Molecular Mechanisms of Plastic Deformation in Sphere-Forming

Article pubs.acs.org/Macromolecules

Molecular Mechanisms of Plastic Deformation in Sphere-Forming 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 ABSTRACT: Plastic deformation of sphere-forming triblock thermoplastic elastomers is studied with molecular dynamics simulations in order to elucidate microscopic mechanisms operative in nanostructured macromolecular materials. Phase separation of linear triblock copolymers is achieved by first equilibrating a melt using soft interactions that are subsequently replaced by a standard bead−spring model to obtain mechanical properties. We compare two deformation modes of both uniaxial stress and strain and vary the polymer chain length from unentangled to moderately entangled chains. Our simulations show that triblocks exhibit significant increase of strain hardening compared to homopolymer elastomers. We analyze several properties such as global chain deformation and local monomer motion, number and shape of spherical domains, and the evolution of the fraction of chains bridging between domains. Results confirm the notion of improved mechanical properties through effectively cross-linking chain ends. Void nucleation at different stages of deformation is observed to occur either at the interface between glassy and rubbery phases or immediately following the breakup of glassy domains and is therefore intimately related to the elastic heterogeneity of the material.

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INTRODUCTION Nanostructured copolymers display complex morphologies which can be exploited to produce materials with enhanced properties.1 The phase diagram of these systems is determined by chain length, the strength of the phase segregation, and the proportion of each type of monomer. The morphologies vary from spherical clusters centered on a body-centered cubic lattice, to cylindrical columns on a square lattice, to gyroidal and lamellar phases.2,3 When in the sphere-forming regime, chain ends are confined to hard glassy minority regions, allowing a rubbery phase to be effectively cross-linked. Normal entanglements in the soft component act as effective cross-links because the ends are securely anchored.4 This forms a thermoplastic elastomer (TPE) combining desirable material properties of rubbers and plastics. Here we focus on linear symmetric ABA triblock copolymers, where the hard component is polystyrene while the soft region is most commonly polybutadiene (SBS), polyisoprene (SIS), or more recently polyisobutylene (SIBS).5 Experimental results show highly nonlinear stress−strain response in styrenic thermoplastic elastomers. Early studies by Holden et al. fit experimental data for SIS and SBS triblocks with a constitutive equation that accounts for the effects of rigid fillers and entanglements.4 This treats the glassy spherical inclusions as hard discrete particles and describes the data up to an extension ratio of 290%, while material failure occurs at much greater elongation of 1000%. At these larger extensions the tensile strength of the triblock copolymers is much greater than vulcanized rubbers. In keeping with this assumption, Prasman and Thomas find that in an SIS/mineral oil blend © XXXX American Chemical Society

deformation is affine up to strains of 300%, and PS domains keep their shape under extension.6 However, other studies find that the microscopic structure can be altered during deformation by chain pullout at temperatures below the glass transition temperature of polystyrene,7 while hysteresis8 and cavitation9,10 were also observed. The proportion of chains bridging between glassy regions controls the structure of the network. This proportion can be varied by introducing diblocks11−13 or cyclic copolymers14,15 to the triblock system and has been shown to affect the macroscopic stress−strain response. On the whole, it is difficult to get a comprehensive picture of how the microstructure of these TPEs determines the macroscopic stress response experimentally. For instance, it is important to understand changes to the morphology during extension to large strains and their dependence on molecular and deformation parameters. We approach this question using molecular dynamics simulations. Simulating block copolymers is a multiscale problem. Obtaining phase-separated morphologies with equilibrated chain conformations presents a set of unique challenges. Chains cannot cross through each other due to hard excluded volume interactions so the time scale of the system is determined by reptation dynamics with a disentanglement time that scales cubically with chain length.16 Additionally, if we start with a random distribution of chains, formation of phaseseparated regions requires matter to move on the length scale Received: June 19, 2015 Revised: October 19, 2015

