Entropy-Driven Segregation and Its Competition with Crystal

Article pubs.acs.org/JPCB

Entropy-Driven Segregation and Its Competition with Crystal Nucleation in the Binary Blends of Stretched and Free Guest Polymers Liyun Zha,† Miaomiao Zhang,† Liangbin Li,‡ and Wenbing Hu*,† †

School of Chemistry and Chemical Engineering, State Key Lab of Coordination Chemistry, Nanjing University, Nanjing 210023, China ‡ Lab of Soft Matter Chemistry, Chinese University of Science and Technology, Hefei 230026, China ABSTRACT: Enhancing the dynamic asymmetry between liquidlike and solid-like components leads to spontaneous segregation. As a typical example, the binary blends of stretched and free guest polymers were investigated by our dynamic Monte Carlo simulations. The results evidenced an entropic driving force for the strain-induced segregation between two components, similar to that for Onsager’s lyotropic liquid crystals. In addition, with the decrease of strain rates, its competition with strain-induced crystal nucleation results in variable compositions in crystal precursors. The scenario helps to settle down the controversial arguments on the flow-induced precursors of shish-kebab crystallites in the melt.

■

INTRODUCTION Viscoelastic segregation between solid-like and liquid-like components appears as a general phenomenon in condensed matter systems.1 For the flow-induced phase separation, Onuki has summarized various systems of dynamic asymmetry as its origin of driving force under the conditions away from the critical point of phase separation in the quiescent mixtures.2 Wolf has derived a thermodynamic equation for the flowinduced phase separation in polymer blends, by considering different viscoelastic responses of the two components.3 Straininduced phase separation in a binary blend of network (solidlike) and free guest (liquid-like) polymers has been observed by Batra et al.,4 who provided a typical example of viscoelastic segregation. Deloche and Samulski ascribed its driving force to the enthalpy gain of oriented network polymers due to shortrange orientation-dependent interactions, similar to Maier− Saupe interactions in thermotropic liquid crystals.5 However, an entropic origin of strain-induced phase separation between stretched short and long polymers has recently been reported.6 Strain-induced segregation in such a dynamic asymmetric system is worthy of further investigation on whether the origin of driving force is enthalpic or entropic. In the extensional flow of polymer solutions, strain-induced phase separation competes with strain-induced crystallization, which appears to be responsible for the time evolution of crystallite morphology toward shish-kebabs.7 The shish-kebab crystallite morphology is unique to polymers8 and holds important implications in the shear flow of polymer processing.9 It is commonly believed that shish formation is induced by the entanglement network formed by a large © 2016 American Chemical Society

enough fraction of long chains that takes over the resistance of shear stress.10−12 In this sense, the stretched long chains should be the dominant component in the shish structures, same as that observed in polymer solutions.13,14 However, recently, short chains have been found to be the dominant component in the early stage of shish formation in polymer melt, raising controversial arguments on the dominant composition in shish precursors.15 The idea of the entangled polymer network to catch the nearby short chains during flow-induced crystal nucleation has been proposed.16 Thus, the scenario of competition between phase separation and crystallization in the binary blends of stretched and free guest polymers serves as a key to understand the composition of the precursor of crystal nucleation in the shear-induced polymer crystallization. In this article, we performed dynamic Monte Carlo (MC) simulations of strain-induced segregation and its competition with crystallization in the homogeneous binary blends of stretched and free guest polymers. The simulation approach of strain-induced polymer crystallization has recently been developed to gain a better understanding from crystal nucleation17 to hierarchical chain-folding,18 comonomer effects,19,20 solvent effects,21 memory effect upon cyclic loading,22 and polydispersity effects.23 Our new results will evidence an entropic origin of strain-induced phase separation in such polymer blends, and the segregation prior to straininduced crystallization provides an interpretation to the variable Received: August 19, 2016 Revised: November 22, 2016 Published: November 28, 2016 12988

DOI: 10.1021/acs.jpcb.6b08399 J. Phys. Chem. B 2016, 120, 12988−12992

Article

The Journal of Physical Chemistry B

polymers. If any chain ends of network polymers were forced to leave the YZ end-plane during chain sliding diffusion, they would be strongly attracted back to the end-plane during the rest relaxation period. As each step added one more X-site to the basic 16 sites, the strain rate of the sample was 6.25%/5000 MC cycles. The samples were alternatively stretched in the +X and −X directions until the two YZ planes of the X axis ends have separately arrived at X = 1 and 80, as demonstrated in Figure 1. By repeating the above steps, the samples were measured at various strains. The acceptance of each trial move was assessed by the conventional Metropolis sampling method, and the energy change could be expressed as

compositions of crystal precursors in the polymer melt and solutions.

