Stretch-Induced Reduction of Friction via Primitive

Sep 24, 2014 - This study examines universality of the mechanism of SORF for narrowly distributed PS, polyisoprene (PI), and poly(n-butyl acrylate) (P...
0 downloads 5 Views 1MB Size
Download PDF
Article pubs.acs.org/Macromolecules

Test of Orientation/Stretch-Induced Reduction of Friction via Primitive Chain Network Simulations for Polystyrene, Polyisoprene, and Poly(n‑butyl acrylate) Yuichi Masubuchi,* Yumi Matsumiya, and Hiroshi Watanabe Institute for Chemical Research, Kyoto University, Gokasyo, Uji, Kyoto 611-0011, Japan ABSTRACT: Entanglement dynamics of polymers under fast elongation has not been fully understood. Namely, the steady-state uniaxial elongational viscosity of entangled polystyrene (PS) solutions increases with an increase of strain rate above the reciprocal Rouse time, whereas the viscosity of PS melts monotonically decreases even at such high rates. This qualitative difference between solution and melt has been hypothesized to result from the orientation/stretchinduced reduction of friction (SORF). This study examines universality of the mechanism of SORF for narrowly distributed PS, polyisoprene (PI), and poly(n-butyl acrylate) (PnBA). For this purpose, SORF was incorporated in the multichain slip-link simulation through an empirical relationship between the magnitude of friction reduction and the stretch/orientation order parameter for PS [reported in Macromolecules 2012, 45, 2773−2782]. This empirical SORF relationship adopted a mean-field view for contacts between polymer backbone and solvent, and the simulation having this mean-field feature was confirmed to reproduce satisfactorily the transient and steady elongational viscosity data for PS melts and solutions. Assuming that the magnitude of SORF is insensitive to the chemical structure of polymers, we further made the simulation with the same SORF relationship for PI and PnBA. For PI, the simulation reasonably described the viscosity data of a melt and solutions. However, for PnBA melt, the data were well described by the simulation but without SORF. This result suggests that the magnitude of SORF is not universal but changes with the chemical structure of polymers. Specifically, for PnBA, the flexible side chains always surround the PnBA backbone and may behave as a solvent to screen the direct interaction between the backbone segments even in melt. For this case, the mean-field type SORF relationship obviously fails. Nevertheless, a nonmean-field type SORF could result in negligible friction reduction even under fast elongation, thereby allowing the simulation without SORF to describe thet data for PnBA.

1. INTRODUCTION Linear viscoelasticity of polymers reflects the equilibrium chain dynamics. Experiments established that such linear viscoelastic properties are universal for polymers of various chemical structures and concentrations, given that those polymers have the same number of entanglements per chain Z and their modulus and time are reduced by the plateau modulus GN and the longest relaxation time τd.1 The universality has been observed also for nonlinear shear properties reflecting the nonequilibrium dynamics under fast shear if the Weissenberg number (shear rate normalized by τ d ) is adequately accommodated. These experimental findings have encouraged theoretical developments of generic molecular models that embed the chemical structure and concentration of polymers into just a few material parameters, i.e., Z, GN, and τd.2 However, progress in experimental techniques has revealed that the universality vanishes under fast elongational flow. For both melts and solutions, the uniaxial elongational viscosity decreases with an increase of strain rate ε̇ when ε̇ is between 1/ τd and 1/τR (reciprocal Rouse relaxation time). Nevertheless, on an increase of ε̇ above 1/τR, the viscosity increases for polystyrene (PS) solutions (with an upturn at ε̇ = 1/τR),3 whereas it keeps monotonically decreasing for PS melts.4 This © XXXX American Chemical Society

nonuniversality between solutions and melts is well established experimentally, but its molecular origin has not been fully elucidated. Leygue et al.5 and Yaoita et al.6 have shown that the nonuniversality cannot be attributed to a difference of the maximum stretch ratio of the entanglement segment in melts and solutions. Marrucci and Ianniruberto7 have proposed the interchain tube pressure effect to introduce a relaxation of the entanglement length (tube diameter). By tuning this tube relaxation time, Wagner et al.8 attained semiquantitative agreement with the PS melt data. However, the physics underlying the difference of the tube relaxation times in solutions and melts remains unclear. Furthermore, the interchain tube pressure effect is conceptually incompatible9 with the convective constraint release, an important molecular mechanism under fast flow.10 A few years ago, Ianniruberto et al.11,12 proposed the orientation/stretch-induced reduction of friction (SORF) to explain the difference of uniaxial elongational behavior of PS solutions and melts. Most of the coarse-grained models adopt Received: August 5, 2014 Revised: September 12, 2014

