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Arenzon JJ, Cugliandolo LF, Picco M. Slicing the three-dimensional Ising model: Critical equilibrium and coarsening dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032142. [PMID: 25871089 DOI: 10.1103/physreve.91.032142] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2014] [Indexed: 06/04/2023]
Abstract
We study the evolution of spin clusters on two-dimensional slices of the three-dimensional Ising model in contact with a heat bath after a sudden quench to a subcritical temperature. We analyze the evolution of some simple initial configurations, such as a sphere and a torus, of one phase embedded into the other, to confirm that their area disappears linearly with time and to establish the temperature dependence of the prefactor in each case. Two generic kinds of initial states are later used: equilibrium configurations either at infinite temperature or at the paramagnetic-ferromagnetic phase transition. We investigate the morphological domain structure of the coarsening configurations on two-dimensional slices of the three-dimensional system, compared with the behavior of the bidimensional model.
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Affiliation(s)
- Jeferson J Arenzon
- Instituto de Física, Universidade Federal do Rio Grande do Sul, C.P. 15051, 91501-970 Porto Alegre, RS, Brazil
| | - Leticia F Cugliandolo
- Sorbonne Universités, Université Pierre et Marie Curie-Paris VI, Laboratoire de Physique Théorique et Hautes Energies UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
| | - Marco Picco
- Sorbonne Universités, Université Pierre et Marie Curie-Paris VI, Laboratoire de Physique Théorique et Hautes Energies UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
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Datt C, Thampi SP, Govindarajan R. Morphological evolution of domains in spinodal decomposition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:010101. [PMID: 25679549 DOI: 10.1103/physreve.91.010101] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2014] [Indexed: 06/04/2023]
Abstract
Domain growth in spinodal decomposition is usually described by a single time-evolving length scale. We show that the evolution of morphology of domains is nonmonotonic. The domains elongate rapidly at first and then, with the help of hydrodynamics, return to a more circular shape. The initial elongation phase does not alter with hydrodynamics. A small deviation from critical composition changes the morphology dramatically.
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Affiliation(s)
- Charu Datt
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, 21 Brundavan Colony, Narsingi, Hyderabad 500075, India
| | - Sumesh P Thampi
- Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, United Kingdom
| | - Rama Govindarajan
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, 21 Brundavan Colony, Narsingi, Hyderabad 500075, India
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Sicilia A, Arenzon JJ, Bray AJ, Cugliandolo LF. Domain growth morphology in curvature-driven two-dimensional coarsening. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:061116. [PMID: 18233823 DOI: 10.1103/physreve.76.061116] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2007] [Indexed: 05/25/2023]
Abstract
We study the distribution of domain areas, areas enclosed by domain boundaries ("hulls"), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, n_{h}(A,t)dA , with enclosed area in the interval (A,A+dA) , is described, for a disordered initial condition, by the scaling function n_{h}(A,t)=2c_{h}(A+lambda_{h}t);{2} , where c_{h}=18pi sqrt[3] approximately 0.023 is a universal constant and lambda_{h} is a material parameter. For a critical initial condition, the same form is obtained, with the same lambda_{h} but with c_{h} replaced by c_{h}2 . For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form n_{d}(A,t)=2c_{d}(lambda_{d}t);{tau'-2}(A+lambda_{d}t);{tau'} , where c_{d} and lambda_{d} are numerically very close to c_{h} and lambda_{h} , respectively, and tau'=18791 approximately 2.055 . For critical initial conditions, one replaces c_{d} by c_{d}2 and the exponent is tau=379187 approximately 2.027 . These results are extended to describe the number density of the length of hulls and domain walls surrounding connected clusters of aligned spins. These predictions are supported by extensive numerical simulations. We also study numerically the geometric properties of the boundaries and areas.
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Affiliation(s)
- Alberto Sicilia
- Université Pierre et Marie Curie-Paris VI, LPTHE UMR 7589, 4 Place Jussieu, Paris Cedex 05, France
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Arenzon JJ, Bray AJ, Cugliandolo LF, Sicilia A. Exact results for curvature-driven coarsening in two dimensions. PHYSICAL REVIEW LETTERS 2007; 98:145701. [PMID: 17501288 DOI: 10.1103/physrevlett.98.145701] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2006] [Indexed: 05/15/2023]
Abstract
We consider the statistics of the areas enclosed by domain boundaries ("hulls") during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area that enclose an area greater than A has, for large time t, the scaling form Nh(A,t)=2c/(A+lambdat), demonstrating the validity of dynamical scaling in this system, where c=1/8pisquare root 3 is a universal constant. Domain areas (regions of aligned spins) have a similar distribution up to very large values of A/lambdat. Identical forms are obtained for coarsening from a critical initial state, but with c replaced by c/2.
