Primitive Chain Network Simulations for Pom-Pom ... - ACS Publications

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Primitive Chain Network Simulations for Pom-Pom Polymers in Uniaxial Elongational Flows Yuichi Masubuchi,*,† Yumi Matsumiya,† Hiroshi Watanabe,† Giuseppe Marrucci,‡ and Giovanni Ianniruberto‡ †

Institute for Chemical Research, Kyoto University, Gokasyo, Uji, Kyoto 611-0011, Japan Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale, Università degli Studi di Napoli “Federico II”, Piazzale Tecchio 80, 80125 Napoli, Italy

‡

ABSTRACT: It is widely accepted that the nonlinear viscoelasticity of polymers with long chain branching can be described by the pom-pom theory [J. Rheol. 1998, 42, 81] that accounts for branchpoint withdrawal (BPW) as a nonlinear relaxation mechanism of the backbone. In spite of the remarkable success attained by refined theories derived from the original pom-pom model, there remain a few questions on the consistency with the theoretical development for linear polymers. For instance, convective constraint release (CCR) is neglected in the pompom theories. In this study, primitive chain network simulations were performed to investigate the details of molecular motion under uniaxial elongation. The simulation automatically includes thermal and convective constraint release via the multichain dynamics. The code that was assembled with finitely extensible nonlinear elasticity (FENE), BPW and the stretch-orientation/induced friction reduction (SORF) shows a reasonable agreement with literature data of linear viscoelasticity and of uniaxial elongational viscosity for monodisperse linear and pom-pom branched polystyrene (PS). Analysis of the simulations reveals that BPW is the dominant mechanism for the backbone relaxation under flow. CCR contributes to accelerate the reptative motion of the branchpoint along the backbone, but this contribution is rather small. SORF reduces the stretch of the arm and of the backbone, but it does not contribute to BPW. These results may help rationalizing the pom-pom theories in view of the dominance of BPW among the relaxation mechanisms.

1. INTRODUCTION Nonlinear viscoelasticity of polymers with long chain branching has been semiquantitatively described by the pom-pom theories. McLeish and Larson1 focused on a simple architecture of branched polymer called pom-pom that consists of two star polymers (arms) connected by a linear one (backbone). To construct the constitutive equation, they employed the tube picture where the effect of entanglements is cast into a meanfield confinement of the polymer motion.2,3 Among the relaxation mechanisms so far proposed in the tube framework, they accounted for (i) arm retraction,4 (ii) tube dilation for the arm retraction,5 (iii) reptation of the backbone after the arm retraction (hierarchical relaxation),6 and (iv) chain stretching of the backbone.7 (They did not consider the stretching of the arms because their focus was on the intermediate flow rates lying between the widely separate relaxation rates of arm and backbone, respectively.) McLeish and Larson also considered (v) the branchpoint withdrawal (BPW) that was originally proposed by Bick and McLeish.8 BPW allows for the relaxation of the backbone owing to the local force balance among subchains at the branchpoint. At flow rates slower than the arm retraction rate, the tension in each arm is kT/a (where a is the undilated tube diameter and a numerical prefactor is neglected). From this value of the arm tension the force © 2014 American Chemical Society

balance between arms and backbone at the branchpoint gives a critical tension of the backbone equal to qkT/a (where q is the number of arms at the branchpoint). If the backbone tension exceeds this critical value, BPW occurs: the branchpoint is withdrawn into the tube of the backbone to maintain the backbone tension at the critical value. BPW then implies that the backbone stretch ratio λ is upper limited by λmax = q, because the tension of the backbone under equilibrium is the same as that of the arms, kT/a. The original pom-pom model qualitatively captures the nonlinear viscoelastic features under shear and extension for low-density polyethylene (LDPE). Inkson et al. 9 accomplished quantitative prediction of commercial LDPE data by a multimode pom-pom model where they regarded the material as a mixture of several pompom molecules with different molecular weights and arm numbers. Blackwell et al.10 improved the pom-pom theory by introducing a parabolic potential for the branchpoint position to eliminate the sharp switching on of BPW at the critical backbone tension. Although this modification introduces an additional fitting parameter (the strength of the potential), Received: February 17, 2014 Revised: May 6, 2014 Published: May 15, 2014 3511

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2. MODEL AND SIMULATIONS In this study, the PCN simulation is modified by incorporating BPW and SORF mechanisms. In the PCN model,18 entangled polymers are replaced by a network, in which sliplinks and network strands represent network nodes and entangled subchains, respectively. Each slip-link bundles two chains reflecting the binary assumption of entanglements. The dynamics of the system is described by a set of kinetic equations for the motion of sliplinks and for the monomer transport between consecutive subchains through sliplinks. The rearrangement of the network topology that corresponds to entanglement/disentanglement between polymers is performed in a stochastic manner. The kinetic equations for the slip-link position R and the monomer number n in the subchain between consecutive sliplinks are given by