A

dx.doi.org/10.1021/ma5016165 | Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

Article

the local equilibrium hypothesis to assume the flowindependent segmental friction ζ. However, this assumption is not straightforwardly verified when the flow rate becomes faster than the stretch relaxation rate of polymers because the local environment formed by the stretched chains should be different from that at equilibrium. (This coupling between the friction and the polymer conformation was discussed also in earlier publications, for example, by Giesekus.13) Ianniruberto et al.11,12 used full-atomic molecular dynamics simulation to show that the friction of PS oligomers decreases under high shear. Acceleration of molecular diffusion under fast flow has been found earlier in united atom molecular dynamics simulations (for C100H202 molecules)14 and nonequilibrium molecular dynamics simulations of bead−spring chains (with various chain length up to 100 beads per chain),15 although the change of friction was not explicitly discussed. Yaoita et al.16 noted the change of friction in the stress relaxation data by Nielsen et al.17 for a PS melt after cessation of fast preelongation. Yaoita et al.16 performed a multimode analysis of those data and showed that the relaxation of the high-order Rouse modes is accelerated according to the rate of preelongation. From those data, they obtained an empirical relationship between the magnitude of friction reduction and the orientation-stretch order parameter: ζ(F ′SO ) = fFENE (1 ζ(0) ⎡ × ⎢β + ⎣

elongational stress relaxation data for PS melt.17 This fitting, cast in the form of eq 1, was made for plots of the experimentally detected magnitude of acceleration of the highest order Rouse relaxation (that coincides with the ζ(F′SO)/ζ(0) factor divided by f FENE) against the data of σEϕp/3GNn0 (= f FENEFSO) on the cessation of the preelongation. (f FENE cannot be separately evauated from the experimental data, so that the inclusion of f FENE in eqs 1 and 2 is inevitable.) Equations 1−5 show that the ζ(F′SO)/ζ(0) ratio stays unity when F′SO < F′*SO, whereas it steeply decreases with an increase of F′SO above F′*SO. This decrease of the ζ(F′SO)/ ζ(0) ratio for large F′SO is attributed to SORF (and partly to FENE). Although eqs 1−5 were obtained from the specific set of experimental data for PS melt, Masubuchi et al.18 reported that the similar behavior is observed in the generic Kremer− Grest simulation.19 It is noteworthy that the other functional forms of the ζ/ζ(0) ratio have been also proposed,12,20 but a general trend (decrease of ζ on chain orientation/stretching) remains qualitatively the same. SORF is still controversial, as similar to the other mechanisms explained earlier. Thus, it needs to be tested against experimental data for various polymer species. The most suitable experiment is to measure the stress relaxation after fast pre-elongation. However, such an experiment is not very easy, and just a limited set of data is available at present. Another strategy is to use the elongational viscosity data. Yaoita et al.16,21 discussed the effect of SORF with the multichain sliplink simulation and with a tube model (toy version of Mead− Larson−Doi model22), both incorporating eqs 1−5. For PS melts and solutions, they switched the SORF in the simulation/ model on and off to demonstrate that the transient/steady elongational viscosities for PS melts at ε̇ > 1/τR cannot be reproduced without SORF. Desai and Larson20 reported similar results using the full version of the Doi−Edwards−Marrucci− Grizzuti model. 23 Although comparison of the model prediction and viscosity data enables just implicit test of SORF, it can be made beneficially for a number of experimental data sets available by now. For example, Masubuchi et al.24 recently examined eqs 1−5 for uniaxial elongational viscosity of a pom-pom PS melt to report that the empirical eqs 1−5 determined for linear PS is compatible even for branch PS. Taking this point of view, this study tests SORF for the experimental data of elongational viscosity of PS,25 polyisoprene (PI),26 and poly(n-butyl acrylate) (PnBA)26 reported recently. Specifically, for PS and PI, a range of samples with different polymer concentrations and molecular weights were examined. As a test of the universality of SORF, we simply implemented eqs 1−5 in the multichain slip-link simulation and utilized the parameter values determined for PS to simulate the elongational viscosity of PS, PI, and PnBA. It turned out that the simulation satisfactorily described the data for PS and PI, but not for PnBA. We conjectured that this difference is owing to the non-mean-f ield type interaction between neighboring backbones of PnBA. Differing from PS and PI, the backbones of PnBA are always surrounded by the rapidly moving/isotropic long flexible side groups, so that the orientational interaction between those backbone could be screened by the side groups even in the melt state and the mean-field type relationship (eq 4) may become invalid. Details of these results are explained below.

