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Menzel AM. Density and concentration field description of nonperiodic structures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:051505. [PMID: 22181420 DOI: 10.1103/physreve.84.051505] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2011] [Revised: 10/05/2011] [Indexed: 05/31/2023]
Abstract
We propose a simple nonlocal energy functional that is suitable for the continuum field characterization of nonperiodic and localized textures. The phenomenological functional is based on the pairwise direction-dependent interaction of field gradients that are separated by a fixed distance. In an appendix, we describe the numerical minimization of our functional. On that basis, we investigate the kinetic evolution of threadlike stripe patterns that are created by the functional when we start from an initially disordered state. At later stages, we find a coarse graining that shows the same scaling behavior as was obtained for the Cahn-Hilliard equation. In fact, the Cahn-Hilliard model is contained in our characterization as a limiting case. A slight modification of our model omits this coarse graining and leads to nonperiodic stripe phases. For the latter case, we investigate the temporal evolution of the defects (end points) of the threadlike stripes. In view of actual applications of this functional, we discuss the characterization of processes observed for polymeric systems and vesicles. The statistics of the growth of the threadlike structures is compared to the case of step-growth polymerization reactions. Furthermore, we demonstrate that the functional may be applied for the study of vesicles in a continuum field description. Basic features, such as the tendency of tank treading in simple shear flows and parachute folding in pipe flows, are reproduced.
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Affiliation(s)
- Andreas M Menzel
- Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
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2
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Chen P. Shear alignments in three-dimensional simulations of lamellar phases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:061503. [PMID: 16089739 DOI: 10.1103/physreve.71.061503] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2004] [Revised: 03/03/2005] [Indexed: 05/03/2023]
Abstract
Experiments studying layer orientation of sheared lamellar phases have consistently observed not only the seemingly obvious parallel orientation (with layers parallel to the shear plane), but also the so called perpendicular orientation (with layer normals along the vorticity direction) at the condition of higher shear frequencies and near the lamellar phase transition. We find that three-dimensional simulations of a deterministic mesoscopic dynamical equation (without thermal fluctuations) under the convection of simple shear flows have shown exactly such a dependence. The simulations show the important role played by the transverse orientation (layer normal along the velocity direction) and highlight the mechanism of the shear alignment being the competition between the shear frequency and the mesoscopic time scale of pattern organization.
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Affiliation(s)
- Peilong Chen
- Department of Physics and Center for Complex Systems, National Central University, Chungli 320, Taiwan
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3
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Rüdiger S. Viscoelastic response of diblock copolymers to oscillatory shear. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2005; 17:45-52. [PMID: 15864726 DOI: 10.1140/epje/i2004-10103-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2004] [Accepted: 02/16/2005] [Indexed: 05/02/2023]
Abstract
A mesoscopic model for diblock copolymers is combined with a simple rheological description of the viscoelastic contrast of its two phases. Under oscillatory shear the contrast generates secondary velocity fields and substantial deviations of volume-averaged flow parameters. The validity of our analytical findings is tested with three-dimensional numerical simulations. Furthermore, we consider the effect of advection by the generated flows on the stability of ordered lamellar states and propose a new criterion for the selection of orientations.
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Affiliation(s)
- S Rüdiger
- Physikalisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany.
