Entropic Network Model for Star Block Copolymer Thermoplastic

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

pubs.acs.org/Macromolecules

Entropic Network Model for Star Block Copolymer Thermoplastic Elastomers Amanda J. Parker*,†,‡,§ and Jörg Rottler†,‡ †

Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada § DATA61 CSIRO, Door 34 Goods Shed, Village St., Docklands, VIC 3008, Australia

Macromolecules Downloaded from pubs.acs.org by UNIV OF WINNIPEG on 11/30/18. For personal use only.

‡

ABSTRACT: Uniaxial tensile deformation of star block polymer elastomers with up to 10 arms is studied with molecular dynamics simulations of a bead−spring representation at two different deformation rates. The mechanical response is rationalized in terms of an entropic network model, which we generalize from linear to star polymers. The model universally accounts for the stress in the entire deformation range well into the nonlinear regime. As the strain rate is increased, we find additional hardening as the number of arms increases, which can be attributed to the breakup of physical cross-links.

■

INTRODUCTION In contrast to vulcanized rubber, thermoplastic elastomers (TPEs) are characterized by the presence of physical rather than chemical cross-links. In linear ABA triblock copolymers, this strategy is realized by end-blocks self-assembling into spherical or cylindrical microdomains. When their glass transition temperature lies above the operation temperature, they anchor the chain ends, thus effectively cross-linking a rubbery midblock.1−5 With the advent of advanced synthesis techniques,6 interest in block copolymer materials with a branched structure has increased tremendously.7−9 (AB)M star polymers have M arms radiating from a central point, and there is a growing body of experimental works which find that TPEs based on a symmetric star topology are stronger than linear triblock elastomers.10−16 By use of asymmetrical miktoarm block copolymers, the fraction of polymers in the hard glassy phase can be raised, thus leading to larger elastic moduli, strength, and toughness.17 Theoretical predictions of the mechanical response of polymer networks have so far mostly focused on permanently cross-linked rubbers. All models attempt, in different ways, to capture the entropic free energy penalty as the polymer chains are being stretched between entanglement and cross-link points. A recently introduced nonaffine network model18 fits experimental data for vulcanized rubber well into the large strain regime. It combines the nonaffine tube19 and the Arruda−Boyce 8-chain20 models so that the former becomes applicable for larger strains. In all these models, approximations and assumptions must be made regarding the microscopic chain configurations, requiring the introduction of fit parameters. Molecular simulations of the mechanical response of elastomeric copolymers have been limited to linear ABA triblocks forming lamellae21,22 or spheres23 and not yet considered branched topologies. © XXXX American Chemical Society

We have recently introduced a variant of an entropic network model, in which the evolution of chain configurations and entanglement points is computed with molecular dynamics simulations of a bead−spring model and then used in the expressions for the strain energy density.24 This combined approach provides an excellent prediction of the stress−strain response of linear sphere-forming ABA triblock TPEs well into the nonlinear regime. In the present contribution, we extend the applicability of our model to symmetric star polymers with M = 2, 3, 5, and 10. To this end, we first generate equilibrated nanostructured morphologies using simulations with soft potentials that allow chain crossings.25 We then compute the variation of bridging ratios in star copolymers that form spherical glassy regions and compare to results from Spencer and Matsen obtained via self-consistent field theory (SCFT).26 We study uniaxial deformation in terms of stretch λ, obtain stress−stretch curves in simulations, and show that they can be accounted for well with our entropic network model for all M. When deformation and breakup of the glassy inclusions become important, our simulations predict a further stiffening of the response of star polymers with increasing number of arms.