1 + β)γ ⎤γ 1 {1 − tanh α(F ′SO − F ′*SO )}⎥ ⎦ 2 (1)

F ′SO = FSOfFENE

(2)

2 FSO = λ ̃ S ̅

(3)

Here, ζ is the segmental friction, and ζ(0) denotes its value at equilibrium. f FENE is the FENE factor, and FSO is the stretch/ orientation order parameter. λ̃ involved in FSO is the average stretch normalized with respect to the maximum stretch, λmax. S̅ is the average orientational anisotropy defined as S ̅ = ϕpS

(4)

where ϕp is the volume fraction of the polymer and S is the anisotropy of polymer: Under uniaxial flow, S is explicitly defined as

S = ⟨ux 2⟩ − ⟨uy 2⟩

(5)

where ux and uy are the components of the unit bond vector of a subchain parallel and perpendicular to the stretching direction (x direction). For a given polymer backbone, the average orientation in the surrounding environment is accounted in eq 4 in a mean-f ield sense (as noted from the ϕp factor simply representing the average probability of finding the backbone of other polymers in that environment). FSO is related to the tensile stress σE on the basis of the decoupling approximation16 as σE =

3G Nn0 f FSO ϕp FENE

(6)

where n0 is the Kuhn segment number between entanglements. In eq 1, the parameter values α (= 20), β (= 5 × 10−9), γ (= 0.15), and F′*SO (= 0.14) were determined via fitting of the B

dx.doi.org/10.1021/ma5016165 | Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

Article

Table 1. Systems Examined and Simulation Parameters polymer

ref

T [K]

M

ϕp

Z0

n0

G0 [MPa]

PS PS PS PS PS PI PI PI PnBA

23 23 23 23 23 24 24 24 24

403 403 403 403 403 294.5 294.5 294.5 294.5

545 000 545 000 285 000 285 000 285 000 145 000 349 000 1 050 000 263 000

1.0 0.58 1.0 0.72 0.44 1.0 0.4 0.14 1.0

50 29 26 19 11 41 41 41 20

15 26 15 21 35 24 59 177 12

0.29 0.095 0.29 0.15 0.054

3.0 0.33 3.0 0.50 0.25

0.18

0.0015

2. MODEL AND SIMULATIONS The simulation in this study is based on the multichain slip-link model called the primitive chain network (PCN) model.27 Since details of the simulation with implementation of SORF have been published earlier,16,24 just a brief description is presented below. The PCN model considers a network consisting of nodes, strands spanning two nodes, and dangling strands (each being constrained only by one node). In the network, consecutive strands including two dangling strands at the end present a polymer chain. The network node represents an entanglement between chains so that a slip-link is located at each node to bundle two chains. The state variables are the position of the slip-links {R}, the number of Kuhn segments in each strand {n}, and the number of slip-links on each polymer chain {Z}. Time evolution of {R} is described by a Langevin-type equation considering the balance of the drag force, elastic force acting on strands, osmotic force suppressing the density fluctuation, and random Brownian force. Kinetic equation for {n} takes account of the forces that are common to the dynamics of {R} but balanced along the chain. The segmental friction ζ involved in the drag force is related to the Brownian force through the fluctuation−dissipation theorem. Taking account of SORF, the simulation allowed ζ to change according to eqs 1−5. FENE is considered in the elastic force with FENE-P approximation. {Z} fluctuates in time due to hooking and unhooking of the dangling strands at the chain ends. If the monomer number in a dangling end strand becomes smaller than 0.5n0 (where n0 is the average value of n at equilibrium), the slip-link separating that end strand and an inner strand is removed to attain unhooking. On the contrary, if the monomer number exceeds 1.5n0, a new slip-link is created on the dangling strand to bundle it with another strand randomly chosen from surrounding segments. In the simulation, the units of the length, energy, and time were taken to be the average strand length at equilibrium a, the thermal energy kBT, and the diffusion time of the slip-link τ0 ≡ ζ(0)n0a2/6kBT, respectively. The unit molecular weight M0 is related to the molecular weight of Kuhn segment MK as M0 = n0MK. The unit modulus G0 is defined as G0 = ρϕpRT/M0 (where ρ is density). The average number of segments per chain at equilibrium, Z0, is defined with respect to the polymer molecular weight M as Z0 = M/M0. The simulation runs were performed with the periodic boundary condition. In the equilibrium calculations for linear viscoelasticity measurement, cubic boxes were used, and the cell size was accommodated to be sufficiently large with respect to the chain dimension. For example, the cell size was 83, 103, and 163 for the chains with Z0 = 11, 19, and 41, respectively. For uniaxial elongation, flat cells were used as the initial cells that

