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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] [What about the content of this article? (0)] [Affiliation(s)] [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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2
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Backofen R, Elder KR, Voigt A. Controlling Grain Boundaries by Magnetic Fields. Phys Rev Lett 2019; 122:126103. [PMID: 30978082 DOI: 10.1103/physrevlett.122.126103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2018] [Indexed: 06/09/2023]
Abstract
The ability to use external magnetic fields to influence the microstructure in polycrystalline materials has potential applications in microstructural engineering. To explore this potential and to understand the complex interactions between electromagnetic fields and solid-state matter transport we consider a phase-field-crystal model. Together with efficient and scalable numerical algorithms this allows the examination of the role that external magnetic fields play on the evolution of defect structures and grain boundaries, on diffusive timescales. Examples for planar and circular grain boundaries explain the essential atomistic processes and large scale simulations in 2D are used to obtain statistical data on grain growth under the influence of external fields.
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Affiliation(s)
- R Backofen
- Institute of Scientific Computing, Technische Universität Dresden, 01062 Dresden, Germany
| | - K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309, USA
| | - A Voigt
- Institute of Scientific Computing, Technische Universität Dresden, 01062 Dresden, Germany
- Dresden Center for Computational Materials Science (DCMS), 01062 Dresden, Germany
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3
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Taha D, Mkhonta SK, Elder KR, Huang ZF. Grain Boundary Structures and Collective Dynamics of Inversion Domains in Binary Two-Dimensional Materials. Phys Rev Lett 2017; 118:255501. [PMID: 28696726 DOI: 10.1103/physrevlett.118.255501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2017] [Indexed: 05/11/2023]
Abstract
Understanding and controlling the properties and dynamics of topological defects is a lasting challenge in the study of two-dimensional materials, and is crucial to achieve high-quality films required for technological applications. Here grain boundary structures, energies, and dynamics of binary two-dimensional materials are investigated through the development of a phase field crystal model that is parametrized to match the ordering, symmetry, energy, and length scales of hexagonal boron nitride. Our studies reveal some new dislocation core structures for various symmetrically and asymmetrically tilted grain boundaries, in addition to those obtained in previous experiments and first-principles calculations. We also identify a defect-mediated growth dynamics for inversion domains governed by the collective atomic migration and defect core transformation at grain boundaries and junctions, a process that is related to inversion symmetry breaking in binary lattice.
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Affiliation(s)
- Doaa Taha
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
| | - S K Mkhonta
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
- Department of Physics, University of Swaziland, Private Bag 4, Kwaluseni M201, Swaziland
| | - K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309, USA
| | - Zhi-Feng Huang
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
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4
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Abstract
We describe a general method to model multicomponent ordered crystals using the phase-field-crystal (PFC) formalism. As a test case, a generic B2 compound is investigated. We are able to produce a line of either first-order or second-order order-disorder phase transitions, features that have not been incorporated in existing PFC approaches. Further, it is found that the only elastic constant for B2 that depends on ordering is C_{11}. This B2 model is then used to study antiphase boundaries (APBs). The APBs are shown to reproduce classical mean-field results. Dynamical simulations of ordering across small-angle grain boundaries predict that dislocation cores pin the evolution of APBs.
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Affiliation(s)
- Eli Alster
- Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USA
- Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
| | - K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309, USA
| | - Jeffrey J Hoyt
- Department of Materials Science and Engineering and Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario, Canada L8S-4L7
| | - Peter W Voorhees
- Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
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Mkhonta SK, Elder KR, Huang ZF. Emergence of Chirality from Isotropic Interactions of Three Length Scales. Phys Rev Lett 2016; 116:205502. [PMID: 27258877 DOI: 10.1103/physrevlett.116.205502] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2015] [Indexed: 06/05/2023]
Abstract
Chirality is known to play a pivotal role in determining material properties and functionalities. However, it remains a great challenge to understand and control the emergence of chirality and the related enantioselective process particularly when the building components of the system are achiral. Here we explore the generic mechanisms driving the formation of two-dimensional chiral structures in systems characterized by isotropic interactions and three competing length scales. We demonstrate that starting from isotropic and rotationally invariant interactions, a variety of chiral ordered patterns and superlattices with anisotropic but achiral units can self-assemble. The mechanisms for selecting specific states are related to the length-scale coupling and the selection of resonant density wave vectors. Sample phase diagrams and chiral elastic properties are identified. These findings provide a viable route for predicting chiral phases and selecting the desired handedness.
