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Le Doussal P, Radzihovsky L. Thermal Buckling Transition of Crystalline Membranes in a Field. PHYSICAL REVIEW LETTERS 2021; 127:015702. [PMID: 34270280 DOI: 10.1103/physrevlett.127.015702] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2021] [Accepted: 06/04/2021] [Indexed: 06/13/2023]
Abstract
Two-dimensional crystalline membranes in isotropic embedding space exhibit a flat phase with anomalous elasticity, relevant, e.g., for graphene. Here we study their thermal fluctuations in the absence of exact rotational invariance in the embedding space. An example is provided by a membrane in an orientational field, tuned to a critical buckling point by application of in-plane stresses. Through a detailed analysis, we show that the transition is in a new universality class. The self-consistent screening method predicts a second-order transition, with modified anomalous elasticity exponents at criticality, while the RG suggests a weakly first-order transition.
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Affiliation(s)
- Pierre Le Doussal
- Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris, France
| | - Leo Radzihovsky
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
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2
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Duzgun A, Selinger JV. Dynamic Theory of Polydomain Liquid Crystal Elastomers. PHYSICAL REVIEW LETTERS 2015; 115:187801. [PMID: 26565497 DOI: 10.1103/physrevlett.115.187801] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2015] [Indexed: 06/05/2023]
Abstract
When liquid crystal elastomers are prepared without any alignment, disordered polydomain structures emerge as the materials are cooled into the nematic phase. These polydomain structures are often attributed to quenched disorder in the cross-linked polymer network. As an alternative explanation, we develop a theory for the dynamics of the isotropic-nematic transition in liquid crystal elastomers, and show that the dynamics can induce a polydomain structure with a characteristic length scale, through a mechanism analogous to the Cahn-Hilliard equation for phase separation.
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Affiliation(s)
- Ayhan Duzgun
- Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA
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Janssen HK, Stenull O. Driven surface diffusion with detailed balance and elastic phase transitions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062145. [PMID: 25019762 DOI: 10.1103/physreve.89.062145] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2014] [Indexed: 06/03/2023]
Abstract
Driven surface diffusion occurs, for example, in molecular beam epitaxy when particles are deposited under an oblique angle. Elastic phase transitions happen when normal modes in crystals become soft due to the vanishing of certain elastic constants. We show that these seemingly entirely disparate systems fall under appropriate conditions into the same universality class. We derive the field-theoretic Hamiltonian for this universality class, and we use renormalized field theory to calculate critical exponents and logarithmic corrections for several experimentally relevant quantities.
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Affiliation(s)
- Hans-Karl Janssen
- Institut für Theoretische Physik III, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
| | - Olaf Stenull
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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Cordoyiannis G, Lebar A, Rožič B, Zalar B, Kutnjak Z, Žumer S, Brömmel F, Krause S, Finkelmann H. Controlling the Critical Behavior of Paranematic to Nematic Transition in Main-Chain Liquid Single-Crystal Elastomers. Macromolecules 2009. [DOI: 10.1021/ma802049r] [Citation(s) in RCA: 36] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
| | | | | | | | | | - Slobodan Žumer
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia and Jožef Stefan Institute, P.O. Box 3000, 1001 Ljubljana, Slovenia
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Xing X, Pfahl S, Mukhopadhyay S, Goldbart PM, Zippelius A. Nematic elastomers: from a microscopic model to macroscopic elasticity theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:051802. [PMID: 18643092 DOI: 10.1103/physreve.77.051802] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2007] [Indexed: 05/26/2023]
Abstract
A Landau theory is constructed for the gelation transition in cross-linked polymer systems possessing spontaneous nematic ordering, based on symmetry principles and the concept of an order parameter for the amorphous solid state. This theory is substantiated with help of a simple microscopic model of cross-linked dimers. Minimization of the Landau free energy in the presence of nematic order yields the neoclassical theory of the elasticity of nematic elastomers and, in the isotropic limit, the classical theory of isotropic elasticity. These phenomenological theories of elasticity are thereby derived from a microscopic model, and it is furthermore demonstrated that they are universal mean-field descriptions of the elasticity for all chemical gels and vulcanized media.
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Affiliation(s)
- Xiangjun Xing
- Physics Department, Syracuse University, Syracuse, New York 13244, USA
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Stenull O. Smectic elastomer membranes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:051702. [PMID: 17677079 DOI: 10.1103/physreve.75.051702] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/05/2007] [Indexed: 05/16/2023]
Abstract
We present a model for smectic elastomer membranes which includes elastic and liquid-crystalline degrees of freedom. Based on our model, we determined the qualitative phase diagram of a smectic elastomer membrane using mean-field theory. This phase diagram is found to comprise five phases, viz., smectic- A -flat, smectic- A -crumpled, smectic- C -flat, smectic- C -crumpled, and smectic- C -tubule phases, where in the latter phase, the membrane is flat in the direction of mesogenic tilt and crumpled in the perpendicular direction. The transitions between adjacent phases are second-order phase transitions. We study in some detail the elasticity of the smectic- C -flat and the smectic- C -tubule phases which are associated with a spontaneous breaking of in-plane rotational symmetry. As a consequence of the Goldstone theorem, these phases exhibit soft elasticity characterized by the vanishing of in-plane shear moduli.