■

SIMULATION DETAILS In the lattice space with an initial size of 16 × 128 × 128 (X × Y × Z) cubic cells, for X = 33−48, we prepared a series of polymer blends with the volume fractions of free guest polymers varied from 0.1 to 0.9, with a step length of 0.1. To this end, a certain amount of network 128-mers was folded 7 times along the X axis with a fold length of 16 extending over the Y axis (or 256-mers folded 15 times); meanwhile, along the same directions, a certain amount of other free guest 128-mers was folded 15 times with a fold length of 8 until the total occupation density of monomers reached up to 0.9375 to mimic the binary blends of bulk polymers. The rest vacancy sites were treated as the free volume for chain motion via a microrelaxation model (local single-site jumping of monomers to its vacancy neighbor, sometimes coupled with partial sliding diffusion along the chain).24 Double monomer occupation and bond intersection were rejected in mimicking the excluded volume of polymers. The coordination number of the cubic lattice is 26, including 6 neighbors along three axes, 12 along face diagonals, and 8 along body diagonals. At the beginning, the sample systems were relaxed under athermal conditions for 5 × 106 MC cycles (MC cycle is the time unit defined as the total trial moves when all the monomers were sampled once on average) into equilibrium coil states. During relaxation, two chain-ends of those network polymers were separately restricted (yet mobile) in the corresponding YZ planes at the ends of the X axis, as demonstrated in the left part of Figure 1.

⎛ Ep cEc + pEp + aEa E ⎞E ΔE = = ⎜c + p + a a⎟ c Ec Ec ⎠ kT kT kT ⎝

(1)

in which Ec is the energy penalty of noncollinear connection between two consecutive bonds along the chain (reflecting the chain flexibility); Ep is the energy benefit for parallel packing of two neighboring bonds (reflecting the driving force for crystallization);25 Ea is the energy change to repel the new vacancy sites (reflecting the normal stress of uniaxial stretching); and c, p, and a are the net changes for noncollinear connection, nonparallel packing, and square distance of the new vacancy site from the YZ center of the sample. For simplicity, the mixing interactions of two components were set as zero; all the old−old, old−new, and new−new mixing interactions of vacancy sites were set as zero; the reduced energy parameter, Ep/Ec, was set as the driving force for crystallization; and Ea/Ec was set as the normal stress to repel the new vacancy sites. kT/ Ec was set as the reduced temperature in our simulations.

■

RESULTS AND DISCUSSION Strain-Induced Phase Separation. We started with the observations of strain-induced phase separation by setting Ep/ Ec = 0 (noncrystallizable polymers) and kT/Ec = 10 to avoid simultaneous crystallization. The volume fraction of free guest 128-mers in their blend of stretched 128-mers was fixed at 0.1. To benefit a complete segregation, the values of Ea/Ec were stepwise increased from 0 to 0.064 with a step length of 0.001 along with the step of stretching. Figure 1 demonstrates two snapshots before and after stretching. One can clearly see the phase separation between free guest polymers (blue) and network polymers (yellow), with the latter accumulating in the center whereas the former surrounding the surface. To monitor the segregation during the stretching process in Figure 1, the demixing parameter was defined as the average occupation fraction of network monomers on the neighboring sites of each network monomer. The strain-evolution curve of the demixing parameter is shown in Figure 2, demonstrating the saturation of spontaneous segregation. As the stretching was performed at a relatively high temperature, kT/Ec = 10, the strain rate of 6.25%/5000 MC cycles has already been high enough for the completion of strain-induced phase separation. For a dynamically symmetric mixture, two components share their initially segregated space with each other, both raising the translational entropy. When one component becomes relatively more rigid in the mixing state, the second component loses its translational entropy because the molecules are trapped in the local region by the surrounding obstacles of the first component. One can imagine that a segregation between two components will return the translational entropy back to the

Figure 1. Snapshots of the initially homogeneous state and the completely segregated state at 400% strain in the network 128-mers blended with 0.1 volume fraction of free guest 128-mers. The stretching is carried out at Ep/Ec = 0 and kT/Ec = 10 with a stepwise increase of Ea/Ec from 0 to 0.064 with a step length of 0.001 along with the stepwise strain rate of 6.25%/5000 MC cycles. The bonds of free guest 128-mers are shown as blue cylinders, and those of network 128-mers, as yellow cylinders with red balls for their chain ends. The frame shows the total lattice space of 80 × 128 × 128 (X × Y × Z) in our simulations.

