1
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Qiu C, Punke M, Tian Y, Han Y, Wang S, Su Y, Salvalaglio M, Pan X, Srolovitz DJ, Han J. Grain boundaries are Brownian ratchets. Science 2024; 385:980-985. [PMID: 39208099 DOI: 10.1126/science.adp1516] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2024] [Accepted: 07/19/2024] [Indexed: 09/04/2024]
Abstract
We demonstrate that grain boundaries (GBs) behave as Brownian ratchets, exhibiting direction-dependent mobilities and unidirectional motion under oscillatory driving forces or cyclic thermal annealing. We observed these phenomena for nearly all nonsymmetric GBs but not for symmetric ones. Our observations build on molecular dynamics and phase-field crystal simulations for a wide range of GB types and driving forces in both bicrystal and polycrystalline microstructures. We corroborate these simulation results through in situ experimental observations. We analyze these results with a Markov chain model and explore the implications of GB ratchet behavior for materials processing and microstructure tailoring.
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Affiliation(s)
- Caihao Qiu
- Department of Materials Science and Engineering, City University of Hong Kong, Hong Kong, China
| | - Maik Punke
- Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany
| | - Yuan Tian
- Department of Materials Science and Engineering, University of California, Irvine, CA 92697, USA
| | - Ying Han
- Department of Materials Science and Engineering, University of California, Irvine, CA 92697, USA
| | - Siqi Wang
- Department of Materials Science and Engineering, City University of Hong Kong, Hong Kong, China
| | - Yishi Su
- State Key Laboratory of Metal Matrix Composites, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Marco Salvalaglio
- Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany
- Dresden Center for Computational Materials Science, TU Dresden, 01062 Dresden, Germany
| | - Xiaoqing Pan
- Department of Materials Science and Engineering, University of California, Irvine, CA 92697, USA
- Irvine Materials Research Institute (IMRI), University of California, Irvine, CA 92697, USA
| | - David J Srolovitz
- Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China
- Materials Innovation Institute for Life Sciences and Energy (MILES), The University of Hong Kong, Shenzhen, China
| | - Jian Han
- Department of Materials Science and Engineering, City University of Hong Kong, Hong Kong, China
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2
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Bottacchiari M, Gallo M, Bussoletti M, Casciola CM. The diffuse interface description of fluid lipid membranes captures key features of the hemifusion pathway and lateral stress profile. PNAS NEXUS 2024; 3:pgae300. [PMID: 39114574 PMCID: PMC11304589 DOI: 10.1093/pnasnexus/pgae300] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/15/2024] [Accepted: 07/09/2024] [Indexed: 08/10/2024]
Abstract
Topological transitions of lipid membranes are ubiquitous in key biological processes for cell life, like neurotransmission, fertilization, morphogenesis, and viral infections. Despite this, they are not well understood due to their multiscale nature, which limits the use of molecular models and calls for a mesoscopic approach such as the celebrated Canham-Helfrich one. Unfortunately, such a model cannot handle topological transitions, hiding the crucial involved forces and the appearance of the experimentally observed hemifused intermediates. In this work, we describe the membrane as a diffuse interface preserving the Canham-Helfrich elasticity. We show that pivotal features of the hemifusion pathway are captured by this mesoscopic approach, e.g. a (meta)stable hemifusion state and the fusogenic behavior of negative monolayer spontaneous curvatures. The membrane lateral stress profile is calculated as a function of the elastic rigidities, yielding a coarse-grained version of molecular models findings. Insights into the fusogenic mechanism are reported and discussed.
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Affiliation(s)
- Matteo Bottacchiari
- Department of Basic and Applied Sciences for Engineering, Sapienza University of Rome, via Antonio Scarpa 16, Rome 00161, Italy
- Department of Mechanical and Aerospace Engineering, Sapienza University of Rome, via Eudossiana 18, Rome 00184, Italy
| | - Mirko Gallo
- Department of Mechanical and Aerospace Engineering, Sapienza University of Rome, via Eudossiana 18, Rome 00184, Italy
| | - Marco Bussoletti
- Department of Mechanical and Aerospace Engineering, Sapienza University of Rome, via Eudossiana 18, Rome 00184, Italy
| | - Carlo M Casciola
- Department of Mechanical and Aerospace Engineering, Sapienza University of Rome, via Eudossiana 18, Rome 00184, Italy
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3
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Burns D, Provatas N, Grant M. Phase field crystal models with applications to laser deposition: A review. STRUCTURAL DYNAMICS (MELVILLE, N.Y.) 2024; 11:014101. [PMID: 38361660 PMCID: PMC10869171 DOI: 10.1063/4.0000226] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/31/2023] [Accepted: 01/10/2024] [Indexed: 02/17/2024]
Abstract
In this article, we address the application of phase field crystal (PFC) theory, a hybrid atomistic-continuum approach, for modeling nanostructure kinetics encountered in laser deposition. We first provide an overview of the PFC methodology, highlighting recent advances to incorporate phononic and heat transport mechanisms. To simulate laser heating, energy is deposited onto a number of polycrystalline, two-dimensional samples through the application of initial stochastic fluctuations. We first demonstrate the ability of the model to simulate plasticity and recrystallization events that follow laser heating in the isothermal limit. Importantly, we also show that sufficient kinetic energy can cause voiding, which serves to suppress shock propagation. We subsequently employ a newly developed thermo-density PFC theory, coined thermal field crystal (TFC), to investigate laser heating of polycrystalline samples under non-isothermal conditions. We observe that the latent heat of transition associated with ordering can lead to long lasting metastable structures and defects, with a healing rate linked to the thermal diffusion. Finally, we illustrate that the lattice temperature simulated by the TFC model is in qualitative agreement with predictions of conventional electron-phonon two-temperature models. We expect that our new TFC formalism can be useful for predicting transient structures that result from rapid laser heating and re-solidification processes.
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Affiliation(s)
- Duncan Burns
- Department of Physics, McGill University, Montréal, Québec H3A 2T8, Canada
| | - Nikolas Provatas
- Department of Physics, McGill University, Montréal, Québec H3A 2T8, Canada
| | - Martin Grant
- Department of Physics, McGill University, Montréal, Québec H3A 2T8, Canada
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4
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Wang K, Chen W, Xiao S, Chen J, Hu W. Pattern Formation under Deep Supercooling by Classical Density Functional-Based Approach. ENTROPY (BASEL, SWITZERLAND) 2023; 25:e25050708. [PMID: 37238463 DOI: 10.3390/e25050708] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2023] [Revised: 04/13/2023] [Accepted: 04/19/2023] [Indexed: 05/28/2023]
Abstract
Solidification patterns during nonequilibrium crystallization are among the most important microstructures in the natural and technical realms. In this work, we investigate the crystal growth in deeply supercooled liquid using the classical density functional-based approaches. Our result shows that the complex amplitude expanded phase-field crystal (APFC) model containing the vacancy nonequilibrium effects proposed by us could naturally reproduce the growth front nucleation (GFN) and various nonequilibrium patterns, including the faceted growth, spherulite, symmetric and nonsymmetric dendrites among others, at the atom level. Moreover, an extraordinary microscopic columnar-to-equiaxed transition is uncovered, which is found to depend on the seed spacing and distribution. Such a phenomenon could be attributed to the combined effects of the long-wave and short-wave elastic interactions. Particularly, the columnar growth could also be predicted by an APFC model containing inertia effects, but the lattice defect type in the growing crystal is different due to the different types of short-wave interactions. Two stages are identified during the crystal growth under different undercooling, corresponding to diffusion-controlled growth and GFN-dominated growth, respectively. However, compared with the second stage, the first stage becomes too short to be noticed under the high undercooling. The distinct feature of the second stage is the dramatic increments of lattice defects, which explains the amorphous nucleation precursor in the supercooled liquid. The transition time between the two stages at different undercooling is investigated. Crystal growth of BCC structure further confirms our conclusions.
