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Jelić A, Ilg P, Ottinger HC. Bridging length and time scales in sheared demixing systems: from the Cahn-Hilliard to the Doi-Ohta model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:011131. [PMID: 20365347 DOI: 10.1103/physreve.81.011131] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2009] [Revised: 12/16/2009] [Indexed: 05/29/2023]
Abstract
We develop a systematic coarse-graining procedure which establishes the connection between models of mixtures of immiscible fluids at different length and time scales. We start from the Cahn-Hilliard model of spinodal decomposition in a binary fluid mixture under flow from which we derive the coarse-grained description. The crucial step in this procedure is to identify the relevant coarse-grained variables and find the appropriate mapping which expresses them in terms of the more microscopic variables. In order to capture the physics of the Doi-Ohta level, we introduce the interfacial width as an additional variable at that level. In this way, we account for the stretching of the interface under flow and derive analytically the convective behavior of the relevant coarse-grained variables, which in the long wavelength limit recovers the familiar phenomenological Doi-Ohta model. In addition, we obtain the expression for the interfacial tension in terms of the Cahn-Hilliard parameters as a direct result of the developed coarse-graining procedure. Finally, by analyzing the numerical results obtained from the simulations on the Cahn-Hilliard level, we discuss that dissipative processes at the Doi-Ohta level are of the same origin as in the Cahn-Hilliard model. The way to estimate the interface relaxation times of the Doi-Ohta model from the underlying morphology dynamics simulated at the Cahn-Hilliard level is established.
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Affiliation(s)
- Asja Jelić
- Polymer Physics, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland.
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2
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Chen XB, Niu LS, Shi HJ. Modeling the phase separation in binary lipid membrane under externally imposed oscillatory shear flow. Colloids Surf B Biointerfaces 2008; 65:203-12. [PMID: 18502621 DOI: 10.1016/j.colsurfb.2008.04.002] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2008] [Revised: 04/09/2008] [Accepted: 04/09/2008] [Indexed: 11/18/2022]
Abstract
By adding external velocity terms, the two-dimensional time-dependent Ginzburg-Landau (TDGL) equations are modified. Based on this, the phase separation in binary lipid membrane under externally imposed oscillatory shear flow is numerically modeled employing the Cell Dynamical System (CDS) approach. Considering shear flows with different frequencies and amplitudes, several aspects of such a phase evolving process are studied. Firstly, visualized results are shown via snapshot figures of the membrane shape. And then, the simulated scattering patterns at typical moments are presented. Furthermore, in order to more quantitatively discuss this phase-separation process, the time growth laws of the characteristic domain sizes in both directions parallel and perpendicular to the flow are investigated for each case. Finally, the peculiar rheological properties of such binary lipid membrane system have been discussed, mainly the normal stress difference and the viscoelastic complex shear moduli.
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Affiliation(s)
- Xiao-Bo Chen
- Key Laboratory of Failure Mechanics, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China
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3
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Fielding SM. Role of inertia in nonequilibrium steady states of sheared binary fluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:021504. [PMID: 18352031 DOI: 10.1103/physreve.77.021504] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2007] [Indexed: 05/26/2023]
Abstract
We study numerically phase separation in a binary fluid subject to an applied shear flow in two dimensions, with full hydrodynamics. To do so, we introduce a mixed finite-differencing and spectral simulation technique, with a transformation to render trivial the implementation of Lees-Edwards sheared periodic boundary conditions. For systems with inertia, we reproduce the nonequilibrium steady states reported in a recent lattice Boltzmann study. The domain coarsening that would occur in zero shear is arrested by the applied shear flow, which restores a finite-domain-size set by the inverse shear rate. For inertialess systems, in contrast, we find no evidence of nonequilibrium steady states free of finite-size effects: Coarsening persists indefinitely until the typical domain size attains the system size, as in zero shear. We present an analytical argument that supports this observation and that furthermore provides a possible explanation for a hitherto puzzling property of the nonequilibrium steady states with inertia.
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Affiliation(s)
- Suzanne M Fielding
- School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Manchester, UK.
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Stratford K, Desplat JC, Stansell P, Cates ME. Binary fluids under steady shear in three dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:030501. [PMID: 17930189 DOI: 10.1103/physreve.76.030501] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2007] [Indexed: 05/25/2023]
Abstract
We simulate by the lattice Boltzmann method the steady shearing of a binary fluid mixture with full hydrodynamics in three dimensions. Contrary to some theoretical scenarios, a dynamical steady state is attained with finite correlation lengths in all three spatial directions. Using large simulations, we obtain at moderately high Reynolds numbers apparent scaling exponents comparable to those found by us previously in two dimensions (2D). However, in 3D there may be a crossover to different behavior at low Reynolds number: accessing this regime requires even larger computational resources than used here.
