Orientational Cross-Correlation in Entangled Binary Blends in

Nov 29, 2016 - Primitive chain network simulations of probe rheology. Yuichi Masubuchi , Yoshifumi Amamoto , Ankita Pandey , Cheng-Yang Liu. Soft Matt...
0 downloads 0 Views 2MB Size
Download PDF
Article pubs.acs.org/Macromolecules

Orientational Cross-Correlation in Entangled Binary Blends in Primitive Chain Network Simulations Yuichi Masubuchi* and Yoshifumi Amamoto National Composite Center, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan ABSTRACT: It has been reported from the molecular dynamics simulation that the orientational cross-correlation (OCC) has a significant contribution to the relaxation modulus of polymers. Remarkably, the OCC contribution has been found to be universal with respect to the molecular weight and even to its distribution for unentangled polymers [Cao; Likhtman Phys. Rev. Lett. 2010, 104, 207801]. In this study, the OCC in entangled bidisperse polymer melts was evaluated by multichain slip-link simulations. The obtained OCC was similar to that in monodisperse systems in its intensity. On the other hand, the time development reflected the motion of short and long chains, and consequently the universality was not observed when the molecular weights of two components are well-separated and the long chain fraction is small. Comparison to the results obtained by the other slip-link model suggests that the cross-correlation is induced by the force balance and the fluctuation at the entanglement.

■

INTRODUCTION It has been rather established that the entangled polymer dynamics can be described by the tube model in which entangled polymer dynamics is replaced by the single chain motion in a tube shaped constraint along the chain backbone.1 In this framework, the dynamics of polymer chain is decomposed into reptation,2,3 contour length fluctuation (CLF),4 and constraint release (CR).5−8 A theoretical description for the former two that depict the motion of the chain inside the tube has been almost established owing to the single chain nature of the problem.9,10 On the other hand, description of CR has been still a matter of discussion owing to the difficulty in casting the multichain dynamics into the single-chain problem. Nevertheless, the modern molecular theories that incorporate all the mechanisms mentioned above have achieved remarkable success to reproduce experiments semiquantitatively. Apart from the success of the tube theory, there remains an unsolved issue called orientational cross-correlation (OCC).11 The tube picture mentioned above intrinsically neglects the cross-correlation between the different polymer chains. However, there are several studies reporting non-negligible contributions of OCC. Kornfield et al.12 measured infrared dichroism and birefringence for bidisperse polyisoprenes to report the OCC between short and long chain components. Ylitalo et al.13 further examined the system for theoretical analysis by the tube theory proposed by Doi et al.14 Graf et al.15 performed H double-quantum NMR measurement for a polybutadiene melt to report that the intermolecular orientational correlation exhibits amount of contribution in the stress relaxation. Cao and Likhtman16 performed molecular dynamics simulations with the Kremer−Grest model to show that the linear relaxation modulus includes the contribution from OCC of around 50% at the terminal relaxation time. Ramirez et al.17 © XXXX American Chemical Society

suggested that the OCC contribution is due to the motional constraint to the test chain by the tube following the theory by Warner and Edwards.18 Masubuchi and Amamoto19 suggested that the OCC is related to the cooperative motion among the chains induced by the incompressibility. An interesting feature of OCC is the universality. Cao and Likhtman16 found that the ratio of OCC contribution to the total relaxation is universal for the bead−spring chains with various molecular weights. Masubuchi and Sukumaran20 performed multichain slip-link simulations to report similar results. Masubuchi and Amamoto21 reported for multichain slip-link simulations of entangled H branch polymers that the OCC contribution is similar to that of linear polymers. Because time development of OCC is also universal and it is rather mild, it has been suggested that the effect of OCC could be embedded into the tuning of plateau modulus in the single chain modeling in which OCC is not considered.16,17 This strategy works given that the universality of OCC is valid entirely. In this respect, it is noted that the effect of molecular weight distribution has not been fully examined yet. Cao and Likhtman16 have reported that the universality of OCC is valid for binary blends of long and short chains. However, due to the computational limitation, their simulations were limited to unentangled and scarcely entangled systems. The blending conditions were also limited. In this study, the OCC contribution was investigated for a data set of linear viscoelasticity of entangled bidisperse polystyrene melts in the literature via multichain slip-link simulations. The fluctuation of subchain orientatinonal tensor of each molecule was recorded separately, and auto- and cross-correlations were Received: August 1, 2016 Revised: October 19, 2016