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Fraser B, Denniston C, Müser MH. Diffusion, elasticity, and shear flow in self-assembled block copolymers: A molecular dynamics study. ACTA ACUST UNITED AC 2005. [DOI: 10.1002/polb.20387] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
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5
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Ho RM, Lin FH, Tsai CC, Lin CC, Ko BT, Hsiao BS, Sics I. Crystallization-Induced Undulated Morphology in Polystyrene-b-Poly(l-lactide) Block Copolymer. Macromolecules 2004. [DOI: 10.1021/ma0492869] [Citation(s) in RCA: 93] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Rong-Ming Ho
- Department of Chemical Engineering, National Tsing-Hua University, Hsinchu 30013, Taiwan, Union Chemical Laboratories, Industrial Technology Research Institute, Hsinchu 30013, Taiwan, Department of Chemical Engineering, National Chung-Hsing University, Taichung 402, Taiwan, Department of Chemistry, National Chung-Hsing University, Taichung 402, Taiwan, and Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400
| | - Fang-Ho Lin
- Department of Chemical Engineering, National Tsing-Hua University, Hsinchu 30013, Taiwan, Union Chemical Laboratories, Industrial Technology Research Institute, Hsinchu 30013, Taiwan, Department of Chemical Engineering, National Chung-Hsing University, Taichung 402, Taiwan, Department of Chemistry, National Chung-Hsing University, Taichung 402, Taiwan, and Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400
| | - Chi-Chun Tsai
- Department of Chemical Engineering, National Tsing-Hua University, Hsinchu 30013, Taiwan, Union Chemical Laboratories, Industrial Technology Research Institute, Hsinchu 30013, Taiwan, Department of Chemical Engineering, National Chung-Hsing University, Taichung 402, Taiwan, Department of Chemistry, National Chung-Hsing University, Taichung 402, Taiwan, and Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400
| | - Chu-Chieh Lin
- Department of Chemical Engineering, National Tsing-Hua University, Hsinchu 30013, Taiwan, Union Chemical Laboratories, Industrial Technology Research Institute, Hsinchu 30013, Taiwan, Department of Chemical Engineering, National Chung-Hsing University, Taichung 402, Taiwan, Department of Chemistry, National Chung-Hsing University, Taichung 402, Taiwan, and Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400
| | - Bao-Tsan Ko
- Department of Chemical Engineering, National Tsing-Hua University, Hsinchu 30013, Taiwan, Union Chemical Laboratories, Industrial Technology Research Institute, Hsinchu 30013, Taiwan, Department of Chemical Engineering, National Chung-Hsing University, Taichung 402, Taiwan, Department of Chemistry, National Chung-Hsing University, Taichung 402, Taiwan, and Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400
| | - Benjamin S. Hsiao
- Department of Chemical Engineering, National Tsing-Hua University, Hsinchu 30013, Taiwan, Union Chemical Laboratories, Industrial Technology Research Institute, Hsinchu 30013, Taiwan, Department of Chemical Engineering, National Chung-Hsing University, Taichung 402, Taiwan, Department of Chemistry, National Chung-Hsing University, Taichung 402, Taiwan, and Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400
| | - Igors Sics
- Department of Chemical Engineering, National Tsing-Hua University, Hsinchu 30013, Taiwan, Union Chemical Laboratories, Industrial Technology Research Institute, Hsinchu 30013, Taiwan, Department of Chemical Engineering, National Chung-Hsing University, Taichung 402, Taiwan, Department of Chemistry, National Chung-Hsing University, Taichung 402, Taiwan, and Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400
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Dwivedi V, Ahluwalia R, Lookman T, Saxena A. Viscoelastic properties of dynamically asymmetric binary fluids under shear flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:011506. [PMID: 15324055 DOI: 10.1103/physreve.70.011506] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2003] [Revised: 04/16/2004] [Indexed: 05/24/2023]
Abstract
We study theoretically the viscoelastic properties of sheared binary fluids that have strong dynamical asymmetry between the two components. The dynamical asymmetry arises due to asymmetry between the viscoelastic stresses, particularly the bulk stress. Our calculations are based on the two-fluid model that incorporates the asymmetric stress distribution. We simulate the phase separation process under an externally imposed shear and compare the asymmetric case with the usual phase separation under shear flow without viscoelastic effects. We also simulate the behavior of phase-separated stable morphologies under applied shear and compute the stress relaxation.
