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Extensional Flows of Solutions of Entangled Polymers Confirm Reduction of Friction Coefficient Giovanni Ianniruberto*

Downloaded by UNIV OF NEW SOUTH WALES on August 28, 2015 | http://pubs.acs.org Publication Date (Web): August 26, 2015 | doi: 10.1021/acs.macromol.5b01401

Department of Chemical, Materials, and Industrial Production Engineering, University Federico II, Piazzale Tecchio 80, 80125 Napoli, Italy ABSTRACT: The extensional-flow data of Huang et al. [Macromolecules 2015, 48, 4158] of several polystyrene systems are here successfully compared with predictions of a recent model of Ianniruberto [J. Rheol. 2015, 59, 211], provided flow-induced friction-reduction effects are accounted for. For the case of solutions, friction reduction must include nematic interactions between the oligomeric solvent and the polymer molecules. It is here found that the coupling interaction parameter ε must be larger than that based on the mean orientation of the solvent molecules, as measured in mildly oriented systems; the larger ε found here is probably due either to the stronger orientation reached in the extensional flows, or to the fact that only the molecules close to the polymer matter in frictional effects. The model here suggested is the first multimode molecular model (aside from simulations) able to describe, qualitatively and even (almost) quantitatively, the nonlinear extensional rheology of entangled polymers, from melts to solutions. the concentration-dependent parameters τe and G0N mentioned above. Unfortunately, this relative simplicity does not extend itself to the nonlinear range, in spite of the fact that the classical theory of Doi and Edwards1 and modifications thereof2,6 predict that such should be the case. Indeed, in the same paper Huang et al.5 show that the response during startup of fast uniaxial elongational flows does not similarly superimpose. In particular, the steady-state values of the elongational viscosity of those systems (although with the same Z) even differ qualitatively, as they change gradually from the thinning behavior of the melt all the way up to the strongly hardening one of the less concentrated solution. The present paper represents an attempt of interpreting this peculiar nonlinear behavior, and of modeling it semiquantitatively. In order to do so, we will refer to our previous works on the possible decrease of the monomeric friction coefficient due to flow-induced monomer alignment (i.e., to an increase of the monomer order parameter) in fast elongational flows of polymers,7−9 further considered in more recent works.10−13 In some of those papers,9,10 it was also suggested that in the case of solutions the relevant order parameter is an average one between the monomers of the polymer and the short molecules of the solvent, and the assumption was made that the solvent molecules remain randomly oriented during flow, so that the mean order parameter was simply that of the polymer monomers, reduced by the polymer volume fraction. As a consequence, in not too concentrated PS solutions in smallmolecule solvents, like the 10% (or less) solutions examined by

1. INTRODUCTION Our current understanding of the rheology of entangled polymeric liquids is invariably based on the tube model developed long ago by Doi and Edwards,1 and then modified by several authors to improve agreement with experiments (see e.g., the review by McLeish2). The linear viscoelasticity (LVE) of monodisperse linear polymers can now be predicted with great accuracy by combining the de Gennes reptation dynamics3 with additional relaxation mechanisms, like constraint release (CR) and contour length fluctuations (CLF).4 Molecular models, together with the molar mass of the polymer, only need to consider two molar-mass-independent parameters, namely a time scale τe related to the monomeric friction coefficient, and a length scale related to the mesh size of the entangled network (expressed either as mean distance a between consecutive entanglements or, equivalently, as molar mass Me of the entangled subchain) that can be obtained from the plateau modulus G0N. The molar mass M enters the picture insomuch as the ratio Z = M/Me (specifying the number of entanglements per chain) determines the “shape” of the LVE response. Recent results of Huang et al.5 add to previous ones by several authors2,6 to fully confirm these predictions in the LVE limit. Indeed, Huang et al.5 prepared several polystyrene (PS) solutions using suitable combinations of polymer molar mass and concentration such that all solutions had the same value of Z ≈ 22, as that of a PS melt of molar mass 285 kg/mol. The LVE response of those systems was determined as dynamic moduli G′ and G″ in a wide range of frequencies, and all such results would superimpose onto master curves of G′ and G″, provided frequencies and moduli are suitably shifted in log scales horizontally and vertically, respectively, thus determining © XXXX American Chemical Society