A

DOI: 10.1021/acs.macromol.6b01642 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

in comparison to blend 1, blend 2 and blend 3 locate at tube reptation regime and chain reptation II regime, respectively. Note that the same data set has been examined for the elongational viscosity,25 and the model parameters have been determined as the unit molecular weight M0 = 11k and the unit time τ0 = 1.0 s at T = 130 °C. From M0, the number of entanglement segment (the number of network strands to form one chain) Z is determined as Z = M/M0 where M is the sample molecular weight. The unit modulus G0 = ρRT/M0 is determined as 0.29 MPa. The bidisperse PS blends reported by Watanabe et al.28 were also simulated for further discussion on the effect of long chain concentration and the short chain molecular weight on OCC. The samples are summarized in Table 2. For these blends the model parameters are M0 = 11k, τ0 = 2.2 × 10−2 s, and G0 = 0.25 MPa at T = 167 °C. The multichain slip-link simulations were performed with a periodic boundary condition for which the box dimension was 163. Here, the unit length is the average strand length that corresponds to the subchain dimension with the molecular weight of M0. The subchain density was fixed at 10, and the osmotic intensity parameter was fixed at ε = 1.0. For statistics, 16 independent simulation runs were made starting with different initial configurations. The stress fluctuation was stored and converted into the linear relaxation modulus by the fluctuation− dissipation theorem (Green−Kubo relation). The obtained linear relaxation function was further converted into the dynamic modulus by the fitting to multimode Maxwell functions. It is fair to note that free energy expression for the primitive chain network model has not been obtained, and the dynamics in the simulation does not fulfill the detailed balance condition.29 Nevertheless, the Green−Kubo relation is practically valid as shown in the Appendix. For comparison, the slip-link simulations developed by Doi and Takimoto30 were performed as well. In this model (hereafter DT model), dynamics of reptating chains is calculated in parallel. Along the chain there exist several entanglement points, and at each entanglement point a slip-link is located. The slip-link is annihilated when the chain slides off due to its reptation motion, and a new slip-link is created when the chain protrudes from the slip-link at the chain end. Because the slip-links are virtually connected with each other to mimic the constraint-release dynamics, at the annihilation the coupled slip-link is removed and at the creation another new slip-link is also created on a randomly chosen segment. As just described, the fundamental picture of the DT model is the same with the multichain slip-link simulation. Nevertheless, the significant difference is the lack of force balance and the fluctuation at the entanglement. For DT DT model, the parameters for blend 1 are MDT 0 = 14k, τ0 = 2.3 × −1 DT 10 s, and G0 = 0.67 MPa. The equilibrium simulation for blend 1 was performed with the number of long and short chains at 10 and 1680, respectively, for 106τDT 0 , and the correlation functions were obtained from the recorded stress.

calculated. The obtained correlation functions were compared to those for monodispese melt of each component. The results revealed that the magnitude of OCC contribution is similar to that for the monodisperse systems. On the other hand, the time development is different from the monodisperse systems and not universal reflecting the dynamics of each component when the difference in molecular weight and in volume fraction is sufficiently large. In addition, comparison to the other slip-link simulation in which the force balance between chains is not considered demonstrated that the origin of OCC is not the coupling induced by constraint release but the force balance. Details are described below.

■

MODEL AND SIMULATIONS In the present study, the multichain slip-link simulation based on the primitive chain network model22−25 was mainly examined. In the model the entangled polymeric liquid is replaced by the network consisting of network strands, nodes, and dangling ends. Each polymer chain corresponds to a path between two dangling ends through the consecutive strands. At each network node there exists a slip-link that mimics the effect of entanglement suppressing the chain dynamics perpendicular to the chain backbone. The dynamics of slip-link is described by a Langevintype equation of motion for the slip-link position with drag force, tension acting on the connected strands, osmotic force induced by density fluctuation, and random force representing thermal agitation. Sliding motion of the chain through the slip-links is described by the kinetic equation for the monomer transport along the chain. The slip-link (and the attached network node) is removed when the connected dangling end slides out from the slip-link. On the contrary, a new slip-link and a network node are created by a dangling end that hooks a randomly chosen surrounding strand when the dangling end protrudes beyond a certain critical amount. The simulation code used in this study is the same with the earlier studies.24,25 A few data sets for bidisperse polystyrene (PS) melts were simulated. Tables 1 and 2 show the summary of samples. The Table 1. Examined Set of Bidisperse Polystyrene Melts I ML MS ZL ZS ϕL

blend 1

blend 2

blend 3

390000 51700 35 5 0.04

390000 51700 35 5 0.14

390000 102800 35 9 0.14

Table 2. Examined Set of Bidisperse Polystyrene Melts II ML MS ZL ZS ϕL

L294/L83

L294/L161

294000 82700 27 7 0.1−0.6

294000 161000 27 14 0.1−0.6

■

CORRELATION FUNCTIONS The linear relaxation modulus G(t) can be obtained from the subchain orientation (i.e., network strand orientation) function S(t) as

data set reported by Nielsen et al.26 are summarized in Table 1. The choice of this specific data set is rationalized by the fact that the blends are reasonably dispersed in the Viovy−Rubinstein− Colby diagram. 27 Namely, for blend 1 the long chain concentration is lower than the overlapping concentration for the long chain (to be categorized as tube Rouse regime).26 Owing to the increase of short chain concentration and its weight

