^{1,2}, Elian M. Masnada

^{3}, Samy Merabia

^{1,a)}, Marc Couty

^{2}and Jean-Louis Barrat

^{3,b)}

### Abstract

We present a numerical study of the slip link model introduced by Likhtman for describing the dynamics of dense polymer melts. After reviewing the technical aspects associated with the implementation of the model, we extend previous work in several directions. The dependence of the relaxation modulus with the slip link density and the slip link stiffness is reported. Then the nonlinear rheological properties of the model, for a particular set of parameters, are explored. Finally, we introduce excluded volume interactions in a mean field such as manner in order to describe inhomogeneous systems, and we apply this description to a simple nanocomposite model. With this extension, the slip link model appears as a simple and generic model of a polymer melt, that can be used as an alternative to molecular dynamics for coarse grained simulations of complex polymeric systems.

J.L.B. is supported by the Institut Universitaire de France.

We thank Juan de Pablo for sharing with us some preliminary results on the nonlinear rheology of a related slip link models.

Samy Merabia wants to thank François Detcheverry for interesting discussions.

I. INTRODUCTION

II. NUMERICAL IMPLEMENTATION OF LIKHTMAN'S MODEL

III. INFLUENCE OF THE MODEL PARAMETERS ON THE RELAXATION MODULUS

IV. NONLINEAR RHEOLOGICAL BEHAVIOR

V. INTRODUCING EXCLUDED VOLUME AND SPACE : A STEP TOWARDS MODELINGNANOCOMPOSITES

VI. SUMMARY

VII. COMMENT

### Key Topics

- Polymers
- 69.0
- Polymer melts
- 22.0
- Shear rate dependent viscosity
- 16.0
- Numerical modeling
- 12.0
- Reptation
- 12.0

##### B82B1/00

## Figures

Rouse chain with slip-links. The ring of one slip-link is located by his curvilinear abscissa x j . From x j the vector is constructed according to Eq. (1) . The anchoring points are distributed around with the following Boltzmann weight:

Rouse chain with slip-links. The ring of one slip-link is located by his curvilinear abscissa x j . From x j the vector is constructed according to Eq. (1) . The anchoring points are distributed around with the following Boltzmann weight:

Stress relaxation modulus as a function of time for different chain lengths N m . N e = 4 and N s = 0.5.

Stress relaxation modulus as a function of time for different chain lengths N m . N e = 4 and N s = 0.5.

Stress relaxation modulus as a function of time for different values of the mean number of monomers between slip links N e . Other parameters are N m = 64 and N s = 0.5.

Stress relaxation modulus as a function of time for different values of the mean number of monomers between slip links N e . Other parameters are N m = 64 and N s = 0.5.

Stress relaxation modulus as a function of time for different slip link stiffness N s . N m = 64 and N e = 4.

Stress relaxation modulus as a function of time for different slip link stiffness N s . N m = 64 and N e = 4.

Fitting procedure to obtain the reptation parameters and τ d from the stress relaxation modulus. The black curve is the simulated relaxation modulus for N m = 64, N e = 1, and N s = 0.5. The red curve is the best fit of G(t) using the reptation model Eqs. (17) and (18) .

Fitting procedure to obtain the reptation parameters and τ d from the stress relaxation modulus. The black curve is the simulated relaxation modulus for N m = 64, N e = 1, and N s = 0.5. The red curve is the best fit of G(t) using the reptation model Eqs. (17) and (18) .

Relaxation time τ d /τ0 as a function of N e for N s = 0.5. For N m = 64, we observe while for N m = 128, .

Relaxation time τ d /τ0 as a function of N e for N s = 0.5. For N m = 64, we observe while for N m = 128, .

Amplitude obtained with the fitting procedure illustrated in Fig. 5 as a function of the parameter N e , for two chain lengths: N m = 64 and N m = 128. The parameter N s = 0.5 is fixed. For N m = 64, we observe , while for N m = 128, .

Amplitude obtained with the fitting procedure illustrated in Fig. 5 as a function of the parameter N e , for two chain lengths: N m = 64 and N m = 128. The parameter N s = 0.5 is fixed. For N m = 64, we observe , while for N m = 128, .

Amplitude as function of the parameter N s and with N e = 4, for N m = 64, the power law obtained is . For N m = 128, we chose N e = 8 the power law is .

Amplitude as function of the parameter N s and with N e = 4, for N m = 64, the power law obtained is . For N m = 128, we chose N e = 8 the power law is .