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Affiliation(s)
- Kun Wang
- College of Materials Science and Engineering, Hunan University, Changsha 410082, China
| | - Wenjin Chen
- College of Materials Science and Engineering, Hunan University, Changsha 410082, China
| | - Shifang Xiao
- Department of Applied Physics, Hunan University, Changsha 410082, China
| | - Jun Chen
- Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
| | - Wangyu Hu
- College of Materials Science and Engineering, Hunan University, Changsha 410082, China
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5
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Wu C, Feng X, Qian L. A Second-Order Crank-Nicolson Leap-Frog Scheme for the Modified Phase Field Crystal Model with Long-Range Interaction. ENTROPY (BASEL, SWITZERLAND) 2022; 24:1512. [PMID: 36359605 PMCID: PMC9689566 DOI: 10.3390/e24111512] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/31/2022] [Revised: 10/07/2022] [Accepted: 10/19/2022] [Indexed: 06/16/2023]
Abstract
In this paper, we construct a fully discrete and decoupled Crank-Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank-Nicolson and to treat non-stiff terms explicitly with Leap-Frog. In addition, the scalar auxiliary variable (SAV) method is used to allow explicit treatment of the nonlinear potential, then, these technique combines with CNLF can lead to the highly efficient, fully decoupled and linear numerical scheme with constant coefficients at each time step. Furthermore, the Fourier spectral method is used for the spatial discretization. Finally, we show that the CNLF scheme is fully discrete, second-order decoupled and unconditionally stable. Ample numerical experiments in 2D and 3D are provided to demonstrate the accuracy, efficiency, and stability of the proposed method.
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Affiliation(s)
- Chunya Wu
- School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China
| | - Xinlong Feng
- College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
| | - Lingzhi Qian
- School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China
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6
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Ankudinov V, Galenko PK. Structure diagram and dynamics of formation of hexagonal boron nitride in phase-field crystal model. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20200318. [PMID: 34974729 DOI: 10.1098/rsta.2020.0318] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2021] [Accepted: 05/21/2021] [Indexed: 06/14/2023]
Abstract
The phase-field crystal (PFC-model) is a powerful tool for modelling of the crystallization in colloidal and metallic systems. In the present work, the modified hyperbolic phase-field crystal model for binary systems is presented. This model takes into account slow and fast dynamics of moving interfaces for both concentration and relative atomic number density (which were taken as order parameters). The model also includes specific mobilities for each dynamical field and correlated noise terms. The dynamics of chemical segregation with origination of mixed pseudo-hexagonal binary phase (the so-called 'triangle phase') is used as a benchmark in two spatial dimensions for the developing model. Using the free energy functional and specific lattice vectors for hexagonal crystal, the structure diagram of co-existence of liquid and three-dimensional hexagonal phase for the binary PFC-model was carried out. Parameters of the crystal lattice correspond to the hexagonal boron nitride (BN) crystal, the values of which have been taken from the literature. The paper shows the qualitative agreement between the developed structure diagram of the PFC model and the previously known equilibrium diagram for BN constructed using thermodynamic functions. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.
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Affiliation(s)
- V Ankudinov
- Vereshchagin Institute of High Pressure Physics, Russian Academy of Sciences, 108840 Moscow (Troitsk), Russia
- Institute of Mathematics, Informatics and Physics, Condensed Matter Physics Lab, Udmurt State University, Izhevsk, Russia
| | - P K Galenko
- Physikalish-Astronomische Fakultät, Otto-Schott-Institut für Materialforschung, Löbdergraben 32, 07743 Jena, Germany
- Laboratory of Multi-scale Mathematical Modeling, Department of Theoretical and Mathematical Physics, Ural Federal University, 620000 Ekaterinburg, Russia
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7
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Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification. CRYSTALS 2021. [DOI: 10.3390/cryst11040437] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
Two-step nucleation and subsequent growth processes were investigated in the framework of the single mode phase-field crystal model combined with diffusive dynamics (corresponding to colloid suspensions) and hydrodynamical density relaxation (simple liquids). It is found that independently of dynamics, nucleation starts with the formation of solid precursor clusters that consist of domains with noncrystalline ordering (ringlike projections are seen from certain angles), and regions that have amorphous structure. Using the average bond order parameter q¯6, we distinguished amorphous, medium range crystallike order (MRCO), and crystalline local orders. We show that crystallization to the stable body-centered cubic phase is preceded by the formation of a mixture of amorphous and MRCO structures. We have determined the time dependence of the phase composition of the forming solid state. We also investigated the time/size dependence of the growth rate for solidification. The bond order analysis indicates similar structural transitions during solidification in the case of diffusive and hydrodynamic density relaxation.
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8
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Simeone D, Garcia P, Bacri CO, Luneville L. Symmetry Breaking Resulting from Long-Range Interactions in Out of Equilibrium Systems: Elastic Properties of Irradiated AgCu. PHYSICAL REVIEW LETTERS 2020; 125:246103. [PMID: 33412047 DOI: 10.1103/physrevlett.125.246103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Revised: 11/03/2020] [Accepted: 11/09/2020] [Indexed: 06/12/2023]
Abstract
This work presents a consistent formulation of the phase-field approach to model the behavior of nonmiscible alloys under irradiation which includes elastic strain fields, an example of a long-range interaction. Simulations show that the spatial isotropy that is characteristic of radiation-induced patterns breaks down as a result of the elastic strain energy. The consequence of this is the emergence of superlattice structures under irradiation liable to modify macroscopic material properties. This approach is assessed against the experimental study of a AgCu alloy under irradiation: we compare our simulation results to measured solubility limits and Young moduli.
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Affiliation(s)
- D Simeone
- CEA, DES, ISAS, DMN, Paris-Saclay, F-91191 Gif sur Yvette, France
| | - P Garcia
- CEA, DES, IRESNE, DEC, F-13108 Saint Paul Lez Durance, France
| | - C O Bacri
- Université Paris-Saclay, CNRS/IN2P3, IJClab, 91405 Orsay, France
| | - L Luneville
- CEA, DES, ISAS, DM2S, Paris-Saclay, F-91191 Gif sur Yvette, France
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9
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Ankudinov V, Elder KR, Galenko PK. Traveling waves of the solidification and melting of cubic crystal lattices. Phys Rev E 2020; 102:062802. [PMID: 33466054 DOI: 10.1103/physreve.102.062802] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/08/2020] [Accepted: 11/04/2020] [Indexed: 11/07/2022]
Abstract
Using the phase field crystal model (PFC model), an analysis of slow and fast dynamics of solid-liquid interfaces in solidification and melting processes is presented. Dynamical regimes for cubic lattices invading metastable liquids (solidification) and liquids propagating into metastable crystals (melting) are described in terms of the evolving amplitudes of the density field. Dynamical equations are obtained for body-centered cubic (bcc) and face-centered cubic (fcc) crystal lattices in one- and two-mode approximations. A universal form of the amplitude equations is obtained for the three-dimensional dynamics for different crystal lattices and crystallographic directions. Dynamics of the amplitude's propagation for different lattices and PFC mode's approximations is qualitatively compared. The traveling-wave velocity is quantitatively compared with data of molecular dynamics simulation previously obtained by Mendelev et al. [Modell. Simul. Mater. Sci. Eng. 18, 074002 (2010)MSMEEU0965-039310.1088/0965-0393/18/7/074002] for solidification and melting of the aluminum fcc lattice.