G(t ) = B

1 S (t ) Γ

(1) DOI: 10.1021/acs.macromol.6b01642 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules ⎛ 1 S(t ) ≡ Nb ⎜⎜ N ⎝ b

⎞⎛ xy ⎟⎜ 1 O ( t + t ′ ) ∑ J ⎟⎜ N ⎠⎝ b J=1 Nc

⎞ xy O ( t ′ ) ∑ J ⎟⎟ ⎠ J=1

later. To see the contribution of SC in the entire relaxation, Cao and Likhtman16 defined the coupling parameter κ(t) written as

Nc

t′

κ (t ) =

(2) ZJ

OJxy(t ) ≡

∑ i=1

■

riJx(t )riJy(t ) niJ (t )

SC(t ) S (t ) =1− A S(t ) S(t )

(13)

RESULTS Figures 1−3 show G*(ω) (black solid curve), GL*(ω) (red solid curve), and GS*(ω) (yellow solid curve) calculated from G(t),

(3)

Here, Γ is the stress-optical coefficient, Oxy J the subchain orientation tensor, Nb the total number of subchains (network strands), Nc the total number of chains in the system, and ZJ the number of subchains for chain J. niJ(t) and riJ = (rxiJ, ryiJ, rziJ) are the Kuhn segment number and the subchain vector for subchain i on chain J. ⟨···⟩t′ indicates the averaging for the origin of time. Note that Nb fluctuates in time, but here we assume that Nb = ∑NJ c∑Zi JniJ(t)/n0 where n0 is the average number of niJ(t). S(t) can be decomposed into autocorrelations within the same chain and cross-correlations between the different chains. S(t ) = ϕLAL (t ) + ϕL 2C LL(t ) + ϕSAS(t ) + ϕS2CSS(t ) + 2ϕLϕSC LS(t )

Aα (t ) = Nb

1 ϕαNb 2

Cαβ(t ) = Nb

(4)

Nc

∑ ⟨OJxy(t + t′)OJxy(t′)⟩t′ (5)

J∈α

1 ϕαϕβ Nb 2

Nc

Nc

∑ ∑ ⟨OIxy(t + t′)OJxy(t′)⟩t′ I∈α ,J∈β I≠J

Figure 1. Linear viscoelasticity of blend 1. Symbols are the experimental data taken from the literature.26 Black, red, and yellow curves are the simulation results for the entire system, long chain contribution, and short chain contribution, respectively. Solid and broken curves are the relaxation with and without the cross-correlation contributions. Solid line exhibits the slope of 1/2.

(6)

Here, Aα is the autocorrelation for component α and Cαβ is the cross-correlation between chains in components α and β. The subscripts L and S mean the long and short chain components, respectively. The total relaxation modulus can be decomposed into the contributions from each component. G(t ) = G L(t ) + GS(t ) =

1 (SL(t ) + SS(t )) Γ

(7)

Here, SL and SS are the orientatinal relaxation of each component defined as Sα(t ) = ϕαAα (t ) + ϕα 2Cαα(t ) + ϕαϕβ Cαβ(t )

(8)

Note that in Sα the cross-correlation terms must be considered for the calculation of the linear relaxation modulus for each component Gα(t) from the fluctuation−dissipation theorem, as reported earlier.17 For convenience, the autocorrelation contribution of each component is denoted hereafter as

GAα(t ) =

1 ϕ A α (t ) Γ α

(9)

Alternatively, the total relaxation modulus can be decomposed into the contributions from auto- and cross-correlations. G(t ) = GA (t ) + GC(t ) =

1 (SA(t ) + SC(t )) Γ

Figure 2. Linear viscoelasticity of blend 2. Symbols are the experimental data taken from the literature.26 Black, red, and yellow curves are the simulation results for the entire system, long chain contribution, and short chain contribution, respectively. Solid and broken curves are the relaxation with and without the cross-correlation contributions.

(10)

SA(t ) ≡ ϕLAL (t ) + ϕSAS(t )

(11)

SC(t ) ≡ ϕL 2C LL(t ) + ϕS2CSS(t ) + 2ϕLϕSC LS(t )

(12)

GL(t), and GS(t), respectively. As reported earlier, G*(ω) is in reasonable agreement with the experimental data. GL*(ω) and GS*(ω) exhibit each contribution clearly. For instance, in Figure 1 for blend 1 in which the long chain concentration is smaller than

It is noted that the effect of CR appears not only in SC but also in SA because the chain dynamics is modulated by CR as shown C

DOI: 10.1021/acs.macromol.6b01642 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

Figure 3. Linear viscoelasticity of blend 3. Symbols are the experimental data taken from the literature.26 Black, red, and yellow curves are the simulation results for the entire system, long chain contribution, and short chain contribution, respectively. Solid and broken curves are the relaxation with and without the cross-correlation contributions.