τ0 [s]

were subjected to the stretch. As long as the steady state was attained, we used the cell with the dimension of 4 × 45 × 45 (that was stretched after equilibration, up to 500 × 4 × 4 to attain the maximum Hencky strain of 4.8). Indeed, the maximum strain of 4.8 is large enough compared to experiments. Nevertheless, that initial cell size did not allow the simulation to fully reach the steady state when ε̇, Z0, and n0 were large. In such cases, we increased the initial cell size until the steady state was attained. For example, for the chain with Z0 = 41 and n0 = 177 under flow at 0.006 < ε̇τ0 < 0.06 (which corresponds to 30 < ε̇τd < 300 and 0.51 < ε̇τR < 5.1), the initial cell dimension was set as 4 × 126 × 126 and its stretch up to a size of 4000 × 4 × 4 (corresponding to the maximum Hencky strain of 6.9) attained the steady state. Table 1 summarizes the systems examined together with the simulation parameters. M0 values for PS and PI melts were reported earlier.28 (Note that M0 is smaller than Me for the tube models due to the fluctuations at the entanglements (slip-links) involved in PCN model.29) We determined M0 for solutions assuming that the ratio M0/Me is independent of polymer concentration ϕp. For the PnBA melt, we determined M0 from the Me value reported by Sridhar et al.,26 assuming that the M0/ Me ratio was the same for PnBA and PS. G0, Z0, and n0 values were then determined from M0 according to the relation G0 = ρϕpRT/M0 and Z0 = M/M0 (with M being the sample molecular weight).28−30 τ0 values for PS samples were determined by fitting the linear viscoelastic data (not shown here), as performed previously.16,24,31 Because neither linear viscoelastic data nor transient viscosity data are available for the PI samples, the simulation was performed in reduced units (σE/ G0 and t/τ0) so that it did not explicitly require the G0 and τ0 values. Furthermore, we chose a few PI solutions having different concentrations but a similar Z0 value as the PI melt, thereby examining most clearly the effect of ϕp and n0. For the PnBA melt, the parameters were obtained by fitting the linear viscoelastic envelope of the elongational viscosity growth reported by Sridhar et al.26

3. RESULTS 3.1. PS Melts and Solutions. Figures 1 and 2 show the transient viscosity data of the PS melts and solutions (symbols) in oligomeric styrene (M = 1920) reported by Huang et al.25 Red dash-dot curves show the linear viscoelastic envelope. For all cases examined, the simulations incorporating SORF (red solid curves) attained good agreement with the data, regardless of the PS concentration and molecular weight. For the melts, the simulation without SORF (dotted thin blue curves) clearly overestimates the viscosity at the highest strain rates. The simulation with SORF attains much better agreement, though the steady-state viscosity is moderately overestimated possibly C

dx.doi.org/10.1021/ma5016165 | Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

Article

Figure 1. Transient uniaxial viscosity of PS with M = 545 000 (a) for the melt at the strain rate of 0.01, 0.003, and 0.001 s−1 (WiR = 3.8, 1.1, and 0.38) from left to right and (b) for the solution with ϕp = 0.58 at the strain rate of 0.45, 0.23, 0.076, 0.023, and 0.0076 s−1 (WiR = 6.4, 3.3, 1.1, 0.33, and 0.11) from left to right. Symbols are experimental data reported by Huang et al.25 Red solid and blue dotted curves are from the simulation with and without SORF, respectively. Red broken curves show the linear viscoelastic envelope.