Received: June 26, 2015 Revised: July 29, 2015

A

DOI: 10.1021/acs.macromol.5b01401 Macromolecules XXXX, XXX, XXX−XXX

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where, instead of the Doi−Edwards tensor, Q is the unit-trace Seth-type tensor given by B1/3/trB1/3, (B being the Finger tensor). In eq 1, τi is the time-dependent orientational relaxation time of mode i, accounting for both thermal (i.e., equilibrium) and convective contributions (CCR), given by

Shridar and co-workers,14 friction reduction is not expected, as indeed proved by Yaoita et al.9 However, subsequent work on PS solutions in different styrene oligomers has shown that the solvent may enter the picture in a more complex way. Particularly, Huang et al.15 invoked nematic-like interactions to explain the different strain hardening behavior of solutions with oligomers of different lengths, ranging in molar mass from 1k to 4k. Nematic interactions between short molecules and polymers have been investigated theoretically,16,17 as well as in many experiments18−21 and simulations,22,23 in both liquid polymers and rubber-like networks. The interaction is measured through the orientational coupling parameter ε, defined as the ratio of the order parameter of the short molecules to the average order parameter of the system. In Kremer−Grest-type molecular dynamics simulations, ε was found to be 0.28 by Baljon et al.,22 and 0.25 by Cao and Likhtman, 23 independently of concentration. However, several experiments show that the value of ε depends on the chemistry. By way of example, for styrene oligomers in PS melts it is ε = 0.26,19 while for butadiene oligomers in polybutadiene melts it is found that ε ranges from 0.4 to 0.9 depending on the mass of the oligomer,21 and for binary blends of exactly alternating ethylene−propylene copolymers ε = 0.45 is found.20 It is further mentioned that the results so far recalled refer to polymeric liquids in the linear range or slightly stretched. In other words, it is not known what the value of ε would be in highly stretched polymers, and another mystery is why in rubber-like networks the value of ε is systematically found to be equal to unity.18,21 It is finally noted that in the linear range the nematic interaction has no effect on rheology.17,24 In the nonlinear range, however, the nematic interaction is expected to become important when the order parameter of the Kuhn segments attains high values, insofar as it reduces the monomeric friction coefficient. The present paper is organized as follows. In section 2, we present the model used to predict the nonlinear behavior of entangled polymers starting from a known LVE response. The model is the same of a previous publication,25 with a few modifications. One simplification is that we here ignore possible effects of flow-induced disentanglement, since we verified that they are negligible in fast elongational flows, which are the focus of the present paper. Conversely, reduction of the monomeric friction coefficient, not considered in the previous paper, is here included as it is in fact central to the interpretation (and semiquantitative prediction) of the fast elongational data of Huang et al.5 Section 3 presents our predictions, compares them with those data, and discusses the results. Conclusions are drawn in section 4.

⎛ 1 1 1 dλ ⎞⎟ = + β ⎜k : S ̅ − ⎝ τi τi,eq λ dt ⎠

where τi,eq is the i-th mode relaxation time obtained from a multimode Maxwell fit of the LVE response (together with the corresponding weight gi to be used later). The bracket in eq 2 gives the frequency of entanglement renewal due to the relative velocity between the chain and its confining tube, the unknown parameter β ruling the CCR effectiveness. In the expression for the CCR frequency, λ is the ratio of the current tube length to its equilibrium value, k is the velocity gradient tensor, and S̅ is an average orientation tensor obtained through an equation like eq 1 where τi is replaced by the average disengagement time τd(t ) =

t

∫−∞ [1/τi(t′)] exp[−∫t′

∑i giτi 2 ∑i giτi

(3)