After athermal relaxation, the nearly random coil polymers were stretched step-by-step.17 In each step, first we chose a random X-site inside the sample space, at which the YZ plane splits the sample into two parts. We moved the +X part one lattice site toward the +X direction and then reconnected all the broken chain segments by local sliding of the rest segments in the −X part. After all the broken chains were reconnected, the sample was allowed to relax for a fixed period, for instance, 5000 MC cycles. During this period, the newly inserted vacancy sites were expelled in mimic to the normal stress of stretched 12989

DOI: 10.1021/acs.jpcb.6b08399 J. Phys. Chem. B 2016, 120, 12988−12992

Article

The Journal of Physical Chemistry B

segregated space; meanwhile, those rigid network polymers slightly sacrifice their mobility. Therefore, we analyzed the monomer mobility of two components before and after straininduced phase separation, with reference to the pure network 128-mers during the same stretching processes. The results of the acceptability difference of the monomer moves of two components from the pure network monomers are also shown in Figure 2. The acceptability difference as reflecting the relative mobility was defined as the averagely accepted moves of each monomer subtracted by the parallel results of pure network 128-mers during the stretching process. One can clearly see that phase separation significantly releases the conformational entropy of free guest polymers, providing them freedom to realize various chain conformations, although their segregated space is limited at the sample surface. Otherwise, they will go with the network polymers to decrease their mobility upon further stretching. In conclusion, the conformational entropy gain of free guest polymers appears as the dominant driving force for the strain-induced phase separation between network and free guest polymers. Competition between Strain-Induced Phase Separation and Crystallization. We then introduced crystallization in competition with the strain-induced phase separation by setting Ep/Ec = 1 at a relatively low temperature, kT/Ec = 4.5. We employed the network 256-mers because longer network polymers allow for larger final strains to conduct the competition of phase transitions. Strain-induced crystallization was monitored by the crystallinity, defined as the fraction of crystalline bonds among the total bonds, and a crystalline bond was assigned as the bond containing more than four parallel neighbors. Here, we chose the criteria of four for the separation between the amorphous bonds and the crystalline bonds (variable from 0 up to 24 neighbors as the total 26 neighbors subtracted by two consecutive bonds along the chain), so as to take account of those less-perfectly-packed bonds at the crystallite edges. The strain-evolution curves of crystallinity for the blends of various compositions are summarized in Figure 4a. One can see an abrupt increase of crystallinity, demonstrating the strain-induced crystallization. With the increase of volume fractions of free guest 128-mers, higher strains are required to induce crystallization. This is because the network polymers that dominate crystal nucleation are diluted by the higher volume fractions of free guest polymers, and, consequently, crystallization got retarded. Figure 4b displays the early-stage difference between the separate crystallinity of network and free guest polymers. The network polymers start to crystallize slightly earlier than the free guest polymers because of the conformational entropy loss of their stretching state. As soon as the network polymers start to crystallize, the free guest polymers join into this process as well, implying that free guest polymers are an important component in the precursor of crystal nucleation. In short, both network and free guest polymers can take part in strain-induced crystal nucleation. Phase separation needs a larger-scale diffusion than crystallization. Therefore, the extent of prior phase separation appears to be sensitive to the decrease of the strain rates, which influences the subsequent crystallization behaviors. We observed the composition dependence of onset strains for crystallization under various strain rates, as summarized in Figure 5a. Under a fixed strain rate, the onset strains increase monotonically with the volume fractions of free guest polymers. With the decrease of strain rates, a horizontal zone occurs near

Figure 2. Strain-evolution curves of the demixing parameter (in the left axis, defined as the mean fraction of the neighboring sites of each network monomer occupied by the same species) and the acceptability difference of monomer motion (in the right axis, separated into network and free guest polymers, defined as the average accepted moves per monomer subtracted by the parallel results of pure network 128-mers) during the stretching process in the blend of network 128mers with a volume fraction of 0.1 of free guest 128-mers.

second component, as illustrated in Figure 3. Although the first component will sacrifice a little translational entropy due to

Figure 3. Illustration of entropy-driven mixing in the conventional dynamically symmetric systems and of entropy-driven segregation in the systems of enhancing dynamic asymmetry. In the first case, both components share a larger space by mixing, whereas in the second case, the higher-mobility components escape from the local surrounding obstacles of the first components and gain a larger space in their concentrated phase by demixing.