due to a problem(s) in our implementation of SORF as discussed earlier16 (e.g., use of scalar ζ that depends on a single parameter, FSO = λ2̃ S)̅ . Further improvement is necessary for this point. Nevertheless, these results for the melts are consistent with our previous study for the other set of melt PS data.16 On the other hand, the simulation for concentrated solutions was attempted for the first time in this study. This is a critical test for eqs 1−5 because the onset of SORF is described according to FSO that includes the effect of polymer concentration ϕp (through both average orientation S̅ and the maximum stretch λmax). As noted in the Figures 1 and 2, the data for the solutions in oligomeric styrene (low-M solvent) are also well described by the simulation incorporating SORF (quantified by eqs 1−5). Figures 3 and 4 show the steady-state viscosity plotted against the strain rate for the PS melts and solutions examined in Figures 1 and 2. As mentioned above, the simulation with SORF (solid curves) reasonably reproduces the data even though some overestimation is noted for the melt data. The simulation without SORF (dotted curves) gives the upturn of viscosity regardless of the polymer concentration. The simulations with SORF also exhibit moderate upturn in particular for the solutions, but the viscosity decreases again at high elongational rates where the chains are highly oriented/ stretched and SORF comes into play. (The peaks in the solid curves correspond to the onset of SORF effect.) Since the experimental data do not show such sharp peaks, further improvement for the implementation of SORF is desirable. (Nevertheless, we note that the tube models with SORF exhibit similar but more pronounced/sharper peaks.20,21) For the melt,

Figure 2. Transient uniaxial viscosity of PS with M = 285 000 (a) for the melt at the strain rate of 0.03, 0.01, 0.003, 0.001, and 0.0003 s−1 (WiR = 3.1, 1.0, 0.31, 0.10, and 0.031) from left to right, (b) for the solution with ϕp = 0.72 at the strain rate of 0.57, 0.29, 0.096, 0.029, and 0.0096 s−1 (WiR = 5.2, 2.7, 0.88, 0.27, and 0.088) from left to right, and (c) for the solution with ϕp = 0.44 at the strain rate of 0.57, 0.29, 0.096, 0.029, and 0.0096 s−1 (WiR = 5.1, 2.6, 0.86, 0.26, and 0.086) from left to right. Symbols are experimental data reported by Huang et al.25 Red solid and blue dotted curves are for the simulation with and without SORF, respectively. Red broken curves show the linear viscoelastic envelope.

the peak is not apparent because the onset of SORF is located at a strain rate close to the inverse Rouse time 1/τR. Nevertheless, the simulation clearly attains much better agreement with the data when SORF is incorporated. 3.2. PI Melts and Solutions. For a PI melt and solutions, Figure 5 shows the data of Trouton ratio (the steady-state elongational viscosity reduced by the zero shear viscosity) plotted against the Weissenberg number (defined with respect to the longest relaxation time) reported by Sridhar et al.26 (see symbols). For the semidilute solution (top panel), the results of the simulations with and without SORF (solid and dotted curves, respectively) are indistinguishable because the polymer concentration is not sufficiently high to activate SORF, as explained later in more detail for Figure 6. The simulated viscosity with/without SORF commonly exhibits the upturn, D

dx.doi.org/10.1021/ma5016165 | Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

Article

Figure 3. Steady-state viscosity plotted against strain rate for PS with M = 545 000. Black and red curves are from the simulation results for the melt (1/τR = 2.6 × 10−3 s−1) and the solution with ϕp = 0.58 (1/τR = 7.0 × 10−2 s−1), respectively. Unfilled symbols are experimental data reported by Huang et al.25 Solid and dotted curves indicate the simulation results with and without SORF, respectively.

Figure 5. Trouton ratio (steady-state viscosity normalized by zero shear viscosity) of PI liquids as a function of Weissenberg number defined with respect to the longest relaxation time. PI molecular weight and polymer concentration are M = 1 050 000, 349 000, and 145 000 and ϕp = 0.14, 0.4, and 1.0 from top to bottom. Unfilled circles indicate the data reported by Sridhar et al.26 Solid and dotted curves show the simulation results with and without SORF, respectively. τd/τR = 59.

Figure 4. Steady-state viscosity plotted against strain rate for PS with M = 285 000. Black, green, and red curves are from the simulation for the melt (1/τR = 9.7 × 10−3 s−1) and the solutions with ϕp = 0.72 (1/ τR = 1.1 × 10−1 s−1) and 0.44 (1/τR = 6.6 × 10−1 s−1), respectively. The 1/τR values (unfilled circles) are experimental data reported by Huang et al.25 Solid and dotted curves indicate the simulation results with and without SORF, respectively.