Moving on to the contribution of chain stretch, the model25 accounts for a distributed friction all along the chain, as well as for possible non-Gaussian effects, by writing the following evolution equation for s(n,t) 4 ⎛ b ∂s ⎞ ∂ 2s ∂s ⎟ h⎜ = k: S ̅ s + ∂t 3π 2τR ⎝ a ∂n ⎠ ∂n2

(4)

where s is the nondimensional curvilinear coordinate along the tube (ratio of current coordinate measured from the chain center to one-half the equilibrium tube length), and n is the monomer number (normalized by taking the ratio to the monomer number in a half chain). In eq 4, h( ·) = {[L‐1( ·)]−2 − sinh−2[L‐1( ·)]}−1with L−1(·) the inverse Langevin function, and τR is the Rouse time of the chain. In the argument of the function h( ·), b is the monomer length, and a is the equilibrium subchain length, often referred to as the tube diameter. Equation 4 is solved with the boundary conditions s = 0 at n = 0, and ∂s/∂n = 1 at n = 1, the latter expressing the fact that chain ends are not stretched. The stress tensor σ is then calculated as σ (t ) = CQ I(t ) ∑i gi Si(t ), I (t ) =

1 L−1(b / a)

1

∫0 dn ∂∂ns L−1( ba ∂∂ns )

(5)

where I(t) accounts for stretch effects (I = 1 when there is no stretch), and the sum over the modes gives the subchain orientation. The scalar factor CQ depends on the choice of tensor Q in eq 1. The value of CQ is determined by the condition that the linear relaxation modulus obtained from eq 5 be G(t) = ∑igi exp(−t/τi,eq). In our case (i.e., Q = B1/3/tr B1/3), it is CQ = 9. It is further noted that the choice of tensor Q also affects the value of the equilibrium entanglement spacing, or tube diameter, a = aFerry(3/CQ)1/2 = aFerry/√3, where aFerry is the entanglement spacing obtained from the entanglement molecular weight Me, itself linked to the plateau modulus through the Ferry equation G0N = cRT/Me, with c the polymer concentration (reducing to the mass density ρ in the case of melts), and R the gas constant. For the case of solutions, since

2. MODEL We use a model for the nonlinear rheology of entangled polymers developed in a recent paper25 that we here recall in its main aspects. The model adopts the classical decoupling approximation between orientation and stretch of the subchains between consecutive entanglements. The orientation is obtained by summing over the modes extracted from the LVE response. The orientational contribution Si of the i-th mode is given by a modified Doi−Edwards equation Si(t ) =

(2)

t

dt ″ /τi(t ″)]Q(t , t ′) dt ′ (1) B


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Macromolecules Table 1. LVE Parameters of the PS Samples from Huang et al.5 PI = Mw/Mn

sample name


PS-285k PS-545k/4k-52 PS-900k/4k-33 PS-1760k/4k-18 PS-3280k/4k-13

1.09 1.12 1.16 1.13 1.38

τe [s]

G0N [MPa] 2.52 6.89 2.73 6.85 2.79

× × × × ×

−1

10 10−2 10−2 10−3 10−3

4.44 2.00 3.20 6.61 1.51

0 ≤ S ≤ Sc ,

ζ = ζeq(S /Sc)−1.25

S > Sc ,

2

λ̅ 2 (Sxx̅ − Syy̅ ), λ ̅ = 2 λmax

∫0

⎛ ∂s ⎞2 dn ⎜ ⎟ ⎝ ∂n ⎠

× × × × ×

102 101 102 102 102

(9)

3. RESULTS AND DISCUSSION Huang et al.5 investigate five PS samples, namely one melt and four solutions. The properties of those samples at 130 °C are reported in Table 1 that reproduces some of the contents of Table 3 in their original paper. In the sample name, both the molar mass of the polymer and the polymer concentration in the 4k oligomer are reported. Table 1 gives the plateau modulus G0N and the Rouse time τe of the entangled subchain (indicated as τc by Huang et al.5), as well as the number Z of entanglements per chain (Z ≈ 22 for all systems), and the Rouse time evaluated as τeZ2. Table 2 reports the discrete Maxwell spectrum of the PS285k melt obtained by fitting the frequency response through Table 2. Nondimensional Relaxation Spectrum for an Entangled Polymer with Z ≈ 22