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Affiliation(s)
- Vinay Dwivedi
- Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, California 93106, USA
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Huang ZF, Viñals J. Shear-induced grain boundary motion for lamellar phases in the weakly nonlinear regime. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:041504. [PMID: 15169022 DOI: 10.1103/physreve.69.041504] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2003] [Indexed: 05/24/2023]
Abstract
We study the effect of an externally imposed oscillatory shear on the motion of a grain boundary that separates differently oriented domains of the lamellar phase of a diblock copolymer. A direct numerical solution of the Swift-Hohenberg equation in shear flow is used for the case of a transverse/parallel grain boundary in the limits of weak nonlinearity and low shear frequency. We focus on the region of parameters in which both transverse and parallel lamellae are linearly stable. Shearing leads to excess free energy in the transverse region relative to the parallel region, which is in turn dissipated by net motion of the boundary toward the transverse region. The observed boundary motion is a combination of rigid advection by the flow and order parameter diffusion. The latter includes break up and reconnection of lamellae, as well as a weak Eckhaus instability in the boundary region for sufficiently large strain amplitude that leads to slow wave number readjustment. The net average velocity is seen to increase with frequency and strain amplitude, and can be obtained by a multiple scale expansion of the governing equations.
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Affiliation(s)
- Zhi-Feng Huang
- School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306-4120, USA.
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Huang ZF, Drolet F, Viñals J. Motion of a Transverse/Parallel Grain Boundary in a Block Copolymer under Oscillatory Shear Flow. Macromolecules 2003. [DOI: 10.1021/ma034083q] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Zhi-Feng Huang
- School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306-4120
| | - François Drolet
- School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306-4120
| | - Jorge Viñals
- School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306-4120
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Guo H, Kremer K, Soddemann T. Nonequilibrium molecular dynamics simulation of shear-induced alignment of amphiphilic model systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:061503. [PMID: 12513288 DOI: 10.1103/physreve.66.061503] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2002] [Indexed: 05/24/2023]
Abstract
We study the shear-induced alignment transition from transverse to the shear plane to parallel and perpendicular in a lamellar or smectics system. A recently proposed simple continuum model for amphiphiles is studied by large-scale nonequilibrium molecular dynamics simulation. We find a shear-rate dependence of the alignment transition under shear flow, parallel at low and perpendicular alignment at high shear rates. To identify the alignment and degree of order of these shear-induced orientations by scattering, a 3D analysis of the scattering data is needed.
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Affiliation(s)
- Hongxia Guo
- Max-Planck-Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany
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10
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Corberi F, Gonnella G, Lamura A. Ordering of the lamellar phase under a shear flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:016114. [PMID: 12241433 DOI: 10.1103/physreve.66.016114] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2001] [Indexed: 05/23/2023]
Abstract
The dynamics of a system quenched into a state with lamellar order and subject to an uniform shear flow is solved in the large-N limit. The description is based on the Brazovskii free energy and the evolution follows a convection-diffusion equation. Lamellas order preferentially with the normal along the vorticity direction. Typical lengths grow as gamma t(5/4) (with logarithmic corrections) in the flow direction and logarithmically in the shear direction. Dynamical scaling holds in the two-dimensional case while it is violated in D=3.
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Affiliation(s)
- F Corberi
- Istituto Nazionale per la Fisica della Materia, Unità di Salerno and Dipartimento di Fisica, Università di Salerno, 84081 Baronissi Salerno, Italy
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11
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Chen P, Viñals J. Lamellar Phase Stability in Diblock Copolymers under Oscillatory Shear Flows. Macromolecules 2002. [DOI: 10.1021/ma011586s] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Peilong Chen
- Department of Physics, National Central University, Chungli 320, Taiwan
| | - Jorge Viñals
- School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306-4120, and Department of Chemical Engineering, FAMU−FSU College of Engineering, Tallahassee, Florida 32310-6046
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12
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Morozov AN, Fraaije JGEM. Orientations of the lamellar phase of block copolymer melts under oscillatory shear flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:031803. [PMID: 11909098 DOI: 10.1103/physreve.65.031803] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2001] [Indexed: 05/23/2023]
Abstract
We develop a theory to describe the reorientation phenomena in the lamellar phase of block copolymer melts under reciprocating shear flow. We show that, similar to the steady shear, the oscillating flow anisotropically suppresses fluctuations and gives rise to the [parallel]--> [perpendicular] transition. The experimentally observed high-frequency reverse transition is explained in terms of interaction between the melt and the shear-cell walls.