Relaxation time τ d /τ0 as a function of N s . For N m = 64, N e = 4 we observe while for N m = 128 and N e = 8, .

Relaxation time τ d /τ0 as a function of N s . For N m = 64, N e = 4 we observe while for N m = 128 and N e = 8, .

Shear stress as a function of time under steady shear flow at several shear rates. The m parameters are: N m = 64, N e = 4, and N s = 0.5. From top to bottom, the shear rates are .

Shear stress as a function of time under steady shear flow at several shear rates. The m parameters are: N m = 64, N e = 4, and N s = 0.5. From top to bottom, the shear rates are .

Time corresponding to the maximum of the stress overshoot (see Fig. 10 ) as a function of the shear rate. The exponent is not sensitive to the definition of the shear stress ( or ). Parameters: N m = 64, N e = 4, N s = 0.5.

Time corresponding to the maximum of the stress overshoot (see Fig. 10 ) as a function of the shear rate. The exponent is not sensitive to the definition of the shear stress ( or ). Parameters: N m = 64, N e = 4, N s = 0.5.

Evolution of the shear plateau as a function of shear rate for Rouse chains of lengths N m = 128 (○); N m = 64 (□), and FENE chains having length N m = 64 (△). Solid lines are guides to the eyes. The other parameters are: N e = 4 and N s = 0.5

Evolution of the shear plateau as a function of shear rate for Rouse chains of lengths N m = 128 (○); N m = 64 (□), and FENE chains having length N m = 64 (△). Solid lines are guides to the eyes. The other parameters are: N e = 4 and N s = 0.5

Viscosity extracted from Fig. 12 for Rouse chains of lengths N m = 128 (○); N m = 64 (□), and FENE chains having length N m = 64 (△). The prediction of the convective constraint release model of Marrucci, 27 is also shown. Our results correspond to with N m = 64. All the results have been obtained using the Rouse expression of the shear stress . The extra contribution of the slip-links to the shear stress changes the shear thinning exponent from 0.66 to 0.67, in the simulations with N m = 64. Same parameters as Fig. 12 .

Viscosity extracted from Fig. 12 for Rouse chains of lengths N m = 128 (○); N m = 64 (□), and FENE chains having length N m = 64 (△). The prediction of the convective constraint release model of Marrucci, 27 is also shown. Our results correspond to with N m = 64. All the results have been obtained using the Rouse expression of the shear stress . The extra contribution of the slip-links to the shear stress changes the shear thinning exponent from 0.66 to 0.67, in the simulations with N m = 64. Same parameters as Fig. 12 .

Slip-link distribution along a chain for (red) and (green). The distribution is uniform for low while it becomes non-uniform under the strong shear flow. The model parameters are: N m = 64, N e = 4, and N s = 0.5.

Slip-link distribution along a chain for (red) and (green). The distribution is uniform for low while it becomes non-uniform under the strong shear flow. The model parameters are: N m = 64, N e = 4, and N s = 0.5.

Typical configuration of a chain (in the frame of the center of mass) under strong shear flow conditions. As the slip-links are advected by the flow, they tend to accumulate at the chain extremities, which explains the non-uniformity observed for large shear rates (see Fig. 14 ).

Typical configuration of a chain (in the frame of the center of mass) under strong shear flow conditions. As the slip-links are advected by the flow, they tend to accumulate at the chain extremities, which explains the non-uniformity observed for large shear rates (see Fig. 14 ).

Evolution of the first normal stress difference ( ) as a function of time. Parameters: N m = 64, N e = 4, N s = 0.5. From top to bottom, the shear rates are equal to .

Evolution of the first normal stress difference ( ) as a function of time. Parameters: N m = 64, N e = 4, N s = 0.5. From top to bottom, the shear rates are equal to .

Evolution of the first normal stress coefficient plateau ( ) as a function of the shear rate. In red, the shear stress is given by while in blue the definition is . We have also shown the theoretical scaling predicted by Marrucci: 27 . In our simulations we obtain . This exponent does not change with the definition of the shear stress. Parameters are: N m = 64, N e = 4, N s = 0.5.

Evolution of the first normal stress coefficient plateau ( ) as a function of the shear rate. In red, the shear stress is given by while in blue the definition is . We have also shown the theoretical scaling predicted by Marrucci: 27 . In our simulations we obtain . This exponent does not change with the definition of the shear stress. Parameters are: N m = 64, N e = 4, N s = 0.5.

Evolution of the second normal stress coefficient plateau ( ) as a function of the shear rate. The scaling law observed experimentally, 23 , is shown for comparison. Parameters: N m = 64, N e = 4, N s = 0.5.