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DOI: 10.1021/acs.macromol.5b01339 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules of these domains. Here we use an approach similar to that currently pursued by several research groups,17−19 which not only retains the notion of discrete chains but also allows for thermodynamically driven phase segregation. The equilibrium morphology is achieved by considering soft interactions that allow chains to cross through each other, while hard sphere monomer interactions are restored for subsequent deformation studies. A few molecular dynamics simulations have studied deformation of phase-separated triblock copolymers so far, but these have relied on building copolymer systems with an imposed morphology. This requires knowledge of a predetermined arrangement of the phase-separated regions. For instance, in the works of Leonforte20 and Makke et al.21,22 samples were constructed by growing individual chains in a fixed lamellar geometry using biased random walk techniques (radical like polymerization).23 These authors varied the fraction of triblocks bridging between different glassy lamellae and explored their role for yield and strain hardening in uniaxial tensile deformation. While decreasing the amount of bridging molecules leads to a rapid reduction of strain hardening, replacing them with loop molecules that return to the same lamella was found to hardly alter the mechanical response. These simulations also show that the lamellar phase exhibits buckling upon yielding and develops a chevron-like structure21 due to the spatially varying elastic properties of the layered material. Deformation of sphere-forming short triblock copolymers was also studied by Aoyagi et al.;24 however, these authors generated a spherical morphology first using selfconsistent field (SCF) simulations before performing molecular simulations with bead−spring chains. Uniaxial elongation of short triblocks shows failure at 350% strain where the minority phase domains break up.24 In an approach similar to ours, Chantawansri et al.25 simulated a dilute gel of physically crosslinked triblock copolymers under uniaxial tension by coupling polymer models with and without chain crossing,26 and determined the chain length dependence of the modulus in accordance with predictions from rubber elasticity models. Other aspects of the plastic deformation of filled elastomers, such as the reinforcement with increasing filler fraction, have also been reproduced with simpler models that consider hard spherical inclusions in a filler matrix modeled implicitly by harmonic springs.27 In this work we present molecular dynamics simulations of coarse-grained sphere forming triblock copolymer systems under uniaxial deformation. We consider both unentangled and moderately entangled chains and compare their mechanical response to homopolymers of the same length for engineering strains of up to 800%. The following section presents the coarse-grained bead-spring model along with our technique for equilibrating initial morphologies and details of the two uniaxial strain deformations. We then discuss the macroscopic stress− strain curves and deformation scenarios, folllowed by analysis of the response of the chains, monomer beads, and phaseseparated regions to deformation. We also investigate density fluctuations that lead to caviation in each deformation mode and correlate the location to the surrounding microstructure.

Neighboring beads along a chain interact with a nonlinear spring potential ⎡ ⎛ r ⎞2 ⎤ 1 2 ⎢ VFENE(r ) = − KR 0 ln 1 − ⎜ ⎟ ⎥ ⎢⎣ 2 ⎝ R 0 ⎠ ⎥⎦

(1)

with R0 = 1.5σ and K = 30ϵ/σ2. A 6−12 Lennard-Jones (LJ) pair potential acts additionally between all beads ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ Vij(r ) = 4ϵij⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r⎠ ⎦ ⎣⎝ r ⎠

(2)

which is truncated at rc = 1.5σ and shifted for continuity. The indices i, j run over the different monomer species. We set ϵAA = ϵ and ϵBB = 0.7ϵ so that the minority A-particles from hard spheres in a soft matrix. ϵAB then controls the segregation strength and is fixed at ϵAB = 0.3ϵ. All chains have the same number of beads N, which is varied from short unentangled chains N = 100 to longer weakly entangled chains N = 300 and N = 500.29 The entanglement length of the homopolymer melt has been determined via primitive path analysis (PPA)29 as Ne ≈ 55. We compare the triblocks to homopolymers of the same chain length N and the same interaction strength as the majority B-type monomers of the triblock, set at 0.7ϵ. All values are reported in reduced simulation units, where length is measured in units of σ, stresses in units of ϵ/σ3, and temperatures in units of ϵ/kB. Similarly, all times are measured in units of τ = (mσ2/ϵ)1/2, where m is the mass of a bead. Simulations have been conducted with LAMMPS30 and HOOMD.31,32 The internal chain conformations and nanostructured morphology are first equilibrated using a coarse-graining strategy, following the method outlined in our previous work.33 In brief, chain connectivity is preserved, but the hard excluded volume interactions of the LJ potential are replaced with a purely repulsive soft potential describing the overlap of spherical volumes. As a result, chains can pass through each other allowing for Rouse-like dynamics, which is sufficiently fast to achieve equilibrated triblock morphologies. The purely repulsive soft potential has the form Vij(r ) =