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Affiliation(s)
- V Ankudinov
- Vereshchagin Institute of High Pressure Physics, Russian Academy of Sciences, 108840 Moscow (Troitsk), Russia
| | - K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309-4487, USA
| | - P K Galenko
- Friedrich Schiller University of Jena, Faculty of Physics and Astronomy, Otto Schott Institute of Materials Research, 07743 Jena, Germany.,Ural Federal University, Theoretical and Mathematical Physics Department, Laboratory of Multi-Scale Mathematical Modeling, 620000 Ekaterinburg, Russia
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10
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Gomez H, Bures M, Moure A. A review on computational modelling of phase-transition problems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20180203. [PMID: 30827214 PMCID: PMC6460062 DOI: 10.1098/rsta.2018.0203] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 12/06/2018] [Indexed: 05/18/2023]
Abstract
Phase-transition problems are ubiquitous in science and engineering. They have been widely studied via theory, experiments and computations. This paper reviews the main challenges associated with computational modelling of phase-transition problems, addressing both model development and numerical discretization of the resulting equations. We focus on classical phase-transition problems, including liquid-solid, gas-liquid and solid-solid transformations. Our review has a strong emphasis on the treatment of interfacial phenomena and the phase-field method. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.
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11
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Zhou W, Wang J, Wang Z, Huang ZF. Mechanical relaxation and fracture of phase field crystals. Phys Rev E 2019; 99:013302. [PMID: 30780269 DOI: 10.1103/physreve.99.013302] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2018] [Indexed: 11/07/2022]
Abstract
A computational method is developed for the study of mechanical response and fracture behavior of phase field crystals (PFC), to overcome a limitation of the PFC dynamics which lacks an effective mechanism for describing fast mechanical relaxation of the material system. The method is based on a simple interpolation scheme for PFC (IPFC) making use of a condition of the displacement field to satisfy local elastic equilibration, while preserving key characteristics of the original PFC model. We conduct a systematic study on the mechanical properties of a sample nanoribbon system with honeycomb lattice symmetry subjected to uniaxial tension, for numerical validation of the IPFC scheme and the comparison with the original PFC and modified PFC methods. Results of mechanical response, in both elasticity and fracture regimes, show the advantage and efficiency of the IPFC method across different system sizes and applied strain rates, due to its effective process of mechanical equilibration. A brittle fracture behavior is obtained in IPFC calculations, where effects of system temperature and chirality on the fracture strength and Young's modulus are also identified, with results agreeing with those found in previous atomistic simulations of graphene. The IPFC scheme developed here is generic and applicable to the mechanical studies using different types of PFC free-energy functionals designed for various material systems.
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Affiliation(s)
- Wenquan Zhou
- State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an 710072, China
| | - Jincheng Wang
- State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an 710072, China
| | - Zhijun Wang
- State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an 710072, China
| | - Zhi-Feng Huang
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
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12
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Skaugen A, Angheluta L, Viñals J. Separation of Elastic and Plastic Timescales in a Phase Field Crystal Model. PHYSICAL REVIEW LETTERS 2018; 121:255501. [PMID: 30608801 DOI: 10.1103/physrevlett.121.255501] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2018] [Revised: 10/15/2018] [Indexed: 06/09/2023]
Abstract
A consistent small-scale description of plasticity and dislocation motion in a crystalline solid is presented based on the phase field crystal description. By allowing for independent mass motion and lattice distortion, the crystal can maintain elastic equilibrium on the timescale of plastic motion. We show that the singular (incompatible) strains are determined by the phase field crystal density, while the smooth distortions are constrained to satisfy elastic equilibrium. A numerical implementation of the model is presented and used to study a benchmark problem: the motion of an edge dislocation dipole in a triangular lattice. The time dependence of the dipole separation agrees with continuum elasticity with no adjustable parameters.
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Affiliation(s)
- Audun Skaugen
- Njord Center, Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway
| | - Luiza Angheluta
- Njord Center, Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway
| | - Jorge Viñals
- School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, Minnesota 55455, USA
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13
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Balakrishna AR, Carter WC. Combining phase-field crystal methods with a Cahn-Hilliard model for binary alloys. Phys Rev E 2018; 97:043304. [PMID: 29758731 DOI: 10.1103/physreve.97.043304] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2017] [Indexed: 06/08/2023]
Abstract
Diffusion-induced phase transitions typically change the lattice symmetry of the host material. In battery electrodes, for example, Li ions (diffusing species) are inserted between layers in a crystalline electrode material (host). This diffusion induces lattice distortions and defect formations in the electrode. The structural changes to the lattice symmetry affect the host material's properties. Here, we propose a 2D theoretical framework that couples a Cahn-Hilliard (CH) model, which describes the composition field of a diffusing species, with a phase-field crystal (PFC) model, which describes the host-material lattice symmetry. We couple the two continuum models via coordinate transformation coefficients. We introduce the transformation coefficients in the PFC method to describe affine lattice deformations. These transformation coefficients are modeled as functions of the composition field. Using this coupled approach, we explore the effects of coarse-grained lattice symmetry and distortions on a diffusion-induced phase transition process. In this paper, we demonstrate the working of the CH-PFC model through three representative examples: First, we describe base cases with hexagonal and square symmetries for two composition fields. Next, we illustrate how the CH-PFC method interpolates lattice symmetry across a diffuse phase boundary. Finally, we compute a Cahn-Hilliard type of diffusion and model the accompanying changes to lattice symmetry during a phase transition process.
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Affiliation(s)
- Ananya Renuka Balakrishna
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - W Craig Carter
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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14
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Xi W, Song X, Hu S, Chen Z. Phase field crystal simulation of stress induced localized solid-state amorphization in nanocrystalline materials. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:475902. [PMID: 28960182 DOI: 10.1088/1361-648x/aa8fee] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In this work, the phase field crystal (PFC) method is used to study the localized solid-state amorphization (SSA) and its dynamic transformation process in polycrystalline materials under the uniaxial tensile deformation with different factors. The impacts of these factors, including strain rates, temperatures and grain sizes, are analyzed. Kinetically, the ultra-high strain rate causes the lattice to be seriously distorted and the grain to gradually collapse, so the dislocation density rises remarkably. Therefore, localized SSA occurs. Thermodynamically, as high temperature increases the activation energy, the atoms are active and prefer to leave the original position, which induce atom rearrangement. Furthermore, small grain size increases the percentage of grain boundary and the interface free energy of the system. As a result, Helmholtz free energy increases. The dislocations and Helmholtz free energy act as the seed and driving force for the process of the localized SSA. Also, the critical diffusion-time step and the percentage of amorphous region areas are calculated. Through this work, the PFC method is proved to be an effective means to study localized SSA under uniaxial tensile deformation.
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Affiliation(s)
- Wen Xi
- State Key Laboratory of Solidification Processing, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China
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15
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Podmaniczky F, Tóth GI, Tegze G, Gránásy L. Hydrodynamic theory of freezing: Nucleation and polycrystalline growth. Phys Rev E 2017; 95:052801. [PMID: 28618608 DOI: 10.1103/physreve.95.052801] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2016] [Indexed: 05/12/2023]
Abstract
Structural aspects of crystal nucleation in undercooled liquids are explored using a nonlinear hydrodynamic theory of crystallization proposed recently [G. I. Tóth et al., J. Phys.: Condens. Matter 26, 055001 (2014)JCOMEL0953-898410.1088/0953-8984/26/5/055001], which is based on combining fluctuating hydrodynamics with the phase-field crystal theory. We show that in this hydrodynamic approach not only homogeneous and heterogeneous nucleation processes are accessible, but also growth front nucleation, which leads to the formation of new (differently oriented) grains at the solid-liquid front in highly undercooled systems. Formation of dislocations at the solid-liquid interface and interference of density waves ahead of the crystallization front are responsible for the appearance of the new orientations at the growth front that lead to spherulite-like nanostructures.
