundamentals to applications. Advances in Physics 69:2, pages 121-247.
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Heike Emmerich, Hartmut Löwen, Raphael Wittkowski, Thomas Gruhn, Gyula I. Tóth, György Tegze & László Gránásy. (2012) Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview. Advances in Physics 61:6, pages 665-743.
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Amanda E. Diegel & Natasha S. Sharma. (2022) A C 0 interior penalty method for the phase field crystal equation . Numerical Methods for Partial Differential Equations 39:3, pages 2510-2537.
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Rainer Backofen & Axel Voigt. (2023) Magnetically Enhanced Thin‐Film Coarsening by a Magnetic XPFC Model Allowing to Decouple Magnetic Anisotropy and Magnetostriction. Advanced Engineering Materials.
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Marcello De Donno, Lucas Benoit–Maréchal & Marco Salvalaglio. (2023) Amplitude expansion of the phase-field crystal model for complex crystal structures. Physical Review Materials 7:3.
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José Causil, Carlos Reales & Iván Velásquez. (2022) A $$C^1$$–$$C^0$$ virtual element discretization for a sixth-order elliptic equation. Calcolo 59:4.
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Xiaofeng Yang & Xiaoming He. (2022) Numerical approximations of flow coupled binary phase field crystal system: Fully discrete finite element scheme with second-order temporal accuracy and decoupling structure. Journal of Computational Physics 467, pages 111448.
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Marco Salvalaglio & Ken R Elder. (2022) Coarse-grained modeling of crystals by the amplitude expansion of the phase-field crystal model: an overview. Modelling and Simulation in Materials Science and Engineering 30:5, pages 053001.
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Ilya Starodumov, Vladimir Ankudinov & Irina Nizovtseva. (2022) A review of continuous modeling of periodic pattern formation with modified phase-field crystal models. The European Physical Journal Special Topics 231:6, pages 1135-1145.
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Fan Wu & Zixian Zhu. (2022) Weak solutions and simulations to a square phase‐field crystal model with Neumann boundary conditions. Mathematical Methods in the Applied Sciences 45:8, pages 4185-4201.
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Veit Krause & Axel Voigt. (2022) Deformable active nematic particles and emerging edge currents in circular confinements. The European Physical Journal E 45:2.
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Min Wang, Qiumei Huang & Cheng Wang. (2021) A Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Square Phase Field Crystal Equation. Journal of Scientific Computing 88:2.
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Karthikeyan Chockalingam & Willy Dörfler. (2021) Implementation of the coupled two‐mode phase field crystal model with Cahn–Hilliard for phase‐separation in battery electrode particles. International Journal for Numerical Methods in Engineering 122:10, pages 2566-2580.
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Rainer Backofen, Lisa Sahlmann, Anna Willmann & Axel Voigt. (2021) A comparison of different approaches to enforce lattice symmetry in two‐dimensional crystals. PAMM 20:1.
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Hayden Nunley, Mikiko Nagashima, Kamirah Martin, Alcides Lorenzo Gonzalez, Sachihiro C. Suzuki, Declan A. Norton, Rachel O. L. Wong, Pamela A. Raymond & David K. Lubensky. (2020) Defect patterns on the curved surface of fish retinae suggest a mechanism of cone mosaic formation. PLOS Computational Biology 16:12, pages e1008437.
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V. Ankudinov & P.K. Galenko. (2020) Growth of different faces in a body centered cubic lattice: A case of the phase-field-crystal modeling. Journal of Crystal Growth 539, pages 125608.
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Simone Assali, Roberto Bergamaschini, Emilio Scalise, Marcel A. Verheijen, Marco Albani, Alain Dijkstra, Ang Li, Sebastian Koelling, Erik P. A. M. Bakkers, Francesco Montalenti & Leo Miglio. (2020) Kinetic Control of Morphology and Composition in Ge/GeSn Core/Shell Nanowires. ACS Nano 14:2, pages 2445-2455.
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R. Backofen & A. Voigt. (2020) Magnetically induced/enhanced coarsening in thin films. Physical Review Materials 4:2.
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Xiaobo Jing & Qi Wang. (2020) Linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints. Computers & Mathematics with Applications 79:3, pages 764-788.
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László Gránásy, Gyula I. Tóth, James A. Warren, Frigyes Podmaniczky, György Tegze, László Rátkai & Tamás Pusztai. (2019) Phase-field modeling of crystal nucleation in undercooled liquids – A review. Progress in Materials Science 106, pages 100569.
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R. Backofen, K. R. Elder & A. Voigt. (2019) Controlling Grain Boundaries by Magnetic Fields. Physical Review Letters 122:12.
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Ying Wei, Chao Yang & Jizu Huang. (2019) Parallel energy-stable phase field crystal simulations based on domain decomposition methods. Computer Physics Communications 234, pages 26-39.
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Jiaoyan Li, Bo Ni, Teng Zhang & Huajian Gao. (2018) Phase field crystal modeling of grain boundary structures and growth in polycrystalline graphene. Journal of the Mechanics and Physics of Solids 120, pages 36-48.
