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Pessot G, Schümann M, Gundermann T, Odenbach S, Löwen H, Menzel AM. Tunable dynamic moduli of magnetic elastomers: from characterization by x-ray micro-computed tomography to mesoscopic modeling. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:125101. [PMID: 29474190 DOI: 10.1088/1361-648x/aaaeaa] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Ferrogels and magnetorheological elastomers are composite materials obtained by embedding magnetic particles of mesoscopic size in a crosslinked polymeric matrix. They combine the reversible elastic deformability of polymeric materials with the high responsivity of ferrofluids to external magnetic fields. These materials stand out, for example, for significant magnetostriction as well as a pronounced increase of the elastic moduli in the presence of external magnetic fields. By means of x-ray micro-computed tomography, the position and size of each magnetic particle can be measured with a high degree of accuracy. We here use data extracted from real magnetoelastic samples as input for coarse-grained dipole-spring modeling and calculations to investigate internal restructuring, stiffening, and changes in the normal modes spectrum. More precisely, we assign to each particle a dipole moment proportional to its volume and set a randomized network of springs between them that mimics the behavior of the polymeric elastic matrix. Extending our previously developed methods, we compute the resulting structural changes in the systems as well as the frequency-dependent elastic moduli when magnetic interactions are turned on. Particularly, with increasing magnetization, we observe the formation of chain-like aggregates. Interestingly, the static elastic moduli can first show a slight decrease with growing amplitude of the magnetic interactions, before a pronounced increase appears upon the chain formation. The change of the dynamic moduli with increasing magnetization depends on the frequency and can even feature nonmonotonic behavior. Overall, we demonstrate how theory and experiments can complement each other to learn more about the dynamic behavior of this interesting class of materials.
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Affiliation(s)
- Giorgio Pessot
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
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Tito NB, Storm C, Ellenbroek WG. Self-Consistent Field Lattice Model for Polymer Networks. Macromolecules 2017; 50:9788-9795. [PMID: 29296030 PMCID: PMC5746847 DOI: 10.1021/acs.macromol.7b01284] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2017] [Revised: 11/18/2017] [Indexed: 11/30/2022]
Abstract
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A lattice model based
on polymer self-consistent field theory is
developed to predict the equilibrium statistics of arbitrary polymer
networks. For a given network topology, our approach uses moment propagators
on a lattice to self-consistently construct the ensemble of polymer
conformations and cross-link spatial probability distributions. Remarkably,
the calculation can be performed “in the dark”, without
any prior knowledge on preferred chain conformations or cross-link
positions. Numerical results from the model for a test network exhibit
close agreement with molecular dynamics simulations, including when
the network is strongly sheared. Our model captures nonaffine deformation,
mean-field monomer interactions, cross-link fluctuations, and finite
extensibility of chains, yielding predictions that differ markedly
from classical rubber elasticity theory for polymer networks. By examining
polymer networks with different degrees of interconnectivity, we gain
insight into cross-link entropy, an important quantity in the macroscopic
behavior of gels and self-healing materials as they are deformed.
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Affiliation(s)
- Nicholas B Tito
- Department of Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands.,Institute for Complex Molecular Systems, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands
| | - Cornelis Storm
- Department of Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands.,Institute for Complex Molecular Systems, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands
| | - Wouter G Ellenbroek
- Department of Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands.,Institute for Complex Molecular Systems, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands
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Lu BS, Naji A, Podgornik R. Pseudo-Casimir stresses and elasticity of a confined elastomer film. SOFT MATTER 2016; 12:4384-4396. [PMID: 27079513 DOI: 10.1039/c6sm00160b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Investigations of the elastic behavior of bulk elastomers have traditionally proceeded on the basis of classical rubber elasticity, which regards chains as thermally fluctuating but disregards the thermal fluctuations of the cross-links. Herein, we consider an incompressible and flat elastomer film of an axisymmetric shape confined between two large hard co-planar substrates, with the axis of the film perpendicular to the substrates. We address the impact that thermal fluctuations of the cross-links have on the free energy of elastic deformation of the system, subject to the requirement that the fluctuating elastomer cannot detach from the substrates. We examine the behavior of the deformation free energy for one case where a rigid pinning boundary condition is applied to a class of elastic fluctuations at the confining surfaces, and another case where the same elastic fluctuations are subjected to soft "gluing" potentials. We find that there can be significant departures (both quantitative and qualitative) from the prediction of classical rubber elasticity theory when elastic fluctuations are included. Finally, we compare the character of the attractive part of the elastic fluctuation-induced, or pseudo-Casimir, stress with the standard thermal Casimir stress in confined but non-elastomeric systems, finding the same power law decay behavior when a rigid pinning boundary condition is applied; for the case of the gluing potential, we find that the leading order correction to the attractive part of the fluctuation stress decays inversely with the inter-substrate separation.