and this prediction is in semiquantitative agreement with the experimental data. For the concentrated solution (middle panel), SORF is activated but the onset of SORF is observed at strain rates much higher than the inverse Rouse time, 1/τR. Because the experimental data are not available in that range of strain rate, the validity of the SORF mechanism cannot be tested for this solution. Nevertheless, the simulated viscosity with/without SORF is in reasonable agreement with the available data. For the melt (bottom panel), an obvious difference is noted for the simulated viscosity with/without SORF, as similar to the case of PS melts. However, the difference of the n0 values of PS and PI melts (larger for PI) leads to a higher strain rate at the onset of SORF for PI so that the simulated viscosity of PI exhibits a pseudoplateau at the strain rate ∼1/τR; see the curves at ε̇τd = 20−100. The experimental data appear to exhibit a similar plateau, but a small upturn, not deduced from the

Figure 6. Friction reduction ratio, orientational anisotropy, stretch factor, and order parameter (from top to bottom) for the PI liquids examined in Figure 5. Green, blue, and red curves indicate the simulated results for PI with molecular weight and polymer concentration of M = 1 050 000, 349 000, and 145 000 and ϕp = 0.14, 0.4, and 1.0, respectively. Solid and dotted curves show the simulated results with and without SORF, respectively. τd/τR = 59.

E

dx.doi.org/10.1021/ma5016165 | Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

Article

simulation, is also noted. Although the simulation with SORF is numerically close to the data in the entire range of concentration (as was the case also for the PS liquids), further tests of SORF are apparently required for PI with various molecular weights and concentrations. The stress relaxation data after cessation of pre-elongation are extremely helpful for this test. In Figure 6, the orientational anisotropy, the stretch factor, and the order parameter obtained from the simulation for the PI systems are plotted against the Weissenberg number, ε̇τd. The friction reduction factor, ζ/ζ̇(0), is also shown in the top panel. The number of entanglements per chain is very similar in those systems having various concentrations, ϕp. Interestingly, the ε̇τd dependence of the orientational anisotropy (second panel from the top) is almost universal regardless of ϕp and is insensitive to SORF incorporated in the simulation. It linearly increases with ε̇τd for low ε̇τd < 1 and gradually levels off at higher ε̇τd. On the other hand, the effect of SORF is clearly observed for the stretch factor (third panel from the top). At low ε̇τd, this factor is independent of ε̇τd and has a smaller value for smaller ϕp (because n0 is larger for smaller ϕp). For high ε̇τd > τd/τR, the stretch increases with increasing ε̇τd, and this increase is suppressed by SORF at high ϕp. The order parameter FSO (bottom panel) reflects both orientation and the stretch thereby showing a two-step increase regardless of ϕp. However, for small ϕp, FSO does not reach the critical value F*SO for the onset of the SORF effect (horizontal dashed line) even in the second plateau regime at high FSO; see the green curve for ϕp = 0.14. For such small ϕp, SORF has no effect on the simulated results, as briefly mentioned for Figure 5. 3.3. PnBA Melt. Figures 7 and 8 respectively compare the simulation results with the data of transient and steady-state

Figure 8. Trouton ratio reported by Sridhar et al.26 for the PnBA melt (cicrles). Red solid and blue dotted curves indicate the simulation results with and without SORF. τd/τR = 25.

excellently reproduces the data whereas the inclusion of SORF (red solid curves) underestimates the viscosity. This result is in contrast to the results for PS and PI. We shall discuss a possible origin of this behavior of PnBA in the next section.

4. DISCUSSION In this section, we discuss changes of SORF with the chemical structure of polymers, specifically for PnBA which has considerably long, flexible alkyl side chains. The flexible side chains should relax very quickly and effectively behave as the solvent, thereby having no contribution to the order parameter. We first examine this hypothesis by tuning ϕp as the volume fraction of the PnBA backbone while keeping the mean-f ield like feature of eq 4. Figure 9 shows results of such a simulation with

Figure 7. Transient elongational viscosity of the PnBA melt at the strain rate of 91, 41, 24, 4.5, and 0.72 s−1 (WiR = 2.7, 1.3, 0.73, 0.14, and 0.041). Red solid and blue dotted curves indicate the simulation results with and without SORF. Black dotted curve shows the linear viscoelastic envelope in the simulation. Symbols indicate experimental data reported by Sridhar et al.26

Figure 9. Trouton ratio reported by Sridhar et al.26 for the PnBA melt (circles). Red, green, and blue curves indicate the simulation results with SORF at ϕp = 1, 1/2, and 0, respectively.

