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King DA. Dynamics of wire frame glasses in two dimensions. J Chem Phys 2023; 159:094903. [PMID: 37668250 DOI: 10.1063/5.0164438] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2023] [Accepted: 08/11/2023] [Indexed: 09/06/2023] Open
Abstract
The dynamics of wire frame particles in concentrated suspension are studied by means of a 2D model and compared to those of rod-like particles. The wire frames have bent or branched structures constructed from infinitely thin, rigid rods. In the model, a particle is surrounded by diffusing points that it cannot cross. We derive a formal expression for the mean squared displacement (MSD) and, by using a self-consistent approximation, we find markedly different dynamics for wire frames and rods. For wire frames, there exists a critical concentration of points above which they become frozen with the long time MSD reaching a plateau. Rods, on the other hand, always diffuse by reptation. We also study the rheology through the elastic stress, and more striking differences are found: the initial magnitude of the stress for wire frames is much larger than for rods, scaling such as the square of the point concentration, and above the critical concentration, the stress for wire frames appears to persist indefinitely while for rods it always decays.
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Affiliation(s)
- David A King
- Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd St., Philadelphia, Pennsylvania 19104, USA
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2
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Müller M. Memory in the relaxation of a polymer density modulation. J Chem Phys 2022; 156:124902. [DOI: 10.1063/5.0084602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Marcus Müller
- Institute for Theoretical Physics, Georg August University Gottingen Faculty of Physics, Germany
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Affiliation(s)
- Rakwoo Chang
- Department of Chemistry, Kwangwoon University, Seoul 01897, Republic of Korea
| | - Arun Yethiraj
- Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, United States
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Ooshida T, Goto S, Matsumoto T, Otsuki M. Insights from Single-File Diffusion into Cooperativity in Higher Dimensions. ACTA ACUST UNITED AC 2016. [DOI: 10.1142/s1793048015400019] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Diffusion in colloidal suspensions can be very slow due to the cage effect, which confines each particle within a short radius on one hand, and involves large-scale cooperative motions on the other. In search of insight into this cooperativity, here the authors develop a formalism to calculate the displacement correlation in colloidal systems, mainly in the two-dimensional (2D) case. To clarify the idea for it, studies are reviewed on cooperativity among the particles in the one-dimensional (1D) case, i.e. the single-file diffusion (SFD). As an improvement over the celebrated formula by Alexander and Pincus on the mean-square displacement (MSD) in SFD, it is shown that the displacement correlation in SFD can be calculated from Lagrangian correlation of the particle interval in the one-dimensional case, and also that the formula can be extended to higher dimensions. The improved formula becomes exact for large systems. By combining the formula with a nonlinear theory for correlation, a correction to the asymptotic law for the MSD in SFD is obtained. In the 2D case, the linear theory gives description of vortical cooperative motion.
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Affiliation(s)
- Takeshi Ooshida
- Department of Mechanical and Aerospace Engineering, Tottori University, Tottori 680-8552, Japan
| | - Susumu Goto
- Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
| | - Takeshi Matsumoto
- Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
| | - Michio Otsuki
- Department of Materials Science, Shimane University, Matsue 690-8504, Japan
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Takeshi O, Goto S, Matsumoto T, Nakahara A, Otsuki M. Analytical calculation of four-point correlations for a simple model of cages involving numerous particles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062108. [PMID: 24483387 DOI: 10.1103/physreve.88.062108] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2013] [Indexed: 06/03/2023]
Abstract
Dynamics of a one-dimensional system of Brownian particles with short-range repulsive interaction (diameter σ) is studied with a liquid-theoretical approach. The mean square displacement, the two-particle displacement correlation, and the overlap-density-based generalized susceptibility are calculated analytically by way of the Lagrangian correlation of the interparticulate space, instead of the Eulerian correlation of density that is commonly used in the standard mode-coupling theory. In regard to the mean square displacement, the linear analysis reproduces the established result on the asymptotic subdiffusive behavior of the system. A finite-time correction is given by incorporating the effect of entropic nonlinearity with a Lagrangian version of mode-coupling theory. The notorious difficulty in derivation of the mode-coupling theory concerning violation of the fluctuation-dissipation theorem is found to disappear by virtue of the Lagrangian description. The Lagrangian description also facilitates analytical calculation of four-point correlations in the space-time, such as the two-particle displacement correlation. The two-particle displacement correlation, which is asymptotically self-similar in the space-time, illustrates how the cage effect confines each particle within a short radius on one hand and creates collective motion of numerous particles on the other hand. As the time elapses, the correlation length grows unlimitedly, and the generalized susceptibility based on the overlap density converges to a finite value which is an increasing function of the density. The distribution function behind these dynamical four-point correlations and its extension to three-dimensional cases, respecting the tensorial character of the two-particle displacement correlation, are also discussed.