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Affiliation(s)
- Frigyes Podmaniczky
- Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary
| | - Gyula I Tóth
- Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary
- Department of Physics, University of Bergen, Allégaten 55, 7005 Bergen, Norway
| | - György Tegze
- Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary
| | - László Gránásy
- Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary
- BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
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16
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Kundin J, Choudhary MA. Numerical determination of the interfacial energy and nucleation barrier of curved solid-liquid interfaces in binary systems. Phys Rev E 2016; 94:012801. [PMID: 27575196 DOI: 10.1103/physreve.94.012801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2016] [Indexed: 06/06/2023]
Abstract
The phase-field crystal (PFC) technique is a widely used approach for modeling crystal growth phenomena with atomistic resolution on mesoscopic time scales. We use a two-dimensional PFC model for a binary system based on the work of Elder et al. [Phys. Rev. B 75, 064107 (2007)PRBMDO1098-012110.1103/PhysRevB.75.064107] to study the effect of the curved, diffuse solid-liquid interface on the interfacial energy as well as the nucleation barrier. The calculation of the interfacial energy and the nucleation barrier certainly depends on the proper definition of the solid-liquid dividing surface and the corresponding nucleus size. We define the position of the sharp interface at which the interfacial energy is to be evaluated by using the concept of equimolar dividing surface (r^{e}) and the minimization of the interfacial energy (r^{s}). The comparison of the results based on both radii shows that the difference r^{e}-r^{s} is always positive and has a limit for large cluster sizes which is comparable to the Tolman length. Furthermore, we found the real nucleation barrier for small cluster sizes, which is defined as a function of the radius r^{s}, and compared it with the classical nucleation theory. The simulation results also show that the extracted interfacial energy as function of both radii is independent of system size, and this dependence can be reasonably described by the nonclassical Tolman formula with a positive Tolman length.
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Affiliation(s)
- Julia Kundin
- Department of Engineering Science, University Bayreuth, D-95440 Bayreuth, Germany
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Heinonen V, Achim CV, Ala-Nissila T. Long-wavelength properties of phase-field-crystal models with second-order dynamics. Phys Rev E 2016; 93:053003. [PMID: 27300969 DOI: 10.1103/physreve.93.053003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2016] [Indexed: 05/11/2023]
Abstract
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 creates an elastic excitation. The dynamics of these excitations poses a challenge: pure diffusive dynamics cannot describe relaxation of elastic stresses that happen through phonon emission. To this end, several different models with fast dynamics have been proposed. In this article we use the amplitude expansion of the PFC model to compare the recently proposed hydrodynamic PFC amplitude model with two simpler models with fast dynamics. We compare these different models analytically and numerically. The results suggest that in order to have proper relaxation of elastic excitations, the full hydrodynamical description of the PFC amplitudes is required.
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Affiliation(s)
- V Heinonen
- COMP Centre of Excellence, Department of Applied Physics, Aalto University, School of Science, P.O. Box 11100, FI-00076, Aalto, Finland
| | - C V Achim
- COMP Centre of Excellence, Department of Applied Physics, Aalto University, School of Science, P.O. Box 11100, FI-00076, Aalto, Finland
| | - T Ala-Nissila
- COMP Centre of Excellence, Department of Applied Physics, Aalto University, School of Science, P.O. Box 11100, FI-00076, Aalto, Finland
- Department of Physics, Brown University, Providence, Rhode Island 02912-1843, USA
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18
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Heinonen V, Achim CV, Kosterlitz JM, Ying SC, Lowengrub J, Ala-Nissila T. Consistent Hydrodynamics for Phase Field Crystals. PHYSICAL REVIEW LETTERS 2016; 116:024303. [PMID: 26824543 DOI: 10.1103/physrevlett.116.024303] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2015] [Indexed: 06/05/2023]
Abstract
We use the amplitude expansion in the phase field crystal framework to formulate an approach where the fields describing the microscopic structure of the material are coupled to a hydrodynamic velocity field. The model is shown to reduce to the well-known macroscopic theories in appropriate limits, including compressible Navier-Stokes and wave equations. Moreover, we show that the dynamics proposed allows for long wavelength phonon modes and demonstrate the theory numerically showing that the elastic excitations in the system are relaxed through phonon emission.
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Affiliation(s)
- V Heinonen
- COMP Centre of Excellence, Department of Applied Physics, Aalto University, School of Science, P.O. Box 11100, FI-00076 Aalto, Finland
| | - C V Achim
- COMP Centre of Excellence, Department of Applied Physics, Aalto University, School of Science, P.O. Box 11100, FI-00076 Aalto, Finland
| | - J M Kosterlitz
- Department of Physics, Brown University, Providence, Rhode Island 02912-1843, USA
| | - See-Chen Ying
- Department of Physics, Brown University, Providence, Rhode Island 02912-1843, USA
| | - J Lowengrub
- Department of Mathematics, University of California, Irvine, California 92697, USA
- Department of Chemical Engineering and Materials Science, University of California, Irvine, California 92697, USA
| | - T Ala-Nissila
- COMP Centre of Excellence, Department of Applied Physics, Aalto University, School of Science, P.O. Box 11100, FI-00076 Aalto, Finland
- Department of Physics, Brown University, Providence, Rhode Island 02912-1843, USA
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19
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Humadi H, Hoyt JJ, Provatas N. Microscopic treatment of solute trapping and drag. Phys Rev E 2016; 93:010801. [PMID: 26871012 DOI: 10.1103/physreve.93.010801] [Citation(s) in RCA: 34] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2014] [Indexed: 11/07/2022]
Abstract
The long wavelength limit of a recent microscopic phase-field crystal (PFC) theory of a binary alloy mixture is used to derive an analytical approximation for the segregation coefficient as a function of the interface velocity, and relate it to the two-point correlation function of the liquid and the thermodynamic properties of solid and liquid phases. Our results offer the first analytical derivation of solute segregation from a microscopic model, and support recent molecular dynamics and numerical PFC simulations. Our results also provide an independent framework, motivated from classical density functional theory, from which to elucidate the fundamental nature of solute drag, which is still highly contested in the literature.
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Affiliation(s)
- Harith Humadi
- Department of Physics, Centre for the Physics of Materials, McGill University, Montreal, QC, Canada.,Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario
| | - J J Hoyt
- Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario
| | - Nikolas Provatas
- Department of Physics, Centre for the Physics of Materials, McGill University, Montreal, QC, Canada
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20
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Grasselli M, Wu H. Robust exponential attractors for the modified phase-field crystal equation. ACTA ACUST UNITED AC 2015. [DOI: 10.3934/dcds.2015.35.2539] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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21
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Tarp JM, Angheluta L, Mathiesen J, Goldenfeld N. Intermittent dislocation density fluctuations in crystal plasticity from a phase-field crystal model. PHYSICAL REVIEW LETTERS 2014; 113:265503. [PMID: 25615353 DOI: 10.1103/physrevlett.113.265503] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2014] [Indexed: 06/04/2023]
Abstract
Plastic deformation mediated by collective dislocation dynamics is investigated in the two-dimensional phase-field crystal model of sheared single crystals. We find that intermittent fluctuations in the dislocation population number accompany bursts in the plastic strain-rate fluctuations. Dislocation number fluctuations exhibit a power-law spectral density 1/f2 at high frequencies f. The probability distribution of number fluctuations becomes bimodal at low driving rates corresponding to a scenario where low density of defects alternates at irregular times with high populations of defects. We propose a simple stochastic model of dislocation reaction kinetics that is able to capture these statistical properties of the dislocation density fluctuations as a function of shear rate.