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Affiliation(s)
- S K Mkhonta
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
- Department of Physics, University of Swaziland, Private Bag 4, Kwaluseni M201, Swaziland
| | - K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309, USA
| | - Zhi-Feng Huang
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
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Elder KR, Chen Z, Elder KLM, Hirvonen P, Mkhonta SK, Ying SC, Granato E, Huang ZF, Ala-Nissila T. Honeycomb and triangular domain wall networks in heteroepitaxial systems. J Chem Phys 2016; 144:174703. [PMID: 27155643 DOI: 10.1063/1.4948370] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A comprehensive study is presented for the influence of misfit strain, adhesion strength, and lattice symmetry on the complex Moiré patterns that form in ultrathin films of honeycomb symmetry adsorbed on compact triangular or honeycomb substrates. The method used is based on a complex Ginzburg-Landau model of the film that incorporates elastic strain energy and dislocations. The results indicate that different symmetries of the heteroepitaxial systems lead to distinct types of domain wall networks and phase transitions among various surface Moiré patterns and superstructures. More specifically, the results show a dramatic difference between the phase diagrams that emerge when a honeycomb film is adsorbed on substrates of honeycomb versus triangular symmetry. It is also shown that in the small deformation limit, the complex Ginzburg-Landau model reduces to a two-dimensional sine-Gordon free energy form. This free energy can be solved exactly for one dimensional patterns and reveals the role of domains walls and their crossings in determining the nature of the phase diagrams.
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Affiliation(s)
- K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309, USA
| | - Z Chen
- Department of Physics, Oakland University, Rochester, Michigan 48309, USA
| | - K L M Elder
- Department of Applied Physics and COMP Centre of Excellence, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Finland
| | - P Hirvonen
- Department of Applied Physics and COMP Centre of Excellence, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Finland
| | - S K Mkhonta
- Department of Physics, University of Swaziland, Private Bag 4, Kwaluseni, Swaziland
| | - S-C Ying
- Department of Physics, Brown University, P.O. Box 1843, Providence, Rhode Island 02912, USA
| | - E Granato
- Department of Physics, Brown University, P.O. Box 1843, Providence, Rhode Island 02912, USA
| | - Zhi-Feng Huang
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
| | - T Ala-Nissila
- Department of Applied Physics and COMP Centre of Excellence, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Finland
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Heinonen V, Achim CV, Elder KR, Buyukdagli S, Ala-Nissila T. Phase-field-crystal models and mechanical equilibrium. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 89:032411. [PMID: 24730856 DOI: 10.1103/physreve.89.032411] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [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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8
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Mkhonta SK, Elder KR, Huang ZF, Grant M. Microphase separation in comblike liquid-crystalline diblock copolymers. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 88:042602. [PMID: 24229201 DOI: 10.1103/physreve.88.042602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2013] [Revised: 07/30/2013] [Indexed: 06/02/2023]
Abstract
The interplay between liquid crystallinity and microphase separation in comblike liquid-crystalline diblock copolymers is examined via a Brazovskii-type phenomenological model using both analytical and numerical calculations. For symmetric diblock copolymers we determine a critical electric field that is required to tilt the orientation of the constituent liquid crystals of the polymer side chains in the microphase-separated lamellar state. Such electrically induced reorientation of the liquid-crystal molecules can lead to substantially large changes of lamellar periodicity. Our numerical results show that highly aligned polymer lamellar domains can self-assemble when the liquid-crystal ordering precedes microphase separation, and that weak electric fields can be used to direct the self-assembly process due to the dielectric anisotropy of the liquid-crystal side chains. We also find that phase separation of asymmetric diblock copolymers can coexist with a network of liquid-crystal nematic orientations, with domain morphology depending on the details of copolymer and liquid-crystal coupling.
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Affiliation(s)
- S K Mkhonta
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA and Department of Physics, University of Swaziland, Private Bag 4, Kwaluseni M201, Swaziland
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9
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Granato E, Ying SC, Elder KR, Ala-Nissila T. Anomalous fast dynamics of adsorbate overlayers near an incommensurate structural transition. Phys Rev Lett 2013; 111:126102. [PMID: 24093278 DOI: 10.1103/physrevlett.111.126102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2013] [Indexed: 06/02/2023]
Abstract
We investigate the dynamics of a compressively strained adsorbed layer on a periodic substrate via a simple two-dimensional model that admits striped and hexagonal incommensurate phases. We show that the mass transport is superfast near the striped-hexagonal phase boundary and in the hexagonal phase. For an initial step profile separating a bare substrate region (or "hole") from the rest of a striped incommensurate phase, the superfast domain wall dynamics leads to a bifurcation of the initial step profile into two interfaces or profiles propagating in opposite directions with a hexagonal phase in between. This yields a theoretical understanding of the recent experiments for the Pb/Si(111) system.