demixing, the total entropy will be increased by this segregation. The scenario is very similar to that for an entropic origin of driving forces for the lyotropic liquid crystals of rodlike particles, as proposed by Onsager in 1949.26 By sacrificing the rotational and translational freedoms of partial anisotropic particles in the concentrated phase and meanwhile releasing more to the others in the diluted phase, the total entropy is increased. Upon stretching, the network polymers become more rigid and they behave like obstacles to “prison” those homogeneously blended free guest polymers into a narrower space and thus to decrease the latter’s capability to move around and to change chain conformation. By phase separation as “prison break”, the free guest polymers gain their high mobility with a higher conformational entropy in their 12990

DOI: 10.1021/acs.jpcb.6b08399 J. Phys. Chem. B 2016, 120, 12988−12992

Article

The Journal of Physical Chemistry B

Figure 4. Strain-evolution curves of (a) crystallinity and (b) component separate crystallinity in the blends of network 256-mers with various volume fractions of free guest 128-mers as labeled, at a strain rate of 6.25%/10000 MC cycles, kT/Ec = 4.5, Ep/Ec = 1, and Ea/Ec = 5000. The crossover of two straight lines demonstrates the onset strain of crystallization in the blend with 0.7 volume fraction of free guest polymers. The initial level of crystallinity can be regarded as a result of thermal fluctuations in the melt state.

Figure 5. (a) Onset strains of crystallization for 256-mers blending with different volume fractions of free guest 128-mers, stretched under three strain rates as denoted, with kT/Ec = 4.5, Ep/Ec = 1, and Ea/Ec = 5000. Each onset strain and its error bar are based on an average of three independent simulations. (b) Strain-evolution curves of demixing parameters for the blends with a volume fraction of 0.5 of free guest 128-mers under three strain rates as denoted. The arrows indicate the corresponding onset strains for crystallization.

entropic origin of strain-induced phase separation, similar to that for Onsager’s lyotropic liquid crystals. The competition between phase separation and crystallization provides a scenario that explains the potentially different compositions of the nucleation precursors of shish-kebab crystallites in the polymer melt and in solutions. Our present simulation results shed light onto the dynamically asymmetric segregation as well as onto the variable compositions of long chains in flowinduced shish precursors.

the lower end of the volume fractions of free guest polymers, implying insensitivity of crystallization to compositions. One can attribute this insensitivity to the completion of prior phase separation. Figure 5b demonstrates the strain-evolution curves of demixing parameters under three strain rates for a blend with a volume fraction of 0.5 of free guest polymers. One can clearly see that at the onset strains of crystallization (indicated by three arrows in the figure), the prior phase separation becomes obvious at the slowest stretching. This result reveals a scenario in which the phase separation has no time to complete either under fast stretching or in the less dynamically asymmetric systems. In this sense, the polydispersed polymer melt is prone to involve less amount of long-chain components in crystal nucleation as compared to the parallel situation in polymer solutions.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: 0086-25-89686667.

■

ORCID

CONCLUSIONS In this work, we used dynamic MC simulations to investigate the strain-induced phase separation between stretched network polymers and free guest polymers, as well as its competition with strain-induced crystal nucleation. The results identified an

Liangbin Li: 0000-0002-1887-9856 Wenbing Hu: 0000-0002-7795-9004 Notes

The authors declare no competing financial interest. 12991

DOI: 10.1021/acs.jpcb.6b08399 J. Phys. Chem. B 2016, 120, 12988−12992

Article

The Journal of Physical Chemistry B

■

(21) Zha, L.; Wu, Y.; Hu, W. Multi-Component Thermodynamics of Strain-Induced Polymer Crystallization. J. Phys. Chem. B 2016, 120, 6890−6896. (22) Guan, X.; Zha, L.; Wu, Y.; Hu, W. Strong Memory of StrainInduced Copolymer Crystallization Revealed by Monte Carlo Simulations. Polymer 2016, 98, 282−286. (23) Zhang, M.; Zha, L.; Gao, H.; Nie, Y.; Hu, W. How Polydispersity of Network Polymers Influences Strain-Induced Crystal Nucleation in a Rubber. Chin. J. Polym. Sci. 2014, 32, 1218−1223. (24) Hu, W.-B. Structural Transformation in the Collapse Transition of the Single Flexible Homopolymer Model. J. Chem. Phys. 1998, 109, 3686−3690. (25) Hu, W.-B.; Frenkel, D. Polymer Crystallization Driven by Anisotropic Interactions. Adv. Polym. Sci. 2005, 191, 1−35. (26) Onsager, L. The Effects of Shape on the Interaction of Colloidal Particles. Ann. N. Y. Acad. Sci. 1949, 51, 627−659.

ACKNOWLEDGMENTS The financial support from National Natural Science Foundation of China (No. 21274061), Program for Changjiang Scholars and Innovative Research Team in University, and Priority Academic Program Development of Jiangsu Higher Education Institutions, is gratefully acknowledged.