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Affiliation(s)
- Jens M Tarp
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
| | - Luiza Angheluta
- Department of Physics, Physics of Geological Processes, University of Oslo, Post Office 1048 Blindern, 0316 Oslo Norway
| | - Joachim Mathiesen
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
| | - Nigel Goldenfeld
- Department of Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, 1110 West Green Street, Urbana, Illinois 61801-3080, USA
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22
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Choudhary MA, Kundin J, Emmerich H, Oettel M. Solid-liquid surface tensions of critical nuclei and nucleation barriers from a phase-field-crystal study of a model binary alloy using finite system sizes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022403. [PMID: 25215738 DOI: 10.1103/physreve.90.022403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2014] [Indexed: 06/03/2023]
Abstract
Phase-field-crystal (PFC) modeling has emerged as a computationally efficient tool to address crystal growth phenomena on atomistic length and diffusive time scales. We use a two-dimensional phase-field-crystal model for a binary system based on Elder et al. [Phys. Rev. B 75, 064107 (2007)] to study critical nuclei and their liquid-solid phase boundaries, in particular the nucleus size dependence of the liquid-solid interface tension as well as of the nucleation barrier. Critical nuclei are stabilized in finite systems of various sizes, however, the extracted interface tension as function of the nucleus radius r is independent of system size. We suggest a phenomenological expression to describe the dependence of the extracted interface tension on the nucleus radius r for the liquid-solid system. Moreover, the numerical PFC results show that this dependency can not be fully described by the nonclassical Tolman formula.
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Affiliation(s)
| | - Julia Kundin
- Lehrstuhl für Material- und Prozesssimulation, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - Heike Emmerich
- Lehrstuhl für Material- und Prozesssimulation, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - Martin Oettel
- Institut für Angewandte Physik, Universität Tübingen, D-72076 Tübingen, Germany
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23
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Heinonen V, Achim CV, Elder KR, Buyukdagli S, Ala-Nissila T. Phase-field-crystal models and mechanical equilibrium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:032411. [PMID: 24730856 DOI: 10.1103/physreve.89.032411] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2013] [Indexed: 05/11/2023]
Abstract
Phase-field-crystal (PFC) models constitute a field theoretical approach to solidification, melting, and related phenomena at atomic length and diffusive time scales. One of the advantages of these models is that they naturally contain elastic excitations associated with strain in crystalline bodies. However, instabilities that are diffusively driven towards equilibrium are often orders of magnitude slower than the dynamics of the elastic excitations, and are thus not included in the standard PFC model dynamics. We derive a method to isolate the time evolution of the elastic excitations from the diffusive dynamics in the PFC approach and set up a two-stage process, in which elastic excitations are equilibrated separately. This ensures mechanical equilibrium at all times. We show concrete examples demonstrating the necessity of the separation of the elastic and diffusive time scales. In the small-deformation limit this approach is shown to agree with the theory of linear elasticity.
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Affiliation(s)
- V Heinonen
- COMP Centre of Excellence at the Department of Applied Physics, Aalto University, School of Science, P. O. Box 11100, FI-00076 Aalto, Finland
| | - C V Achim
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany
| | - K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309, USA
| | - S Buyukdagli
- COMP Centre of Excellence at the Department of Applied Physics, Aalto University, School of Science, P. O. Box 11100, FI-00076 Aalto, Finland
| | - T Ala-Nissila
- COMP Centre of Excellence at the Department of Applied Physics, Aalto University, School of Science, P. O. Box 11100, FI-00076 Aalto, Finland and Department of Physics, Brown University, Providence, Rhode Island 02912-1843, USA
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24
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Tóth GI, Gránásy L, Tegze G. Nonlinear hydrodynamic theory of crystallization. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2014; 26:055001. [PMID: 24334547 DOI: 10.1088/0953-8984/26/5/055001] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We present an isothermal fluctuating nonlinear hydrodynamic theory of crystallization in molecular liquids. A dynamic coarse-graining technique is used to derive the velocity field, a phenomenology which allows a direct coupling between the free energy functional of the classical density functional theory and the Navier-Stokes equation. In contrast to the Ginzburg-Landau type amplitude theories, the dynamic response to elastic deformations is described by parameter-free kinetic equations. Employing our approach to the free energy functional of the phase-field crystal model, we recover the classical spectrum for the phonons and the steady-state growth fronts. The capillary wave spectrum of the equilibrium crystal-liquid interface is in good qualitative agreement with the molecular dynamics simulations.
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Affiliation(s)
- Gyula I Tóth
- Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, PO Box 49, H-1525 Budapest, Hungary
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25
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Menzel AM, Ohta T, Löwen H. Active crystals and their stability. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022301. [PMID: 25353466 DOI: 10.1103/physreve.89.022301] [Citation(s) in RCA: 39] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2013] [Indexed: 06/04/2023]
Abstract
A recently introduced active phase field crystal model describes the formation of ordered resting and traveling crystals in systems of self-propelled particles. Increasing the active drive, a resting crystal can be forced to perform collectively ordered migration as a single traveling object. We demonstrate here that these ordered migrating structures are linearly stable. In other words, during migration, the single-crystalline texture together with the globally ordered collective motion is preserved even on large length scales. Furthermore, we consider self-propelled particles on a substrate that are surrounded by a thin fluid film. We find that in this case the resulting hydrodynamic interactions can destabilize the order.
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Affiliation(s)
- Andreas M Menzel
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany and Department of Physics, Kyoto University, Kyoto 606-8502, Japan
| | - Takao Ohta
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan and Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany
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26
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Huang Z, Xiong H, Wu Z, Conway P, Davies H, Dinsdale A, En Y, Zeng Q. Microstructure-based multiphysics modeling for semiconductor integration and packaging. ACTA ACUST UNITED AC 2014. [DOI: 10.1007/s11434-013-0103-7] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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27
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Jou D, Galenko PK. Coarse graining for the phase-field model of fast phase transitions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042151. [PMID: 24229159 DOI: 10.1103/physreve.88.042151] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2013] [Indexed: 05/11/2023]
Abstract
Fast phase transitions under lack of local thermalization between successive elementary steps of the physical process are treated analytically. Non-Markovian master equations are derived for fast processes, which do not have enough time to reach energy or momentum thermalization during rapid phase change or freezing of a highly nonequilibrium system. These master equations provide a further physical basis for evolution and transport equations of the phase-field model used previously in the analyses of fast phase transitions.
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Affiliation(s)
- D Jou
- Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain
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28
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Biswas S, Grant M, Samajdar I, Haldar A, Sain A. Micromechanics of emergent patterns in plastic flows. Sci Rep 2013; 3:2728. [PMID: 24056757 PMCID: PMC3779853 DOI: 10.1038/srep02728] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2013] [Accepted: 08/19/2013] [Indexed: 11/11/2022] Open
Abstract
Crystalline solids undergo plastic deformation and subsequently flow when subjected to stresses beyond their elastic limit. In nature most crystalline solids exist in polycrystalline form. Simulating plastic flows in polycrystalline solids has wide ranging applications, from material processing to understanding intermittency of earthquake dynamics. Using phase field crystal (PFC) model we show that in sheared polycrystalline solids the atomic displacement field shows spatio-temporal heterogeneity spanning over several orders of length and time scales, similar to that in amorphous solids. The displacement field also exhibits localized quadrupolar patterns, characteristic of two dislocations of the opposite sign approaching each other. This is a signature of crystallinity at microscopic scale. Polycrystals being halfway between single crystals and amorphous solids, in terms of the degree of structural order, descriptions of solid mechanics at two widely different scales, namely continuum plastic flow and discrete dislocation dynamics turns out to be necessary here.