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Affiliation(s)
- Enzo Granato
- Laboratório Associado de Sensores e Materiais, Instituto Nacional de Pesquisas Espaciais, 12245-970 São José dos Campos, São Paulo, Brazil and Department of Physics, Post Office Box 1843, Brown University, Providence, Rhode Island 02912-1843, USA
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10
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Faghihi N, Provatas N, Elder KR, Grant M, Karttunen M. Phase-field-crystal model for magnetocrystalline interactions in isotropic ferromagnetic solids. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 88:032407. [PMID: 24125276 DOI: 10.1103/physreve.88.032407] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2013] [Indexed: 06/02/2023]
Abstract
An isotropic magnetoelastic phase-field-crystal model to study the relation between morphological structure and magnetic properties of pure ferromagnetic solids is introduced. Analytic calculations in two dimensions were used to determine the phase diagram and obtain the relationship between elastic strains and magnetization. Time-dependent numerical simulations in two dimensions were used to demonstrate the effect of grain boundaries on the formation of magnetic domains. It was shown that the grain boundaries act as nucleating sites for domains of reverse magnetization. Finally, we derive a relation for coercivity versus grain misorientation in the isotropic limit.
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Affiliation(s)
- Niloufar Faghihi
- Department of Applied Mathematics, The University of Western Ontario, 1151 Richmond St. N., London, Ontario, Canada N6A 5B7 and Department of Physics, McGill University, 3600 rue University, Montréal, Québec, Canada H3A 2T8
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11
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Mkhonta SK, Elder KR, Huang ZF. Exploring the complex world of two-dimensional ordering with three modes. Phys Rev Lett 2013; 111:035501. [PMID: 23909335 DOI: 10.1103/physrevlett.111.035501] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2013] [Indexed: 05/11/2023]
Abstract
The world of two-dimensional crystals is of great significance for the design and study of structural and functional materials with novel properties. Here we examine the mechanisms governing the formation and dynamics of these crystalline or polycrystalline states and their elastic and plastic properties by constructing a generic multimode phase field crystal model. Our results demonstrate that a system with three competing length scales can order into all five Bravais lattices, and other more complex structures including honeycomb, kagome, and other hybrid phases. In addition, nonequilibrium phase transitions are examined to illustrate the complex phase behavior described by the model. This model provides a systematic path to predict the influence of lattice symmetry on both the structure and dynamics of crystalline and defected systems.
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Affiliation(s)
- S K Mkhonta
- Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA
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12
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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. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 88:013310. [PMID: 23944586 DOI: 10.1103/physreve.88.013310] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [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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13
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Elder KR, Rossi G, Kanerva P, Sanches F, Ying SC, Granato E, Achim CV, Ala-Nissila T. Patterning of heteroepitaxial overlayers from nano to micron scales. Phys Rev Lett 2012; 108:226102. [PMID: 23003626 DOI: 10.1103/physrevlett.108.226102] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2011] [Indexed: 05/11/2023]
Abstract
Thin heteroepitaxial overlayers have been proposed as templates to generate stable, self-organized nanostructures at large length scales, with a variety of important technological applications. However, modeling strain-driven self-organization is a formidable challenge due to different length scales involved. In this Letter, we present a method for predicting the patterning of ultrathin films on micron length scales with atomic resolution. We make quantitative predictions for the type of superstructures (stripes, honeycomb, triangular) and length scale of pattern formation of two metal-metal systems, Cu on Ru(0001) and Cu on Pd(111). Our findings are in excellent agreement with previous experiments and call for future experimental investigations of such systems.
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Affiliation(s)
- K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309, USA.
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14
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Granato E, Ramos JAP, Achim CV, Lehikoinen J, Ying SC, Ala-Nissila T, Elder KR. Glassy phases and driven response of the phase-field-crystal model with random pinning. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 84:031102. [PMID: 22060323 DOI: 10.1103/physreve.84.031102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2011] [Revised: 07/10/2011] [Indexed: 05/31/2023]
Abstract
We study the structural correlations and the nonlinear response to a driving force of a two-dimensional phase-field-crystal model with random pinning. The model provides an effective continuous description of lattice systems in the presence of disordered external pinning centers, allowing for both elastic and plastic deformations. We find that the phase-field crystal with disorder assumes an amorphous glassy ground state, with only short-ranged positional and orientational correlations, even in the limit of weak disorder. Under increasing driving force, the pinned amorphous-glass phase evolves into a moving plastic-flow phase and then, finally, a moving smectic phase. The transverse response of the moving smectic phase shows a vanishing transverse critical force for increasing system sizes.