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Affiliation(s)
- K Stratford
- SUPA, School of Physics, The University of Edinburgh, JCMB The King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom
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5
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LUO Y, OUYANG W, YANG Q, LI G. STUDY ON THE PHASE SEPARATION KINETICS OF PS/PVME BLENDS UNDER OSCILLATORY SHEAR——THE TIME DEPENDENCE AND STRESS RESPONSE OF PHASE SEPARATION. ACTA POLYM SIN 2006. [DOI: 10.3724/sp.j.1105.2006.00557] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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6
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Stansell P, Stratford K, Desplat JC, Adhikari R, Cates ME. Nonequilibrium steady states in sheared binary fluids. PHYSICAL REVIEW LETTERS 2006; 96:085701. [PMID: 16606196 DOI: 10.1103/physrevlett.96.085701] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2005] [Indexed: 05/08/2023]
Abstract
We simulate by lattice Boltzmann the steady shearing of a binary fluid mixture undergoing phase separation with full hydrodynamics in two dimensions. Contrary to some theoretical scenarios, a dynamical steady state is attained with finite domain lengths L(x,y) in the directions (x,y) of velocity and velocity gradient. Apparent scaling exponents are estimated as Lx approximately gamma (-2/3) and Ly approximately gamma(-3/4). We discuss the relative roles of diffusivity and hydrodynamics in attaining steady state.
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Affiliation(s)
- P Stansell
- SUPA, School of Physics, University of Edinburgh, JCMB Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom
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Berti S, Boffetta G, Cencini M, Vulpiani A. Turbulence and coarsening in active and passive binary mixtures. PHYSICAL REVIEW LETTERS 2005; 95:224501. [PMID: 16384226 DOI: 10.1103/physrevlett.95.224501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2005] [Indexed: 05/05/2023]
Abstract
Phase separation between two fluids in two dimensions is investigated by means of direct numerical simulations of coupled Navier-Stokes and Cahn-Hilliard equations. We study the phase ordering process in the presence of an external stirring acting on the velocity field. For both active and passive mixtures we find that, for a sufficiently strong stirring, coarsening is arrested in a stationary dynamical state characterized by a continuous rupture and formation of finite domains. Coarsening arrest is shown to be independent of the chaotic or regular nature of the flow.
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Affiliation(s)
- S Berti
- Dipartimento di Fisica Generale and INFN Università di Torino, Via Pietro Giuria 1, I-10125 Torino, Italy
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Cirillo ENM, Gonnella G, Saracco GP. Monte Carlo results for the Ising model with shear. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:026139. [PMID: 16196676 DOI: 10.1103/physreve.72.026139] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2005] [Indexed: 05/04/2023]
Abstract
We study the kinetics of domain growth in the Ising model with nonconserved dynamics under the action of a stochastic driving field that mimics the action of a shear flow. At late times, we found multistriped configurations with constant transversal size and linear growth in the direction of the flow. In cases with weak shear, a regime characterized by the decreasing of the transversal size is found that could correspond to previous theoretical investigations. This behavior is confirmed by the analysis of the structure factor patterns.
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Affiliation(s)
- E N M Cirillo
- Dipartimento de Metodi e Modelli Matematici, Facoltà di Ingegneria, Universita di Roma La Sapienza, via A. Scarpa 16, 00161 Rome, Italy
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Travasso RDM, Buxton GA, Kuksenok O, Good K, Balazs AC. Modeling the morphology and mechanical properties of sheared ternary mixtures. J Chem Phys 2005; 122:194906. [PMID: 16161616 DOI: 10.1063/1.1903883] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Through a combination of simulation techniques, we determine both the structural evolution and mechanical properties of blends formed from immiscible ternary mixtures. In this approach, we first use the lattice Boltzmann method to simulate the phase separation dynamics of A/B/C fluid mixtures for varying compositions within the spinodal region. We also investigate the effect of an imposed shear on the phase ordering of the mixture. We assume that the fluid is quenched sufficiently rapidly that the phase-separated structure is preserved in the resultant solid. Then, the output from our morphological studies serves as the input to the lattice spring model, which is used to simulate the elastic response of solids to an applied deformation. These simulations reveal how the local stress and strain fields and the global Young's modulus depend on the composition of the blend and the stiffness of the components. By comparing the results for the sheared and unsheared cases, we can isolate optimal processing conditions for enhancing the mechanical performance of the blends. Overall, the findings provide fundamental insight into the relationship between structure, processing, and properties for heterogeneous materials and can yield guidelines for formulating blends with the desired macroscopic mechanical behavior.