(7)

gi/G0N

where Sx̅ x and S̅yy are components of the average orientation tensor S̅ in the stretching direction, and orthogonally to it, respectively, and λmax = a/b is the value of λ when the chain is fully stretched. Also λmax is chemistry dependent; indeed for PS it is b = 18 Å, and for PS melts it is aFerry = 85 Å,27 while (as mentioned above) for PS solutions a increases proportionally to c−0.6. For the case of solutions, the order parameter S becomes an average value between that of the monomers in the polymer chain and that of the solvent molecules S = ϕSp + (1 − ϕ)Ss

2.03 9.33 1.62 3.44 7.44

In the following, predictions of the above model in fast elongational flows will be compared to the data of Huang et al.5 In order to do so, we will use the LVE spectrum of the various systems that, to within the known horizontal and vertical shift factors, is the same for all. Following the suggestion of Huang et al.,5 from the LVE spectrum we also inherit the Rouse time values (see Table 1). Finally, we need to assign a value to the CCR parameter β, and to the just mentioned coupling interaction parameter ε.

(6)

1

21.4 21.6 22.5 22.8 22.2

S = [ϕ + (1 − ϕ)ε]Sp

where S is the order parameter of the system, and Sc is the critical value of S below which the friction coefficient remains fixed at the equilibrium value. The threshold for the onset of friction-coefficient reduction is set to the value Sc = 0.14, as in Yaoita et al.,9 while the complex formula of Yaoita et al. containing a hyperbolic tangent is replaced by a simple power law. In the case of melts, S coincides with the order parameter Sp of the polymer monomers (actually Kuhn segments), calculated as9 Sp =

10 10−1 10−1 10−1 100

those “touching” the polymer itself (and not those far away from it), we will assume that the relevant order parameter of the solvent to be used in eq 8 is directly proportional to Sp through a concentration-independent nematic interaction parameter ε (i.e., Ss = ε Sp), so that eq 8 becomes

typically G0N is proportional to c2.2, there follows that a scales as c−0.6. The elongational experiments analyzed in this paper start from an equilibrated sample; hence the initial condition to be used in the model recalled above is that all state variables take up their equilibrium values. Specifically, all orientational tensors equal one-third the unit tensor I, and the nondimensional curvilinear coordinate s(n,0) of the monomers is s = n. As mentioned in the Introduction, we here ignore flowinduced disentanglement effects discussed in previous papers.25,26 Conversely, we want to account for flow-induced orientation effects on the monomeric friction coefficient ζ, which gets reduced with respect to that prevailing in the isotropic case (i.e., in the LVE range), ζeq. We expect that reduction of the friction coefficient is chemistry dependent.8 For the case here considered of PS systems, we will use the simple recipe ζ = ζeq

× × × × ×

τR = τeZ2 [s]

Z −1

1.16 7.27 2.21 9.11 3.85 1.80 1.57 1.85 1.95 2.39 3.73

× × × × × × × × × × ×

103 100 100 10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−3

τi,eq/τe 1.56 1.10 6.37 3.20 1.78 1.15 7.44 4.23 2.10 8.77 5.96

× × × × × × × × × × ×

10−4 10−2 10−2 10−1 100 101 101 102 103 103 104

(8)

where ϕ is the polymer volume fraction, and Ss is the order parameter of the solvent molecules. Now, while Yaoita et al.9 assumed that the solvent remained randomly oriented (Ss = 0), we will here somehow account for the nematic interactions along the lines summarized in the Introduction. More specifically, by considering that the solvent molecules interacting with the polymer in frictional events are only

the open source software Reptate.28 Results in Table 2 are actually given in nondimensional form {gi/G0N,τi,eq/τe} by using the melt data of G0N and τe in Table 1. Such nondimensional spectrum applies to all five samples, to within the minor differences in the number Z of entanglements per chain. Needless to say, the predictions for the polymeric solutions reported in the following are obtained by taking the C