ϕp = 0, 0.5, and 1. The cases of ϕp = 1 and 0 correspond to the simulation with and without SORF shown in Figure 8, and ϕp = 0.5 is an estimate of the backbone volume fraction in PnBA melt. The choice of ϕp = 0.5, based on the mean-field hypothesis regarding the side chains as solvent molecules that can freely escape/approach the closest vicinity of the backbone, improves the simulation result to some extent; see the increase of the strain rate at the onset of the decrease of the Trouton ratio (onset of significant SORF effect). However, there

viscosities of PnBA melt reported by Sridhar et al.24 The simulated linear viscoelastic envelope (black dotted curve in Figure 7) is in good agreement with the data, indicating that the values of the basic parameters (G0 and τ0) used in the simulation are reasonable. Consequently, the transient and the steady viscosities at ε̇ < 1/τR are well reproduced by the simulation regardless of the inclusion of SORF. At higher ε̇ > 1/τR, the simulation without SORF (blue dotted curves) F

dx.doi.org/10.1021/ma5016165 | Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

Article

remains a clear discrepancy between the result with ϕp = 0.5 and the data. This fact strongly suggests that the effect of chemistry on the magnitude of SORF is more complicated than cast in the mean-field type expression (eqs 1−5). Concerning this complication, we note that the side chains are always attached to the PnBA backbone and are not fully equivalent to solvent molecules that can freely move out the close vicinity of the chain backbone. The probability for a backbone to find another neighboring backbone in the close vicinity would be significantly reduced, to a level well below the mean-field probability ϕp, by such side chains. In other words, the side chains appear to screen considerably the direct orientational interaction between the backbones. For this reason, the simulation based on the average orientation, S̅ = ϕpSp (eq 4) with ϕp being the mean-field probability, could fail for PnBA melt, as notes in Figures 7−9. (In contrast, eq 4 may reasonably work for the solutions where the real solvent molecules freely escape/approach the closest vicinity of the chain backbone, as noted for PS and PI solutions.) In an extreme case for PnBA, we may have to assume that the effective concentration of the side chain (bound solvent) in the vicinity of the chain backbone is close to unity irrespective of the ϕp value. For this case, the mean-field probability ϕp appearing in eq 4 should be effectively reduced to a much smaller value, so that the simulation with very small ϕp should describe the data. This could be the case for PnPA examined in Figure 9. It is desirable to further test this non-mean-field view of friction reduction from theoretical and experimental aspects. Theoretically, we need to define a correlation length within which the orientational interaction between the chain backbones effectively determines/reduces the friction. From an experimental point of view, the non-mean-field type view could be tested via comparison between polymer melt and solution, the latter having the side groups in the melt as nonbounded solvents. These tests are considered to be important subjects in future work.

SORF relationship determined for PS to chemically different PI solutions and melts. On the other hand, for the PnBA melt, the data were well described by simulation without SORF, but not with SORF. This result suggests that the flexible side chains of PnBA screen the direct orientational interaction among the chain backbones to suppress the friction reduction (compared to the mean-field type reduction considered in the SORF relationship). A further study is desired for this issue.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Tel +81-774-38-3136; Fax +81774-38-3139 (Y.M.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Y.M. is financially supported by Grant-in-Aid for Scientific Research (B) No. 26288059 from JSPS.