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Francesco Alaimo & Axel Voigt. (2018) Microscopic field-theoretical approach for mixtures of active and passive particles. Physical Review E 98:3.
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Lixiu Dong, Wenqiang Feng, Cheng Wang, Steven M. Wise & Zhengru Zhang. (2018) Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation. Computers & Mathematics with Applications 75:6, pages 1912-1928.
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J. Zhang, Z. Chen, C. Cheng & Y. X. Wang. (2017) A phase field crystal model simulation of morphology evolution and misfit dislocation generation in nanoheteroepitaxy. Modern Physics Letters B 31:30, pages 1750283.
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Yibao Li & Junseok Kim. (2017) An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation. Computer Methods in Applied Mechanics and Engineering 319, pages 194-216.
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Shi Hu, Zheng Chen, Wen Xi & Ying-Ying Peng. (2017) Phase-field-crystal study on the evolution behavior of microcracks initiated on grain boundaries under constant strain. Journal of Materials Science 52:10, pages 5641-5651.
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Gu Ji-Wei, Wang Jin-Cheng, Wang Zhi-Jun, Li Jun-Jie, Guo Can & Tang Sai. (2017) Phase-field crystal modelling the nucleation processes of graphene structures on different substrates. Acta Physica Sinica 66:21, pages 216101.
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Hyun Geun Lee & Junseok Kim. (2016) A simple and efficient finite difference method for the phase-field crystal equation on curved surfaces. Computer Methods in Applied Mechanics and Engineering 307, pages 32-43.
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F Alaimo, S Praetorius & A Voigt. (2016) A microscopic field theoretical approach for active systems. New Journal of Physics 18:8, pages 083008.
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Christian Köhler, Rainer Backofen & Axel Voigt. (2016) Stress Induced Branching of Growing Crystals on Curved Surfaces. Physical Review Letters 116:13.
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Monica Conti, Andrea Giorgini & Maurizio Grasselli. (2016) Phase-field crystal equation with memory. Journal of Mathematical Analysis and Applications 436:2, pages 1297-1331.
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Can Guo, Jincheng Wang, Zhijun Wang, Junjie Li, Yaolin Guo & Yunhao Huang. (2016) Interfacial free energy adjustable phase field crystal model for homogeneous nucleation. Soft Matter 12:20, pages 4666-4673.
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Mehdi Dehghan & Vahid Mohammadi. (2016) The numerical simulation of the phase field crystal (PFC) and modified phase field crystal (MPFC) models via global and local meshless methods. Computer Methods in Applied Mechanics and Engineering 298, pages 453-484.
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T. Witkowski, S. Ling, S. Praetorius & A. Voigt. (2015) Software concepts and numerical algorithms for a scalable adaptive parallel finite element method. Advances in Computational Mathematics 41:6, pages 1145-1177.
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Hyun Geun Lee, Jaemin Shin & June-Yub Lee. (2015) First and second order operator splitting methods for the phase field crystal equation. Journal of Computational Physics 299, pages 82-91.
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P. Vignal, L. Dalcin, D.L. Brown, N. Collier & V.M. Calo. (2015) An energy-stable convex splitting for the phase-field crystal equation. Computers & Structures 158, pages 355-368.
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Simon Praetorius & Axel Voigt. (2015) A Navier-Stokes phase-field crystal model for colloidal suspensions. The Journal of Chemical Physics 142:15.
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Maurizio Grasselli & Hao Wu. (2015) Robust exponential attractors for the modified phase-field crystal equation. Discrete & Continuous Dynamical Systems - A 35:6, pages 2539-2564.
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Simon Praetorius & Axel Voigt. (2015) Development and Analysis of a Block-Preconditioner for the Phase-Field Crystal Equation. SIAM Journal on Scientific Computing 37:3, pages B425-B451.
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Ebrahim Asadi & Mohsen Asle Zaeem. (2014) A Review of Quantitative Phase-Field Crystal Modeling of Solid–Liquid Structures. JOM 67:1, pages 186-201.
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Teng Zhang, Xiaoyan Li & Huajian Gao. (2014) Designing graphene structures with controlled distributions of topological defects: A case study of toughness enhancement in graphene ruga. Extreme Mechanics Letters 1, pages 3-8.
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Jian Long, Shuai Zhang, YuLong Zhao, QingHua Long, Tao Yang & Zheng Chen. (2014) Deformation study of bicrystalline and nano-polycrystalline structures using phase field crystal method. Science China Physics, Mechanics & Astronomy 57:6, pages 1046-1056.
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R. Backofen & A. Voigt. (2014) A phase field crystal study of heterogeneous nucleation – application of the string method. The European Physical Journal Special Topics 223:3, pages 497-509.
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Sai Tang, Yan-Mei Yu, Jincheng Wang, Junjie Li, Zhijun Wang, Yaolin Guo & Yaohe Zhou. (2014) Phase-field-crystal simulation of nonequilibrium crystal growth. Physical Review E 89:1.