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Affiliation(s)
- A N Morozov
- Instituut-Lorentz for Theoretical Physics, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands.
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13
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Morozov AN, Zvelindovsky AV, Fraaije JG. Influence of confinement on the orientational phase transitions in the lamellar phase of a block-copolymer melt under shear flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:051803. [PMID: 11735954 DOI: 10.1103/physreve.64.051803] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/18/2001] [Indexed: 05/23/2023]
Abstract
In this paper, we incorporate some real-system effects into the theory of orientational phase transitions under shear flow [M. E. Cates and S. T. Milner, Phys. Rev. Lett. 62 1856 (1989) and G. H. Fredrickson, J. Rheol. 38, 1045 (1994)]. In particular, we study the influence of the shear-cell boundaries on the orientation of the lamellar phase. We predict that at low shear rates, the parallel orientation appears to be stable. We show that there is a critical value of the shear rate at which the parallel orientation loses its stability and the perpendicular one appears immediately below the spinodal. We associate this transition with a crossover from the fluctuation to the mean-field behavior. At lower temperatures, the stability of the parallel orientation is restored. We find that the region of stability of the perpendicular orientation rapidly decreases as shear rate increases. This behavior might be misinterpreted as an additional perpendicular to parallel transition recently discussed in literature.
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Affiliation(s)
- A N Morozov
- Faculty of Mathematics and Natural Sciences, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands.
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14
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Ren SR, Hamley IW, Teixeira PI, Olmsted PD. Cell dynamics simulations of shear-induced alignment and defect annihilation in stripe patterns formed by block copolymers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:041503. [PMID: 11308844 DOI: 10.1103/physreve.63.041503] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2000] [Revised: 12/05/2000] [Indexed: 05/23/2023]
Abstract
The effect of large amplitude oscillatory shear on two-dimensional stripe patterns formed by block copolymers was investigated using cell dynamics simulations. A global orientational order parameter S and a correlation function for stripe normals G(r-r(')) were used to characterize the degree of stripe orientation under shear. The kinetics of stripe alignment, quantified by S, at various shear and quench conditions were studied as a function of strain amplitude, shear frequency, and temperature. The mechanisms of shear alignment and defect annihilation were investigated. A critical shear condition for complete alignment of stripes along the shear direction was also identified.
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Affiliation(s)
- S R Ren
- School of Chemistry, University of Leeds, Leeds LS2 9JT, United Kingdom
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Zvelindovsky AV, Sevink GJ, Fraaije JG. Shear-induced transitions in a ternary polymeric system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:R3063-R3066. [PMID: 11088879 DOI: 10.1103/physreve.62.r3063] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/1999] [Indexed: 05/23/2023]
Abstract
The first three-dimensional simulation of shear-induced phase transitions in a polymeric system has been performed. The method is based on dynamic density-functional theory. The pathways between a bicontinuous phase with developing gyroid mesostructure and a lamellar/cylinder phase coexistence are investigated for a mixture of flexible triblock ABA copolymer and solvent under simple steady shear.
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Affiliation(s)
- AV Zvelindovsky
- Faculty of Mathematics and Natural Sciences, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands and Leiden Institute of Chemistry, Leiden University, Einsteinweg 55, 2300 RA Leiden, The Netherlands*
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Cohen Y, Albalak RJ, Dair BJ, Capel MS, Thomas EL. Deformation of Oriented Lamellar Block Copolymer Films. Macromolecules 2000. [DOI: 10.1021/ma000513q] [Citation(s) in RCA: 144] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Yachin Cohen
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973
| | - Ramon J. Albalak
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973
| | - Benita J. Dair
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973
| | - Malcolm S. Capel
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973
| | - Edwin L. Thomas
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973
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