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only, while in fast extensional flows (ε̇τR ≥ 1) CCR becomes insignificant. Although here CCR effects are not important, CCR (with β = 0.25) has been maintained in the predictions reported below. The key ingredient in fast extensional flows of PS melts is the friction reduction effect, as already emphasized by Yaoita et al.9 with their Brownian simulations, and here confirmed in Figure 1b, where predictions with and without the flow-induced friction-reduction mechanism are compared with the PS-285k data. When ε̇τR ≥ 1, chain-stretch-induced strain hardening is observed (and classically predicted) at short times. However, chain stretch also induces monomer alignment (i.e., the monomer order parameter increases with time), and above a critical value of the order parameter (Sc = 0.14 in our model) the monomeric friction coefficient starts decreasing, i.e., flow “loses grip” on the polymer molecule, and the stretch saturates at a lower value. As shown in Figure 1b, predictions obtained by ignoring such an effect, i.e., by assuming that the monomeric friction coefficient stays constant at its equilibrium value, are not consistent with PS melt data: the steady state observed in the dashed lines of Figure 1b (due to finite extensibility of the polymer chains) is higher than shown by the data, by nearly an order of magnitude at the highest strain rate. Figure 1c shows the effect of different proposals for the friction reduction effect in PS melts, within the context of the present model. The original formula by Yaoita et al.9 appears to overestimate the reduction, therefore leading to too low a value of the steady-state elongational viscosity. On the contrary, both formulas suggested by Desai and Larson12 underestimate friction reduction; i.e., they predict larger values of the elongational viscosity than shown by the data. As mentioned above, the message conveyed by Figure 1 is not new since it was already emphasized by Yaoita et al. by resorting to Brownian simulations,9 as well as by performing calculations with the toy (single-mode) Mead−Larson−Doi model.11 Steady-state predictions were also reported by Desai and Larson,12 based on their single-mode model. Here, however, for the first time, a quantitative comparison is successfully attempted by using a multimode model accounting for flow-induced friction reduction effects. To be fair, it should be mentioned that a multimode model fitting similar PS melt data nearly quantitatively was proposed by Wagner and coworkers,29 based however on a completely different physical picture. Such a picture, although rooted in the interchain pressure effect advanced by us some years ago,30 does not appear convincing since the proposed explanation for the different behavior of PS melts and solutions seems artificial.31 Good agreement with other PS melt data of different molar mass by Hassager and co-workers32,33 has also been checked (results not shown for conciseness). Overall, we can conclude that the power-law rule given by eq 6 (simpler than the original one suggested by Yaoita et al.9) reasonably accounts for the friction-reduction effect in PS melts. It is fair to note that agreement between data and predictions in Figure 1 deteriorates somewhat in moving toward slow flows. In that region, however, monomeric friction coefficient reduction is absent, and possible reasons for the discrepancies are discussed at the end of this section. Moving on to PS solutions, where the order parameter of the solvent is needed (see eqs 8 and 9), we need to specify the value of the nematic interaction parameter ε. For that purpose we start by performing a comparison with the most concentrated (ϕ = 0.52) and most diluted (ϕ = 0.13) solutions


nondimensional spectrum in Table 2, and making it dimensional through the (G0N,τe)-values reported in Table 1. It should also be mentioned that only the slowest 7 modes (out of 11) are in fact needed to obtain predictions in the startup of uniaxial extensional flows reported below, since the filament stretching data were collected for times larger than 1 s, and for extension rates not larger than 1 s−1. Hence the shortest relaxation times of the spectrum are irrelevant. As regards nonlinear parameters, in the case of melts we only need to specify the CCR parameter β. However, as previously mentioned with reference to flow-induced disentanglement, we similarly expect that CCR does not play a significant role in fast extensional flows. This is confirmed in Figure 1a, where results