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Affiliation(s)
- Ooshida Takeshi
- Department of Mechanical and Aerospace Engineering, Tottori University, Tottori 680-8552, Japan
| | - Susumu Goto
- Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
| | - Takeshi Matsumoto
- Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
| | - Akio Nakahara
- Laboratory of Physics, College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan
| | - Michio Otsuki
- Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 229-8558, Japan
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6
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Sussman DM, Schweizer KS. Microscopic Theory of Quiescent and Deformed Topologically Entangled Rod Solutions: General Formulation and Relaxation after Nonlinear Step Strain. Macromolecules 2012. [DOI: 10.1021/ma300006s] [Citation(s) in RCA: 44] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Daniel M. Sussman
- Department
of Physics, ‡Department of Materials Science, and §Frederick Seitz Materials Research
Laboratory, University of Illinois, 1304 W. Green Street, Urbana, Illinois 61801, United States
| | - Kenneth S. Schweizer
- Department
of Physics, ‡Department of Materials Science, and §Frederick Seitz Materials Research
Laboratory, University of Illinois, 1304 W. Green Street, Urbana, Illinois 61801, United States
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7
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Tse YLS, Andersen HC. A lattice model of the translational dynamics of nonrotating rigid rods. J Chem Phys 2012; 136:024904. [DOI: 10.1063/1.3673791] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Sussman DM, Schweizer KS. Microscopic theory of the tube confinement potential for liquids of topologically entangled rigid macromolecules. PHYSICAL REVIEW LETTERS 2011; 107:078102. [PMID: 21902432 DOI: 10.1103/physrevlett.107.078102] [Citation(s) in RCA: 33] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2011] [Indexed: 05/31/2023]
Abstract
We formulate and apply a microscopic self-consistent theory for the dynamic transverse confinement field in solutions of zero-excluded-volume rods based solely on topological entanglements. In agreement with the phenomenological tube model, an infinitely deep potential is predicted. However, strong anharmonicities are found to qualitatively soften localization, in quantitative agreement with experiments on heavily entangled biopolymer solutions. Predictions are also made for the effect of rod alignment on the transverse diffusion constant, tube diameter, and confinement force.
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Affiliation(s)
- Daniel M Sussman
- Department of Physics, University of Illinois, Urbana, 61801, USA
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Sussman DM, Schweizer KS. Microscopic theory of topologically entangled fluids of rigid macromolecules. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061501. [PMID: 21797366 DOI: 10.1103/physreve.83.061501] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/16/2011] [Indexed: 05/31/2023]
Abstract
We present a first-principles theory for the slow dynamics of a fluid of entangling rigid crosses of zero excluded volume based on a generalization of the dynamic mean-field approach of Szamel for infinitely thin nonrotating rods. The latter theory exactly includes topological constraints at the two-body collision level and self-consistently renormalizes an effective diffusion tensor to account for many-body effects. Remarkably, it predicts scaling laws consistent with the phenomenological reptation-tube predictions of Doi and Edwards for the long-time diffusion and the localization length in the heavily entangled limit. We generalize this approach to a different macromolecular architecture, infinitely thin three-dimensional crosses, and also extend the range of densities over which a dynamic localization length can be calculated for rods. Ideal gases of nonrotating crosses have recently received attention in computer simulations and are relevant as a simple model of both a strong-glass former and entangling star-branched polymers. Comparisons of our theory with these simulations reveal reasonable agreement for the magnitude and reduced density dependence of the localization length and also the self-diffusion constant if the consequences of local density fluctuations are taken into account.