■

SIMULATION METHODOLOGY AND POLYMER TOPOLOGIES We use a molecular dynamics approach with a typical bead− spring representation of polymers.27 Lennard-Jones (LJ) pair potentials act between all beads of the same diameter within a cutoff distance rc = 1.5, constant for all monomer species. To drive the phase separation, the energy scale ϵ of the LJ potential is varied, ϵAB = 0.2, ϵAA = 1.0, and ϵBB = 0.5, to give a Received: July 1, 2018 Revised: October 12, 2018

A

DOI: 10.1021/acs.macromol.8b01348 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

reveals a spherical structure for a 3-arm star polymer simulation, while Figure 1b depicts a single star polymer and defines the end-to-end distance REE and the center-to-end distance RCE. To quantitatively analyze any variation in phaseseparated morphology, we calculate the static structure factor

glass transition temperature that differs by a factor of 2. A nonlinear elastic spring potential acts along the backbone of the chains. Deformations are performed at reduced temperature T = 0.29 in between the glass transition temperatures of the hard and soft phases. Equilibration at high temperature is required to achieve phase-separated regions and relaxed chain networks. The time scale for the equilibration of star polymers is even more challenging than for reptating linear chains, as the arms essentially need to retract to the center point of the star to explore a new path. As a result, center-of-mass diffusion is very slow.28 We therefore implement a coarse-grained equilibration step, which has been successful in the equilibration of entangled linear triblock chains.25 The LJ pair potential is temporarily replaced with a softer potential that allows for chain crossing and quicker equilibration by virtue of the resulting Rouse dynamics. After chain conformations have relaxed, LJ interactions are reintroduced with parameters that match the overall thermodynamics. Here we use the same parameters determined in ref 24 for linear chains, as direct equilibration of star polymers even at short chain length becomes prohibitive. We consider (AB)M stars, with M arms each of 300 monomers in length and an outer glassy part of 10%. We equilibrate stars with 3−10 arms and a linear triblock (M = 2) for comparison. Each system contains a total of 480000 monomers (1600 star arms). We first consider the phase-separated morphologies of the glassy and soft regions of the equilibrated structures. Figure 1a

SAA(q) =

1 ⟨ NA

NA

2

∑ eiq ⃗·ri⃗ ⟩ i=1

(1)

where the sum runs over the number of glassy monomers NA. Figure 1c demonstrates that in our model there is no variation in the structure factor for stars with 3−10 arms or in comparison to the linear triblock. This trend agrees qualitatively with recent experiments that report unchanged morphologies between linear and M = 6 star polymers forming cylinders.16 Note that new equilibrium phases could still emerge at a larger number of arms, as has been observed previously.29 This result provides an excellent opportunity to use star polymers to study diverse network topologies with a consistent phase-separated morphology. We can thereby isolate the effects of changing bridging ratios from changing morphology. The network structure of a linear triblock TPE can be summarized by considering the proportion of chains with both ends embedded in the same glassy region (looping chains) and those with the chain forming a link between two glassy regions (bridging chains). For the present model, the proportion of bridging chains is 88% for the M = 2 linear triblock. We analyze the more complex bridging statistics of star polymers by two metrics. First, we compute the probability p(x) that a glassy cluster containing one arm from a star will contain a total of x arms from that star (see Figure 2a). We find that the probability of there being a single arm in each cluster drops from 88% for the linear triblock (the percentage of bridging chains) to 60% for M = 5 stars. We can directly compare this measure obtained in molecular dynamics to the self-consistent field theory (SCFT) results of Spencer and Matsen.26 We find that for M = 3 the probabilities for (1, 2, 3) arms in each cluster are 75%, 21%, and 3% compared to their 64%, 30%, and 5%. This is very good agreement considering the two completely different equilibration methods. We find a greater p(1) (75% compared to 64%) and a steeper drop-off in probability. M = 5 stars also show good agreement to these SCFT results, though again p(1) is higher in our case (60% compared to 48%). We also consider the probability p(y) that a star’s arms will be in y different glassy clusters (see Figure 2b). This indicates that for a small number of arms (M = 2, 3) the most likely configurations are with all arms in separate clusters. However, stars with a larger number of arms (M = 5, 10) reach a peak in probability at y = 4, 7 clusters. The values for p(y) = 1 give the probability of all chains in a star looping back to the same cluster. This drops dramatically from 12% looping chains in the linear triblock to 3% for 3-arm stars. Therefore, even a star with 5 arms changes the topology of our network so that there are no longer any completely looping chains. Given this variation in the bridging statistics as a result of increasing the number of star arms, we now compare the corresponding stress responses for these star polymers in uniaxial deformation.