(

3kBT κ0 ± 8πrs 3ρ0

χ0 2

) ⎛⎜2 + ⎝

2 r ⎞⎛ r ⎞ ⎟⎜1 − ⎟ 2rs ⎠⎝ 2rs ⎠

(3)

that is truncated at rc = 2rs. As suggested by Müller,34 we set the size of the soft beads to rs = (3π/4)1/3b = 1.25σ, where b = 0.97σ is the equilibrium bond length given by eq 1. The degree of phase separation is set by χ0 where + (−) applies to unlike (like) particles. The parameter κ0 can be related to the inverse isothermal compressibility κT35 and determines the softness of the beads. The equilibration is performed in the melt at temperature T = 1 with segment density ρ0 = 0.85. To match the segregation strength of the soft model to that of the bead−spring model, we perform a brute force equilibration of short N = 100 chains following the standard method of Auhl et al.36 The segregration strength can be estimated from the radial distribution functions via the expression37

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MODEL AND METHODS We model generic linear symmetric ABA triblock copolymers, where the A-type minority monomers make up 10% of the chain, with a standard coarse-grained bead−spring model.28

χ = ρ0

∫ d3r[gAB(r)VAB(r) − (gAA (r)VAA(r) + gBB(r)VBB(r))/2] (4)

B


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Macromolecules where gij(r) are the radial distribution functions (rdf) of the different monomer species at the end of the equilibration procedure. After equilibration, a short reverse coarse-graining step switches to the bead−spring Lennard-Jones model with minimal disruption to the distribution of covalent bonds. Subsequently, we quench the temperature from 1.0 to 0.3 with quench rate 10−4 at fixed positive pressure. This temperature lies between the glass transition temperature Tg determined for homopolymers consisting of A (Tg = 0.39) and B (Tg = 0.27) type monomers. Values for Tg were estimated from the intersection of linear fits to the temperature dependence of the specific volume during cooling. This results in glassy A-type inclusions, while the B-monomers are still in the rubbery phase. Figure 1 shows equilibrated morphologies before deformation for the N = 100 and N = 500 triblock systems. Hard glassy

averaged over three different replicas for each chain length, and the number of glassy spheres varies by less than 5%. Static structure factors reveal a characteristic spacing of 27 (N = 500)33 but give no indication of long-range order. We perform two uniaxial deformations, pure strain and pure stress, both with a constant engineering strain rate ε̇zz = 10−4 in the z-direction. In the pure uniaxial strain deformation only the εzz component of the strain tensor is nonzero, and the box dimensions in the two perpendicular directions are unchanged. As a result, εxx = εyy = 0, and the cross-sectional area perpendicular to the deformation is conserved. By contrast, in the pure stress deformation the only nonzero stress component is σzz while σxx and σyy are held at zero using a Nosé−Hoover barostat with barostat time constant 7.5τ. At the same time, the temperature is controlled via a Langevin thermostat with time constant 1τ. The equations of motion are integrated with a velocity-Verlet formulation with a time step of 0.005τ and the streaming velocity of the box resize removed for thermostatting. We simulate large strains up to εzz = 8 to investigate polymeric effects in the strain hardening regime.

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RESULTS We first consider the macroscopic response during deformation before relating back to the evolution of the microstructure. Figure 2 shows the stress−strain curves for each deformation mode. In the pure stress case (Figure 2a) at small strain we observe initially a cavitation peak followed by softening and drawing for both homopolymers and triblocks. The peak stress at cavitation is highest for the homopolymers, where it is also independent of chain length. For both the unentangled N = 100 triblock and homopolymers, the stress tends to zero at large

Figure 1. Snapshots of morphologies before deformation in a box of size L = 80σ. Top: 1600 triblock copolymer chains in a 5−90−5 configuration. Bottom: 960 triblock copolymer chains in 25−450−25 configuration. Minority monomers are in blue and majority monomers in red.