agreement with experiments becomes better than for the original model. Keeping this parabolic potential, Verbeeten et al.11 derived the extended pom-pom model where they dropped off the maximum stretch condition of the backbone. The extended pom-pom model is useful for computational fluid dynamics owing to the differential form of the constitutive equation. Wagner and Rolón-Garrido12 proposed another implementation of BPW within their molecular stress function model to examine the effects of backbone dilution and of finite chain extensibility. The remarkable success of the pom-pom models brings up a few questions about consistency with the theories developed for linear polymers. It is worth noting that quantitative agreement of the pom-pom model with data has been attained only for the multimode varieties, even for samples with fairly monodisperse architecture. The multimode fitting process may hide the effect of unaccounted relaxation mechanisms, unless the relaxation modes used in the fitting are assigned to specific dynamics of the molecule. Indeed, the existing pom-pom theories do not account for convective constraint release (CCR). CCR is the convectively induced disentanglement13 that adds to the thermally driven constraint release (TCR). The latter is accounted for in the pom-pom theories as a tube dilation process. In the modern theories for linear polymers, CCR is one of the essential relaxation mechanisms under fast flow; it reduces excessive orientation along the flow direction, improving agreement with experimental data. Read14 mentioned a possible implementation of CCR into the pom-pom framework, but to our knowledge such improvement has so far not been reported. Because CCR acts to reduce stretch and orientation of the polymer, it might suppress BPW. Another mechanism that may affect BPW is the stretch/orientationinduced reduction of molecular friction (SORF). Huang et al.15 reported that polystyrene (PS) melts and solution behave differently under uniaxial flow, even when equally entangled; for solutions, the steady state elongational viscosity shows an upturn when the stretch rate exceeds the Rouse relaxation rate, whereas for melts it monotonically decreases. To explain this difference, we have proposed SORF16 that occurs in the melt due to high orientation. Since SORF reduces chain stretch, it might also suppress BPW. Nevertheless, SORF has never been considered for branched polymers. (The difference between melts and solutions is less significant for other polymers such as polyisoprene,17 but it is well established for PS.) In this paper, we investigate the dynamics of pom-pom polymers under uniaxial elongational flow by multichain sliplink simulations, called primitive chain network (PCN) simulations,18 to separate effects of BPW and SORF. Owing to the multichain nature, PCN automatically accounts for TCR and CCR. We modified the simulation code to implement BPW, and accomplished reasonable agreement with literature data of uniaxial elongational viscosity of a pom-pom PS melt. Then we analyzed the simulation data and found that the reptation motion of the branchpoint is accelerated under flow owing to CCR. However, BPW was found to affect the backbone relaxation much more significantly compared to branchpoint diffusion. We also noted that SORF contributes to a reduction of the stretch of the arm and the backbone, but not to the backbone relaxation. These results may rationalize the pom-pom theories not incorporating CCR and SORF. Details are explained in the following sections.

4

(ζs1 + ζs2)(Ṙ − κ·R) =

∑ Fi + Ff

+ FB

i

ζs n ̇ = (Fi − Fi − 1) + ff + fB ρ

(1)

(2)

Here, the left-hand sides of the equations are the drag forces and ζs is the subchain friction coefficient that is related to the monomeric one ζ as ζs = nζ. In eqs 1 and 2, ζs1 and ζs2 are the friction coefficients of the two subchains forming the entanglement, κ is the velocity gradient tensor, ρ the linear density of the monomers, Ṙ the time derivative of R, and ṅ the rate of change of ni due to the monomer sliding from the (i − 1)th to the ith subchain. In the right-hand sides, the first terms of the equations indicate the net effect of the subchain elastic forces, Fi and Fi (with Fi the modulus of Fi). The second terms, Ff and f f , are the osmotic forces that limit density fluctuations. The third terms, FB and f B, are the random forces accounting for thermal agitation. The FENE effect is considered for the elastic forces. The FENE factor f FENE is determined via the FENE-P approximation19,20 to eliminate numerical difficulties generated by a sudden change of the subchain length in the entanglement/ disentanglement algorithm. Because the stretch of the backbone becomes much higher than that of the arm, f FENE was separately calculated for backbones and arms as f C FENE =

1 2

1 − ⟨λ ̃ ⟩C

(3)

where the superscript/subscript C distinguishes between subchains belonging to either the arm or the backbone class, and ⟨...⟩C means the ensemble average over each class. The stretch λ̃ is normalized with respect to the maximum stretch, 2 λmax = (n)1/2, and the average λ ̃ in eq 3 is taken after calculating λmax for each subchain. In order to insert SORF into the PCN simulation, we expressed the monomeric friction coefficient ζ as a function of the stretch/orientation factor FSO as16 ζ(FSO) 1 = fFENE (1 + β)γ ζ(0) ⎤γ ⎡ 1 * ′ {1 tanh ( F F )} β + − α ′ − SO SO ⎥ ⎢⎣ ⎦ 2 3512

(4)

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Macromolecules F ′SO = FSOfFENE 2

FSO = λ ̃ S ̅

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(5) (6)