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Affiliation(s)
- Jeferson J Arenzon
- Instituto de Física, Universidade Federal do Rio Grande do Sul, CP 15051, 91501-970 Porto Alegre RS, Brazil
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Kwon Y, Thornton K, Voorhees PW. Coarsening of bicontinuous structures via nonconserved and conserved dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:021120. [PMID: 17358326 DOI: 10.1103/physreve.75.021120] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2006] [Indexed: 05/14/2023]
Abstract
Coarsening subsequent to phase separations occurs in many two-phase mixtures. While unique scaled particle size distributions have been determined for highly asymmetric mixtures in which spherical particles form in a matrix, it is not known if a unique scaled structure exists for symmetric mixtures, which yield bicontinuous structures having intricately interpenetrating phase domains. Using large-scale simulations, we have established that unique scaled microstructures exist in symmetric mixtures evolving via nonconserved and conserved dynamics. We characterize their morphologies by the interfacial shape distribution, a counterpart to the particle size distribution, and their topologies by the genus. We find that the two dynamics result in unique, but different, scaled interfacial shape distributions, with conserved dynamics yielding a narrower distribution around zero mean curvature. In contrast, the two scaled structures are topologically similar, having nearly equal values of the scaled genus.
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Affiliation(s)
- Yongwoo Kwon
- Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, Illinois 60208-3108, USA
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Cybulski O, Matysiak D, Babin V, Holyst R. Pattern formation in nonextensive thermodynamics: selection criterion based on the Renyi entropy production. J Chem Phys 2005; 122:174105. [PMID: 15910021 DOI: 10.1063/1.1886728] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We analyze a system of two different types of Brownian particles confined in a cubic box with periodic boundary conditions. Particles of different types annihilate when they come into close contact. The annihilation rate is matched by the birth rate, thus the total number of each kind of particles is conserved. When in a stationary state, the system is divided by an interface into two subregions, each occupied by one type of particles. All possible stationary states correspond to the Laplacian eigenfunctions. We show that the system evolves towards those stationary distributions of particles which minimize the Renyi entropy production. In all cases, the Renyi entropy production decreases monotonically during the evolution despite the fact that the topology and geometry of the interface exhibit abrupt and violent changes.
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Affiliation(s)
- Olgierd Cybulski
- Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw.
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Fiałkowski M, Hołyst R. Global symmetry breaking in the nonconserved order parameter system during phase ordering. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2005; 16:247-251. [PMID: 15726246 DOI: 10.1140/epje/e2005-00025-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2004] [Indexed: 05/24/2023]
Abstract
We study global symmetry breaking in the 2D system of scalar nonconserved order parameter following a quench to zero temperature. We show that the instant of time when the symmetry is broken and the final morphology is chosen corresponds to the saturation of the order parameter inside the domains. There are three possible final morphologies: the positive and negative order parameter final morphology, and the state of the coexisting positive and negative order parameter subsystems with a flat interface between them. We find also that each type of the final morphology constitutes about 1/3 of all cases, what agrees with the results obtained recently by Spirin et al. [Phys. Rev. E 65, 016119 (2001)]. Our results are pertinent for the two dimensional systems, but we suspect that there is also a way to apply similar arguments for the three dimensional ones.
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Affiliation(s)
- M Fiałkowski
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland.
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Fiałkowski M, Nowakowski R, Hołyst R. A Morphological Study of the Formation of PdHx on Thin Palladium Films. J Phys Chem B 2004. [DOI: 10.1021/jp034593m] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Marcin Fiałkowski
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland, and Department of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszyński University, Dewajtis 5, 01-815 Warsaw, Poland
| | - Robert Nowakowski
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland, and Department of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszyński University, Dewajtis 5, 01-815 Warsaw, Poland
| | - Robert Hołyst
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland, and Department of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszyński University, Dewajtis 5, 01-815 Warsaw, Poland
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Fiałkowski M, Grzeszczak P, Nowakowski R, Hołyst R. Absorption of Mercury in Gold Films and Its Further Desorption: Quantitative Morphological Study of the Surface Patterns. J Phys Chem B 2004. [DOI: 10.1021/jp0365413] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Marcin Fiałkowski
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland, and Department of Mathematics and Natural Sciences - College of Science, Cardinal Stefan Wyszyński University, Dewajtis 5, 01-815 Warsaw, Poland
| | - Patrycja Grzeszczak
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland, and Department of Mathematics and Natural Sciences - College of Science, Cardinal Stefan Wyszyński University, Dewajtis 5, 01-815 Warsaw, Poland
| | - Robert Nowakowski
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland, and Department of Mathematics and Natural Sciences - College of Science, Cardinal Stefan Wyszyński University, Dewajtis 5, 01-815 Warsaw, Poland
| | - Robert Hołyst
- Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland, and Department of Mathematics and Natural Sciences - College of Science, Cardinal Stefan Wyszyński University, Dewajtis 5, 01-815 Warsaw, Poland
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Graca M, Wieczorek SA, Hołyst R. Growth of Polystyrene Domains in Isotropic, Nematic and Smectic Phase of 8CB Liquid Crystal. Macromolecules 2003. [DOI: 10.1021/ma034135b] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- M. Graca
- Institute of Physical Chemistry PAS, Department III, Kasprzaka 44/52, 01-224 Warsaw, Poland, and WMP-SNS, Cardinal Stefan Wyszyṅski University, Dewajtis 5, Warsaw, Poland
| | - S. A. Wieczorek
- Institute of Physical Chemistry PAS, Department III, Kasprzaka 44/52, 01-224 Warsaw, Poland, and WMP-SNS, Cardinal Stefan Wyszyṅski University, Dewajtis 5, Warsaw, Poland
| | - R. Hołyst
- Institute of Physical Chemistry PAS, Department III, Kasprzaka 44/52, 01-224 Warsaw, Poland, and WMP-SNS, Cardinal Stefan Wyszyṅski University, Dewajtis 5, Warsaw, Poland
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