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Affiliation(s)
- Rui D M Travasso
- Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, USA
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Xu A, Gonnella G, Lamura A. Phase-separating binary fluids under oscillatory shear. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:056105. [PMID: 12786218 DOI: 10.1103/physreve.67.056105] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/05/2002] [Indexed: 05/24/2023]
Abstract
We apply the lattice Boltzmann methods to study the segregation of binary fluid mixtures under oscillatory shear flow in two dimensions. The algorithm allows to simulate systems whose dynamics is described by the Navier-Stokes and the convection-diffusion equations. The interplay between several time scales produces a rich and complex phenomenology. We investigate the effects of different oscillation frequencies and viscosities on the morphology of the phase separating domains. We find that at high frequencies the evolution is almost isotropic with growth exponents 2/3 and 1/3 in the inertial (low viscosity) and diffusive (high viscosity) regimes, respectively. When the period of the applied shear flow becomes of the same order of the relaxation time T(R) of the shear velocity profile, anisotropic effects are clearly observable. In correspondence with nonlinear patterns for the velocity profiles, we find configurations where lamellar order close to the walls coexists with isotropic domains in the middle of the system. For particular values of frequency and viscosity it can also happen that the convective effects induced by the oscillations cause an interruption or a slowing of the segregation process, as found in some experiments. Finally, at very low frequencies, the morphology of domains is characterized by lamellar order everywhere in the system resembling what happens in the case with steady shear.
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Affiliation(s)
- Aiguo Xu
- Istituto Nazionale per la Fisica della Materia, Unità di Bari, Università di Bari, 70126 Bari, Italy
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11
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Zhu YJ, Ma YQ. Phase separation of a polymer blend driven by oscillating particles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:041503. [PMID: 12786366 DOI: 10.1103/physreve.67.041503] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2002] [Indexed: 05/24/2023]
Abstract
We study the possible formation of ordered structures of a binary polymer blend by introducing mobile particles in a periodically oscillating driving field. The particles which have a preferential attraction to one of the immiscible phases, will significantly perturb the phase separation of the system and breakup the isotropy of the system, so that some interesting structures such as lamellar and cylinder phases are observed by appropriate selection of the simulation parameters. We examine in detail the dependence of formed morphology and domain size on the oscillating fields, the relative composition of mixtures, the diffusion coefficient, and quench depth, and then discuss how to realize stable and highly ordered structures.
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Affiliation(s)
- Yue-jin Zhu
- National Laboratory of Solid State Microstructures, Nanjing University, China
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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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13
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Luo K, Yang Y. Lamellar Orientation and Corresponding Rheological Properties of Symmetric Diblock Copolymers under Steady Shear Flow. Macromolecules 2002. [DOI: 10.1021/ma010889j] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Kaifu Luo
- Department of Macromolecular Science, Key Laboratory of Molecular Engineering of Polymers, SMEC, Fudan University, Shanghai 200433, China
| | - Yuliang Yang
- Department of Macromolecular Science, Key Laboratory of Molecular Engineering of Polymers, SMEC, Fudan University, Shanghai 200433, China
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14
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Kiwata H. Phase separation under two-dimensional Poiseuille flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:051505. [PMID: 11414907 DOI: 10.1103/physreve.63.051505] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2000] [Revised: 12/04/2000] [Indexed: 05/23/2023]
Abstract
The spinodal decomposition of a two-dimensional binary fluid under Poiseuille flow is studied by numerical simulation. We investigated time dependence of domain sizes in directions parallel and perpendicular to the flow. In an effective region of the flow, the power-law growth of a characteristic length in the direction parallel to the flow changes from the diffusive regime with the growth exponent alpha=1/3 to a new regime. The scaling invariance of the growth in the perpendicular direction is destroyed after the diffusive regime. A recurrent prevalence of thick and thin domains which determines log-time periodic oscillations has not been observed in our model. The growth exponents in the infinite system under two-dimensional Poiseuille flow are obtained by the renormalization group.
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Affiliation(s)
- H Kiwata
- Division of Natural Science, Osaka Kyoiku University, Kashiwara, Osaka 582-8582, Japan
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Berthier L. Phase separation in a homogeneous shear flow: morphology, growth laws, and dynamic scaling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:051503. [PMID: 11414905 DOI: 10.1103/physreve.63.051503] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2000] [Indexed: 05/23/2023]
Abstract
We numerically investigate the influence of a homogeneous shear flow on the spinodal decomposition of a binary mixture by solving the Cahn-Hilliard equation in a two-dimensional geometry. Several aspects of this much studied problem are clarified. Our numerical data show unambiguously that, in the shear flow, the domains have on average an elliptic shape. The time evolution of the three parameters describing this ellipse is obtained for a wide range of shear rates. For the lowest shear rates investigated, we find the growth laws for the two principal axis R perpendicular (t) approximately const, R parallel (t) approximately t, while the mean orientation of the domains with respect to the flow is inversely proportional to the strain. This implies that when hydrodynamics is neglected, a shear flow does not stop the domain growth process. We also investigate the possibility of dynamic scaling, and show that only a nontrivial form of scaling holds, as predicted by a recent analytical approach to the case of a nonconserved order parameter. We show that a simple physical argument may account for these results.
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Affiliation(s)
- L Berthier
- Laboratoire de Physique, Ecole Normale Supérieure, 46 Allée d'Italie, F-69007 Lyon, France
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