Here, α, β, γ and F′*SO are parameters whose values for PS were fixed at 20, 5 × 10−9, 0.15, and 0.14, respectively. S̅ is the order parameter of the average orientation defined as S ̅ = ϕpS

(7)

where ϕP is the volume fraction of the polymer. The polymer contribution S is obtained from the subchain orientation vector u (having |u| = 1) as S = ⟨uxux − uyuy⟩

(8)

where ux and uy are the components parallel and perpendicular to the stretching direction, respectively. The functional form of eq 4 and the above parameter values were determined from experimental data, and the PCN simulation utilizing eq 4 with those parameters was found to quantitatively describe the tensile stress data of linear PS melts and solutions.16 Thus, we use eq 4 also for pom-pom PS because the friction reduction is a local event, not affected by the chain architecture. The rearrangement of the network in the PCN simulation, which corresponds to entanglement/disentanglement between chains, is attained at the chain ends and around branchpoints in the following way. If the number n of monomers in the subchain at the chain end exceeds 3n0/2, with n0 the average n value at equilibrium, this subchain hooks another subchain that is chosen randomly from the surroundings to create a new sliplink. On the contrary, if n becomes less than n0/2, the slip-link at the chain end is removed and the bundled partner chain is released. For the case of pom-pom chains, this rearrangement occurs only at the free end of the arms. For branched polymers, additional rearrangement is necessary to attain the reptation motion of the branchpoint as proposed in the hierarchical picture.6,21 Figure 1 shows a schematic of such an event for a pom-pom polymer with 3-arms (blue curves) and one backbone (black curve) that diverge from a branch point (circle) (Figure 1a). (The other branch point is omitted for clarity.) When the number of sliplinks on an arm becomes zero as a result of the fluctuation as shown in Figure 1b (this corresponds to the deep retraction in the tube picture), this arm (with zero entanglements, red curve) penetrates a slip-link either of another arm or of the backbone. The latter case is shown in Figure 1c. In this case, the slip-link also hops beyond the branchpoint thus generating an elementary curvilinear sliding event (reptation) of the backbone (see Figure 1d). The slip-link is allowed to hop back over the branchpoint via thermal fluctuations (in this case the polymer goes back to the state shown in Figure 1c), but in some cases another slip-link is inserted between the branch point and the hopping slip-link by the hooking of a surrounding arm as shown in Figure 1e. Then the exchange of the topological position between branchpoint and slip-link is finalized. Hereafter, we denote this event as branchpoint reptation (BPR). Further details have been reported previously.21 Even though there is no tuning parameter such as the p2 parameter in the tube models,22 we have shown that the PCN simulation incorporating BPR quantitatively describes experimental results for asymmetric star polymers23 and comb polymers.24 The network rearrangement mentioned above is performed periodically with a period of the order of the Rouse time of a subchain, specifically:

Figure 1. Schematic representation of the branchpoint reptation (BPR) implemented in the PCN simulation. (a) Configuration around a branchpoint (circle) is shown with 3-arms (blue curves) and a backbone (black curve). (b) One of the arms (red curve) looses all the sliplinks due to fluctuations. (c) The relaxed arm (red curve) penetrates the slip-link on the backbone next to the branchpoint (red ring). (d) The shared slip-link hops over the branchpoint to attain the reptation of the backbone. (e) The topological exchange between the branchpoint and the slip-link is finalized when another slip-link is inserted in between.

τ=

ζn0a 2 6kT

(9)

Here a is the average distance between sliplinks at equilibrium. This time interval may vary under flow according to the change of ζ if SORF is turned on. In fact, τ marks the time resolution of the coarse-grained simulation. In addition to the network rearrangement mentioned above, we also accounted for BPW. Let us consider the case depicted in Figure 2a where three arms and one backbone converge to a branchpoint. As a result of stretch under flow, the backbone tension Fb eventually becomes higher than the sum of the arm tensions, ∑Fai. When Fb > ∑Fai, BPW occurs through a change of the branchpoint location relative to the neighboring slip-link. Namely, the slip-link on the backbone slides beyond the branchpoint and sucks in the arms as shown in Figure 2b. The friction of the sucking slip-link is determined from the number of bundled subchains (for instance, four subchains in Figure 2b) with respect to the normal slip-link that bundles only two 3513

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(used to obtain linear viscoelasticity) by our serial code (NAPLES code) on a workstation equipped with a Xeon E52667v2 3.3 GHz. Nevertheless, owing to the size of the simulation (number of the chains in the box), reproducibility of the elongation simulation is sufficient. For example, in our earlier publication,20 we showed correspondence of the results obtained with different box dimensions that are filled with different initial configurations.