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Verena Schmid & Axel Voigt. (2014) Crystalline order and topological charges on capillary bridges. Soft Matter 10:26, pages 4694.
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Arvind Baskaran, Zhengzheng Hu, John S. Lowengrub, Cheng Wang, Steven M. Wise & Peng Zhou. (2013) Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation. Journal of Computational Physics 250, pages 270-292.
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Zhengru Zhang, Yuan Ma & Zhonghua Qiao. (2013) An adaptive time-stepping strategy for solving the phase field crystal model. Journal of Computational Physics 249, pages 204-215.
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P. K. Galenko, H. Gomez, N. V. Kropotin & K. R. Elder. (2013) Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation. Physical Review E 88:1.
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Uwe Thiele, Andrew J. Archer, Mark J. Robbins, Hector Gomez & Edgar Knobloch. (2013) Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity. Physical Review E 87:4.
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A. Baskaran, J. S. Lowengrub, C. Wang & S. M. Wise. (2013) Convergence Analysis of a Second Order Convex Splitting Scheme for the Modified Phase Field Crystal Equation. SIAM Journal on Numerical Analysis 51:5, pages 2851-2873.
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Philippe A. Vignal, Nathan Collier & V.M. Calo. (2013) Phase Field Modeling Using PetIGA. Procedia Computer Science 18, pages 1614-1623.
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Sebastian Aland, John Lowengrub & Axel Voigt. (2012) Particles at fluid-fluid interfaces: A new Navier-Stokes-Cahn-Hilliard surface- phase-field-crystal model. Physical Review E 86:4.
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A. J. Archer, M. J. Robbins, U. Thiele & E. Knobloch. (2012) Solidification fronts in supercooled liquids: How rapid fronts can lead to disordered glassy solids. Physical Review E 86:3.
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Simon Praetorius & Axel Voigt. (2011) A Phase Field Crystal Approach for Particles in a Flowing Solvent. Macromolecular Theory and Simulations 20:7, pages 541-547.
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Sebastian Aland, John Lowengrub & Axel Voigt. (2011) A continuum model of colloid-stabilized interfaces. Physics of Fluids 23:6.
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Yan-Mei Yu, Rainer Backofen & Axel Voigt. (2011) Morphological instability of heteroepitaxial growth on vicinal substrates: A phase-field crystal study. Journal of Crystal Growth 318:1, pages 18-22.
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Morgan Pierre & Arnaud Rougirel. (2011) Stationary solutions to phase field crystal equations. Mathematical Methods in the Applied Sciences 34:3, pages 278-308.
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C. Wang & S. M. Wise. (2011) An Energy Stable and Convergent Finite-Difference Scheme for the Modified Phase Field Crystal Equation. SIAM Journal on Numerical Analysis 49:3, pages 945-969.
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R Backofen & A Voigt. (2010) A phase-field-crystal approach to critical nuclei. Journal of Physics: Condensed Matter 22:36, pages 364104.
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Zhi-Feng Huang, K. R. Elder & Nikolas Provatas. (2010) Phase-field-crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures. Physical Review E 82:2.
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Francisco Bernal, Rainer Backofen & Axel Voigt. (2010) Elastic interactions in phase-field crystal models: numerics and postprocessing. International Journal of Materials Research 101:4, pages 467-472.
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Rainer Backofen, Axel Voigt & Thomas Witkowski. (2010) Particles on curved surfaces: A dynamic approach by a phase-field-crystal model. Physical Review E 81:2.
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T. Hirouchi, T. Takaki & Y. Tomita. (2010) Effects of temperature and grain size on phase-field-crystal deformation simulation. International Journal of Mechanical Sciences 52:2, pages 309-319.
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K. R. Elder, Zhi-Feng Huang & Nikolas Provatas. (2010) Amplitude expansion of the binary phase-field-crystal model. Physical Review E 81:1.
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R Backofen & A Voigt. (2009) Solid–liquid interfacial energies and equilibrium shapes of nanocrystals. Journal of Physics: Condensed Matter 21:46, pages 464109.
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Z. Hu, S.M. Wise, C. Wang & J.S. Lowengrub. (2009) Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation. Journal of Computational Physics 228:15, pages 5323-5339.
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Sven van Teeffelen, Rainer Backofen, Axel Voigt & Hartmut Löwen. (2009) Derivation of the phase-field-crystal model for colloidal solidification. Physical Review E 79:5.
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S. M. Wise, C. Wang & J. S. Lowengrub. (2009) An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation. SIAM Journal on Numerical Analysis 47:3, pages 2269-2288.
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Ji Hee Kim & Stephen H. Garofalini. (2008) Modeling microstructural evolution using atomic density function and effective pair potentials. Physical Review B 78:14.
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Benjamin Berkels, Andreas Rätz, Martin Rumpf & Axel Voigt. (2007) Extracting Grain Boundaries and Macroscopic Deformations from Images on Atomic Scale. Journal of Scientific Computing 35:1, pages 1-23.
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