Figure 1. Transient elongational viscosity ηel+ vs time for PS-285k at ε̇ = 3 × 10−5, 3 × 10−4, 10−3, 3 × 10−3, 10−2, and 3 × 10−2 s−1 from right to left. Symbols are data from Huang et al.,5 lines (with the corresponding colors) are model predictions. In panel a, predictions (accounting for friction reduction) are obtained with β = 0.25 and β = 0, full and dashed lines, respectively. In panel b, predictions (accounting for CCR with β = 0.25) are obtained with and without friction reduction, with full and dashed lines, respectively. In panel c, predictions (only for the highest rate) are reported for several friction reduction formulas: eqs 40 and 41 in Desai and Larson12 (dashed and dotted lines, respectively), eq 6 of this paper (full line), and eq 7 in Yaoita et al.9 (dash-dotted line).

with and without CCR are compared with the data for the PS285k melt. More specifically, in Figure 1a we report model predictions obtained with β = 0 and β = 0.25, the latter choice being made in a previous paper25 on the basis of a comparison with several shear data on PS melts. Figure 1a clearly shows that CCR has some (minor) effect at intermediate strain rates D


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examined by Huang et al.5 Figure 2 compares their data on the PS-545k/4k-52 and PS-3280k/4k-13 samples with our

Figure 3. Transient elongational viscosity ηel+ vs time for the two intermediate concentrations, ϕ = 0.33 (a), and ϕ = 0.18 (b). Symbols are data from Huang et al.,5 lines are model predictions. Strain rates (from right to left) are ε̇ = 7 × 10−4, 2 × 10−3, 7 × 10−3, 2 × 10−2, 4 × 10−2, 7 × 10−2, and 2 × 10−1 s−1 and ε̇ = 10−3, 3 × 10−3, 10−2, 3 × 10−2, and 6 × 10−2s−1, in parts a and b, respectively. Figure 2. Effect of the nematic interaction parameter ε on predictions of ηel+ vs time for the most concentrated (a) and most diluted (b) PS solutions. Symbols are data from Huang et al.,5 and dotted, dashed, and full lines are model predictions with ε = 0, 0.26, 0.5, respectively. Strain rates (from right to left) are ε̇ = 10−3, 3 × 10−3, 10−2, 3 × 10−2, 10−1, and 2 × 10−1 s−1 and ε̇ = 3 × 10−4, 10−3, 3 × 10−3, 6 × 10−3, 10−2, and 3 × 10−2 s−1, in parts a and b, respectively.

predictions obtained by using different values of ε. Figure 2 includes the case ε = 0 to explicitly show that nematic interactions are indeed indispensable to explain results in sufficiently long PS oligomers, as already emphasized by Huang et al.15 by comparing elongational data in different PS oligomers (with molar mass ranging from 4k to 1k). Figure 2 shows that a reasonable fit of the data (again only at high rates) can be obtained by choosing ε = 0.5, i.e., with a value of the coupling interaction parameter larger than the one (ε = 0.26) suggested by Tassin et al.19 on the basis of their experiments on PS blends, where, however, the Hencky strain was much smaller than that reached by Huang et al.5 It is also noted that the choice ε = 0.5 refers to the 4k PS oligomers here considered. As proved by the data by Huang et al.15 mentioned above, we expect smaller ε values for the case of shorter oligomers. As shown by Yaoita et al.,9 no nematic effects (ε = 0) needs to be invoked when the solvent is a small molecule. With the choice ε = 0.5, we examine in Figure 3 the behavior of the two remaining solutions, at the intermediate concentrations ϕ = 0.33 and ϕ = 0.18. Here again we find reasonable agreement at high strain rates, i.e., when a large chain stretch and the resulting friction-coefficient reduction play a significant role. The model predictions for the steady-state viscosity are reported in Figure 4a for all the systems, together with the corresponding data from Huang et al.5 At high rates, predictions are in qualitative and even semiquantitative agreement with data. Indeed, for all solutions one may note

Figure 4. Normalized steady-state extensional viscosity for the five PS systems of Table 1. (a) Symbols are data from Huang et al.5 Full and dashed lines are model predictions with and without friction reduction, respectively. The dotted horizontal lines on the left and on the right indicate asymptotic values, at low and high strain rates, respectively. (b) Solid lines are the same as in panel a, i.e., with ε = 0.5; dashed lines with ε = 0 show the effect of suppressing the nematic interaction.