3. RESULTS 3.1. Linear Viscoelasticity. For a basic test of the simulation, top panel of Figure 3 compares the linear

Figure 2. Schematic representation of the branchpoint withdrawal (BPW) implemented in the PCN simulation. Configuration around a branchpoint is shown with a backbone (red curve), three arms (blue curves), slip-link next to the branchpoint (black circle), and a chain bundled with the backbone by the slip-link (black curve). (a) Force balance check before BPW. (b) Configuration after BPW.

subchains. After the BPW event, the spatial position of the branchpoint is annealed according to the local elastic force balance. In our simulation, BPW prevails over BPR. When checking on network topology, we perform BPW for the branchpoints that fulfill the force balance condition. For the rest of the branchpoints, we perform BPR if one of the arms is found with zero entanglements. At this stage, some withdrawn branchpoints can go back to the nonwithdrawn configuration unless the backbone tension is sufficiently large. In this study, literature data were simulated for a fairly monodisperse pom-pom PS melt with arm molecular weight Ma = 28k, backbone molecular weight Mb = 140k, and number of arms at each branchpoint q = 3.25 (Although q is reported as 2.5, we set q = 3 for simplicity. The simulation results for q = 2.5 are shown in Appendix A.) We chose the unit molecular weight (average subchain molecular weight at equilibrium) as M0 = 11k, which gives for the average subchain number at equilibrium Za = 3 and Zb = 13 for the arm and the backbone, respectively. The unit molecular weight M0 also specifies the unit modulus through the relation G0 = ρRT/M0 = 0.29 MPa, where the mass density ρ was taken at the experimental temperature T = 403 K. The unit time, corresponding to the equilibrium value of τ given by eq 9, was set to τ0 ≡ (ζ(0)n0a2/ 6kT) = 2.5 s. We confirmed that the simulation with these parameter values can reasonably reproduce linear viscoelasticity and transient uniaxial elongational viscosities of linear PS melts, as shown in Appendix B. The simulations were performed using periodic boundary conditions. For linear viscoelasticity, we performed simulations without flow for a sufficiently long time in a cubic unit cell with the dimension of (8a)3, and the results were converted to the linear relaxation modulus with the aid of the Green−Kubo relation. For the simulations under uniaxial stretch, we started the simulations from a flat unit cell with the dimension of (45 × 45 × 4)a3 and stretched the cell up to (4 × 4 × 500)a3 to attain a maximum Hencky strain of 4.8. Artifacts due to the use of a flat cell and of a limited cell size were negligible, as discussed previously.20 The subchain density was fixed at 10/a3, and the number of pom-pom molecules in the simulation box was 165 for the equilibrium calculation and 2,581 for the elongational simulations. The linear relaxation modulus was obtained from 8 independent runs whereas the elongational data were made with a single simulation run for each condition. Computation time to perform a simulation for 4000 timesteps (that corresponds to 104 s for PS melt at 403 K) is around 2 weeks for the large box simulation (used for elongation) and around 40 min for the small box simulation

Figure 3. Linear viscoelasticity of the pom-pom PS melt. Comparison between experimental data (symbols) extracted from ref.25 and simulation (black curves) (top panel), and comparison between contributions of arms (blue curves) and backbones (red curves) (bottom panel). Solid and dotted curves indicate G′ and G″, respectively, in both panels. In the top panel, filled and unfilled symbols indicate G′ and G″ data, respectively. In the bottom panel, the overall relaxation is also shown for comparison (black curves).

viscoelasticity obtained from the simulation (black curves) with the data (symbols). The simulation does not consider the distribution in the molecular structure (q = 3 instead of q = 2.5) but describes the data very well. In bottom panel of Figure 3, the contribution of the arm (blue curves) and the backbone (red curves) in the simulation are separately examined. Although the blue curves suggest that arm relaxation is essentially complete around ω = 10−2 s−1, comparison of the overall G″ curve (black dotted) with the backbone contribution (red dotted curve) shows that arm relaxation is in fact complete at ω ≈ 10−3 s−1. A possibly slower arm relaxation in pom-poms with respect to star-like polymers is attributable to entanglements between arms and backbones. 3.2. Startup of Elongational Flows. Figure 4 shows a series of snapshots (at progressively increasing Hencky strains) for a molecule during the uniaxial elongation at the strain rate 3514

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Figure 4. Typical dynamics of the pom-pom polymer during uniaxial elongation at the strain rate of ε̇ = 0.001 s−1. Blue cylinders and blue spheres show subchains and branchpoints, respectively. Sliplinks are located at the kinks between consecutive cylinders. Red thin lines are other subchains bundled by sliplinks. The Hencky strain ε for each snapshot is shown in the figure.