Figure 1. (a) Phase-separated morphology with minority monomers in blue and majority monomers in red. Only minority monomers are shown on the right half of the cube. (b) Cartoon representation of a 3-arm star polymer defining the end-to-end distance REE and the center-to-end distance RCE. (c) Static structure factor for star polymers with varying numbers of arms M: (green ×) linear triblock (M = 2); (teal □) M = 3; (blue ○) M = 5; (red △) M = 10. B


Article

Macromolecules

stretch λk. We use the Arruda−Boyce 8-chain model20 to write each of these stretches in terms of Cartesian components, i.e. λc =

1 (λcx 2 + λcy 2 + λcz 2) 3

thus building a model that is based on the behavior of chains directly rather than on the applied global stretch. For linear polymers described in terms of monomer number density ρ, entanglement length Ne, number of entanglement points Nk, and the chain length N = NkNe, we showed in ref 24 that the stress invariant can be expressed in terms of the stretch components between entanglement points λki and cross-link points λci as ÄÅ −1 ÉÑ −1 ÅÅ 3 (h ) ÑÑ 1 3 (hc) k σz − σx = GeÅÅÅÅ (λkz 2 − λkx 2) + (λcz 2 − λcx 2)ÑÑÑÑ ÅÅ 3hk ÑÑ Nk 3hc ÅÇ ÑÖ

(2)

where hc = REE/Nb and hk = Rk/Neb. Here REE and Rk denote the distances between chain ends and entanglement points, and b is the bond length. The inverse Langevin function 3−1 results from considering the entropy of chains with finite extensibility constraints. When we consider adapting this model for application to star polymers there are two important considerations: the calculation of REE and the treatment of the star centers as entanglements or cross-links. The network functionality ϕ averages the number of chains which meet at each cross-link. An extension to affine network models assumes chains are not perfectly coupled to the global deformation but are connected via a phantom network of chains. This leads to a phantom network correction factor for the cross-link stress contribution dependent on the network functionality of cross-links (1 − 2/ ϕ). However, ϕ for the triblock copolymer TPEs is very high in comparison to the typical ϕ = 4 assumed for vulcanized rubbers. In our simulations, ∼50 chain ends are cross-linked by each glassy region. This network structure negates the impact of any phantom network correction factor (1 − 2/ϕ) for the cross-link stress contribution, and consequently this factor is not included in the model above. By the same observation, the network functionality of the star centers (ϕ = 2 → 10) is far closer to the entanglements in this system (ϕ ≥ 4) than to a cross-link (ϕ ≈ 50). Therefore, our approach is to consider the star polymer networks as a combination of linear chain contributions (with the center defined as an entanglement point) rather than constructing a model with a third additive stress contribution. REE could be calculated by averaging the distances between the M chain ends in a given star. However, in entropic network models that treat REE as an indication of the chain statistics, REE2 ∝ Nb2 is expected as the initial state of the network. In the case of looping chains, REE is a very poor representation of the chain statistics. Looping chain ends are very close together, and in any regime with deformation of the glassy regions this separation will increase dramatically. Given the number of looping chains varies with M, we instead define the center-toend distance RCE (see Figure 1b) as representative of the chain statistics. Any sufficiently large chain segments could be substituted for REE by this approach. Taking the center-to-end distance is a choice dictated by convenience and has no equivalence to defining the center as a new type of cross-linked point. By adjustment of the model for this redefinition with the

Figure 2. (a) Probability that a glassy cluster containing one arm from a star will contain a total of x arms from that star. (b) Probability that a star’s arms will be in x different glassy clusters. Star polymers with varying numbers of arms M: (green ×) linear triblock (M = 2); (teal □) M = 3; (blue ○) M = 5; (red △) M = 10. Filled symbols are data reproduced from Figure 4 in ref 26: (■) M = 3; (●) M = 5.

