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Numerical study of a slip-link model for polymer melts and nanocomposites
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10.1063/1.4799263
/content/aip/journal/jcp/138/19/10.1063/1.4799263
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/19/10.1063/1.4799263

Figures

Image of FIG. 1.
FIG. 1.

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

Image of FIG. 2.
FIG. 2.

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

Image of FIG. 3.
FIG. 3.

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

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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

Image of FIG. 6.
FIG. 6.

Relaxation time τ as a function of for = 0.5. For = 64, we observe while for = 128, .

Image of FIG. 7.
FIG. 7.

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

Image of FIG. 8.
FIG. 8.

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

Image of FIG. 9.
FIG. 9.

Relaxation time τ as a function of . For = 64, = 4 we observe while for = 128 and = 8, .

Image of FIG. 10.
FIG. 10.

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

Image of FIG. 11.
FIG. 11.

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: = 64, = 4, = 0.5.

Image of FIG. 12.
FIG. 12.

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

Image of FIG. 13.
FIG. 13.

Viscosity extracted from Fig. 12 for Rouse chains of lengths = 128 (○); = 64 (□), and FENE chains having length = 64 (△). The prediction of the convective constraint release model of Marrucci, is also shown. Our results correspond to with = 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 = 64. Same parameters as Fig. 12 .

Image of FIG. 14.
FIG. 14.

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: = 64, = 4, and = 0.5.

Image of FIG. 15.
FIG. 15.

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 ).

Image of FIG. 16.
FIG. 16.

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

Image of FIG. 17.
FIG. 17.

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: . In our simulations we obtain . This exponent does not change with the definition of the shear stress. Parameters are: = 64, = 4, = 0.5.

Image of FIG. 18.
FIG. 18.

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

Image of FIG. 19.
FIG. 19.

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 = 15 (data taken from Ref. ). The two fitting parameters used here are = 30.5 Å and τ = 3 × 10 s. The other slip-link parameters are = 4, = 64, and = 0.5.

Image of FIG. 20.
FIG. 20.

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

Image of FIG. 21.
FIG. 21.

Monomer density distribution in cells of length δ = 4.37b for a polymer melt with excluded volume interaction κ = 50, ρ = 6b, and with slip links ( = 64, = 4, = 0.5). The black curve displays the theoretical distribution, Eq. (30) .

Image of FIG. 22.
FIG. 22.

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 = 8 nodes (blue curve) and = 64 nodes (black curve). See text for further detail on the density discretization. The parameters are ρ = 6b, κ = 50. The other parameters retained are: = 64, = 4, = 0.5.

Image of FIG. 23.
FIG. 23.

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 while the brown line corresponds to the effective radius σ = σ + . We have considered = 8 fillers dispersed on a cubic lattice. The polymer parameters are = 32, κ = 50, ρ = 5.98, = 4, and = 0.5.

Image of FIG. 24.
FIG. 24.

Viscosity of the model nanocomposite as a function of the filler volume fraction ϕ. The = 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 η and β are the two fitting parameters (η = 889 ± 33 / τ and β = 2.9 ± 1.2).

Tables

Generic image for table
Table I.

Main parameters that define the slip link model.

Generic image for table
Table II.

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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/content/aip/journal/jcp/138/19/10.1063/1.4799263
2013-05-15
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Numerical study of a slip-link model for polymer melts and nanocomposites
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/19/10.1063/1.4799263
10.1063/1.4799263
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