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Affiliation(s)
- Santidan Biswas
- Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai-400 076, India
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29
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Schwalbach EJ, Warren JA, Wu KA, Voorhees PW. Phase-field crystal model with a vapor phase. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:023306. [PMID: 24032965 DOI: 10.1103/physreve.88.023306] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2013] [Indexed: 06/02/2023]
Abstract
Phase-field crystal (PFC) models are able to resolve atomic length scale features of materials during temporal evolution over diffusive time scales. Traditional PFC models contain solid and liquid phases, however many important materials processing phenomena involve a vapor phase as well. In this work, we add a vapor phase to an existing PFC model and show realistic interfacial phenomena near the triple point temperature. For example, the PFC model exhibits density oscillations at liquid-vapor interfaces that compare favorably to data available for interfaces in metallic systems from both experiment and molecular-dynamics simulations. We also quantify the anisotropic solid-vapor surface energy for a two-dimensional PFC hexagonal crystal and find well-defined step energies from measurements on the faceted interfaces. Additionally, the strain field beneath a stepped interface is characterized and shown to qualitatively reproduce predictions from continuum models, simulations, and experimental data. Finally, we examine the dynamic case of step-flow growth of a crystal into a supersaturated vapor phase. The ability to model such a wide range of surface and bulk defects makes this PFC model a useful tool to study processing techniques such as chemical vapor deposition or vapor-liquid-solid growth of nanowires.
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Affiliation(s)
- Edwin J Schwalbach
- Materials Science and Engineering Division, Material Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
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30
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Bjerre M, Tarp JM, Angheluta L, Mathiesen J. Rotation-induced grain growth and stagnation in phase-field crystal models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:020401. [PMID: 24032765 DOI: 10.1103/physreve.88.020401] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2013] [Indexed: 06/02/2023]
Abstract
We consider grain growth and stagnation in polycrystalline microstructures. From the phase-field crystal modeling of the coarsening dynamics, we identify a transition from a grain-growth stagnation upon deep quenching below the melting temperature T(m) to a continuous coarsening at shallower quenching near T(m). The grain evolution is mediated by local grain rotations. In the deep quenching regime, the grain assembly typically reaches a metastable state where the kinetic barrier for recrystallization across boundaries is too large and grain rotation with subsequent coalescence or boundary motion is infeasible. For quenching near T(m), we find that the grain growth depends on the average rate of grain rotation, and follows a power-law behavior with time, with a scaling exponent that depends on the quenching depth.
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Affiliation(s)
- Mathias Bjerre
- Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
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31
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Galenko PK, Gomez H, Kropotin NV, Elder KR. Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:013310. [PMID: 23944586 DOI: 10.1103/physreve.88.013310] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2013] [Indexed: 05/11/2023]
Abstract
The phase-field crystal model (PFC model) resolves systems on atomic length scales and diffusive time scales and lies in between standard phase-field modeling and atomistic methods. More recently a hyperbolic or modified PFC model was introduced to describe fast (propagative) and slow (diffusive) dynamics. We present a finite-element method for solving the hyperbolic PFC equation, introducing an unconditionally stable time integration algorithm. A spatial discretization is used with the traditional C^{0}-continuous Lagrange elements with quadratic shape functions. The space-time discretization of the PFC equation is second-order accurate in time and is shown analytically to be unconditionally stable. Numerical simulations are used to show a monotonic decrease of the free energy during the transition from the homogeneous state to stripes. Benchmarks on modeling patterns in two-dimensional space are carried out. The benchmarks show the applicability of the proposed algorithm for determining equilibrium states. Quantitatively, the proposed algorithm is verified for the problem of lattice parameter and velocity selection when a crystal invades a homogeneous unstable liquid.
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Affiliation(s)
- P K Galenko
- Friedrich-Schiller-Universität Jena, Physikalisch-Astronomische Fakultät, D-07737 Jena, Germany.
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32
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Adland A, Xu Y, Karma A. Unified theoretical framework for polycrystalline pattern evolution. PHYSICAL REVIEW LETTERS 2013; 110:265504. [PMID: 23848896 DOI: 10.1103/physrevlett.110.265504] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2013] [Revised: 05/14/2013] [Indexed: 06/02/2023]
Abstract
The rate of curvature-driven grain growth in polycrystalline materials is well known to be limited by interface dissipation. We show analytically and by simulations that, for systems forming modulated phases or nonequilibrium patterns with crystal ordering, growth is limited by bulk dissipation associated with lattice translation, which dramatically slows down grain coarsening. We also show that bulk dissipation is reduced by thermal noise and that this reduction leads to faster coarsening behavior dominated by interface dissipation for a high Peierls-Nabarro barrier to dislocation motion and high noise. Those results provide a unified theoretical framework for understanding and modeling polycrystalline pattern evolution in diverse systems over a broad range of length and time scales.
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Affiliation(s)
- Ari Adland
- Physics Department and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Massachusetts 02115, USA
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33
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Thiele U, Archer AJ, Robbins MJ, Gomez H, Knobloch E. Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:042915. [PMID: 23679497 DOI: 10.1103/physreve.87.042915] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2013] [Revised: 03/21/2013] [Indexed: 05/11/2023]
Abstract
The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.
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Affiliation(s)
- Uwe Thiele
- Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom.
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34
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Menzel AM, Löwen H. Traveling and resting crystals in active systems. PHYSICAL REVIEW LETTERS 2013; 110:055702. [PMID: 23414036 DOI: 10.1103/physrevlett.110.055702] [Citation(s) in RCA: 60] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2012] [Revised: 12/18/2012] [Indexed: 05/07/2023]
Abstract
A microscopic field theory for crystallization in active systems is proposed which unifies the phase-field-crystal model of freezing with the Toner-Tu theory for self-propelled particles. A wealth of different active crystalline states are predicted and characterized. In particular, for increasing strength of self-propulsion, a transition from a resting crystal to a traveling crystalline state is found where the particles migrate collectively while keeping their crystalline order. Our predictions, which are verifiable in experiments and in particle-resolved computer simulations, provide a starting point for the design of new active materials.
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Affiliation(s)
- Andreas M Menzel
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstrasse 1, D-40225 Düsseldorf, Germany.
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35
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Humadi H, Hoyt JJ, Provatas N. Phase-field-crystal study of solute trapping. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022404. [PMID: 23496523 DOI: 10.1103/physreve.87.022404] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2012] [Indexed: 06/01/2023]
Abstract
In this study we have incorporated two time scales into the phase-field-crystal model of a binary alloy to explore different solute trapping properties as a function of crystal-melt interface velocity. With only diffusive dynamics, we demonstrate that the segregation coefficient, K as a function of velocity for a binary alloy is consistent with the model of Kaplan and Aziz where K approaches unity in the limit of infinite velocity. However, with the introduction of wavelike dynamics in both the density and concentration fields, the trapping follows the kinetics proposed by Sobolev [Phys. Lett. A 199, 383 (1995)], where complete trapping occurs at a finite velocity.