clusters are identified with a cluster algorithm that investigates the local neighborhood of each particle and links all particles that are separated by less than a cutoff of 1.2. The cutoff is chosen to be larger than the nearest-neighbor shell and less than the shortest separated clusters. The shorter the chains, the greater the number Nc of these phase-separated regions, Nc = 318, 76, and 31 for N = 100, 300, and 500, respectively. For all chain lengths the simulation box contains 480 000 monomers so that there are at least 30 phase-separated minority monomer regions to ensure good configurational averaging. Results are

Figure 2. True stress vs engineering strain for (a) pure uniaxial strain and (b) pure uniaxial stress deformation. Solid lines: triblocks. Dashed lines: homopolymers. Chains lengths N are given by the color key. C


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Macromolecules strains, indicating material failure through chain pullout. As the chain length and hence number of entanglements increase, we observe an increasing amount of strain hardening as an upturn in the stress above εzz = 4.5 (εzz = 6) for the N = 500 (N = 300) chains. This trend is enhanced once chain ends are physically cross-linked by the glassy spheres. As a result, the stress−strain curves of the triblock systems lie above those for the corresponding homopolymers. These trends are in qualitative agreement with experiments.8 In the case of pure stress deformation (Figure 2b) the caviation peak is absent as the simulation box can adjust laterally to the applied strain, and both the entangled homopolymers and triblocks display strain hardening. However, this hardening is more dramatic in the triblock system. First, the stress response is steeper at intermediate strains. Second, there is again a further increase of the stress at large strains. This hardening regime is related to the deformation and breakup of hard minority phase-separated regions (see below). The small strain (≤1%) elastic response in the pure stress protocol permits the direct measurement of Young’s modulus Y and the Poisson ratio ν for the different polymers. For the homopolymers we find a ratio YA/YB ≈ 100, while νA = 0.41 and νB = 0.48 independent of chain length. Figures 3 and 4 show several snapshots of the N = 500 triblock system during each deformation mode. In Figure 3 we

Figure 4. Snapshots of the 25−450−25 triblock system during pure uniaxial stress deformation at strains 1, 3, 5, and 7. Minority monomers are in blue and majority monomers in red. Minority monomers are not displayed in the upper portion.

given chain. In order to see whether the chains are being deformed more or less than the simulation box as a whole, we subtract the affine deformation εR ̅ e0, where Re0 is the end-toend vector in the undeformed configuration and ε̅ is the global strain tensor. Figure 5 shows the nonaffine change of Re relative to Re0 in the undeformed material for different deformations. First we observe that in all cases the deformation is more affine in the triblock than in the corresponding homopolymer case. For each chain length there are similar trends for the corresponding triblock and homopolymer. In the pure strain deformations (Figure 5a) none of the systems deform affinely, which is consistent with the large increase in box volume and consequent void formation. For the entangled chains the nonaffine strain of |Re| is positive while for the N = 100 homopolymer it is negative for strains larger than two. At these strains, the untangled elastomer is failing through chain pullout as the stress drops monotonically (see Figure 2a). In the pure stress deformations (Figure 5b) the entangled triblock systems deform affinely as well as the N = 500 homopolymer. In general, the deformation becomes more affine with increasing chain length, which is consistent with the notion of a deforming entanglement network. The presence of glassy inclusions promotes this trend further as chain ends are spatially immobilized. To understand the dynamics of the soft and hard components on a more local level, we consider the mean nonaffine displacement (NAD) D na of the monomers themselves. The components Dna m are given by

Figure 3. Snapshots of the 25−450−25 triblock system during pure uniaxial strain deformation at strains 1, 3, 5, and 7. Minority monomers are in blue and majority monomers in red. Here the average density decreases as ρ(ε) = ρinitial/(ε + 1).

observe cavitation and drawing in the pure strain deformation. During the formation of the fibril network, there is significant breakup of the glassy inclusions. In Figure 4, by contrast, we see the minority clusters primarily stretch and deform during pure stress deformation. We now quantify what is happening to the microstructure during these deformations. We can consider how the chains are deforming globally by studying the end-to-end distance Re = r(N) − r(1), where r(1) and r(N) are the positions of the first and last monomer on a D


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Figure 6. Nonaffine monomer displacement magnitudes calculated between snapshots separated by a strain of 2.5%. Top: filled symbols, homopolymers; open symbols, triblocks. Bottom: crosses, A-type beads in triblock; filled triangles, B-type beads in triblocks. Left: pure uniaxial strain. Right: pure uniaxial stress. Figure 5. Nonaffine strain of the end-to-end distance during (a) pure uniaxial strain deformation and (b) pure uniaxial stress deformation. Solid lines: triblocks. Dashed lines: homopolymers. Chains lengths N are given by the legend.