Figure 5. Transient elongational viscosities at the strain rates of 0.1, 0.03, 0.01, 0.003, 0.001, 0.0003, and 0.0001 s−1 from left to right. Experimental data extracted from ref25 are shown with symbols. Simulation results with Gaussian spring (green dotted), Gaussian + BPW (green solid), FENE only (black), FENE + BPW (blue) and FENE + BPW + SORF (red) are shown by curves. The curves coincide with the red curves at low strain rates. The linear viscoelastic envelope obtained from the data in Figure 2 is shown as a dotted curve.

of ε̇ = 0.001 s−1, as obtained in the simulation with FENE + BPW + SORF. The snapshots clearly show that the backbone of the pom-pom molecule becomes highly stretched and oriented toward the elongation direction. This is consistent with the conventional view in the pom-pom theories. On the other hand, the behavior of the arms is somewhat different from that assumed in the conventional theories. Since the stretch rate is lower than the relaxation rate of the arm (see Figure 3), those theories assume weakly oriented and nonstretched arms. However, as shown in the snapshots for high Hencky strains, the arms are apparently oriented and partly stretched. This is because of the entanglements between arms and backbones, orientation and stretch of the arms being coupled to those of the backbone by the local force balance. Indeed, there are several elongated subchains (thin red lines) entangled with the arms. Furthermore, since the backbone is strongly stretched, the arms are far away from the friction center of the molecule, and therefore subjected to large friction forces due to large relative velocities. In Figure 5, the predictions of our simulations (curves) are compared with the transient uniaxial viscosity data (symbols). Thin black, blue, and red curves indicate the transient elongational viscosity with FENE only, FENE + BPW, and FENE + BPW + SORF, respectively. Thin green curves are for the results without FENE (with Gaussian spring). Solid and dotted curves are with and without BPW. The linear viscoelastic envelope (obtained from the data in Figure 3) is shown by the thick dotted curve. Some discrepancy between experiments and simulations is noted for this envelope at long times, which corresponds to the discrepancy at low ω seen in Figure 3. Nevertheless, strain hardening at high stretch rates is well captured in Figure 5, and the onset of deviation from the linear behavior is reproduced regardless of FENE, BPW and SORF in the simulation. This result is reasonable because FENE, BPW and SORF do not contribute to the polymer dynamics in the vicinity of the linear regime. Effects of these mechanisms only appear close to the steady state that is attained in the highly stretched situation (at long times). In Figure 5, we note that the simulations with Gaussian spring (green curves) show diverging viscosities at high strain rates (ε̇ ≥ 0.01 s−1). For the simulation without BPW (green dotted curve), the backbone stretches when the strain rate

becomes higher than BPR rate. BPW reduces such a backbone stretch, as well as the viscosity for some cases. However, unlike the pom-pom theories BPW in our implementation (fully accounting for friction) does not introduce a maximum stretch ratio of the backbone. Consequently the chain is infinitely stretched if the strain rate is higher than the contraction rate of the chain. According to the theory proposed by Ianniruberto and Marrucci26 (who consider BPW) the stretch relaxation time of the examined pom-pom polymer is to be 2 × 102 s, corresponding to a critical strain rate for chain stretch of about 5 × 10−3 s−1. On the other hand, our simulation with Gaussian spring and BPW attains the steady state up to ε̇ = 3 × 10−3 s−1. In Figure 5, we also note that the viscosity diverges at high strain rates (ε̇ = 0.1 and 0.03 s−1, black curves) if the simulation includes just FENE. This divergence occurs because BPR is not sufficient to relax the backbone at those rates. With BPW (blue curves), the divergence is eliminated and a steady state viscosity is obtained, which clearly demonstrates that BPW controls backbone relaxation at high ε̇. In addition, BPW decreases the viscosity even at lower ε̇, down to 0.001 s−1, which indicates that BPW is active even at such low ε̇ owing to the stochastic nature of the simulation (molecular individualism27,28). On the other hand, the effect of SORF is rather limited but becomes detectable only at the two highest stretch rates, as revealed by comparing the simulated viscosities (with BPW) for the cases with and without SORF. 3.3. Steady State Elongational Viscosity. Hereafter, we focus on the steady behavior. Thus, we omit the results from the simulations with Gaussian spring, which do not show a steady state for most of the examined strain rates. Figure 6 shows the steady state viscosity against strain rate. In the top panel, the simulated viscosities (curves) are compared with the experimental data (symbols). The simulation somewhat overestimates the steady state viscosity, even when including FENE + BPW + SORF (solid curve). This discrepancy could be partly due to the polydispersity of the material: As discussed in Appendix A, a mixture of pom-poms 3515

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3.4. Decoupling Analysis. We attempt to analyze the backbone and arm viscosities on the basis of a decoupling approximation that factorizes the stress into the orientational anisotropy (S) and the square stretch ratio λ2. Following this 2 strategy, Figure 7 shows plots of S, λ ̃ (square stretch ratio

Figure 6. Simulated steady-state elongational viscosities plotted against strain rate (black curves). Simulation results with FENE only, FENE + BPW and FENE + BPW + SORF are shown as dotted, dashed and solid curves, respectively. In the top panel, experimental data extracted from ref 25 shown as symbols. In the bottom panel, the contributions from arms (blue curves) and backbones (red curves) are also reported. 2 Figure 7. Orientation anisotropy S, square stretch ratio λ ̃ , FENE factor f FENE, and subchain number per chain Z at steady state plotted against strain rate in simulations with FENE only (dotted curves), FENE + BPW (dashed curves) and FENE + BPW + SORF (solid curves). Values for arms (blue curves) and backbones (red curves) are indicated separately. In the top panel, the solid straight line shows the slope of unity.