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Affiliation(s)
- Olaf Stenull
- Fachbereich Physik, Universität Duisburg-Essen, Campus Duisburg, Duisburg, Germany
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Stenull O, Lubensky TC. Soft elasticity in biaxial smectic and smectic-C elastomers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:051709. [PMID: 17279928 DOI: 10.1103/physreve.74.051709] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2006] [Revised: 09/05/2006] [Indexed: 05/13/2023]
Abstract
Ideal (monodomain) smectic-A elastomers cross-linked in the smectic-A phase are simply uniaxial rubbers, provided deformations are small. From these materials smectic-C elastomers are produced by a cooling through the smectic-A to smectic-C phase transition. At least in principle, biaxial smectic elastomers could also be produced via cooling from the smectic-A to a biaxial smectic phase. These phase transitions, respectively, from Dinfinityh to C2h and from Dinfinityh to D2h symmetry, spontaneously break the rotational symmetry in the smectic planes. We study the above transitions and the elasticity of the smectic-C and biaxial phases in three different but related models: Landau-like phenomenological models as functions of the Cauchy-Saint-Laurent strain tensor for both the biaxial and the smectic-C phases and a detailed model, including contributions from the elastic network, smectic layer compression, and smectic-C tilt for the smectic-C phase as a function of both strain and the c-director. We show that the emergent phases exhibit soft elasticity characterized by the vanishing of certain elastic moduli. We analyze in some detail the role of spontaneous symmetry breaking as the origin of soft elasticity and we discuss different manifestations of softness like the absence of restoring forces under certain shears and extensional strains.
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Affiliation(s)
- Olaf Stenull
- Fachbereich Physik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany
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Petridis L, Terentjev EM. Nematic-isotropic transition with quenched disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:051707. [PMID: 17279926 DOI: 10.1103/physreve.74.051707] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2006] [Indexed: 05/13/2023]
Abstract
Nematic elastomers do not show the discontinuous, first-order, phase transition that the Landau-De Gennes mean field theory predicts for a quadrupolar ordering in three dimensions. We attribute this behavior to the presence of network crosslinks, which act as sources of quenched orientational disorder. We show that the addition of weak random anisotropy results in a singular renormalization of the Landau-De Gennes expression, adding an energy term proportional to the inverse quartic power of order parameter Q. This reduces the first-order discontinuity in Q. For sufficiently high disorder strength the jump disappears altogether and the phase transition becomes continuous, in some ways resembling the supercritical transitions in external field.
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Affiliation(s)
- L Petridis
- Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom
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Brand HR, Pleiner H, Martinoty P. Selected macroscopic properties of liquid crystalline elastomers. SOFT MATTER 2006; 2:182-189. [PMID: 32646144 DOI: 10.1039/b512693m] [Citation(s) in RCA: 54] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
In this short review we give an overview of selected macroscopic properties of sidechain liquid crystalline elastomers (LCEs) focusing on three closely related topics (a) the influence of relative rotations between the director and the strain field on various reorientation instabilities, (b) the nonlinear stress-strain curves for the polydomain-monodomain transition and for the reorientation transition in LCE monodomains and (c) the shear mechanical response of LCEs in the linear regime. We consider only already existing real materials and do not discuss hypothetical "ideal" systems. We conclude that all observations reported to date can be accounted for without invoking the concept of soft elasticity, but instead relying on macroscopic dynamics in the linear and the nonlinear domain.
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Affiliation(s)
- Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany.
| | - Harald Pleiner
- Max-Planck-Institute for Polymer Research, 55021 Mainz, Germany.
| | - Philippe Martinoty
- Institut de Mécanique des Fluides et des Solides UMR 7507 CNRS-ULP, 67000 Strasbourg, France.
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Xing X, Radzihovsky L. Phases and transitions in phantom nematic elastomer membranes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:011802. [PMID: 15697623 DOI: 10.1103/physreve.71.011802] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2004] [Indexed: 05/24/2023]
Abstract
Motivated by recently discovered unusual properties of bulk nematic elastomers, we study a phase diagram of liquid-crystalline polymerized phantom membranes, focusing on in-plane nematic order. We predict that such membranes should generically exhibit five phases, distinguished by their conformational and in-plane orientational properties: namely, isotropic-crumpled, nematic-crumpled, isotropic-flat, nematic-flat, and nematic-tubule phases. In the nematic-tubule phase, the membrane is extended along the direction of spontaneous nematic order and is crumpled in the other. The associated spontaneous symmetries breaking guarantees that the nematic tubule is characterized by a conformational-orientational soft (Goldstone) mode and the concomitant vanishing of the in-plane shear modulus. We show that long-range orientational order of the nematic tubule is maintained even in the presence of harmonic thermal fluctuations. However, it is likely that tubule's elastic properties are qualitatively modified by these fluctuations, which can be studied using a nonlinear elastic theory for the nematic tubule phase that we derive at the end of this paper.