Dmna = rm − rm0 − εmnrn0

A similar pattern is seen in the pure stress deformation (panel d). Here the crossover happens later since Dna for the minority monomers increases more slowly. We now analyze changes in shape and number of the minority monomer phase-separated regions during deformation. The average asphericity of minority monomer clusters 1 A = λ1 − (λ 2 + λ3) (6) 2 is calculated from an average of the ordered eigenvalues λi of the mean radius of gyration tensor for each cluster j

(5)

where r0 denotes the monomer position at an earlier time and m, n denote Cartesian components. Here we focus on the NAD between small strain intervals of 2.5%. Panels a and b of Figure 6 show the magnitude of the NAD averaged over all monomers as a function of strain. This quantity serves as an indicator for plastic activity. For pure strain deformation (panel a) the NAD rapidly increases from zero to a maximum value over a strain of 10% and even faster for pure shear deformation (panel b). Note that in amorphous materials, nonaffine displacements already occur in the elastic (reversible) loading regime due to local elastic heterogeneities. In both cases, Dna decreases as the deformation progresses, and the decrease is more pronounced for longer chain lengths. For the unentangled homopolymer, Dna remains close to constant while for the longer N = 500 chains Dna decreases by 20% for the homopolymers and 40% for the triblocks at εzz = 8. The magnitude of Dna is always less for the triblock monomers than for monomers of the equivalent chain length homopolymers. This difference becomes more pronounced during the deformation. Panels c and d of Figure 6 average Dna over majority and minority monomers in the triblock systems. For the pure strain deformation (panel c) we see that initially the soft majority beads (triangles) have a higher Dna ≈ 1.3, which is the same for all chain lengths. However, the monomers in the hard beads (crosses) show a lower Dna the longer the chains become, but this value increases during deformation. For the N = 500 (N = 300) triblocks, there is a crossover at εzz = 7.5 (εzz = 7) where the minority beads first have a larger Dna than the majority beads. For the shorter N = 100 chains no crossover is reached.

Smn =

1 Nj

Nj (j) (j) (i) ∑ (rm(i) − rCM m)(rn − rCMn) i=1

(7)

The sums run over the number of beads in each cluster Nj, and coordinates r(j) CM are with respect to the center of mass for each cluster. Figure 7a shows that the asphericity increases during pure strain deformation; however, a threshold is reached where this value plateaus or decreases. This indicates that the clusters can only deform to a threshold point before they break up into more spherical clusters. In the pure stress deformation shown in Figure 7b, A increases much more dramatically than the pure strain case; i.e., the spheres are being deformed much more strongly as they are staying intact longer. This effect can also be seen qualitatively in the snapshots Figures 3 and 4. The number of clusters Nc shown in Figure 7c,d exhibits opposite trends. Nc rises much faster to up to 3.5 times their initial number Nc0 during pure strain deformation, while Nc rises to only about 1.5Nc0 during pure stress deformation. Interestingly, in the latter case Nc first decreases before rising at larger strains as clusters that are close in the xy-plane first merge and then break up later. Clusters therefore stay more spherical but break up more easily during pure strain E


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Figure 8. Percentage of chains which switch from (a, b) bridges to loops and (c, d) loops to bridges in a strain interval of 0.5 in pure uniaxial strain deformation (a, c) and pure uniaxial stress deformation (b, d). Green squares: 25−450−25 triblocks; red circles: 15−270−15 triblocks; blue triangles: 5−90−5 triblocks. All data reflect averaging over three simulation boxes containing 480 000 monomers.

Figure 7. Asphericity of minority monomer domains (a, b), number of minority monomer domains (c, d), and proportion bridging chains (e, f) during pure uniaxial strain deformation (left: a, c, e) and pure uniaxial stress deformation (right: b, d, e). Green: 25−450−25 triblocks; red: 15−270−15 triblocks; blue: 5−90−5 triblocks. All data reflect averaging over three simulation boxes containing 480 000 monomers.