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Affiliation(s)
- E Granato
- Laboratório Associado de Sensores e Materiais, Instituto Nacional de Pesquisas Espaciais, 12227-010 São José dos Campos, SP, Brazil
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15
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Abstract
In this paper the liquid phase heteroepitaxial growth of two-dimensional strained islands on nanomembranes is examined via an amplitude expansion of a binary phase field crystal model. The maximum size that the islands can grow to coherently is shown to be strongly dependent on the nanomembrane thickness and to a lesser extent on the flux rate. For a large membrane-island misfit of about 10%, islands were found to be able to grow coherently with the membrane to almost twice as large as those grown on thick membranes (or infinite substrates). It was also found that when islands are growing on both sides of the membrane, strain relaxation in the membrane leads to more and less favorable growth regions. For thinner membranes this effect increases the degree of ordering of the islands, a result consistent with recent experimental findings.
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Affiliation(s)
- K R Elder
- Department of Physics, Oakland University, Rochester, MI 48309, USA
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16
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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. Phys Rev E Stat Nonlin Soft Matter Phys 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] [What about the content of this article? (0)] [Affiliation(s)] [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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17
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Mkhonta SK, Elder KR, Grant M. Novel mechanical properties in lamellar phases of liquid-crystalline diblock copolymers. Eur Phys J E Soft Matter 2010; 32:349-355. [PMID: 20680381 DOI: 10.1140/epje/i2010-10636-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2010] [Accepted: 07/06/2010] [Indexed: 05/29/2023]
Abstract
Structural properties of flexible nematic diblock copolymers in the lamellar phase are investigated using a mean-field model. We address two complementary questions on the mechanics of the system: 1) How does the nematic order affect the elasticity of the one-dimensional solid? 2) What effect does the block copolymer microstructure has on the orientation of the nematic director? In the limit when the microstructure does not influence the nematic director orientation we predict a soft lamellar compression mode. When the microstructure does influence the nematic director orientation, small compressions lead to conventional elasticity, until a critical strain is reached, where there is then a transition to a softer response. On the other hand, we show that an identifiable lamellar symmetry provides a direction along which the nematic director prefers to align. Our model provides avenues to explore nonlinear properties of flexible diblock copolymers in which the monomers on both sides have mesogenic side groups.
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Affiliation(s)
- S K Mkhonta
- Department of Physics, University of Swaziland, Kwaluseni, Swaziland.
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18
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Elder KR, Huang ZF, Provatas N. Amplitude expansion of the binary phase-field-crystal model. Phys Rev E Stat Nonlin Soft Matter Phys 2010; 81:011602. [PMID: 20365379 DOI: 10.1103/physreve.81.011602] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2009] [Indexed: 05/11/2023]
Abstract
Amplitude representations of a binary phase-field-crystal model are developed for a two-dimensional triangular lattice and three-dimensional bcc and fcc crystal structures. The relationship between these amplitude equations and the standard phase-field models for binary-alloy solidification with elasticity are derived, providing an explicit connection between phase-field-crystal and phase-field models. Sample simulations of solute migration at grain boundaries, eutectic solidification, and quantum dot formation on nanomembranes are also presented.
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Affiliation(s)
- K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309, USA
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19
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Ramos JAP, Granato E, Ying SC, Achim CV, Elder KR, Ala-Nissila T. Dynamical transitions and sliding friction of the phase-field-crystal model with pinning. Phys Rev E Stat Nonlin Soft Matter Phys 2010; 81:011121. [PMID: 20365337 DOI: 10.1103/physreve.81.011121] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2009] [Indexed: 05/11/2023]
Abstract
We study the nonlinear driven response and sliding friction behavior of the phase-field-crystal (PFC) model with pinning including both thermal fluctuations and inertial effects. The model provides a continuous description of adsorbed layers on a substrate under the action of an external driving force at finite temperatures, allowing for both elastic and plastic deformations. We derive general stochastic dynamical equations for the particle and momentum densities including both thermal fluctuations and inertial effects. The resulting coupled equations for the PFC model are studied numerically. At sufficiently low temperatures, we find that the velocity response of an initially pinned commensurate layer shows hysteresis with dynamical melting and freezing transitions for increasing and decreasing applied forces at different critical values. The main features of the nonlinear response in the PFC model are similar to the results obtained previously with molecular dynamics simulations of particle models for adsorbed layers.
