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Simple Models for Complex Nonequilibrium Uids

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Simple Models for Complex Nonequilibrium Uids Available online at www.sciencedirect.com Physics Reports 390 (2004) 453–551 www.elsevier.com/locate/physrep Simple models for complex nonequilibrium &uids Martin Kr)ogera;b;∗ aPolymer Physics, Materials Science, ETH Zentrum, 8092 Zurich, Switzerland bInstitut f. Theoretische Physik, Techn. Univ. Berlin, 10623 Berlin, Germany Accepted 15 October 2003 editor: E. Sackmann Abstract This review is concerned with the nonequilibrium dynamics and structure ofcomplex &uids based on simple micro- and mesoscopic physical models which are not rigorously solvable by analytic methods. Special emphasis is placed on the ÿnitely extendable nonlinear elastic (FENE) chain models which account for molecular stretch, bending, and topology. More coarse-grained descriptions such as primitive path models, and elongated particle models are reviewed as well. We focus on their inherently anisotropic material—in particular rheological—properties via deterministic and stochastic approaches. A number ofrepresentative examples are given on how simple (often high-dimensional) models can, and have been implemented in order to enable the analysis ofthe microscopic origins ofthe nonlinear viscoelastic behavior ofpolymeric materials. These examples are shown to provide us with a number ofroutes fordeveloping and establishing coarse-grained (low-dimensional) models devoted to the prediction ofa reduced number ofsigniÿcant material properties. At this stage approximations which allow for an analytical treatment are discussed as well. Concerning the types ofcomplex &uids, we cover the range from&exible to semi&exible polymers in melts and solutions, wormlike micelles, structural suspensions including ferro&uids in ÿeld-induced anisotropic or liquid crystalline phases. c 2003 Elsevier B.V. All rights reserved. PACS: 05.10.−a; 83.10.Mj; 05.20.Dd Keywords: Polymer; Non-Newtonian rheology; Flow; Structure; Dynamics; Viscosities; Orientation; Microscopic models; Mesoscopic models; Computer simulation; Molecular dynamics; Brownian dynamics; Stochastic di?erential equation; Constitutive equation; Kinetic theory; Liquid crystal; Ferro&uid; Semi&exible chain; Network; Tube model; Primitive path; Elongated particle; Order parameter ∗ Corresponding author. Polymer physics, Materials Science, ETH Zentrum, 8092, Zurich, Switzerland. E-mail address: [email protected] (M. Kr)oger). 0370-1573/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2003.10.014 454 M. Kroger / Physics Reports 390 (2004) 453–551 Contents 1. Introduction ........................................................................................ 456 2. FENE dumbbell models in inÿnitely diluted solution ..................................................... 463 2.1. FENE-PMF dumbbell in ÿnitely diluted solution .................................................... 464 2.2. Introducing a mean ÿeld potential ................................................................ 465 2.3. Relaxation equation for the tensor of gyration ...................................................... 465 2.4. Symmetry adapted basis ......................................................................... 466 2.5. Stress tensor and material functions ............................................................... 469 2.6. Reduced description ofkinetic models ............................................................ 471 3. FENE chain in dilute solution including hydrodynamic interactions ........................................ 471 3.1. Long chain limit, Cholesky decomposition ......................................................... 473 3.2. NEBD simulation details ........................................................................ 473 3.3. Universal ratios ................................................................................ 474 4. FENE chains in melts ............................................................................... 476 4.1. NEMD simulation method ....................................................................... 478 4.2. Stress tensor ................................................................................... 478 4.3. Lennard-Jones (LJ) units ........................................................................ 479 4.4. Flow curve and dynamical crossover .............................................................. 479 4.5. Characteristic lengths and times .................................................................. 479 4.6. Origin ofthe stress-optic rule (SOR) and its failures ................................................ 482 4.7. Interpretation ofdimensionless simulation numbers .................................................. 485 5. FENE-CB chains .................................................................................... 486 5.1. Conformational statistics of wormlike chains (WLC) ................................................ 486 5.1.1. Functional integrals for WLCs ............................................................. 487 5.1.2. Properties ofWLCs, persistence length, radius ofgyration .................................... 488 5.1.3. Scattering functions ...................................................................... 488 5.2. FENE-C wormlike micelles ...................................................................... 489 5.2.1. Flow-induced orientation and degradation ................................................... 490 5.2.2. Length distribution ....................................................................... 491 5.2.3. FENE-C theory vs simulation, rheology, &ow alignment ...................................... 492 5.3. FENE-B semi&exible chains, actin ÿlaments ....................................................... 493 5.4. FENE-B liquid crystalline polymers ............................................................... 499 5.4.1. Static structure factor ..................................................................... 503 5.5. FENE-CB transient semi&exible networks, ring formation ............................................ 505 6. Primitive paths ...................................................................................... 508 6.1. Doi–Edwards tube model and its improvements .................................................... 509 6.2. Reÿned tube model with anisotropic &ow-induced tube renewal ...................................... 511 6.2.1. Linear viscoelasticity ofmelts and concentrated solutions ..................................... 512 6.3. Nonlinear viscoelasticity, particular closure ........................................................ 513 6.3.1. Example: reÿned tube model, stationary shear &ow ........................................... 514 6.3.2. Example: transient viscosities for rigid polymers ............................................. 514 6.3.3. Example: Doi–Edwards model as a special case ............................................. 515 6.4. Nonlinear viscoelasticity without closure, Galerkin’s principle ........................................ 516 7. Elongated particles .................................................................................. 519 7.1. Director theory ................................................................................. 520 7.2. Structural theories ofsuspensions ................................................................. 520 7.2.1. Semi-dilute suspensions ofelongated particles ............................................... 522 7.2.2. Concentrated suspensions ofrod-like polymers ............................................... 522 7.3. Uniaxial &uids, micro-macro correspondence ....................................................... 522 7.3.1. Application: concentrated suspensions ofdisks, spheres, rods .................................. 523 M. Kroger / Physics Reports 390 (2004) 453–551 455 7.3.2. Example: tumbling ....................................................................... 524 7.3.3. Example: Miesowicz viscosities ............................................................ 524 7.4. Uniaxial &uids: decoupling approximations ......................................................... 526 7.4.1. Decoupling with correct tensorial symmetry ................................................. 527 7.5. Ferro&uids: dynamics and rheology ............................................................... 528 7.6. Liquid crystals: periodic and irregular dynamics .................................................... 531 7.6.1. Landau–de Gennes potential ............................................................... 531 7.6.2. In-plane and out-of-plane states ............................................................ 531 8. Connection between di?erent levels ofdescription ....................................................... 533 8.1. Boltzmann equation ............................................................................. 533 8.2. Generalized Poisson structures ................................................................... 534 8.3. GENERIC equations ............................................................................ 534 8.4. Dissipative particles ............................................................................. 535 8.5. Langevin and Fokker–Planck equation, Brownian dynamics .......................................... 536 8.6. Projection operator methods ...................................................................... 536 8.7. Stress tensors: Giesekus–Kramers–GENERIC ....................................................... 538 8.8. Coarse-graining: from atomistic chains
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