■

ENTROPIC NETWORK MODEL AS APPLIED TO STAR POLYMERS Typically, entropic network models focus on relating experimentally measurable quantities to a model and backing out the microscopic parameters through fitting. This involves assuming some relationship between macroscopic deformation and deformation of chain entanglements to ultimately calculate the resulting stress. However, through simulation, we have access to the details of the microscopic scale stretches and can decouple these two effects. Entropic network models of rubbers usually have a form with additive stress contributions due to entanglements and cross-links and are a function of the global applied deformation. The two elastic moduli Ge (entanglements) and Gc (cross-links) are then fitted to experimental or simulation data. However, the moduli are related, and we can avoid fitting both Ge = ρkBT /Ne , Gc = ρkBT /N = Ge /Nk where Ne is the entanglement length and Nk denotes the number of entanglement points or kinks. Our recently introduced model for linear triblocks takes advantage of this simplification.24 The model considers stretches on two scales in the system: chain end-to-end stretch λc and entanglement C


Article

Macromolecules mappings REE → RCE, N → N/2, Ne → Ne, and Nk → Nk/2, the final form of eq 2 remains the same for star polymers.

while a topological minimization is applied to all chains simultaneously. Kinks in the resulting shortest minimally connected path identify entanglement points. Figure 4 shows

■

RESULTS To test this model, we perform volume conserving uniaxial strain simulations at two fixed strain rates ε̇zz = 10−4τ−1 and ε̇zz = 10−5τ−1. The resulting stress response as a function of stretch is shown in Figure 3 and is qualitatively similar for both strain

Figure 4. Evolution of the entanglement length Ne during volume conserving uniaxial strain deformation with strain rates (a) 10−4τ−1 and (b) 10−5τ−1: (green ×) linear triblock (M = 2); (teal □) M = 3; (blue ○) M = 5; (red △) M = 10.

how the entanglement length Ne increases from approximately 40 to 60 with increasing applied strain. Most notably, M has no effect on the evolution of these entanglement statistics. This trend is consistent across both strain rates, although the overall entanglement loss is smaller for the slower deformation. Variation in Ne can therefore be discounted as an explanation for the enhanced strain hardening with increasing M. However, since there is still a significant change of Ne during deformation, we do not treat Ne as constant but rather identify the entanglement points at every stretch analyzed. Entanglement stretch components in the Cartesian directions λkx, λky, and λkz are therefore calculated as RMS values. For application of the entropic network model we must also track the crosslink stretch components λcx, λcy, and λcz defined in terms of the center-to-end distances. Figure 5 demonstrates the success and limitations of this network model when applied to star polymers. The simulated stress is compared to the model stress calculated from eq 2. In this representation, a linear relationship indicates agreement with the model, and the modulus Ge can then be read off as the slope. We first observe that the curves collapse for all M, indicating that the model accounts for the effects of varying star functionality. For the lower strain rate 10−5τ−1 shown in

Figure 3. Volume conserving uniaxial strain deformation for star polymers with strain rates (a) 10−4τ−1 and (b) 10−5τ−1 and varying numbers of arms M: (green ×) linear triblock (M = 2); (teal □) M = 3; (blue ○) M = 5; (red △) M = 10. The stress is averaged over 10 LJ time units.

rates. For small extensions λ < 3, we see no dependence of the tensile stress on the number of star arms. However, further into the nonlinear stress regime we find enhanced strain hardening with increasing M. The stresses of the star polymers with the higher M start to rise at lower strains. Our molecular simulation qualitatively reproduces experimental trends, where the stress responses are consistent for small strains, but the 6-arm stars display stronger strain hardening than linear copolymers.16 Deformation at the lower strain rate ε̇zz = 10−5τ−1 leads to smaller stresses as viscous effects are greatly reduced. What then is the microscopic mechanism of the enhanced strain hardening with increasing M? We first consider the behavior of the network of entanglements between chains. To identify entanglement points as the applied strain is increased, chain conformations are analyzed using the Krö ger Z1 method.30 For each snapshot, chain end-points are anchored D


Article

Macromolecules

Figure 6. (a) Asphericity and (b) number of majority clusters scaled by the initial number of clusters for volume conserving uniaxial strain deformation with strain rate 10−4τ−1. Star polymers with varying numbers of arms: (green ×) linear triblock (M = 2); (teal □) M = 3; (blue ○) M = 5; (red △) M = 10. Colored vertical lines indicate the strain at which the number of clusters is minimal for that M.

