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 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.
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, .
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 .
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 .
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
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.
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 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.
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.
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.
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).
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.
Article metrics loading...
Full text loading...