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**Entanglements and stress correlations in coarsegrained molecular dynamics**

Alexei E. Likhtman, Sathish K. Sukumuran, Jorge Ramirez Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

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**Hierarchical modelling in polymer dynamics**

Constitutive equations Tube theories Single chain models Coarse-grained many-chains models Atomistic simulations > Quantum mechanics simulations Traditional rheology CR Tube Model? Traditional physics The weakest link Kremer-Grest MD, Padding-Briels Twentanglemets, NAPLES Well established coarse- graining procedures, force-fields, commercial packages

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**The missing link Many chains system One chain model The ultimate goal:**

Stochastic equation of motion for the chain in self-consistent entanglement field Many chains system + self-consistent field One chain model

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Is there a tube model? Best definition of the tube model: one-dimensional Rouse chain projected onto three-dimensional random walk tube. Open questions: Can I have expression for the tube field, please? How to “measure” tube in MD? Is the tube semiflexible? Diameter = persistence length? Branch point motion How does the contour length changes with deformation? Tube parameters for different polymers? Tube parameters for different concentrations?

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**Rubinstein-Panyukov network model**

Rubinstein and Panyukov, Macromolecules 2002, 6670

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**Construction of the model**

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**Shanbhag, Larson, Takimoto, Doi 2001**

Constraint release Hua and Schieber 1998 Shanbhag, Larson, Takimoto, Doi 2001

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**A.E.Likhtman, Macromolecules 2005**

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**Relaxation of dilute long chains (36K) in a short matrix: constraint release**

Mwmat labeled Rouse M.Zamponi et al, PRL 2006

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**Molecular Dynamics -- Kremer-Grest**

Polymers – Bead-FENE spring chains k = 30/2 R0=1.5 With excluded volume – Purely repulsive Lennard-Jones interaction between beads Density, = 0.85 Friction coefficent, = 0.5 Time step, dt = 0.012 Temperature, T = /k K.Kremer, G. S. Grest JCP (1990)

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g1(t) from MD for N=100,350 1 d 0.5 1/4 0.5 R e

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g1(i,t)/t0.5 from MD for N=350 ends g1(i,t)/t0.5 middle t

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**G(t) from MD for N=50,100,200,350 (Ne~50)**

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**G(t) from MD for N=50,100,200,350 (Ne~70)**

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**g1(i,t) -- MD vs sliplinks mapping 1:1 (N=200)**

Lines - MD Points - slip-links d 1 1 0 g1(i,t)/t0.5 e t

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**G(t) -- MD vs sliplinks mapping 1:1 (N=200)**

0 1 5 G(t)*t1/2 Lines - MD Points - slip-links e t

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**Questions for discussion**

Binary nature of entanglements? Can one propose an experiment which contradicts this? Non-linear flows: do entanglements appear in the middle of the chain? Is there an instability in monodisperse linear polymers?

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