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Affiliation(s)
- J A P Ramos
- Departamento de Ciências Exatas, Universidade Estadual do Sudoeste da Bahia, Vitória da Conquista, BA, Brazil
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Jaatinen A, Achim CV, Elder KR, Ala-Nissila T. Thermodynamics of bcc metals in phase-field-crystal models. Phys Rev E Stat Nonlin Soft Matter Phys 2009; 80:031602. [PMID: 19905118 DOI: 10.1103/physreve.80.031602] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2009] [Revised: 07/16/2009] [Indexed: 05/28/2023]
Abstract
We examine the influence of different forms of the free-energy functionals used in the phase-field-crystal (PFC) model, and compare them with the second-order density-functional theory (DFT) of freezing, by using bcc iron as an example case. We show that there are large differences between the PFC and the DFT and it is difficult to obtain reasonable parameters for existing PFC models directly from the DFT. Therefore, we propose a way of expanding the correlation function in terms of gradients that allows us to incorporate the bulk modulus of the liquid as an additional parameter in the theory. We show that this functional reproduces reasonable values for both bulk and surface properties of bcc iron, and therefore it should be useful in modeling bcc materials. As a further demonstration, we also calculate the grain boundary energy as a function of misorientation for a symmetric tilt boundary close to the melting transition.
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Affiliation(s)
- A Jaatinen
- Department of Applied Physics, COMP Center of Excellence, Helsinki University of Technology, PO Box 1100, Helsinki FIN-02015 TKK, Finland
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Achim CV, Ramos JAP, Karttunen M, Elder KR, Granato E, Ala-Nissila T, Ying SC. Nonlinear driven response of a phase-field crystal in a periodic pinning potential. Phys Rev E Stat Nonlin Soft Matter Phys 2009; 79:011606. [PMID: 19257044 DOI: 10.1103/physreve.79.011606] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2008] [Revised: 12/22/2008] [Indexed: 05/27/2023]
Abstract
We study numerically the phase diagram and the response under a driving force of the phase field crystal model for pinned lattice systems introduced recently for both one- and two-dimensional systems. The model describes the lattice system as a continuous density field in the presence of a periodic pinning potential, allowing for both elastic and plastic deformations of the lattice. We first present results for phase diagrams of the model in the absence of a driving force. The nonlinear response to a driving force on an initially pinned commensurate phase is then studied via overdamped dynamic equations of motion for different values of mismatch and pinning strengths. For large pinning strength the driven depinning transitions are continuous, and the sliding velocity varies with the force from the threshold with power-law exponents in agreement with analytical predictions. Transverse depinning transitions in the moving state are also found in two dimensions. Surprisingly, for sufficiently weak pinning potential we find a discontinuous depinning transition with hysteresis even in one dimension under overdamped dynamics. We also characterize structural changes of the system in some detail close to the depinning transition.
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Affiliation(s)
- C V Achim
- Department of Applied Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 TKK, Espoo, Finland
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Abstract
The instability of strained films for island formation is examined through an approach incorporating both discrete microscopic details and continuum mechanics. A linear relationship between the island wave number and misfit strain is found for large strains, while only in the small strain limit is a crossover to the continuum elasticity result obtained. A universal scaling relation accommodating all range of misfit strains is identified. Our results indicate that continuum mechanics may break down even at relatively small misfit stress due to the discrete nature of crystalline surfaces.
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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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Ramos JAP, Granato E, Achim CV, Ying SC, Elder KR, Ala-Nissila T. Thermal fluctuations and phase diagrams of the phase-field crystal model with pinning. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 78:031109. [PMID: 18850995 DOI: 10.1103/physreve.78.031109] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2008] [Indexed: 05/26/2023]
Abstract
We study the influence of thermal fluctuations in the phase diagram of a recently introduced two-dimensional phase field crystal model with an external pinning potential. The model provides a continuum description of pinned lattice systems allowing for both elastic deformations and topological defects. We introduce a nonconserved version of the model and determine the ground-state phase diagram as a function of lattice mismatch and strength of the pinning potential. Monte Carlo simulations are used to determine the phase diagram as a function of temperature near commensurate phases. The results show a rich phase diagram with commensurate, incommensurate, and liquidlike phases with a topology strongly dependent on the type of ordered structure. A finite-size scaling analysis of the melting transition for the c(2x2) commensurate phase shows that the thermal correlation length exponent nu and specific heat behavior are consistent with the Ising universality class as expected from analytical arguments.