■

REFERENCES

(1) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley & Sons: New York, 1980. (2) Masubuchi, Y. Annu. Rev. Chem. Biomol. Eng. 2014, 11−33. (3) Bhattacharjee, P. K.; Oberhauser, J. P.; McKinley, G. H.; Leal, L. G.; Sridhar, T. Macromolecules 2002, 35, 10131−10148. (4) Bach, A.; Almdal, K.; Rasmussen, H. K.; Hassager, O. Macromolecules 2003, 36, 5174−5179. (5) Leygue, A.; Bailly, C.; Keunings, R. J. Non-Newtonian Fluid Mech. 2006, 136, 1−16. (6) Yaoita, T.; Isaki, T.; Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Marrucci, G. Macromolecules 2011, 44, 9675−9682. (7) Marrucci, G.; Ianniruberto, G. Macromolecules 2004, 37, 3934− 3942. (8) Wagner, M. H.; Rolon-Garrido, V. H. Korea-Australia Rheol. J. 2009, 21, 203−211. (9) Dhole, S.; Leygue, A.; Bailly, C.; Keunings, R. J. Non-Newtonian Fluid Mech. 2009, 161, 10−18. (10) Ianniruberto, G.; Marrucci, G. J. Non-Newtonian Fluid Mech. 1996, 65, 241−246. (11) Ianniruberto, G., Brasiello, A., Marrucci, G. In 7th Annual European Rheology Conference; 2011; p 61. (12) Ianniruberto, G.; Brasiello, A.; Marrucci, G. Macromolecules 2012, 45, 8058−8066. (13) Giesekus, H. J. Non-Newtonian Fluid Mech. 1982, 11, 69−109. (14) Moore, J. D.; Cui, S. T.; Cochran, H. D.; Cummings, P. T. J. Non-Newtonian Fluid Mech. 2000, 93, 83−99. (15) Hunt, T. a; Todd, B. D. J. Chem. Phys. 2009, 131, 054904. (16) Yaoita, T.; Isaki, T.; Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Marrucci, G. Macromolecules 2012, 45, 2773−2782. (17) Nielsen, J. K.; Rasmussen, H. K.; Hassager, O. J. Rheol. 2008, 52, 885−899. (18) Masubuchi, Y.; Yaoita, T.; Matsumiya, Y.; Watanabe, H.; Ianniruberto, G.; Marrucci, G. Nihon Reoroji Gakkaishi 2013, 41, 35− 37. (19) Kremer, K.; Grest, G. S. J. Chem. Phys. 1990, 92, 5057−5086. (20) Desai, P. S.; Larson, R. G. J. Rheol. 2014, 58, 255−279. (21) Yaoita, T.; Masubuchi, Y.; Watanabe, H. Nihon Reoroji Gakkaishi 2014, 42, 207−213. (22) Mead, D. W.; Larson, R. G.; Doi, M. Macromolecules 1998, 31, 7895−7914. (23) Marrucci, G.; Grizzuti, N. Gazz. Chim. Ital. 1988, 118, 179−185. (24) Masubuchi, Y.; Matsumiya, Y.; Watanabe, H.; Marrucci, G.; Ianniruberto, G. Macromolecules 2014, 47, 3511−3519.

5. CONCLUSION We examined the concept of SORF for recently published data sets of uniaxial elongational viscosity for PS, PI, and PnBA with the aid of the multichain slip-link simulation. For the implementation of SORF into the simulation, we utilized the previously reported empirical relationship (determined from literature data of PS melt) between the magnitude of friction reduction and the stretch/orientation order parameter. For the new data set for PS melts and solutions (different from those underlying the empirical relationship), the simulation reproduces the data satisfactory. This result validates the functional form of the SORF relation for PS melts and solutions. Because the SORF relationship for other chemical species has not been specified experimentally, the relationship obtained for PS was assumed to be valid for the PI and PnBA without any modification in the functional form and even in the parameter values. For the PI solutions, the simulation thus conducted suggested that SORF suppresses the polymer stretching and reduces the viscosity in a way similar to PS. In a limited range of moderately low strain rates where the data for PI solutions are available, the viscosities simulated with/without SORF were indistinguishable and in reasonable agreement with the data. Nevertheless, for the PI melt, the simulation with SORF gave better agreement with the data compared to that without SORF. This result suggests reasonable applicability of the G

dx.doi.org/10.1021/ma5016165 | Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

Article

(25) Huang, Q.; Mednova, O.; Rasmussen, H. K.; Alvarez, N. J.; Skov, A. L.; Almdal, K.; Hassager, O. Macromolecules 2013, 46, 5026− 5035. (26) Sridhar, T.; Acharya, M.; Nguyen, D.; Bhattacharjee, P. K. Macromolecules 2014, 47, 379−386. (27) Masubuchi, Y.; Takimoto, J.; Koyama, K.; Ianniruberto, G.; Marrucci, G.; Greco, F. J. Chem. Phys. 2001, 115, 4387−4394. (28) Masubuchi, Y.; Ianniruberto, G.; Greco, F.; Marrucci, G. J. NonNewtonian Fluid Mech. 2008, 149, 87−92. (29) Masubuchi, Y.; Ianniruberto, G.; Greco, F.; Marrucci, G. J. Chem. Phys. 2003, 119, 6925−6930. (30) Masubuchi, Y.; Uneyama, T.; Watanabe, H.; Ianniruberto, G.; Greco, F.; Marrucci, G. J. Chem. Phys. 2010, 132, 134902. (31) Yaoita, T.; Isaki, T.; Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Marrucci, G. Macromolecules 2011, 44, 9675−9682.

H

dx.doi.org/10.1021/ma5016165 | Macromolecules XXXX, XXX, XXX−XXX