has a lower steady state viscosity. Nevertheless, the agreement seen in Figure 6 is reasonable if all the mechanisms are considered. With just FENE (dotted curve), the viscosity shows an upturn at ε̇ = 2 × 10−2 s−1. This reflects the fact that without BPW the backbone does not relax at high ε̇. Indeed, results of the simulation with FENE + BPW (dashed curve) show no upturn, confirming that backbone relaxation is due to BPW. As mentioned for the transient viscosity, BPW also reduces the viscosity at lower ε̇ because BPW occurs stochastically even at these ε̇. On the other hand, SORF (solid curve) has some limited effect only at the highest ε̇, as already shown in the transient behavior (Figure 5). The bottom panel in Figure 6 compares the contributions from arms (blue curves) and backbones (red curves) to the total viscosity (black curves), showing that the backbone contribution dominates the total viscosity. Hence it is confirmed that the backbone viscosity is significantly reduced by BPW (red dashed curve), compared to the viscosity obtained with just FENE (red dotted curve). The bottom panel also confirms that SORF has a minor effect on the backbone viscosity (red solid curve). Concerning the curves for the arm viscosity, it is remarkable that their shape comes out similar to that of backbones. Specifically, for the simulation with just FENE, the upturn occurs at the same ε̇, while BPW suppresses the upturn of the arm viscosity and generates a plateau from ε̇ = 10−3 to 10−2 s−1, followed by a decrease. The effect of SORF is present only at high ε̇, as expected, but it predominantly appears in the arm viscosity contribution. At high ε̇, the arm viscosity is not negligible compared to the backbone viscosity, and thus, SORF also somewhat reduces the total viscosity.

normalized by λ2max), FENE factor f FENE, and number of entangled segments Z against stretch rate as obtained from the simulation. Each quantity is separately shown for the backbone (red) and the arm (blue), for simulations implementing FENE only (dotted curves), FENE + BPW (dashed curves) and FENE + BPW + SORF (solid curves). If ε̇ is below the terminal 2 relaxation rate, S increases linearly with increasing ε̇, while λ ̃ , f FENE, and Z stay at their equilibrium levels. However, no such linear behavior is observed, not even for the arm, in the rather wide range of ε̇ examined. Specifically, for the arm (blue 2 curves) λ ̃ , f FENE, and Z are virtually constant at their equilibrium value for ε̇ < 10−3 s−1, but S increases more strongly than in the linear range even at such low ε̇. This nonlinearity of S is responsible for the nonlinearity of the arm viscosity in this range of ε̇ (see Figure 6). As previously mentioned, the enhanced orientation of the arm (larger S) results from the entanglements between arms and stretched backbones, and from the larger friction forces due to an increased distance of the arms from the friction center following stretching of the backbone. However, the arm orientation is always lower than the backbone orientation even with BPW, due to the free ends of arms. Specifically, the examined pom-pom molecule has only three segments on the 3516

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arm, and thus, on average, 1/3 of the arm portion is always isotropic. Concerning the effect of BPW, Figure 7 shows that the arm orientation is slightly reduced by BPW. The reason is that, since BPW reduces the backbone stretch, the arms fall back closer to the friction center of the molecule, where the velocity of the medium surrounding the arms is smaller. This effect is more effective on arm orientation than the sucking-in of BPW. The simulation also reveals that multibundled arm subchains exist even without BPW, as a result of BPR. (For instance, in the configuration shown in Figure 1e, it is possible that the arm located at the bottom relaxes and penetrates into the red ring to attain a configuration where three arms share one slip-link.) In a steady elongational flow at ε̇ = 0.01 s−1, the fractions of arm subchains occupying the same slip-link two at the same time, or three at the same time, are 0.28 and 0.042, respectively, for FENE + BPW, becoming 0.28 and 0.015, respectively, for FENE only. This result indicates that BPW increases the multibundled arms but not significantly in comparison to BPR, at least for this particular pom-pom molecule. This result may vary if the arm molecular weight becomes larger. For the backbone (red curves in Figure 7), the stretch rates examined are larger than the terminal relaxation rate at equilibrium (∼10−4 s−1 as noted in Figure 3), and thus all parameters show a nonlinear behavior. The backbone viscosity (and the overall viscosity) shows a pseudolinear behavior in the range of ε̇ from 4 × 10−4 to 4 × 10−3 s−1 but this is due to a 2 compensation of opposite effects, since λ ̃ and f FENE increase, while S nearly saturates and Z decreases. For the simulation 2 with just FENE (dotted curves), λ ̃ continuously increases with increasing ε̇, approaching unity at ε̇ ∼ 2 × 10−2 s−1. Consequently, f FENE and viscosity diverge. For the simulations incorporating BPW and SORF (dashed and solid curves, 2 respectively), λ ̃ and f FENE saturate at ε̇ ∼ 1 × 10−2 s−1, and the viscosity decreases accordingly. Among the implemented mechanisms, BPW is responsible for the reduction of orientation and stretch, both for arms and backbones. The effect of SORF on orientation and stretch is rather limited. SORF also affects the subchain number Z that, in the absence of SORF, monotonically decreases with increasing ε̇. SORF weakens the decrease of Z because the reduced friction enhances the chance to recreate entanglements at the free end of the arms. (We recall that the time interval between slip-link creation/destruction events decreases according to the reduction of friction, as described by eq 9.) However, such revived entanglements do not contribute much to the stress/ viscosity (see Figure 6). 3.5. Frequency of BPR and BPW. In order to better understand the effect of BPW on backbone relaxation, we monitored sliplinks that moved from the backbone to the arm, and calculated separately the frequency of this event as induced by BPR and BPW, respectively. Figure 8 shows the frequency pBPR (per branchpoint) of BPR events (black curves) and that of BPW (pBPW, red curves) ones as functions of ε̇. The pBPR frequency stays at its equilibrium value at low ε̇, then increases at higher ε̇. This increase of pBPR is attributable to CCR, the disentanglement induced by the convective motion of the polymers (being pulled apart by the flow). Such convective motion enhances the probability of attaining zero sliplinks on the arms, thereby triggering BPR events. This increase of pBPR is suppressed by FENE and BPW at high ε̇ to result in a saturation of pBPR at a value of ∼2 × 10−2 s−1. This value seems