Figure 4. Long chain contribution in blend 1 (top), blend 2 (middle), and blend 3 (bottom). Black solid curve is the normalized long chain contribution in the blends, SL(t)/ϕL. Red, blue, and green solid curves are the components of SL(t)/ϕL: AL(t), ϕLCLL(t), and ϕSCLS(t), respectively. Dotted curves show S(t) and A(t) in the monodisperse melt for comparison. The Rouse relaxation time of the long chain τLR is indicated by an arrow.

the overlapping concentration, G*L (ω) exhibits ω1/2 dependence in the short time region, indicating the Rouse behavior of the long chain.5,31 In Figures 2 and 3, the slope of GL*(ω) is apparently smaller than 1/2, reflecting the entanglement between long chains. Nevertheless, the comparison to the experimental data validates the simulation, and the good agreement allows further analysis for the correlation functions. Broken curves in Figures 1−3 show G*A (ω) (black broken curve), G*AL(ω) (red broken curve), and G*AS(ω) (yellow broken curve), calculated from GA(t), GAL(t), and GAS(t), respectively. As reported earlier, GA*(ω) is similar to G*(ω), and the difference seems accommodated by a tuning of the unit modulus. This observation lends support the strategy for implementation of OCC in single chain models by tuning the plateau modulus. On the other hand, for each component the difference between full relaxation and autocorrelation is rather complicated. For the long chain component (red curves), in blend 1 G*L (ω) and G*AL(ω) are close to each other, reflecting the dilute condition for the long chain that rarely entangles with each other. In blend 2, there exists a slight difference between G*L (ω) and G*AL(ω). This difference is due to the terms ϕL2CLL(t) and ϕLϕSCLS(t) in eq 8. In particular, the difference between GL*(ω) and GAL * (ω) seen in the time domain of the short chain relaxation is owing to the third term ϕLϕSCLS(t). Nevertheless, the difference might be accommodated by tuning the modulus for the long chain component because the relaxation functions are similar to each other in their shape. However, for blend 3 there is a significant difference between G*L (ω) and G*AL(ω), reflecting an intense contribution of ϕLϕSCLS(t). Indeed, G*L (ω) shows a clear twostep relaxation owing to the relaxation of each component whereas the short chain relaxation is not apparent in GAL * (ω). The contribution of cross-correlation is not negligible also for the short chains (yellow curves) for all the systems examined, as shown below. Figures 4 and 5 show Sα(t)/ϕα (black curves) with its components, Aα(t) (red curves), ϕαCαα(t) (green curves), and ϕβCαβ(t) (blue curves) for long and short chains. (For the definition of each function, see eqs 5 and 6.) For comparison, S(t) and A(t) for monodisperse melts are also shown (broken black and red curves). The Rouse relaxation time of each

Figure 5. Short chain contribution in blend 1 (top), blend 2 (middle), and blend 3 (bottom). Black solid curve is the normalized short chain contribution in the blends, SS(t)/ϕS. Red, blue, and green solid curves are the components of SS(t)/ϕS: AS(t), ϕSCSS(t), and ϕLCLS(t), respectively. Dotted curves show S(t) and A(t) in the monodisperse melt for comparison. The Rouse relaxation times of the short chains τSR are indicated by arrows.

component is calculated as τR = Z2/(2π2) (as discussed earlier32) and shown by arrows. For the long chain component (Figure 4), the contribution of cross-correlation, i.e., the difference between D

DOI: 10.1021/acs.macromol.6b01642 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules SL(t)/ϕL (black curve) and AL(t) (red curve), is dominated by ϕSCLS(t) (blue curve) in the short time region, reflecting the short chain relaxation and the large ϕS value. On the other hand, the relaxation time of CLL(t) is comparable to AL(t) as observed in monodisperse melts, and thus, after the short chain relaxation ϕLCLL(t) becomes relevant to induce the crossover between ϕSCLS(t) and ϕLCLL(t). This crossover is seen as a bump in SL(t)/ ϕL. For the short chain component (Figure 5), in the crosscorrelation contribution ϕSCSS(t) (green curve) is dominant in the short time range due to the large ϕS. After the short chain relaxation, ϕLCLS(t) comes into play. In particular, for blends 2 and 3 the long time tail of SS(t)/ϕS is apparently due to ϕLCLS(t). Actually, AS(t) is almost the same with A(t) in the monodisperse short chain (see red solid and broken curves in Figure 5). This observation demonstrates that the slowdown of short chain component in the blends is not due to constraint release but due to the cross-correlation. Figure 6 shows the coupling parameter κ(t) for blend 1, blend 2, and blend 3 (colored solid curves) and the monodisperse