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Affiliation(s)
- Harith Humadi
- Department of Materials Science and Engineering and Brockhouse Institute for Materials Research, McMaster University, 1280 Main Street West, Hamilton, Canada L8S-4L7
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Pisutha-Arnond N, Chan VWL, Iyer M, Gavini V, Thornton K. Classical density functional theory and the phase-field crystal method using a rational function to describe the two-body direct correlation function. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:013313. [PMID: 23410466 DOI: 10.1103/physreve.87.013313] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2012] [Revised: 11/30/2012] [Indexed: 06/01/2023]
Abstract
We introduce a new approach to represent a two-body direct correlation function (DCF) in order to alleviate the computational demand of classical density functional theory (CDFT) and enhance the predictive capability of the phase-field crystal (PFC) method. The approach utilizes a rational function fit (RFF) to approximate the two-body DCF in Fourier space. We use the RFF to show that short-wavelength contributions of the two-body DCF play an important role in determining the thermodynamic properties of materials. We further show that using the RFF to empirically parametrize the two-body DCF allows us to obtain the thermodynamic properties of solids and liquids that agree with the results of CDFT simulations with the full two-body DCF without incurring significant computational costs. In addition, the RFF can also be used to improve the representation of the two-body DCF in the PFC method. Last, the RFF allows for a real-space reformulation of the CDFT and PFC method, which enables descriptions of nonperiodic systems and the use of nonuniform and adaptive grids.
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Affiliation(s)
- N Pisutha-Arnond
- Materials Science and Engineering Department, University of Michigan, Ann Arbor, Michigan 48109, USA.
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37
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Ofori-Opoku N, Hoyt JJ, Provatas N. Phase-field-crystal model of phase and microstructural stability in driven nanocrystalline systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:066706. [PMID: 23368077 DOI: 10.1103/physreve.86.066706] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/18/2012] [Indexed: 06/01/2023]
Abstract
We present a phase-field-crystal model for driven systems which describes competing effects between thermally activated diffusional processes and those driven by externally imposed ballistic events. The model demonstrates how the mesoscopic Enrique and Bellon [Phys. Rev. Lett. 84, 2885 (2000)] model of externally induced ballistic mixing can be incorporated into the atomistic phase-field-crystal formalism. The combination of the two approaches results in a model capable of describing the microstructural and compositional evolution of a driven system while incorporating elastoplastic effects. The model is applied to the study of grain growth in nanocrystalline materials subjected to an external driving.
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Affiliation(s)
- Nana Ofori-Opoku
- Department of Materials Science and Engineering and Brockhouse Institute for Materials Research, McMaster University, 1280 Main Street West, Hamilton, Canada L8S 4L7.
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38
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Kharchenko VO, Kharchenko DO. Nanosize pattern formation in overdamped stochastic reaction-diffusion systems with interacting adsorbate. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:041143. [PMID: 23214565 DOI: 10.1103/physreve.86.041143] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2012] [Revised: 09/17/2012] [Indexed: 05/28/2023]
Abstract
We study overdamped stochastic model describing adsorption or desorption processes with nonequilibrium chemical reactions on the surface. It is shown that internal noise satisfying the fluctuation-dissipation relation at small intensities governs transitions between ordered thermodynamical dense and diluted phases. These phase transitions are characterized by an increase of fluctuations of the coverage filed and correlation radius of spatial modulation. At large noise intensity a transition towards disordered phase with chaotic spatial configuration is realized. We have shown that organized stationary patterns are of nanometer range. We define that both period of stationary structures and corresponding correlation radius depend on the noise intensity.
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Affiliation(s)
- Vasyl O Kharchenko
- Institute of Applied Physics, National Academy of Sciences of Ukraine, 58 Petropavlivska St., 40000 Sumy, Ukraine.
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39
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Choudhary MA, Li D, Emmerich H, Löwen H. DDFT calibration and investigation of an anisotropic phase-field crystal model. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2011; 23:265005. [PMID: 21666297 DOI: 10.1088/0953-8984/23/26/265005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
The anisotropic phase-field crystal model recently proposed and used by Prieler et al (2009 J. Phys.: Condens. Matter 21 464110) is derived from microscopic density functional theory for anisotropic particles with fixed orientation. Its morphology diagram is also explored. In particular we have investigated the influence of anisotropy and undercooling on the process of nucleation and microstructure formation from the atomic to the microscale. To that end numerical simulations were performed varying those dimensionless parameters which represent anisotropy and undercooling in our anisotropic phase-field crystal model. The results from these numerical simulations are summarized in terms of a morphology diagram of the stable state phases. These stable phases are also investigated with respect to their kinetics and characteristic morphological features.
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40
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Ohnogi H, Shiwa Y. Nucleation, growth, and coarsening of crystalline domains in order-order transitions between lamellar and hexagonal phases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:011611. [PMID: 21867186 DOI: 10.1103/physreve.84.011611] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2011] [Revised: 04/04/2011] [Indexed: 05/31/2023]
Abstract
Using the numerical solution of the time-dependent Ginzburg-Landau equation, we study the entire process of transformation between the lamellar and the hexagonal phases from the early-stage nucleation and growth to the late-stage coarsening regime. The metastable crystalline structure that nucleates first is identified in terms of the mean-field theory under the single-wave-number approximation. This has been borne out by the numerically efficient preparation of single-crystal structure developed via the noise-induced self-organization. We also present results for the scaling of the late-time domain growth, which is quantified by two measures: the structure factor and the orientational correlation function. In particular, the growth exponent is shown to be robust and indifferent to conservation of the order parameter.
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Affiliation(s)
- H Ohnogi
- Statistical Mechanics Laboratory, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan
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41
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Berry J, Grant M. Modeling multiple time scales during glass formation with phase-field crystals. PHYSICAL REVIEW LETTERS 2011; 106:175702. [PMID: 21635050 DOI: 10.1103/physrevlett.106.175702] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2011] [Revised: 04/01/2011] [Indexed: 05/30/2023]
Abstract
The dynamics of glass formation in monatomic and binary liquids are studied numerically using a microscopic field theory for the evolution of the time-averaged atomic number density. A stochastic framework combining phase-field crystal free energies and dynamic density functional theory is shown to successfully describe several aspects of glass formation over multiple time scales. Agreement with mode coupling theory is demonstrated for underdamped liquids at moderate supercoolings, and a rapidly growing dynamic correlation length is found to be associated with fragile behavior.
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Affiliation(s)
- Joel Berry
- Physics Department, McGill University, 3600 rue University, Montréal, Québec, Canada H3A 2T8
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42
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Greenwood M, Rottler J, Provatas N. Phase-field-crystal methodology for modeling of structural transformations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:031601. [PMID: 21517507 DOI: 10.1103/physreve.83.031601] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2010] [Indexed: 05/30/2023]
Abstract
We introduce and characterize free-energy functionals for modeling of solids with different crystallographic symmetries within the phase-field-crystal methodology. The excess free energy responsible for the emergence of periodic phases is inspired by classical density-functional theory, but uses only a minimal description for the modes of the direct correlation function to preserve computational efficiency. We provide a detailed prescription for controlling the crystal structure and introduce parameters for changing temperature and surface energies, so that phase transformations between body-centered-cubic (bcc), face-centered-cubic (fcc), hexagonal-close-packed (hcp), and simple-cubic (sc) lattices can be studied. To illustrate the versatility of our free-energy functional, we compute the phase diagram for fcc-bcc-liquid coexistence in the temperature-density plane. We also demonstrate that our model can be extended to include hcp symmetry by dynamically simulating hcp-liquid coexistence from a seeded crystal nucleus. We further quantify the dependence of the elastic constants on the model control parameters in two and three dimensions, showing how the degree of elastic anisotropy can be tuned from the shape of the direct correlation functions.