Figure 5. Model collapse using chain center-to-end distance during volume conserving uniaxial strain deformation with strain rates (a) 10−4τ−1 and (b) 10−5τ−1 of star polymers: (green ×) linear triblock (M = 2); (teal □) M = 3; (blue ○) M = 5; (red △) M = 10. Black lines indicate (a) Ge = 0.013 and (b) Ge = 0.006. Colored vertical lines in (a) indicate the snapshot with the smallest number of clusters for that M (see Figure 6b).

= 5.5, 6.5, and 7 for M = 10, 5, and 3, respectively. The vertical lines of corresponding colors indicate these minima in Figures 5 and 6. We can therefore see that the breakup of glassy regions corresponds to the onset on enhanced strain hardening for varied M. The entropic network model does not account for deformation or breakup of the glassy clusters, and the nonlinearity in Figure 5 corresponds to increasing asphericity. The deformation behavior of the glassy inclusions is notably different at the slower strain rate 10−5τ−1, as evidenced in Figure 7. At this deformation rate, the asphericity (panel a) rises to only about 1/3 of the values seen at the faster deformation rate. Similarly, the number of clusters increases hardly at all (see panel b), which indicates that most glassy inclusions remain intact. This observation explains the much greater range of validity of the network model found in Figure 5b.

panel b, we find overall good agreement with a linear trend over the entire deformation range, while for the higher strain rate 10−4τ−1, some deviations become apparent: At higher strains the curves become nonlinear, indicating a hardening effect unaccounted for in the entropic network model. We have not included in our model a stress contribution from the deformation of the glassy regions themselves. We do, however, account for how RCE changes as a result of triblock midblock chains being “attached” to the glassy regions. In Figure 5a, the M = 10 star stress response deviates from the linear triblock curve at the smallest strain, followed by M = 5 and M = 3. Figure 6 describes the deformation of the glassy regions and offers us an explanation for this variation in hardening response. Figure 6a plots the average asphericity 1 A = λ1 − 2 (λ 2 + λ3) of the glassy regions (clusters) with increasing applied stretch, which is calculated from the ordered eigenvalues λi of the mean radius of gyration tensor for each glassy region.23 These curves indicate substantial plastic deformation of the domains for extensions λ > 5. Figure 6b shows the number of these regions normalized by their number in the undeformed sample. The numbers decrease initially as adjacent regions come into contact. Then, as the asphericity increases and the clusters are increasingly deformed, they start to divide, except in the linear triblock case where this point is not reached. The minima in the number of clusters occur at λ

■

CONCLUSIONS Through application of an entropic network model, we have added to the understanding of variation in TPE material properties with an increasing number of star arms M. We observed that well into the strain hardening regime (λ ≤ 5) there is no difference in stress response between linear triblocks and stars with M ≤ 10. Our bead−spring polymer simulations also reproduce enhanced strain hardening with increasing M due to an increasing number of star arms at larger deformations (λ > 5). In this regime, it is important to also consider deformation rate effects. For our slow strain rate of 10−5τ−1, the tensile stresses are small enough to avoid breakup E


Article

Macromolecules ORCID

Amanda J. Parker: 0000-0003-2207-744X Jörg Rottler: 0000-0002-5273-7480 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Discovery Grant program of the Natural Sciences and Engineering Research Council of Canada.