(in both data and model predictions) that, after the upturn envisaged in the classical theory at ε̇τR ≈ 1 as due to onset of chain stretch, a leveling off or even a decrease of the steadystate elongational viscosity appears. This is due to frictionreduction effects that become increasingly dominant with increasing polymer concentration. E


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Macromolecules Such leveling off should not be confused with the plateau arising from finite extensibility, which runs much higher than the data. Indeed, it can be shown that, should the chains be completely extended, the elongational viscosity obeys the relationship ηel/G0Nτe = (π2/4)ZM/MK, with MK the molar mass of a Kuhn segment, equal to 0.72k for PS.27 Such asymptotic values of the elongational viscosity are shown on the right of Figure 4a as dotted horizontal lines for each of the five systems. The dashed curves in Figure 4a show the rise toward those asymptotic viscosity levels as predicted by the model in the absence of frictional reduction effects. The data of Huang et al.5 in Figure 4a are in sharp contradiction with predictions obtained by ignoring frictionreduction effects. The plateau-like behavior clearly indicated by the data of the PS-545k/4k-52 and PS-900k/4k-33 samples is quantitatively predicted by the model accounting for friction reduction effects. On the contrary, the plateau associated with the fully extended conformation is about an order of magnitude higher than that shown by the data. At low extension rates, the curves in Figure 4a predicted by the model approach the LVE viscosity, as they should. The data, however, run systematically somewhat higher. A possible explanation for this discrepancy might be the polydispersity of the polymer (see Table 1), insofar as the longer chains might be somewhat stretched even at low stretching rates. For the case of solutions, another source of discrepancy is perhaps to be found in the minor deviations in the terminal region between the normalized G′-G″ data of the solutions and those of the melt (see Figure 2 of Huang et al.5). As a final comment on Figure 4a, we wish to point out that the cusps appearing in the full curves correspond to the abrupt transition from a constant value of the friction coefficient to one decreasing with a power law (see eq 6), occurring when S = Sc. The cusps are expected to disappear should one use a smooth transition. Finally, Figure 4b shows the effect of suppressing the nematic interaction between solvent and polymer, i.e., of setting ε = 0. For all solutions, the difference between ε = 0.5 and ε = 0 is significant, and setting ε = 0 strongly deteriorates comparison with data. In particular, at the lower concentrations (ϕ = 0.18 and ϕ = 0.13), setting ε = 0 is equivalent to altogether suppressing friction reduction, since for those systems the average order parameter always remains below the threshold value, Sc = 0.14, if the nematic interaction is suppressed. As detailed in section 2, and as it is in fact well-known, both “tube” orientation and chain stretch contribute to the stress, and the question then arises on whether friction reduction and nematic interactions act mostly on orientation or on stretch, or more or less equally on both. The answer to such question is reported in Figure 5, where the steady-state average tube segment orientation Sx̅ x − Sy̅ y, and chain stretch λ are reported as a function of the extension rate ε̇, for ε = 0.5 and ε = 0. By comparing solid and dashed lines in Figure 5, it is apparent that the nematic interaction affects chain stretch, and has no significant effect on orientation. Moreover, since friction reduction is more effective in the melt, and much less effective in solutions, the quasi-superposition of the curves for all PS systems in Figure 5a indicates that, more generally, the very phenomenon of friction reduction has virtually no effect on orientation. Before closing this section it is worth discussing the role played by the chemistry-dependent parameter λmax which has often been considered relevant in fast extensional flows (see,

Figure 5. Steady-state values of the average tube segment orientation (a) and of the average chain stretch (b) for all PS systems. Solid and dashed lines are for ε = 0.5 and ε = 0, respectively.