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Affiliation(s)
- Bing-Sui Lu
- Department of Theoretical Physics, J. Stefan Institute, 1000 Ljubljana, Slovenia.
| | - Ali Naji
- School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran
| | - Rudolf Podgornik
- Department of Theoretical Physics, J. Stefan Institute, 1000 Ljubljana, Slovenia.
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Neuefeind J, Skov AL, Daniels JE, Honkimäki V, Jakobsen B, Oddershede J, Poulsen HF. A multiple length scale description of the mechanism of elastomer stretching. RSC Adv 2016. [DOI: 10.1039/c6ra22802j] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022] Open
Abstract
Conventionally, the stretching of rubber is modeled exclusively by rotations of segments of the embedded polymer chains; i.e. changes in entropy.
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Affiliation(s)
- J. Neuefeind
- Chemical and Engineering Materials Division
- Oak Ridge National Laboratory
- Oak Ridge
- USA
| | - A. L. Skov
- Danish Polymer Centre
- DTU Chemical Engineering
- 2800 Kongens Lyngby
- Denmark
| | - J. E. Daniels
- School of Materials Science and Engineering
- UNSW Australia
- Kensington 2052
- Australia
- European Synchrotron Radiation Facility
| | - V. Honkimäki
- European Synchrotron Radiation Facility
- 38043 Grenoble
- France
| | - B. Jakobsen
- Glass and Time
- Department of Sciences
- Roskilde University
- 4000 Roskilde
- Denmark
| | - J. Oddershede
- Neutrons and X-rays for Materials Physics
- DTU Physics
- 2800 Kongens Lyngby
- Denmark
| | - H. F. Poulsen
- Neutrons and X-rays for Materials Physics
- DTU Physics
- 2800 Kongens Lyngby
- Denmark
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5
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Wu R, Li T, Nies E. Langevin Dynamics Simulation of Chain Crosslinking into Polymer Networks. MACROMOL THEOR SIMUL 2012. [DOI: 10.1002/mats.201100088] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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Xing X. Biaxial deformations of rubber: a comparison between entanglement theory and elastic fluctuation theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:021801. [PMID: 21929009 DOI: 10.1103/physreve.84.021801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2010] [Indexed: 05/31/2023]
Abstract
The classical theory of rubber elasticity fails in the regime of large deformation. The underlying physical mechanism has been under debate for a long time. In this work, we test the recently proposed mechanism of thermal elastic fluctuations by Xing, Goldbart, and Radzihovsky (XGR) against the biaxial stress-strain data of three distinct polymer networks with very different network structures, synthesized by Urayama and Kawabata, respectively. We find that both the two-parameter version and the one-parameter version of the XGR theory provide a satisfactory description of the elasticity in whole deformation range. For comparison, we also fit the same sets of data using the slip-link model by Edwards and Vilgis with four parameters. The fitting qualities of two theories are found to be comparable.
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Affiliation(s)
- Xiangjun Xing
- Institute of Natural Science and Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China.
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Blundell JR, Terentjev EM. The influence of disorder on deformations in semiflexible networks. Proc Math Phys Eng Sci 2011. [DOI: 10.1098/rspa.2010.0600] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We present a model that assesses the different elastic responses of a semiflexible network, which either (i) is constrained to deform in an affine way or (ii) is permitted to thermally fluctuate and deviate from affine response. The thermal, non-affine response of the network is achieved using a Metropolis Monte Carlo algorithm with dynamic step size. We find that non-affine deformations soften the network dramatically at low strains and make the eventual nonlinear strain stiffening far more pronounced. We show that the effect of these non-affine deformations are very sensitive to the degree variation in the lengths of filaments connecting cross-links. Where there is high variation, non-affine deformations allow internal stresses to relax, giving rise to a smaller range of tensile forces in filaments and a dramatic reduction of network stiffness. This highlights that non-affine deformations are crucial in small strain response of stiff polymer networks.