Figure 7. Linear viscoelasticity (upper panel) and coupling parameter (lower panel) for L294/L83 blends. Symbols are experimental data taken from the literature.28 Solid and broken curves are the simulation results. The long chain concentration is 0.1 (red), 0.2 (blue), 0.4 (green), and 0.6 (black) for the blends. The results for monodisperse systems are shown by broken curves.

not distinguishable from that of the monodisperse long chain. Cao and Likhtman16 reported that κ(t) is universal for binary blends of Kremer−Grest chains with the bead number per chain at 10 and 100 for ϕL = 0.3−1.0. Our results suggest that κ(t) would be dependent on the blending condition for blends with lower ϕL values. Figure 8 shows the results for the L294/L161 blends.28 Again, the simulation semiquantitatively reproduces G*(ω) as shown in the upper panel, and the contribution of cross-correlation is not negligible as indicated by κ(t) in the lower panel. In comparison to the other blends, κ(t) is not apparently sensitive to the blending condition because the molecular weights of long and short chains are not well-separated. Figure 9 shows the result of DT model for blend 1. As shown in the upper panel, linear viscoelasticity is nicely reproduced for this model as well as the multichain model. However, in the stress there is virtually no contribution from the cross-correlation. Namely, S(t) as a whole is well-described by the autocorrelations (as shown in the lower panel). This result means that the crosscorrelation in the multichain model is induced by the force balance and the fluctuation at the entanglement rather than the constraint-release coupling.

Figure 6. Time development of coupling parameter. Broken curves are for monodisperse melts with Z = 5, 9, and 35 from left to right. Red, blue, and green curves are for blend 1, blend 2, and blend 3, respectively.

systems (black broken curves). In the time range of t/τ0 ≤ 10, κ(t) is almost universal regardless of the molecular weight and its distribution. The nature of logarithmic plot does not allow to indicate that at t = 0 κ(t) is smaller than 0.05 but still exists. Nevertheless, this result is consistent with the bead−spring molecular dynamics simulations. However, in the long time range, the κ(t) behavior for blends is different from that of monodisperse systems. κ(t) exhibits a peak at a certain characteristic time comparable to the short chain relaxation time (see Figure 5). Then κ(t) decreases after the peak, but it revives to the level of monodisperse systems with a growth time comparable to the long chain relaxation time (see Figure 4). Figure 7 shows the results for L294/L83 blends reported by Watanabe et al.,28 who systematically changed the long chain concentration. As reported earlier,33 even for these blends the simulation semiquantitatively reproduces G*(ω) as shown in the upper panel. In this G*(ω) there exists the contribution from cross-correlation as in the blends reported by Nielsen et al.,34 and the coupling parameter κ(t) is shown in the lower panel. Just as in Figure 6, κ(t) is not universal and it depends on the blending condition; it shows a peak reflecting the short time relaxation, and afterward it revives to the monodisperse behavior in the time range of long chain relaxation. However, the peak in the short time becomes less apparent when the long chain concentration becomes larger. For instance, for the blend with ϕL = 0.6, κ(t) is

■

DISCUSSION It so appears that in the multichain simulations the contribution of cross-correlation between chains is not negligible for bidisperse blends, and it depends on the blending conditions. Such effects are not considered in the single chain models, although both approaches can reproduce linear viscoelasticity in similar accuracies, as shown in Figure 9 for example. These results demonstrate that the cross-correlation in the multichain E

DOI: 10.1021/acs.macromol.6b01642 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

Figure 10. Relaxation modulus of blend 1 obtained under step-shear deformations (symbols) at the initial strain of 1.0 (black), 0.5 (red) and 0.25 (blue), and via the fluctuation−dissipation theorem (red solid line). The contribution from autocorrelation GA(t) is shown by red broken curve for comparison.

where GN is the unit modulus, φ(t) is the tube survival probability, and R(t) is the contribution of constraint release (CR). (Although there are different implementations of CR,5−8 the difference can be embedded into the description of R(t).) To cast this expression to our results in which G(t) has the form G(t) = GA(t) + GC(t), φ(t)R(t) must be separated into the auto- and cross-correlation contributions. Such a discrimination among the relaxation modes is not straightforward. A possible route for the mapping is via the single-chain slip-spring model in which a Rouse chain anchored by some virtual springs is considered. Ramirez et al.17 obtained the correlation functions of stress fluctuations for the model with varying the intensity of virtual springs to report that the cross-correlation between the Rouse chain stress and the virtual spring stress increases with increasing spring constant for the virtual springs. This result suggests that the cross-correlation between the backbone stress and the virtual spring stress corresponds to OCC in the multichain models, and it can be accommodated by tuning the fluctuation at the bead to reproduce the results in the multichain models.35 On the other hand, the single-chain slip-spring model would be mapped to the other single-chain models on the basis of the fluctuation at entanglement.36−39 Although the fluctuation in such models is to incorporate CR and not OCC, the fluctuation might be accommodated to that in the single-chain slip-spring model for the evaluation of OCC. The mapping of obtained results to those reported for the bead−spring simulations is also difficult. Masubuchi and Sukumaran20 reported that κ(t) (for monodisperse melts) in the multichain slip-link simulation is similar to that observed in the Kremer−Grest simulations16 but different for the time development in which κ(t) for the Kremer−Grest simulation exhibits the growth in a time range much shorter than the sliplink simulation. This difference in κ(t) might be due to the excluded volume interaction, but further investigation is necessary. For such a purpose, the OCC in the other multichain models is worth discussing. There have been reported several multichain models with different descriptions for entanglement. In some models,40−43 the virtual spring for entanglement is incorporated in addition to the interparticle interactions. The effect of incompressibility on OCC could be separately discussed in such models from that induced by the virtual spring given that the intensities of virtual spring and interparticle interactions are systematically varied. The OCC in the multichain model44 where