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Affiliation(s)
- Daniel M Sussman
- Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
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Yatsenko G, Schweizer KS. Glassy dynamics and kinetic vitrification of isotropic suspensions of hard rods. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2008; 24:7474-7484. [PMID: 18547074 DOI: 10.1021/la8002492] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
A nonlinear Langevin equation (NLE) theory for the translational center-of-mass dynamics of hard nonspherical objects has been applied to isotropic fluids of rigid rods. The ideal kinetic glass transition volume fraction is predicted to be a monotonically decreasing function beyond an aspect ratio of two. The functional form of the decrease is weaker than the inverse aspect ratio. Vitrification occurs at lower volume fractions for corrugated tangent bead rods compared to their smooth spherocylinder analogs. The ideal glass transition signals a crossover to activated dynamics, which is estimated to be observable before the nematic phase boundary is encountered if the aspect ratio is less than roughly 25. Calculations of the glassy elastic shear modulus and absolute yield stress reveal a roughly exponential growth with volume fraction. The dependence of entropic barriers and mean barrier hopping times on concentration for rods of variable aspect ratios can be collapsed quite well based on a difference volume fraction variable that quantifies the distance from the ideal glass boundary. Full numerical solution of the NLE theory via stochastic trajectory simulation was performed for tangent bead rods, and the results were compared to their hard sphere analogs. With increasing shape anisotropy the characteristic length scales of the nonequilibrium free energy increase and the magnitude of the localization well and entropic barrier curvatures decreases. These changes result in a significant aspect ratio dependence of dynamical properties and time correlation functions including weaker intermediate time subdiffusive transport, stronger two-step decay of the incoherent dynamic structure factor, longer mean alpha relaxation time, and stronger wavevector-dependent decoupling of relaxation times and the self-diffusion constant. The theoretical results are potentially testable via computer simulation, confocal microscopy, and dynamic light scattering.
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Affiliation(s)
- Galina Yatsenko
- Department of Materials Science and Frederick Seitz Materials Research Laboratory, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801, USA
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11
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Miyazaki K, Bagchi B, Yethiraj A. Self-consistent mode-coupling theory for the viscosity of rodlike polyelectrolyte solutions. J Chem Phys 2004; 121:8120-7. [PMID: 15485276 DOI: 10.1063/1.1797193] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A self-consistent mode-coupling theory is presented for the viscosity of solutions of charged rodlike polymers. The static structure factor used in the theory is obtained from polymer integral equation theory; the Debye-Huckel approximation is inadequate even at low concentrations. The theory predicts a nonmonotonic dependence of the reduced excess viscosity eta(R) on concentration from the behavior of the static structure factor in polyelectrolyte solutions. The theory predicts that the peak in eta(R) occurs at concentrations slightly lower than the overlap threshold concentration, c*. The peak height increases dramatically with increasing molecular weight and decreases with increased concentrations of added salt. The position of the peak, as a function of concentration divided by c*, is independent of salt concentration or molecular weight. The predictions can be tested experimentally.
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Affiliation(s)
- Kunimasa Miyazaki
- Department of Chemistry, Columbia University, New York, NY 10027, USA.
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Szamel G. Colloidal glass transition: beyond mode-coupling theory. PHYSICAL REVIEW LETTERS 2003; 90:228301. [PMID: 12857344 DOI: 10.1103/physrevlett.90.228301] [Citation(s) in RCA: 48] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2003] [Indexed: 05/24/2023]
Abstract
A new theory for the dynamics of concentrated colloidal suspensions and the colloidal glass transition is proposed. The starting point is the memory function representation of the density correlation function. The memory function can be expressed in terms of a time-dependent pair-density correlation function. An exact, formal equation of motion for this function is derived and a factorization approximation is applied to its evolution operator. In this way a closed set of equations for the density correlation function and the memory function is obtained. The theory predicts an ergodicity breaking transition similar to that predicted by mode-coupling theory, but at a higher density.
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Affiliation(s)
- Grzegorz Szamel
- Department of Chemistry, Colorado State University, Fort Collins, Colorado 80525, USA
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