Figure 8. Frequencies of BPR and BPW events (black and red curves, respectively) plotted against strain rate in the simulations with FENE only (dotted curves), FENE + BPW (dashed curves) and FENE + BPW + SORF (solid curves). The solid curves overlap the dashed and dotted curves in ranges where those curves are not visible. Note also that the dotted black curve (pBPR for the simulation with FENE only) ends at ε̇ ∼ 2 × 10−2 s−1 due to the divergence of stress.

to be the upper limit of pBPR for the examined system and corresponds to the critical ε̇ for the viscosity divergence observed in the simulation with just FENE. Modification on the BPR algorithm may change these results quantitatively. Nevertheless, BPR is not sufficient to allow for the backbone relaxation at high ε̇, even with the help of CCR. For the simulations with BPW, pBPW is much smaller than pBPR at low ε̇ but steeply increases with increasing ε̇, and finally becomes comparable to pBPR at ε̇ ∼ 2 × 10−3 s−1 (see red curves in Figure 8). This increase of pBPW is not related to CCR but due to the flow-induced enhancement of the backbone tension. The viscosity reduction due to BPW (Figure 6) is observed even at low ε̇ where pBPR is larger than pBPW. In this regime, however, the value of pBPW becomes already comparable to ε̇, and thus, BPW tends to reduce the backbone stretch and orientation. At higher ε̇, the increase of pBPW is much more significant than that of pBPR, and pBPW becomes larger than pBPR. In this regime BPW is the dominant relaxation mechanism for the backbone. 3.6. Effect of SORF. Since the simulations with SORF do not show significant differences from the simulations without SORF, one may argue that SORF is not activated in the conditions examined. However, SORF is in fact active as shown in Figure 9 where the friction coefficient is plotted against ε̇. Since arms are not sufficiently oriented and stretched, the friction coefficient keeps its equilibrium value up to ε̇ ∼ 4 ×

Figure 9. Friction coefficient, normalized to its equilibrium value, as a function of stretch rate. 3517

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10−3 s−1. For larger ε̇, friction decreases due to arm orientation and stretch (see Figure 7, blue curves). At these ε̇, SORF comes into play by reducing the stretch of the arm and the backbone, although not significantly, as shown in Figure 7.

the stretch of the arm and the backbone, but not significantly. The BPW-dominance in the backbone relaxation somehow rationalizes the current pom-pom theories not incorporating CCR and SORF. The analysis confirmed that strain hardening of branched polymers arises from the backbone stretch. The above result for SORF appears to be at variance with the suggestion of some of us26 whereby branched and linear PS melts behave similarly in the steady state of fast elongational flows. Indeed, in fast elongational flows of linear PS melts the role of SORF is very much significant,16 while here we have shown that SORF plays a relatively minor role. This difference might be due to the relatively small arm molecular weight of the examined pom-pom, which suppresses the average orientation. Nevertheless, further studies are required to explore the role played by the assumptions made in this and in previous works. Further data on well characterized branched polymers would also be useful.