■

(1) Holden, G.; Bishop, E.; Legge, N. R. Thermoplastic elastomers. J. Polym. Sci., Part C: Polym. Symp. 1969, 26, 37−57. (2) Chen, Y.-D.; Cohen, R. The influence of molecular weight on the large deformation behavior of SBS triblock copolymer elastomers. J. Appl. Polym. Sci. 1977, 21, 629−643. (3) Pakula, T.; Saijo, K.; Kawai, H.; Hashimoto, T. Deformation behavior of styrene-butadiene-styrene triblock copolymer with cylindrical morphology. Macromolecules 1985, 18, 1294−1302. (4) Kennedy, J. P.; Price, J. L.; Koshimura, K. Novel thermoplastic elastomer triblocks of a soft polyisobutylene midblock connected to two hard PMMA stereocomplex outer blocks. Macromolecules 1991, 24, 6567−6571. (5) Roos, A.; Creton, C. Effect of the presence of diblock copolymer on the nonlinear elastic and viscoelastic properties of elastomeric triblock copolymers. Macromolecules 2005, 38, 7807−7818. (6) Hadjichristidis, N.; Pispas, S.; Floudas, G. Block Copolymers: Synthetic Strategies, Physical Properties, and Applications; John Wiley & Sons: 2003. (7) Guo, A.; Liu, G.; Tao, J. Star Polymers and Nanospheres from Cross-Linkable Diblock Copolymers. Macromolecules 1996, 29, 2487−2493. (8) Bates, F. S.; Hillmyer, M. A.; Lodge, T. P.; Bates, C. M.; Delaney, K. T.; Fredrickson, G. H. Multiblock polymers: panacea or pandoras box? Science 2012, 336, 434−440. (9) Burns, A. B.; Register, R. A. Strategies for the Synthesis of WellDefined Star Polymers by Anionic Polymerization with Chlorosilane Coupling and Preservation of the Star Architecture during Catalytic Hydrogenation. Macromolecules 2016, 49, 2063−2070. (10) Puskas, J.; Antony, P.; el Fray, M.; Altstädt, V. The effect of hard and soft segment composition and molecular architecture on the morphology and mechanical properties of polystyrenepolyisobutylene thermoplastic elastomeric block copolymers. Eur. Polym. J. 2003, 39, 2041−2049. (11) Kwee, T.; Taylor, S. J.; Mauritz, K. A.; Storey, R. F. Morphology and mechanical and dynamic mechanical properties of linear and star poly(styrene-b-isobutylene-b-styrene) block copolymers. Polymer 2005, 46, 4480−4491. (12) Jeusette, M.; Leclère, P.; Lazzaroni, R.; Simal, F.; Vaneecke, J.; Lardot, T.; Roose, P. New All-Acrylate Block Copolymers:? Synthesis and Influence of the Architecture on the Morphology and the Mechanical Properties. Macromolecules 2007, 40, 1055−1065. (13) Nese, A.; Mosnácĕ k, J.; Juhari, A.; Yoon, J. A.; Koynov, K.; Kowalewski, T.; Matyjaszewski, K. Synthesis, Characterization, and Properties of Starlike Poly(n-butyl acrylate)-b-poly(methyl methacrylate) Block Copolymers. Macromolecules 2010, 43, 1227−1235. (14) Pakula, T.; Koynov, K.; Boerner, H.; Huang, J.; Lee, H.-i.; Pietrasik, J.; Sumerlin, B.; Matyjaszewski, K. Effect of chain topology on the self-organization and the mechanical properties of poly(n-butyl acrylate)-b-polystyrene block copolymers. Polymer 2011, 52, 2576− 2583. (15) Kong, J. F.; Lipik, V.; Abadie, M. J. M.; Deen, G. R.; Venkatraman, S. S. Characterization and degradation of elastomeric fourarmed star copolymers based on caprolactone and L-lactide. J. Biomed. Mater. Res., Part A 2012, 100A, 3436−3445.