Figure 8. Linear viscoelasticity (upper panel) and coupling parameter (lower panel) for L294/L161 blends. Symbols are experimental data taken from the literature.28 Solid and broken curves are the simulation results. The long chain concentration is 0.1 (red), 0.2 (blue), 0.4 (green), and 0.6 (black) for the blends. The results for monodisperse systems are shown by broken curves.

Figure 9. Linear viscoelasticity (upper panel) and correlation functions (lower panel) for blend 1 obtained by the DT model. For upper panel, symbols are experimental data taken from the literature.26 Solid and broken curves are the simulation results. For lower panel, unfilled circle is S(t) as a whole, and dotted curves are SA(t) (thick black curve), ϕSAS(t) (thin yellow curve), and ϕLAL(t) (thin red curve).

model is compensated somehow in the single chain model by certain relaxation modes, but the mapping is difficult. In the tube theories, the relaxation modulus is written as G(t) = GNφ(t)R(t), F

DOI: 10.1021/acs.macromol.6b01642 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

and by Council for Science, Technology and Innovation, Crossministerial Strategic Innovation Promotion Program, “Structural Materials for Innovation” from JST.

the entangled dynamics is attained by uncrossability between chains and not by virtual springs is also to be evaluated.

■

■

CONCLUSIONS The orientational cross-correlation between different chains was observed for bidisperse entangled polymer melts in multichain slip-link simulations. The relaxation modulus was decomposed into the contributions from auto- and cross-correlations of stress for the same chain and between different chains. The coupling parameter that is the ratio of the cross-correlation contribution to the entire relaxation modulus was similar to that for monodisperse polymers in its magnitude. On the other hand, its time development did not trace the universal curve for monodisperse systems but exhibited a two-step behavior reflecting the relaxations of short and long chains, when the molecular weights of two components are well-separated and the long chain fraction is small. In such cases, the autocorrelation function of the long chain component was modulated by the short chain dynamics. On the contrary, for the short chain component the autocorrelation function was almost identical to that of the monodisperse melt regardless of the long chain component. The short chain contribution in the total relaxation modulus was modulated by the long chain through the crosscorrelation. Comparison to the results obtained by the other sliplink model suggests that the cross-correlation is induced by the force balance and the fluctuation at the entanglement.