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Affiliation(s)
- J A P Ramos
- Departamento de Ciências Exatas, Universidade Estadual do Sudoeste da Bahia, 45000-000 Vitória da Conquista, Ba, Brasil
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Jiang F, Munkholm A, Wang RV, Streiffer SK, Thompson C, Fuoss PH, Latifi K, Elder KR, Stephenson GB. Spontaneous oscillations and waves during chemical vapor deposition of InN. Phys Rev Lett 2008; 101:086102. [PMID: 18764640 DOI: 10.1103/physrevlett.101.086102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2008] [Indexed: 05/26/2023]
Abstract
We report observations of self-sustaining spatiotemporal chemical oscillations during metal-organic chemical vapor deposition of InN onto GaN. Under constant supply of vapor precursors trimethylindium and NH3, the condensed-phase cycles between crystalline islands of InN and elemental In droplets. Propagating fronts between regions of InN and In occur with linear, circular, and spiral geometries. The results are described by a model in which the nitrogen activity produced by surface-catalyzed NH3 decomposition varies with the exposed surface areas of GaN, InN, and In.
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Affiliation(s)
- F Jiang
- Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
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Berry J, Elder KR, Grant M. Simulation of an atomistic dynamic field theory for monatomic liquids: freezing and glass formation. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 77:061506. [PMID: 18643271 DOI: 10.1103/physreve.77.061506] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2007] [Revised: 03/30/2008] [Indexed: 05/26/2023]
Abstract
We examine a phase field crystal model for simple liquid-solid systems consisting of a free energy functional related to the Ramakrishnan-Yussouff free energy of classical density functional theory and an equation of motion capable of describing long-time-scale behavior in the deeply supercooled regime. The thermodynamics and dynamics of freezing and glass formation in this model system are studied through large-scale three-dimensional Langevin simulations. At low cooling rates bcc crystals are formed by nucleation and growth from the melt. At large cooling rates no clear glass transition is observed, but a kinetically driven first-order transition from supercooled liquid to a disordered glasslike solid does occur. Despite the peculiarities of the transition, the structure and properties of the resulting disordered solid are shown to strongly resemble those of a typical glass. Consequences of pseudocritical behavior and heterogeneity near the liquid spinodal are also discussed.
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Affiliation(s)
- Joel Berry
- Physics Department, McGill University, Rutherford Building, 3600 rue University, Montréal, Québec, Canada
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Achim CV, Karttunen M, Elder KR, Granato E, Ala-Nissila T, Ying SC. Phase diagram and commensurate-incommensurate transitions in the phase field crystal model with an external pinning potential. Phys Rev E Stat Nonlin Soft Matter Phys 2006; 74:021104. [PMID: 17025390 DOI: 10.1103/physreve.74.021104] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2006] [Indexed: 05/12/2023]
Abstract
We study the phase diagram and the commensurate-incommensurate transitions in a phase field model of a two-dimensional crystal lattice in the presence of an external pinning potential. The model allows for both elastic and plastic deformations and provides a continuum description of lattice systems, such as for adsorbed atomic layers or two-dimensional vortex lattices. Analytically, a mode expansion analysis is used to determine the ground states and the commensurate-incommensurate transitions in the model as a function of the strength of the pinning potential and the lattice mismatch parameter. Numerical minimization of the corresponding free energy shows reasonable agreement with the analytical predictions and provides details on the topological defects in the transition region. We find that for small mismatch the transition is of first order, and it remains so for the largest values of mismatch studied here. Our results are consistent with results of simulations for atomistic models of adsorbed overlayers.
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Affiliation(s)
- C V Achim
- Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 TKK, Finland
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Berry J, Grant M, Elder KR. Diffusive atomistic dynamics of edge dislocations in two dimensions. Phys Rev E Stat Nonlin Soft Matter Phys 2006; 73:031609. [PMID: 16605539 DOI: 10.1103/physreve.73.031609] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2005] [Indexed: 05/08/2023]
Abstract
The fundamental dislocation processes of glide, climb, and annihilation are studied on diffusive time scales within the framework of a continuum field theory, the phase field crystal model. Glide and climb are examined for single edge dislocations subjected to shear and compressive strain, respectively, in a two-dimensional hexagonal lattice. It is shown that the natural features of these processes are reproduced without any explicit consideration of elasticity theory or ad hoc construction of microscopic Peierls potentials. Particular attention is paid to the Peierls barrier for dislocation glide or climb and the ensuing dynamic behavior as functions of strain rate, temperature, and dislocation density. It is shown that the dynamics are accurately described by simple viscous motion equations for an overdamped point mass, where the dislocation mobility is the only adjustable parameter. The critical distance for the annihilation of two edge dislocations as a function of separation angle is also presented.