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Affiliation(s)
- Michael Greenwood
- Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T1Z1, Canada
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43
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Lebedev V, Sysoeva A, Galenko P. Unconditionally gradient-stable computational schemes in problems of fast phase transitions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:026705. [PMID: 21405928 DOI: 10.1103/physreve.83.026705] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2010] [Indexed: 05/11/2023]
Abstract
Equations of fast phase transitions, in which the phase boundaries move with velocities comparable with the atomic diffusion speed or with the speed of local structural relaxation, are analyzed. These equations have a singular perturbation due to the second derivative of the order parameter with respect to time, which appears due to phenomenologically introduced local nonequilibrium. To develop unconditionally stable computational schemes, the Eyre theorem [D. J. Eyre, unpublished] proved for the classical equations, based on hypotheses of local equilibrium, is used. An extension of the Eyre theorem for the case of equations for fast phase transitions is given. It is shown that the expansion of the free energy on contractive and expansive parts, suggested by Eyre for the classical equations of Cahn-Hilliard and Allen-Cahn, is also true for the equations of fast phase transitions. Grid approximations of these equations lead to gradient-stable algorithms with an arbitrary time step for numerical modeling, ensuring monotonic nonincrease of the free energy. Special examples demonstrating the extended Eyre theorem for fast phase transitions are considered.
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Affiliation(s)
- Vladimir Lebedev
- Department of Theoretical Physics, Udmurt State University, 426034 Izhevsk, Russia
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44
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Jaatinen A, Ala-Nissila T. Eighth-order phase-field-crystal model for two-dimensional crystallization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:061602. [PMID: 21230677 DOI: 10.1103/physreve.82.061602] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2010] [Revised: 11/19/2010] [Indexed: 05/30/2023]
Abstract
We present a derivation of the recently proposed eighth-order phase-field crystal model [A. Jaatinen, Phys. Rev. E 80, 031602 (2009)] for the crystallization of a solid from an undercooled melt. The model is used to study the planar growth of a two-dimensional hexagonal crystal, and the results are compared against similar results from dynamical density functional theory of Marconi and Tarazona, as well as other phase-field crystal models. We find that among the phase-field crystal models studied, the eighth-order fitting scheme gives results in good agreement with the density functional theory for both static and dynamic properties, suggesting it is an accurate and computationally efficient approximation to the density functional theory.
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Affiliation(s)
- A Jaatinen
- Department of Applied Physics, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Finland
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45
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Provatas N, Majaniemi S. Phase-field-crystal calculation of crystal-melt surface tension in binary alloys. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:041601. [PMID: 21230281 DOI: 10.1103/physreve.82.041601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2010] [Indexed: 05/30/2023]
Abstract
A phase field crystal (PFC) density functional for binary mixtures is coarse grained and a formalism for calculating the simultaneous concentration, temperature, and density dependence of the surface energy anisotropy of a solid-liquid interface is developed. The methodology systematically relates bulk free energy coefficients arising from coarse graining to thermodynamic data, while gradient energy coefficients are related to molecular properties. Our coarse-grained formalism is applied to the determination of surface energy anisotropy in two-dimensional Zn-Al films, a situation relevant for quantitative phase field simulations of dendritic solidification in zinc coatings.
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Affiliation(s)
- Nikolas Provatas
- Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7.
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46
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Muralidharan S, Haataja M. Phase-field crystal modeling of compositional domain formation in ultrathin films. PHYSICAL REVIEW LETTERS 2010; 105:126101. [PMID: 20867659 DOI: 10.1103/physrevlett.105.126101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2010] [Indexed: 05/29/2023]
Abstract
Bulk-immiscible binary systems often form stress-induced miscible alloy phases when deposited on a substrate. Both alloying and surface dislocation formation lead to the decrease of the elastic strain energy, and the competition between these two strain-relaxation mechanisms gives rise to the emergence of pseudomorphic compositional nanoscale domains, often coexisting with a partially coherent single phase. In this work, we develop a phase-field crystal model for compositional patterning in monolayer aggregates of binary metallic systems. We first demonstrate that the model naturally incorporates the competition between alloying and misfit dislocations, and quantify the effects of misfit and line tension on equilibrium domain size. Then, we quantitatively relate the parameters of the phase-field crystal model to a specific system, CoAg/Ru(0001), and demonstrate that the simulations capture experimentally observed morphologies.
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Affiliation(s)
- Srevatsan Muralidharan
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA.
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47
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Majaniemi S, Provatas N, Nonomura M. Effective model hierarchies for dynamic and static classical density functional theories. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2010; 22:364111. [PMID: 21386527 DOI: 10.1088/0953-8984/22/36/364111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
The origin and methodology of deriving effective model hierarchies are presented with applications to solidification of crystalline solids. In particular, it is discussed how the form of the equations of motion and the effective parameters on larger scales can be obtained from the more microscopic models. It will be shown that tying together the dynamic structure of the projection operator formalism with static classical density functional theories can lead to incomplete (mass) transport properties even though the linearized hydrodynamics on large scales is correctly reproduced. To facilitate a more natural way of binding together the dynamics of the macrovariables and classical density functional theory, a dynamic generalization of density functional theory based on the nonequilibrium generating functional is suggested.
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Affiliation(s)
- S Majaniemi
- Department of Applied Physics, Aalto University School of Science and Technology, PO Box 11100, FI-00076 Aalto, Finland.
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48
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Huang ZF, Elder KR, Provatas N. Phase-field-crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:021605. [PMID: 20866824 DOI: 10.1103/physreve.82.021605] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2010] [Indexed: 05/11/2023]
Abstract
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, zeroth-mode 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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Affiliation(s)
- Zhi-Feng Huang
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
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49
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Greenwood M, Provatas N, Rottler J. Free energy functionals for efficient phase field crystal modeling of structural phase transformations. PHYSICAL REVIEW LETTERS 2010; 105:045702. [PMID: 20867862 DOI: 10.1103/physrevlett.105.045702] [Citation(s) in RCA: 38] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2009] [Revised: 06/20/2010] [Indexed: 05/11/2023]
Abstract
The phase field crystal (PFC) method is a promising technique for modeling materials with atomic resolution on mesoscopic time scales. While numerically more efficient than classical density functional theory (CDFT), its single mode free energy limits the complexity of structural transformations that can be simulated. We introduce a new PFC model inspired by CDFT, which uses a systematic construction of two-particle correlation functions that allows for a broad class of structural transformations. Our approach considers planar spacings, lattice symmetries, planar atomic densities, and atomic vibrational amplitudes in the unit cell, and parameterizes temperature and anisotropic surface energies. The power of our approach is demonstrated by two examples of structural phase transformations.
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Affiliation(s)
- Michael Greenwood
- Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road Vancouver, British Columbia, V6T 1Z1, Canada
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50
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Chan PY, Tsekenis G, Dantzig J, Dahmen KA, Goldenfeld N. Plasticity and dislocation dynamics in a phase field crystal model. PHYSICAL REVIEW LETTERS 2010; 105:015502. [PMID: 20867460 DOI: 10.1103/physrevlett.105.015502] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2010] [Indexed: 05/29/2023]
Abstract
The critical dynamics of dislocation avalanches in plastic flow is examined using a phase field crystal model. In the model, dislocations are naturally created, without any ad hoc creation rules, by applying a shearing force to the perfectly periodic ground state. These dislocations diffuse, interact and annihilate with one another, forming avalanche events. By data collapsing the event energy probability density function for different shearing rates, a connection to interface depinning dynamics is confirmed. The relevant critical exponents agree with mean field theory predictions.
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Affiliation(s)
- Pak Yuen Chan
- Department of Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, 1110 West Green Street, Urbana, Illinois, 61801-3080, USA
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