(1) Masubuchi, Y. Simulating the Flow of Entangled Polymers. Annu. Rev. Chem. Biomol. Eng. 2014, 5 (1), 11−33. (2) de Gennes, P. G. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. J. Chem. Phys. 1971, 55 (2), 572. (3) Doi, M.; Edwards, S. F. Dynamics of Concentrated Polymer Systems. Part 1. Brownian Motion in the Equilibrium State. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1789. (4) Doi, M. Explanation for the 3.4-Power Law for Viscosity of Polymeric Liquids on the Basis of the Tube Model. J. Polym. Sci., Polym. Phys. Ed. 1983, 21 (5), 667−684. (5) Graessley, W. W. Entangled Linear, Branched and Network Polymer Systems  Molecular Theories. In Advances in Polymer Science; Springer-Verlag: Berlin, 1982; Vol. 47, pp 67−117. (6) Marrucci, G. Relaxation by Reptation and Tube Enlargement: A Model for Polydisperse Polymers. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 159−177. (7) Tsenoglou, C. Viscoelasticity of Binary Polymer Blends. ACS Polym. Prepr. 1987, 28, 185−186. (8) des Cloizeaux, J. Double Reptation vs. Simple Reptation in Polymer Melts. Europhys. Lett. 1988, 5 (5), 437−442. (9) Likhtman, A. E.; McLeish, T. C. B. Quantitative Theory for Linear Dynamics of Linear Entangled Polymers. Macromolecules 2002, 35 (16), 6332−6343. (10) Milner, S. T.; McLeish, T. C. B. Reptation and Contour-Length Fluctuations in Melts of Linear Polymers. Phys. Rev. Lett. 1998, 81 (3), 725−728. (11) Likhtman, A. E. Whither Tube Theory: From Believing to Measuring. J. Non-Newtonian Fluid Mech. 2009, 157 (3), 158−161. (12) Kornfield, J. A.; Fuller, G. G.; Pearson, D. S. Infrared Dichroism Measurements of Molecular Relaxation in Binary Blend Melt Rheology. Macromolecules 1989, 22 (3), 1334−1345. (13) Ylitalo, C. M.; Kornfield, J. A.; Fuller, G. G.; Pearson, D. S. Molecular Weight Dependence of Component Dynamics in Bidisperse Melt Rheology. Macromolecules 1991, 24 (3), 749−758. (14) Doi, M.; Graessley, W. W.; Helfand, E.; Pearson, D. S. Dynamics of Polymers in Polydisperse Melts. Macromolecules 1987, 20 (8), 1900− 1906. (15) Graf, R.; Heuer, A.; Spiess, H. W. Chain-Order Effects in Polymer Melts Probed by 1H Double-Quantum NMR Spectroscopy. Phys. Rev. Lett. 1998, 80 (26), 5738−5741. (16) Cao, J.; Likhtman, A. E. Time-Dependent Orientation Coupling in Equilibrium Polymer Melts. Phys. Rev. Lett. 2010, 104 (20), 207801. (17) Ramírez, J.; Sukumaran, S. K.; Likhtman, A. E. Significance of Cross Correlations in the Stress Relaxation of Polymer Melts. J. Chem. Phys. 2007, 126 (24), 244904. (18) Warner, M.; Edwards, S. F. Neutron Scattering from Strained Polymer Networks. J. Phys. A: Math. Gen. 1978, 11 (8), 1649−1655. (19) Masubuchi, Y.; Amamoto, Y. Effect of Osmotic Force on Orientational Cross-Correlation in Primitive Chain Network Simulation. Nihon Reoroji Gakkaishi (J. Soc. Rheol. Jpn.) 2016, 44 (4), 219− 222. (20) Masubuchi, Y.; Sukumaran, S. K. Cross-Correlation Contributions to Orientational Relaxations in Primitive Chain Network Simulations. Nihon Reoroji Gakkaishi (J. Soc. Rheol. Jpn) 2013, 41 (1), 1−6. (21) Masubuchi, Y.; Amamoto, Y. Orientational Cross Correlation of H Branch Polymers in Primitive Chain Network Simulations. Nihon Reoroji Gakkaishi (J. Soc. Rheol. Jpn), submitted. (22) Masubuchi, Y.; Takimoto, J.-I.; Koyama, K.; Ianniruberto, G.; Marrucci, G.; Greco, F. Brownian Simulations of a Network of Reptating Primitive Chains. J. Chem. Phys. 2001, 115 (9), 4387. (23) Masubuchi, Y.; Ianniruberto, G.; Greco, F.; Marrucci, G. Quantitative Comparison of Primitive Chain Network Simulations

■

APPENDIX. TEST OF GREEN−KUBO RELATION We performed multichain slip-link simulations for stress relaxation after step-shear deformations and compared the relaxation modulus to that obtained from the stress autocorrelation under equilibrium. After the equilibration for a long time sufficiently longer than the longest relaxation time, step shear deformations were applied affinely under Lees−Edwards boundary condition for which the simulation box size was 163. For statistics, eight independent simulations were made from different initial configurations and obtained stress relaxations were averaged. The result for blend 1 is shown in Figure 10. Apparently, the short time response corresponds to each other given that G(t) is calculated by eq 10 in which the crosscorrelation is included. For comparison, the contribution from autocorrelation, GA(t), is shown by a dotted curve which is apparently lower than the step deformation results. Although the signal under the deformation becomes weaker than the stress fluctuation in the long time regime, we conclude that the Green− Kubo relation is valid for the multichain slip-link simulation, and the contribution of cross-correlation is included in the stress under deformation as reported for the single-chain slip-spring simulation.17

■

REFERENCES

AUTHOR INFORMATION

Corresponding Author

*(Y.M.) E-mail: [email protected]. ORCID

Yuichi Masubuchi: 0000-0002-1306-3823 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors greatly thank Prof. J.-I. Takimoto for his help for the simulation of DT model. This work is supported in part by Grant-in-Aid for Scientific Research (B) (26288059) from JSPS G