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Affiliation(s)
- J Berry
- Physics Department, Rutherford Building, 3600 rue University, McGill University, Montréal, Québec, Canada H3A 2T8
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Elder KR, Grant M. Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys Rev E Stat Nonlin Soft Matter Phys 2004; 70:051605. [PMID: 15600626 DOI: 10.1103/physreve.70.051605] [Citation(s) in RCA: 50] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/26/2003] [Revised: 05/07/2004] [Indexed: 05/11/2023]
Abstract
A continuum field theory approach is presented for modeling elastic and plastic deformation, free surfaces, and multiple crystal orientations in nonequilibrium processing phenomena. Many basic properties of the model are calculated analytically, and numerical simulations are presented for a number of important applications including, epitaxial growth, material hardness, grain growth, reconstructive phase transitions, and crack propagation.
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Affiliation(s)
- K R Elder
- Department of Physics, Oakland University, Rochester, MI 48309-4487, USA
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Karttunen M, Elder KR, Tarlie MB, Grant M. Instabilities and resistance fluctuations in thin accelerated superconducting rings. Phys Rev E Stat Nonlin Soft Matter Phys 2002; 66:026115. [PMID: 12241245 DOI: 10.1103/physreve.66.026115] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2001] [Indexed: 11/07/2022]
Abstract
The nonequilibrium properties of a driven quasi-one-dimensional superconducting ring subjected to a constant electromotive force (emf) is studied. The emf accelerates the superconducting electrons until the critical current is reached and a dissipative phase slip occurs that lowers the current. The phase-slip phenomena is examined as a function of the strength of the emf, thermal noise, and normal state resistivity. Numerical and analytic methods are used to make detailed predictions for the magnitude of phase slips and subsequent dissipation.
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Affiliation(s)
- Mikko Karttunen
- Department of Physics and Centre for the Physics of Materials, McGill University, 3600 rue University, Montréal, Québec, Canada H3A 2T8
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Abstract
A new model of crystal growth is presented that describes the phenomena on atomic length and diffusive time scales. The former incorporates elastic and plastic deformation in a natural manner, and the latter enables access to time scales much larger than conventional atomic methods. The model is shown to be consistent with the predictions of Read and Shockley for grain boundary energy, and Matthews and Blakeslee for misfit dislocations in epitaxial growth.
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Affiliation(s)
- K R Elder
- Department of Physics, Oakland University, Rochester, MI 48309-4487, USA
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Dubé M, Majaniemi S, Rost M, Alava MJ, Elder KR, Ala-Nissila T. Interface pinning in spontaneous imbibition. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 64:051605. [PMID: 11735937 DOI: 10.1103/physreve.64.051605] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/01/2001] [Indexed: 05/23/2023]
Abstract
Evaporation and gravity induced pinning in spontaneous imbibition are examined within a phase field formalism. Evaporation is introduced via a nonconserving term and gravity through a convective term that constrains the influx of liquid. Their effects are described by dimensionless coupling constants epsilon and g, respectively. From liquid conservation, the early time behavior of the average interface position follows H(t) approximately t1/2 until a crossover time t*(g,epsilon). After that the pinning height Hp(g,epsilon) is approached exponentially in time, in accordance with mean field theory. The statistical roughness of the interface is described by an exponent chi approximately 1.25 at all stages of the rise, but the dynamic length scale controlling roughness crosses over from xi(x) approximately H1/2 to a time independent pinning length scale xi(p)(epsilon,g).