4.5, and 6 for N = 100, 300, and 500, respectively. These strains correspond well to the upturn in the number of A-type clusters seen in Figure 7d as well as the decrease of asphericity in Figure 7b. The behavior of the transition rates also explains the observed nonmonotonic trends of the bridging fraction pb. Since the transition rate for loops to bridges is always smaller than the transition rate for bridges to loops up to a threshold strain, pb initially decreases. However, at large strains loops transform into bridges faster than vice versa, and pb rises again. We also consider how the bridging and looping chains are deforming separately. Figure 9 shows the mean-squared end-toend distance scaled by chain length, Ree2/N, for chains that were classified as bridges (panels a and b) or loops (panels c and d) in the undeformed systems. Initially, we find Ree2/N ≈ 1.8−2.0 for bridging chains, whilefor looping chains Ree2/N ≈ 0.12. These differences reflect the fact that the loop chain ends by definition are located in the same cluster. During deformation, the end-to-end distance of the bridges increases monotonically for all chain lengths in both pure strain (Figure 9a) and pure stress deformations (Figure 9b). For the loop chains by contrast, Ree2/N initially remains constant and increases only later once a threshold strain is reached. By comparison with Figure 8c,d and Figure 7c,d, respectively, we see that these strains again closely correspond to those where the rate of loops switching to bridges and the number of clusters Nc increase. At large strains, where the breakup of hard minority monomer regions has occurred, we observe some void formation in the long chain triblock systems undergoing pure stress deformation. This is not seen in the rubbery homopolymer systems. To visualize this process, we divide the simulation box volume into ∼8000 cubic bins (∼4 monomer diameters in length) and consider the local density of monomers ρ relative to the average density ρ̅ in each. Figure

deformation, while they tend to stay intact but deform more strongly in the pure stress deformation. We also consider the proportion of bridging chains pb in Figure 7e,f. Chain ends terminating in the same cluster are said to loop, while chain ends ending in two different clusters are said to bridge. The bridging fraction ranges between 80% and 90%, which is very close to values reported in a previous simulation study.24 In experiments values ranging between 40%38 and 90%14 are reported and may be specific to material and preparation protocol. Interestingly, the bridging fraction changes little throughout the deformations. Despite the changes in the number of clusters they remain connected by bridging chains. In general, there is an increasing trend of pb when the number of clusters is increasing and vice versa. This means that loops are turning into bridges if clusters divide rather than isolated chains pulling out of clusters. To further understand the variation in A, Nc, and pb, we consider the rate at which bridging chains are changing to looping chains and vice versa. Figure 8 shows the percentage of chains that switch in strain intervals of 0.5. In panels a and b, we see that the rate of bridges switching to loops is fairly constant but nonzero for both pure strain (a) and pure stress (b) deformations. The rate of chains changing from loops to bridges (panels c and d) shows more pronounced trends. For both deformation modes there is initially a low rate of transition from loops to bridges which eventually increases, but the increase happens at larger strains for longer chains. For the pure stress deformations (panel d), the increase in transition from loops to bridges happens at strains of approximately 2, F


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Figure 9. Mean-squared end-to-end distance scaled by chain length N for chains that were (a, b) bridges at t = 0 and (c, d) loops at the beginning of the deformation for pure strain deformation (a, c) and pure stress deformation (b, d). Green: 25−450−25 triblocks; red: 15− 270−15 triblocks; blue: 5−90−5 triblocks. All data reflect averaging over a simulation box containing 480 000 monomers.

10 shows snapshots of the density binned in the x = 0 plane. We see the development of voids at εzz = 6.5. Also shown is the density of the minority monomers ρA relative to the total density. In these snapshots we can see that hard minority clusters start to deform above εzz = 3.5 and break up at εzz = 6.5. The voids appear to form primarily where minority clusters are being pulled apart, while little correlation was found with the local pressure distribution. This is consistent with earlier simulation studies on caviation in glassy homopolymers, which identified instead low local elastic moduli as the best predictor for the formation of cavities.39 Figure 11 shows the same visualization of the density during the early stages of pure strain deformation where cavitation occurs. Figure 11c shows that in the N = 500 homopolymer there is no variation in density at εzz = 0.05, but at greater strains localized round voids form. Figure 11b shows the same snapshot for the N = 500 triblock. Here variations in density already become visible at εzz = 0.05. Voids develop first at the surfaces of the minority regions (shown in Figure 11a). The minority regions remain undeformed, and the voids develop in a nonsymmetrical way around them. Indeed, the stress−strain curves of Figure 2a indicate that cavitation occurs at smaller strains in the 25−450−25 triblock than in the homopolymers. Irrespective of chain length all homopolymers display the same behavior at small strain, while for the triblocks the stress response is chain length dependent.