DOI: 10.1021/acs.macromol.6b01642 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules with Literature Data of Linear Viscoelasticity for Polymer Melts. J. NonNewtonian Fluid Mech. 2008, 149 (1−3), 87−92. (24) Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Greco, F.; Marrucci, G. Comparison among Slip-Link Simulations of Bidisperse Linear Polymer Melts. Macromolecules 2008, 41 (21), 8275−8280. (25) Takeda, K.; Sukumaran, S. K.; Sugimoto, M.; Koyama, K.; Masubuchi, Y. Primitive Chain Network Simulations for Elongational Viscosity of Bidisperse Polystyrene Melts. Adv. Model. Simul. Eng. Sci. 2015, 2 (1), 11. (26) Nielsen, J. K.; Rasmussen, H. K.; Hassager, O.; McKinley, G. H. Elongational Viscosity of Monodisperse and Bidisperse Polystyrene Melts. J. Rheol. (Melville, NY, U. S.) 2006, 50 (4), 453. (27) Viovy, J. L.; Rubinstein, M.; Colby, R. H. Constraint Release in Polymer Melts. Tube Reorganization versus Tube Dilation. Macromolecules 1991, 24 (12), 3587−3596. (28) Watanabe, H.; Sakamoto, T.; Kotaka, T. Viscoelastic Properties of Binary Blends of Narrow Molecular Weight Distribution Polystyrenes. 2. Macromolecules 1985, 18 (5), 1008−1015. (29) Uneyama, T.; Masubuchi, Y. Detailed Balance Condition and Effective Free Energy in the Primitive Chain Network Model. J. Chem. Phys. 2011, 135 (18), 184904. (30) Doi, M.; Takimoto, J. Molecular Modelling of Entanglement. Philos. Trans. R. Soc., A 2003, 361 (1805), 641−650. (31) Rubinstein, M.; Colby, R. H. Self-Consistent Theory of Polydisperse Entangled Polymers: Linear Viscoelasticity of Binary Blends. J. Chem. Phys. 1988, 89 (8), 5291. (32) Furuichi, K.; Nonomura, C.; Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Greco, F.; Marrucci, G. Entangled Polymer Orientation and Stretch under Large Step Shear Deformations in Primitive Chain Network Simulations. Rheol. Acta 2008, 47 (5−6), 591−599. (33) Masubuchi, Y.; Watanabe, H.; Ianniruberto, G.; Greco, F.; Marrucci, G. Comparison among Slip-Link Simulations of Bidisperse Linear Polymer Melts. Macromolecules 2008, 41 (21), 8275−8280. (34) van Ruymbeke, E.; Nielsen, J.; Hassager, O. Linear and Nonlinear Viscoelastic Properties of Bidisperse Linear Polymers: Mixing Law and Tube Pressure Effect. J. Rheol. (Melville, NY, U. S.) 2010, 54 (5), 1155. (35) Sukumaran, S. K.; Likhtman, A. E. Modeling Entangled Dynamics: Comparison between Stochastic Single-Chain and Multichain Models. Macromolecules 2009, 42 (12), 4300−4309. (36) Khaliullin, R. N.; Schieber, J. D. Self-Consistent Modeling of Constraint Release in a Single-Chain Mean-Field Slip-Link Model. Macromolecules 2009, 42 (19), 7504−7517. (37) Khaliullin, R. N.; Schieber, J. D. Application of the Slip-Link Model to Bidisperse Systems. Macromolecules 2010, 43 (14), 6202− 6212. (38) Schieber, J. D.; Andreev, M. Entangled Polymer Dynamics in Equilibrium and Flow Modeled Through Slip Links. Annu. Rev. Chem. Biomol. Eng. 2014, 5 (1), 367−381. (39) Schieber, J. D.; Indei, T.; Steenbakkers, R. J. A. Fluctuating Entanglements in Single-Chain Mean-Field Models. Polymers (Basel, Switz.) 2013, 5 (2), 643−678. (40) Uneyama, T.; Masubuchi, Y. Multi-Chain Slip-Spring Model for Entangled Polymer Dynamics. J. Chem. Phys. 2012, 137 (15), 154902. (41) Chappa, V. C.; Morse, D. C.; Zippelius, A.; Müller, M. Translationally Invariant Slip-Spring Model for Entangled Polymer Dynamics. Phys. Rev. Lett. 2012, 109 (14), 148302. (42) Ramírez-Hernández, A.; Detcheverry, F. A.; Peters, B. L.; Chappa, V. C.; Schweizer, K. S.; Müller, M.; de Pablo, J. J. Dynamical Simulations of Coarse Grain Polymeric Systems: Rouse and Entangled Dynamics. Macromolecules 2013, 46 (15), 6287−6299. (43) Langeloth, M.; Masubuchi, Y.; Böhm, M. C.; Müller-plathe, F. Recovering the Reptation Dynamics of Polymer Melts in Dissipative Particle Dynamics Simulations via Slip-Springs. J. Chem. Phys. 2013, 138 (2013), 104907. (44) Padding, J. T.; Briels, W. J. Uncrossability Constraints in Mesoscopic Polymer Melt Simulations: Non-Rouse Behavior of C120H242. J. Chem. Phys. 2001, 115 (6), 2846.

H

DOI: 10.1021/acs.macromol.6b01642 Macromolecules XXXX, XXX, XXX−XXX