4. DISCUSSION 4.1. Comment on Pom-Pom Theories. The current pompom theories do not incorporate CCR and SORF and yet describe the data reasonably well. Such an agreement should be related to the dominant effect of BPW on the backbone relaxation. Indeed, as shown in Figure 8, the backbone relaxation is dominated by BPW even at high ε̇, notwithstanding BPR is accelerated by CCR. Figure 7 also shows that the backbone stretch is controlled by BPW and is not significantly sensitive to SORF. Rather, we have shown that CCR and SORF mainly affect the arm relaxation. Thus, apart from the rather small arm contribution to the stress, the nonlinear rheology of pom-pom polymers appears to be dominated by BPW. This lends support to pom-pom theories not incorporating CCR and SORF. 4.2. Strain Hardening of Branched Polymers. It has been experimentally established that polymer melts with long chain branching show strong strain hardening (a significant upturn in the curve ηE vs ε̇) in comparison to linear polymers.29 This behavior of branched polymers has been attributed to the backbone stretch in pom-pom theories. Although true also for our simulations, here we wish to further discuss such an aspect. We recall that linear PS shows significant strain hardening if in entangled solutions, but not in the melt.30,31 We have proposed SORF as a possible mechanism to explain this difference between PS melts and solutions.16 In melts, the Kuhn segment orientation becomes so large in fast flows as to reduce the friction coefficient. This SORF effect mitigates chain stretch and strain hardening. On the other hand, in solutions, the isotropically oriented solvent molecules reduce the average orientation in the system, weakening the SORF effect and thereby allowing a significant strain hardening to emerge. This explanation suggests that SORF has an important role in the strain hardening of linear polymers. Concerning the pom-pom polymer examined in this study, some significant strain hardening is observed at low ε̇, due to stretching of the backbone not yet mitigated by BPW. At high ε̇, strain softening occurs because both orientation and stretch of the backbone essentially saturate (see Figure 6), the latter because of BPW. In this ε̇ range, there is a minor effect of SORF in further reducing the elongational viscosity because also arms become more oriented and somewhat stretched.

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APPENDIX A: EFFECT OF STRUCTURAL DISTRIBUTION To demonstrate the effect of a distribution in the branching structure, Figure 10 shows the linear viscoelasticity and the

5. CONCLUSIONS The primitive chain network (PCN) simulation was extended to pom-pom polymers by incorporating the BPW and SORF mechanisms. The simulation results were compared with literature data of transient uniaxial elongational viscosity reported for a fairly monodisperse pom-pom PS melt. The simulation with BPW and SORF were in very good agreement with data of linear viscoelasticity and of nonlinear uniaxial elongational stress. Motivated by this agreement, we analyzed in some detail the molecular dynamics of the pom-pom chain as deduced from the simulation. The analysis suggested that (1) BPW governs the backbone relaxation, (2) although CCR increases the branch-point mobility, its contribution to the backbone relaxation is relatively small, and (3) SORF weakens

Figure 10. Linear viscoelasticity (upper panel) and transient uniaxial viscosity (lower panel) for an equimolar mixture of pom-pom polymers with q = 2 and q = 3. Green and red curves indicate the results for the mixture and for the monodisperse pom-pom with q = 3, respectively. Experimental data extracted from ref 25 are shown as symbols.

transient uniaxial viscosity for a mixture of pom-pom polymers. To attain the experimental value of q = 2.5, we mixed equimolarly two pom-pom polymers with q = 2 and q = 3. The chain stretch should be different for these pom-pom molecules, so that we calculated f FENE separately for arms and backbones of each polymer. The linear viscoelasticity (upper panel, green curves) does not change much, only showing a small deviation 3518

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from that obtained for monodisperse pom-pom with q = 3 (red curves) in the ω range [10−3, 2 × 10−2] s−1. In particular, the terminal region is virtually unchanged. The elongational viscosity (bottom panel) is smaller than that in the monodisperse case, in particular at low ε̇. This difference of elongational viscosity suggests that the fitting to the experiment is somewhat sensitive to the detailed structural distribution. One may note that the value q = 2.5 could also imply asymmetric pom-poms with two and three arms on opposite sides.

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AUTHOR INFORMATION

Corresponding Author

*(Y. Masubuchi) E-mail: [email protected]. Telephone: +81-774-38-3136. Fax: +81-774-38-3139. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study is financially supported by a Grant-in-Aid for Scientific Research (B), 23350113 from Japan Society for the Promotion of Science.

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APPENDIX B: PREDICTIONS FOR A LINEAR PS MELT PCN simulations with FENE and SORF were run in the linear and nonlinear viscoelastic ranges for a linear PS melt30 with a molecular weight M = 200k (Z = 18), similar to that of the pom-pom molecule examined in this study. The parameter values for the simulations in Figure 11 are common to those of

REFERENCES

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Figure 11. Linear viscoelasticity (upper panel) and transient uniaxial viscosity (lower panel) for a linear PS melt with M = 200k. Simulation results and experimental data30 are shown as red curves and symbols, respectively. The strain rates of uniaxial elongation are 0.1, 0.03, 0.01, 0.003, and 0.001 s−1, from left to right.

Figures 3 and 4. Although the linear viscoelasticity is quantitatively captured, the steady state elongational viscosity is overestimated for large ε̇ even with SORF. This is possibly due to our implementation of SORF in the simulation, as observed in our earlier publication on linear PS,16 and thus the viscosity is overestimated also for pom-poms. Nevertheless, the PCN simulation reasonably predicts, in a consistent way, the linear and nonlinear viscoelasticity of linear and branched polystyrenes. 3519

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