Figure 10. Monomer density relative to average density (black, red, yellow scale) and minority monomer density relative to average density (white to blue scale) in a plane of width ∼4σ located at x = 0. Five different values of strain = 2, 3.5, 5, 6.5, and 8 are shown during pure uniaxial stress deformation of a 25−450−25 triblock system. Voids (black) begin to appear for strains ≥6.5 and reach a volume fraction of ∼1% for ϵ = 8. Red dashed lines highlight one minority monomer domain breaking apart.

therefore mimicking the situation in a thermoplastic elastomer. Under pure stress uniaxial deformation, both unentangled homopolymer and triblock chains show yielding and chain pullout at large strains, which is consistent with previous simulation results for short chains.24 For entangled polymers, we find stress−strain curves that are qualitatively similar to styrenic block copolymer experiments:4,10,40 the triblocks exhibit a much stronger hardening response with stresses that exceed those in the corresponding homopolymer elastomer by up to 6 times. The global chain deformation becomes affine as the chain length increases. An analysis of the NAD as a measure of the degree of local plastic activity reveals a gradual transition of plastic activity from the rubbery regions to the glassy regions at very large strains. This transition coincides with the strain at which the asphericity of the spheres is maximal. Only upon further straining do spheres begin to break up and their number increases, while the bridging fraction remains constant during the entire deformation. These observations are consistent with the interpretation that the trapping of chain ends in the hard glassy regions is key to forming a cross-linked rubbery phase in which entanglements in the soft background are preserved. Under pure strain deformation by contrast, the elastomer first fails through cavitation followed by a fibril drawing process. Systems composed of short chains fail through chain pullout

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CONCLUSION We have performed molecular simulations of the structure and response of model ABA triblock copolymers in a spherical phase-separated morphology under two uniaxial deformation modes. Model parameters were chosen so that the A (B) phases were below (above) the glass transition temperatures, G


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more complex topologies such as star or miktoarm polymers. The potential of these polymers to further improve mechanical properties is currently being explored in experiments.40,42

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (A.J.P.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Natural Science and Engineering Research Council of Canada (NSERC) for financial support. REFERENCES

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Figure 11. Minority monomer density relative to average density (white to blue scale) and monomer density relative to average density (black, red, yellow scale) during the early stages of pure strain deformation of the 25−450−25 triblock (a) and (b) and in the N = 500 homopolymer (c). Densities are shown in a plane of width ∼4σ in the x = 0 plane at strains = 0, 0.05, 0.1, 0.15, and 0.2.

following cavitation, in close similarity to simulations of crazing in glassy polymers.41 Longer triblock chains exhibit a stress plateau followed by strain hardening. In contrast to the pure stress deformation, glassy spheres do break up much earlier in the deformation process and therefore remain much more spherical. The slight increase of the bridging fraction with sphere breakup indicates that chain pullout is not dominant in this deformation mode either. Plastic activity is initially concentrated in the rubbery phase but increases in the glassy phase as strain hardening progresses. Localized density variations during deformation of triblock elastomers have been observed in experiments. Inoue et al. have found using small-angle X-ray scattering of SIS polymers that at higher elongations regions of low density form in the isoprene region, stating that microvoids could lead to material failure.10 Our study shows that void formation both in the early stages of pure strain deformation and in the late stages of pure stress deformation is intimately related to the presence of glassy inclusions. Cavities are observed preferentially either at the interface between glassy and rubbery regions or between a glassy region that fragmented just before cavitation. Our simulations are consistent with experimental trends and provide insight into microscopic deformation mechanisms of nanostructured macromolecular materials. The simulation methodology is flexible and can be extended beyond linear polymers to H


Article

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Molecular Mechanisms of Plastic Deformation in Sphere-Forming

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