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Affiliation(s)
- Xiangjun Xing
- Department of Physics, University of Illinois, Urbana-Champaign, Illinois 61801, USA
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Selinger JV, Ratna BR. Isotropic-nematic transition in liquid-crystalline elastomers: lattice model with quenched disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:041707. [PMID: 15600425 DOI: 10.1103/physreve.70.041707] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2004] [Indexed: 05/24/2023]
Abstract
When liquid-crystalline elastomers pass through the isotropic-nematic transition, the orientational order parameter and the elastic strain vary rapidly but smoothly, without the expected first-order discontinuity. This broadening of the phase transition is an important issue for applications of liquid-crystalline elastomers as actuators or artificial muscles. To understand this behavior, we develop a lattice model of liquid-crystalline elastomers, with local directors coupled to a global strain variable. In this model, we can consider either random-bond disorder (representing chemical heterogeneity) or random-field disorder (representing heterogeneous local stresses). Monte Carlo simulations show that both types of disorder cause the first-order isotropic-nematic transition to broaden into a smooth crossover, consistent with the experiments. For random-field disorder, the smooth crossover into an ordered state can be attributed to the long-range elastic interaction.
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Affiliation(s)
- Jonathan V Selinger
- Center for Bio/Molecular Science and Engineering, Naval Research Laboratory, Code 6900, 4555 Overlook Avenue, SW, Washington, DC 20375, USA
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Stenull O, Lubensky TC. Commentary on "Mechanical properties of monodomain side chain nematic elastomers" by P. Martinoty et al. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2004; 14:333-340. [PMID: 15365857 DOI: 10.1140/epje/i2004-10028-x] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We discuss the rheology experiments on nematic elastomers by Martinoty et al. in the light of theoretical models for the long-wavelength low-frequency dynamics of these materials. We review these theories and discuss how they can be modified to provide a phenomenological description of the non-hydrodynamic frequency regime probed in the experiments. Moreover, we review the concepts of soft and semi-soft elasticity and comment on their implications for the experiments.
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Affiliation(s)
- O Stenull
- Department of Physics and Astronomy, University of Pennsylvania, PA 19104, Philadelphia, USA
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Stenull O, Lubensky TC. Dynamics of nematic elastomers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:051801. [PMID: 15244840 DOI: 10.1103/physreve.69.051801] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2003] [Indexed: 05/24/2023]
Abstract
We study the low-frequency, long-wavelength dynamics of soft and semisoft nematic elastomers using two different but related dynamic theories. Our first formulation describes the pure hydrodynamic behavior of nematic elastomers in which the nematic director has relaxed to its equilibrium value in the presence of strain. We find that the sound-mode structure for soft elastomers is identical to that of columnar liquid crystals. Our second formulation generalizes the derivation of the equations of nematohydrodynamics by Forster et al. to nematic elastomers. It treats the director explicitly and describes slow modes beyond the hydrodynamic limit.
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Affiliation(s)
- Olaf Stenull
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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Stenull O, Lubensky TC. Anomalous elasticity of nematic and critically soft elastomers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:021807. [PMID: 14995483 DOI: 10.1103/physreve.69.021807] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2003] [Indexed: 05/24/2023]
Abstract
Uniaxial elastomers are characterized by five elastic constants. If their elastic modulus C5 describing the energy of shear strains in planes containing the anisotropy axis vanishes, they are said to be soft. In spatial dimensions d less than or equal to 3, soft elastomers exhibit anomalous elasticity with certain length-scale-dependent bending moduli that diverge and shear moduli that vanish at large length scales. Using renormalized field theory at d=3 and to first order in epsilon=3-d, we calculate critical exponents and other properties characterizing the anomalous elasticity of two soft systems: (i) nematic elastomers in which softness is a manifestation of a Goldstone mode induced by the spontaneous symmetry breaking associated with a transition from an isotropic state to a nematic state, and (ii) a particular version of what we call a critically soft elastomer in which C(5)=0 corresponds to a critical point terminating the stability regime of a uniaxial elastomer with C5>0.
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Affiliation(s)
- Olaf Stenull
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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Wojciechowski KW. Non-chiral, molecular model of negative Poisson ratio in two dimensions. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/47/005] [Citation(s) in RCA: 164] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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