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Affiliation(s)
- M Dubé
- Centre for the Physics of Materials, McGill University, Montréal, Québec, Canada H3A 2T8
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Elder KR, Grant M, Provatas N, Kosterlitz JM. Sharp interface limits of phase-field models. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 64:021604. [PMID: 11497600 DOI: 10.1103/physreve.64.021604] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/25/2000] [Indexed: 05/23/2023]
Abstract
The use of continuum phase-field models to describe the motion of well-defined interfaces is discussed for a class of phenomena that includes order-disorder transitions, spinodal decomposition and Ostwald ripening, dendritic growth, and the solidification of eutectic alloys. The projection operator method is used to extract the "sharp-interface limit" from phase-field models which have interfaces that are diffuse on a length scale xi. In particular, phase-field equations are mapped onto sharp-interface equations in the limits xi(kappa)<<1 and xi(v)/D<<1, where kappa and v are, respectively, the interface curvature and velocity and D is the diffusion constant in the bulk. The calculations provide one general set of sharp-interface equations that incorporate the Gibbs-Thomson condition, the Allen-Cahn equation, and the Kardar-Parisi-Zhang equation.
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Affiliation(s)
- K R Elder
- Department of Physics, Oakland University, Rochester, Michigan 48309-4487, USA.
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Dubé M, Rost M, Elder KR, Alava M, Majaniemi S, Ala-Nissila T. Comment on "Pipe network model for scaling of dynamic interfaces in porous media". Phys Rev Lett 2001; 86:6046-6047. [PMID: 11415425 DOI: 10.1103/physrevlett.86.6046] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2000] [Indexed: 05/23/2023]
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Drolet F, Elder KR, Grant M, Kosterlitz JM. Phase-field modeling of eutectic growth. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 2000; 61:6705-6720. [PMID: 11088365 DOI: 10.1103/physreve.61.6705] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/1998] [Revised: 10/08/1999] [Indexed: 05/23/2023]
Abstract
A phase-field model of eutectic growth is proposed in terms of a free energy F, which is a functional of a liquid-solid order parameter psi, and a conserved concentration field c. The model is shown to recover the important features of a eutectic phase diagram and to reduce to the standard sharp-interface formulation of nonequilibrium growth. It is successfully applied to the study of directional solidification when the solid phase is a single or two phase state. The crystallization of a eutectic compound under isothermal conditions is also considered. For that process, the transformed volume fraction and psi-field structure factor, both measured during numerical simulations, closely match theoretical predictions. Three possible growth mechanisms are also identified: diffusion-limited growth, lamellar growth, and spinodal decomposition.
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Affiliation(s)
- F Drolet
- Supercomputer Computations Research Institute, Florida State University, Tallahassee, Florida 32306-4052, USA
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Elder KR, Gunton JD, Grant M. Nonisothermal eutectic crystallization. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1996; 54:6476-6484. [PMID: 9965869 DOI: 10.1103/physreve.54.6476] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Provatas N, Ala-Nissila T, Grant M, Elder KR, Piché L. Flame propagation in random media. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1995; 51:4232-4236. [PMID: 9963134 DOI: 10.1103/physreve.51.4232] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Morin B, Elder KR, Grant M. Theory for quenches from ordered states in nonconserved systems. Phys Rev B Condens Matter 1993; 47:2487-2492. [PMID: 10006299 DOI: 10.1103/physrevb.47.2487] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Elder KR, Viñals J, Grant M. Dynamic scaling and quasiordered states in the two-dimensional Swift-Hohenberg equation. Phys Rev A 1992; 46:7618-7629. [PMID: 9908113 DOI: 10.1103/physreva.46.7618] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Elder KR, Viñals J, Grant M. Ordering dynamics in the two-dimensional stochastic Swift-Hohenberg equation. Phys Rev Lett 1992; 68:3024-3027. [PMID: 10045588 DOI: 10.1103/physrevlett.68.3024] [Citation(s) in RCA: 30] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Elder KR, Morin B, Grant M, Desai RC. Late-time theory for the effects of a conserved field on the kinetics of an order-disorder transition. Phys Rev B Condens Matter 1991; 44:6673-6688. [PMID: 9998540 DOI: 10.1103/physrevb.44.6673] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Thomson JR, Cowan WM, Elder KR, Daviet P, Soga G, Zhang Z, Grant M, Zuckermann MJ. Neural networks with constrained inputs as models for pattern formation in primate visual cortex. J Biol Phys 1991. [DOI: 10.1007/bf00417810] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022] Open
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Elder KR, Desai RC. Role of nonlinearities in off-critical quenches as described by the Cahn-Hilliard model of phase separation. Phys Rev B Condens Matter 1989; 40:243-254. [PMID: 9990908 DOI: 10.1103/physrevb.40.243] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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50
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Rogers TM, Elder KR, Desai RC. Droplet growth and coarsening during heterogeneous vapor condensation. Phys Rev A Gen Phys 1988; 38:5303-5309. [PMID: 9900252 DOI: 10.1103/physreva.38.5303] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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