e.g., Huang et al.33). It is worth noting that the values of λmax = a/b used in this work are smaller than the classical values because of our choice of tensor Q. Indeed, with the Doi− Edwards tensor, a is smaller than aFerry by the factor 4/5; conversely, with our Q tensor a gets reduced with respect to aFerry by √3. Smaller values of λmax decrease the steady state elongational viscosity (see also below). However, in spite of a smaller λmax, friction reduction effects were needed to fit the data. The necessity of friction reduction effects is further emphasized by the results reported in Figure 6, where, for all

Figure 6. Transient elongational viscosity ηel+ vs time for all PS systems, each at a single strain rate (ε̇ = 0.03, 0.2, 0.04, 0.03, 0.01 s−1 for ϕ = 1, 0.52, 0.33, 0.18, 0.13, respectively). Symbols are data from Huang et al.,5 and lines are model predictions, both with colors indicating the systems as in Figure 5. Solid lines are as in Figures 1−3, dashed lines are obtained by reducing λmax by a factor of 2 and simultaneously suppressing friction reduction effects.

five PS systems here considered, λmax was artificially further reduced by a factor of 2, while simultaneously switching off friction reduction effects. Figure 6 shows that the predictions in such artificial situation are substantially similar to (though somewhat less accurate than) those so far obtained. F


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The decrease of the steady state elongational viscosity with decreasing λmax is in line with the fact that, for a fully extended chain, energy dissipation (hence viscosity) is proportional to the square of the molecular length. Comparison of the PS-285k curves without friction reduction in Figure 6 and in Figure 1b shows that, by artificially reducing λmax by a factor of 2, dissipation for not fully stretched molecules gets reduced even more, by as much as a factor of nearly 5.

4. CONCLUSIONS We have here shown that the uniaxial extensional data of Huang et al.5 on several PS systems with the same number Z of entanglements per chain can be modeled almost quantitatively by suitably accounting for the flow-induced reduction of the monomeric friction coefficient. It is also confirmed that the solvent/polymer nematic interactions play a crucial role for the long oligomeric solvent used by Huang et al.5 Indeed, as shown in Figure 2, should the solvent be assumed to remain randomly oriented, the model predictions are significantly off. Another possibly significant result is that the order parameter of the solvent to be used for the friction-reduction effect is not that of the solvent in the bulk, corresponding to a nematic interaction parameter ε = 0.26 for long PS oligomers (10k and 27k) in mildly stretched samples.19 A much larger ε value is here required, ε ≈ 0.5, that we interpret as the coupling interaction parameter of the solvent in close proximity of the highly stretched polymer molecules. Smaller ε are however expected for the shorter oligomers (down to 1k), as implicitly shown by the data of Huang et al.15 It would be interesting to explore these aspects by suitable molecular dynamics simulations. The results reported here confirm that the gradual change with concentration of the steady-state extensional viscosity, from the strain-rate thinning of the melt to the strain-rate thickening of the more dilute solutions, is due to the gradual loss of friction-coefficient reduction with decreasing polymer concentration. The plateau-like behavior of the elongational viscosity shown by some of the systems in fast flows (see Figure 4) is due to friction reduction, not to finite extensibility.

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AUTHOR INFORMATION

Corresponding Author

*(G.I.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Useful discussions with G. Marrucci are acknowledged. I am also grateful to Q. Huang and O. Hassager for sending me the data used in this paper. I also thank them for sending me ref. 5 before publication.

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REFERENCES

(1) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics. Clarendon Press: Oxford, U.K., 1986. (2) McLeish, T. C. B. Adv. Phys. 2002, 51, 1379. (3) de Gennes, P. G. J. Chem. Phys. 1971, 55, 572. (4) Likhtman, A. E.; McLeish, T. C. B. Macromolecules 2002, 35, 6332. (5) Huang, Q.; Hengeller, L.; Alvarez, N. J.; Hassager, O. Macromolecules 2015, 48, 4158. (6) Watanabe, H. Prog. Polym. Sci. 1999, 24, 1253. G


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