Figure 7. (a) Asphericity and (b) number of majority clusters scaled by the initial number of clusters for volume conserving uniaxial strain deformation with strain rate 10−5τ−1. Star polymers with varying numbers of arms: (green ×) linear triblock (M = 2); (teal □) M = 3; (blue ○) M = 5; (red △) M = 10.

of the glassy domains. The entropic network model is applicable up to very large stretches and fully accounts for the variation with M and nonlinear strain hardening. The good agreement with the model trend shown in Figure 5b is strong evidence that all relevant deformation mechanisms that contribute to the stress have been accounted for in eq 2. We also gain insight into material behavior beyond this regime, in particular into the microscopic mechanisms that become active when the model begins to fail. For deformation at the faster strain rate 10−4τ−1, we find that the network structure does impact the deformation, and glassy regions become strongly deformed and break up. Stars with more arms have a higher proportion of bridging chains, and this corresponds to deformation and breakup of glassy regions at smaller strains. This in turn corresponds to an additional mechanism that contributes to enhanced strain hardening of star polymers. Still, the entropic network model does agree with the smaller deformation range where the chain ends are still anchored in the glassy cross-links. Our study confirms that to take advantage of varying material properties with the design of star polymers it is important to consider the response of the glassy (typically styrenic) regions. Differing stress response arises when these physical cross-links can no longer be considered unbreakable, which in turn can be sensitive to the applied deformation rate.

■

REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. F


Article

Macromolecules (16) Burns, A. B.; Register, R. A. Mechanical Properties of Star Block Polymer Thermoplastic Elastomers with Glassy and Crystalline End Blocks. Macromolecules 2016, 49, 9521−9530. (17) Shi, W.; Lynd, N. A.; Montarnal, D.; Luo, Y.; Fredrickson, G. H.; Kramer, E. J.; Ntaras, C.; Avgeropoulos, A.; Hexemer, A. Toward Strong Thermoplastic Elastomers with Asymmetric Miktoarm Block Copolymer Architectures. Macromolecules 2014, 47, 2037−2043. (18) Davidson, J. D.; Goulbourne, N. A nonaffine network model for elastomers undergoing finite deformations. J. Mech. Phys. Solids 2013, 61, 1784−1797. (19) Rubinstein, M.; Panyukov, S. Elasticity of polymer networks. Macromolecules 2002, 35, 6670−6686. (20) Arruda, E. M.; Boyce, M. C. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 1993, 41, 389−412. (21) Aoyagi, T.; Honda, T.; Doi, M. Microstructural study of mechanical properties of the ABA triblock copolymer using selfconsistent field and molecular dynamics. J. Chem. Phys. 2002, 117, 8153−8161. (22) Makke, A.; Lame, O.; Perez, M.; Barrat, J.-L. Influence of tie and loop molecules on the mechanical properties of lamellar block copolymers. Macromolecules 2012, 45, 8445−8452. (23) Parker, A. J.; Rottler, J. Molecular mechanisms of plastic deformation in sphere-forming thermoplastic elastomers. Macromolecules 2015, 48, 8253−8261. (24) Parker, A. J.; Rottler, J. Nonlinear Mechanics of Triblock Copolymer Elastomers: From Molecular Simulations to Network Models. ACS Macro Lett. 2017, 6, 786−790. (25) Parker, A. J.; Rottler, J. Using soft potentials for the simulation of block copolymer morphologies. Macromol. Theory Simul. 2014, 23, 401−409. (26) Spencer, R. K.; Matsen, M. W. Domain Bridging in Thermoplastic Elastomers of Star Block Copolymer. Macromolecules 2017, 50, 1681−1687. (27) Auhl, R.; Everaers, R.; Grest, G. S.; Kremer, K.; Plimpton, S. J. Equilibration of long chain polymer melts in computer simulations. J. Chem. Phys. 2003, 119, 12718−12728. (28) De Gennes, P. Reptation of stars. J. Phys. (Paris) 1975, 36, 1199−1203. (29) Alward, D. B.; Kinning, D. J.; Thomas, E. L.; Fetters, L. J. Effect of arm number and arm molecular weight on the solid-state morphology of poly(styrene-isoprene) star block copolymers. Macromolecules 1986, 19, 215−224. (30) Kröger, M. Shortest multiple disconnected path for the analysis of entanglements in two-and three-dimensional polymeric systems. Comput. Phys. Commun. 2005, 168, 209−232.

G


Entropic Network Model for Star Block Copolymer Thermoplastic

Recommend Documents