■

REFERENCES

(1) Tanaka, H. Viscoelastic Phase Separation. J. Phys.: Condens. Matter 2000, 12, R207−R264. (2) Onuki, A. Phase Transitions of Fluids in Shear Flow. J. Phys.: Condens. Matter 1997, 9, 6119−6157. (3) Wolf, B. A. Thermodynamic Theory of Flowing Polymer Solutions and its Application to Phase Separation. Macromolecules 1984, 17, 615−618. (4) Batra, A.; Hedden, R. C.; Schofield, P.; Barnes, A.; Cohen, C.; Duncan, T. M. Conformational Behavior of Guest Chains in Uniaxially Stretched Poly(diethylsiloxane) Elastomers: 2H NMR and SANS. Macromolecules 2003, 36, 9458−9466. (5) Deloche, B.; Samulski, T. Short-Range Nematic-like Orientational Order in Strained Elastomers: A Deuterium Magnetic Resonance Study. Macromolecules 1981, 14, 575−581. (6) Lei, Q.-L.; Feng, J.-W.; Ding, H.-M.; Ren, C.-L.; Ma, Y.-Q. Modeling Stretching-Induced Immiscibility in Nonmonodisperse Polymer Systems. ACS Macro Lett. 2015, 4, 1033−1038. (7) Hashimoto, T.; Murase, H.; Ohta, Y. A New Scenario of FlowInduced Shish-Kebab Formation in Entangled Polymer Solutions. Macromolecules 2010, 43, 6542−6548. (8) Keller, A. Polymer Crystals. Rep. Prog. Phys. 1968, 31, 623−704. (9) Zhou, Q.; Liu, F.; Guo, C.; Fu, Q.; Shen, K.; Zhang, J. ShishKebab-like Cylindrulite Structures Resulted from Periodical ShearInduced Crystallization of Isotactic Polypropylene. Polymer 2011, 52, 2970−2978. (10) Keller, A. Unusual Orientation Phenomena in Polyethylene Interpreted in Terms of the Morphology. J. Polym. Sci. 1955, 15, 31− 49. (11) Keller, A.; Machin, M. J. Oriented Crystallization in Polymers. J. Macromol. Sci., Part B: Phys. 1967, 1, 41−91. (12) Keller, A.; Willmouth, F. M. On the Morphology and Origin of the Fibers Observed in Nascent Ziegler Polyethylene. Makromol. Chem. 1969, 121, 42−50. (13) Seki, M.; Thurman, D. W.; Oberhauser, J. P.; Kornfield, J. A. Shear-Mediated Crystallization of Isotactic Polypropylene. The Role of Long Chain-Long Chain Overlap. Macromolecules 2002, 35, 2583− 2594. (14) Somani, R. H.; Yang, L.; Hsiao, B. S.; Agarwal, P. K.; Fruitwala, H. A.; Tsou, A. H. Shear-Induced Precursor Structures in Isotactic Polypropylene Melt by in-situ Rheo-SAXS and Rheo-WAXD Studies. Macromolecules 2002, 35, 9096−9104. (15) Kimata, S.; Sakurai, T.; Nozue, Y.; Kasahara, T.; Yamaguchi, N.; Karino, T.; Shibayama, M.; Kornfield, J. A. Molecular Basis of the Shish-Kebab Morphology in Polymer Crystallization. Science 2007, 316, 1014−1017. (16) Zhao, B.; Li, X.; Huang, Y.; Li, L. Inducing Crystallization of Polymer through Stretched Network. Macromolecules 2009, 42, 1428− 1432. (17) Nie, Y.; Gao, H.; Yu, M.; Hu, Z.; Reiter, G.; Hu, W. Competition of Crystal Nucleation to Fabricate the Oriented SemiCrystalline Polymers. Polymer 2013, 54, 3402−3407. (18) Nie, Y.; Gao, H.; Hu, W. Variable Trends of Chain-Folding in Separate Stages of Strain-Induced Crystallization of Bulk Polymers. Polymer 2014, 55, 1267−1272. (19) Nie, Y.; Gao, H.; Wu, Y.; Hu, W. Thermodynamics of StrainInduced Crystallization of Random Copolymers. Soft Matter 2014, 10, 343−347. (20) Nie, Y.; Gao, H.; Wu, Y.; Hu, W. Effect of Comonomer Sizes on the Strain-Induced Crystal Nucleation of Random Copolymers. Eur. Polym. J. 2016, 81, 34−42. 12992

DOI: 10.1021/acs.jpcb.6b08399 J. Phys. Chem. B 2016, 120, 12988−12992