Evolution of the second normal stress coefficient plateau ( ) as a function of the shear rate. The scaling law observed experimentally, 23 , is shown for comparison. Parameters: N m = 64, N e = 4, N s = 0.5.

Comparison between the transient viscosities under steady shear flow obtained in the slip-link model and the experimental curves corresponding to polystyrene with a comparable number of entanglements per chain Z = 15 (data taken from Ref. 23 ). The two fitting parameters used here are b = 30.5 Å and τ0 = 3 × 10−5 s. The other slip-link parameters are N e = 4, N m = 64, and N s = 0.5.

Comparison between the transient viscosities under steady shear flow obtained in the slip-link model and the experimental curves corresponding to polystyrene with a comparable number of entanglements per chain Z = 15 (data taken from Ref. 23 ). The two fitting parameters used here are b = 30.5 Å and τ0 = 3 × 10−5 s. The other slip-link parameters are N e = 4, N m = 64, and N s = 0.5.

Comparison between the steady values of Ψ1 and the experimental data of polystyrene having the same degree of entanglement (from Ref. 23 ). Same fitting and simulation parameters as in Fig. 19 .

Monomer density distribution in cells of length δ = 4.37b for a polymer melt with excluded volume interaction κ0 N m = 50, ρ0 = 6b−3, and with slip links (N m = 64, N e = 4, N s = 0.5). The black curve displays the theoretical distribution, Eq. (30) .

Monomer density distribution in cells of length δ = 4.37b for a polymer melt with excluded volume interaction κ0 N m = 50, ρ0 = 6b−3, and with slip links (N m = 64, N e = 4, N s = 0.5). The black curve displays the theoretical distribution, Eq. (30) .

Stress relaxation modulus against time for a melt of ghost polymer chains with slip links (red curve) and for a melt of interacting chains. In this latter case, we have compared the result when a monomer contributes to the density of P 3 = 8 nodes (blue curve) and P 3 = 64 nodes (black curve). See text for further detail on the density discretization. The parameters are ρ0 = 6b−3, κ0 N m = 50. The other parameters retained are: N m = 64, N e = 4, N s = 0.5.

Stress relaxation modulus against time for a melt of ghost polymer chains with slip links (red curve) and for a melt of interacting chains. In this latter case, we have compared the result when a monomer contributes to the density of P 3 = 8 nodes (blue curve) and P 3 = 64 nodes (black curve). See text for further detail on the density discretization. The parameters are ρ0 = 6b−3, κ0 N m = 50. The other parameters retained are: N m = 64, N e = 4, N s = 0.5.

Monomer density as a function of the distance to the center of the filler for different filler volume fractions, ϕ = 10%, ϕ = 20%, and ϕ = 30%. In blue, we have represented the filler radius σ f while the brown line corresponds to the effective radius σ eff = σ f + b. We have considered n f = 8 fillers dispersed on a cubic lattice. The polymer parameters are N m = 32, κ0 N m = 50, ρ0 = 5.98, N e = 4, and N s = 0.5.

Monomer density as a function of the distance to the center of the filler for different filler volume fractions, ϕ = 10%, ϕ = 20%, and ϕ = 30%. In blue, we have represented the filler radius σ f while the brown line corresponds to the effective radius σ eff = σ f + b. We have considered n f = 8 fillers dispersed on a cubic lattice. The polymer parameters are N m = 32, κ0 N m = 50, ρ0 = 5.98, N e = 4, and N s = 0.5.

Viscosity of the model nanocomposite as a function of the filler volume fraction ϕ. The n f = 8 fillers are distributed on a cubic lattice. The different parameters used are summarized in Table II . We have also represented the fit obtained from the expression where η0 and β are the two fitting parameters (η0 = 889 ± 33 k B T/b 3τ0 and β = 2.9 ± 1.2).

Viscosity of the model nanocomposite as a function of the filler volume fraction ϕ. The n f = 8 fillers are distributed on a cubic lattice. The different parameters used are summarized in Table II . We have also represented the fit obtained from the expression where η0 and β are the two fitting parameters (η0 = 889 ± 33 k B T/b 3τ0 and β = 2.9 ± 1.2).

## Tables

Main parameters that define the slip link model.

Main parameters that define the slip link model.

Main parameters that define the slip link model applied to a nanocomposite. We have also indicated the values of the parameters used in this work.

Main parameters that define the slip link model applied to a nanocomposite. We have also indicated the values of the parameters used in this work.

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