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Mao X, Goldbart PM, Xing X, Zippelius A. Soft random solids and their heterogeneous elasticity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:031140. [PMID: 19905095 DOI: 10.1103/physreve.80.031140] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2008] [Indexed: 05/28/2023]
Abstract
Spatial heterogeneity in the elastic properties of soft random solids is examined via vulcanization theory. The spatial heterogeneity in the structure of soft random solids is a result of the fluctuations locked-in at their synthesis, which also brings heterogeneity in their elastic properties. Vulcanization theory studies semimicroscopic models of random-solid-forming systems and applies replica field theory to deal with their quenched disorder and thermal fluctuations. The elastic deformations of soft random solids are argued to be described by the Goldstone sector of fluctuations contained in vulcanization theory, associated with a subtle form of spontaneous symmetry breaking that is associated with the liquid-to-random-solid transition. The resulting free energy of this Goldstone sector can be reinterpreted as arising from a phenomenological description of an elastic medium with quenched disorder. Through this comparison, we arrive at the statistics of the quenched disorder of the elasticity of soft random solids in terms of residual stress and Lamé-coefficient fields. In particular, there are large residual stresses in the equilibrium reference state, and the disorder correlators involving the residual stress are found to be long ranged and governed by a universal parameter that also gives the mean shear modulus.
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Affiliation(s)
- Xiaoming Mao
- Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
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Structure–mechanical property correlations of model siloxane elastomers with controlled network topology. POLYMER 2009. [DOI: 10.1016/j.polymer.2008.10.027] [Citation(s) in RCA: 79] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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Svaneborg C, Everaers R, Grest GS, Curro JG. Connectivity and Entanglement Stress Contributions in Strained Polymer Networks. Macromolecules 2008. [DOI: 10.1021/ma800018f] [Citation(s) in RCA: 46] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Carsten Svaneborg
- Department of Chemistry and Interdisciplinary Nanoscience Center (iNANO), University of Aarhus, Langelandsgade 140, DK-8000 Århus, Denmark, Laboratoire de Physique, École Normale Supérieure de Lyon, CNRS UMR 5672, Université de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France, Sandia National Laboratories, Albuquerque, New Mexico 87185, and Department of Chemical & Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131
| | - Ralf Everaers
- Department of Chemistry and Interdisciplinary Nanoscience Center (iNANO), University of Aarhus, Langelandsgade 140, DK-8000 Århus, Denmark, Laboratoire de Physique, École Normale Supérieure de Lyon, CNRS UMR 5672, Université de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France, Sandia National Laboratories, Albuquerque, New Mexico 87185, and Department of Chemical & Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131
| | - Gary S. Grest
- Department of Chemistry and Interdisciplinary Nanoscience Center (iNANO), University of Aarhus, Langelandsgade 140, DK-8000 Århus, Denmark, Laboratoire de Physique, École Normale Supérieure de Lyon, CNRS UMR 5672, Université de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France, Sandia National Laboratories, Albuquerque, New Mexico 87185, and Department of Chemical & Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131
| | - John G. Curro
- Department of Chemistry and Interdisciplinary Nanoscience Center (iNANO), University of Aarhus, Langelandsgade 140, DK-8000 Århus, Denmark, Laboratoire de Physique, École Normale Supérieure de Lyon, CNRS UMR 5672, Université de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France, Sandia National Laboratories, Albuquerque, New Mexico 87185, and Department of Chemical & Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131
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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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Hansen R, Skov AL, Hassager O. Constitutive equation for polymer networks with phonon fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:011802. [PMID: 18351868 DOI: 10.1103/physreve.77.011802] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2007] [Indexed: 05/26/2023]
Abstract
Recent research by Xing [Phys. Rev. Lett. 98, 075502 (2007)] has provided an expression for the Helmholtz free energy related to phonon fluctuations in polymer networks. We extend this result by constructing the corresponding nonlinear constitutive equation, usable for entirely general, volume conserving deformation fields. Constitutive equations for the sliplink model and the tube model are derived and the three models are examined by comparison with each other and with data from Xu and Mark [Rubber Chem. Technol. 63, 276 (1990)] and Wang and Mark [J. Polym. Sci., Part B: Polym. Phys. 30, 801 (1992)]. Elastic moduli are derived for the three models and compared with the moduli determined from the chemical stoichiometry. We conclude that the sliplink model and the phonon fluctuation model are relatively consistent with each other and with the data. The tube model seems consistent neither with the other models nor with the data.
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Affiliation(s)
- Rasmus Hansen
- Center for Fluid Dynamics at DTU, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
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