Sawtooth Tensile Response of Model Semiflexible and Block

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Sawtooth Tensile Response of Model Semiflexible and Block Copolymer Elastomers Bernardo M. Aguilera-Mercado, Claude Cohen, and Fernando A. Escobedo* School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14853, United States S Supporting Information *

ABSTRACT: We study via coarse-grained molecular modeling the elastic response of semiflexible elastomer networks with idealized diamond connectivity. Under strain-driven uniaxial deformation, these networks are found to exhibit first a perfect soft elastic behavior (including a buckling instability preceding hardening) up to moderate extension ratios ( 21/6 ⎩

2. DIAMOND NETWORK MODEL AND SIMULATION METHODS 2.1. Model Network. We adopted networks with idealized diamond-like regular connectivity wherein the chain ends connect to the tetrafunctional cross-links which, in a maximally swollen system, would lie on the lattice points of a diamond lattice (see Figure 1). Such a network consists of a single

(2)

and UFENE(r) is UFENE(r ) = −(KFENER 0 2/2) ln(1 − r 2/R 0 2)

(3)

where KFENE and R0 are, respectively, the spring force constant and the maximum length of the FENE bond, which are set to KFENE= 30ε/σ2LJ and R0 = 1.5σLJ.36 Equations 1−3 apply to 841

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Figure 2. (a) Strain-driven stress−elongation ratio (σ* vs α) curves for 20-mer diamond networks with different values of chain stiffness KBend. Coupling between mesomorphic behavior and chain deformation gives rise to “sawtooth-shaped” behavior for KBend > 3.75. (b) Depiction of the “staircaselike” σ* vs α curve for the same system under stress-driven deformation (dashed) mapped from the strain-driven deformation curve (full line). Boxes delimit the portions of the strain-driven deformation curve that are likely to be visited in stress-driven runs, and arrows illustrating the evolution of α for a given range of applied stress. Stress-driven deformation hence occurs through “yield” events where the system behaves liquid-like while yielding and solid-like while resisting yield. Stress σ* is in kBT/σLJ3 units.

fixed melt-like number density of 0.9 sites/σLJ3. Newton’s equations of motion are integrated using a velocity−Verlet algorithm40,41 and the Andersen thermostat40 is employed to impose the desired temperature, which is T*= kBT/ε = 2.0 in most of the studied cases. Periodic boundary conditions are used in all directions of the orthorhombic simulation box. The time step used for the integration is Δt = 0.005τ, where τ =σLJ (mLJ/ε)1/2 is the LJ unit time (mLJ denotes the mass of the LJ bead, and the collision frequency of the Andersen thermostat is ν = 10.0τ−1. The network is initially created at a very low number density [O(10−3)σLJ−3], slowly compressed in ∼106 steps to its final melt-like number density, and then equilibrated at constant volume for 2 × 108 time steps until converged values are observed for such properties as average potential energy, stress, and bending angle. The resulting structure is the unstrained state (α = 1) used as starting point for the straindriven simulations (see Figure 1, parts f and g, for sample snapshots of such structures for homopolymer and triblock copolymer networks). 2.2. Uniaxial Strain-Driven Deformation. The deformation is conducted via molecular dynamics (MD) simulations in a quasi-continuous manner.42 The extension ratio α, defined as quotient of the box length to the initial undeformed box length along the strain axis, is changed gradually by Δα=10−4 increments every 5000 MD time steps, i.e., with a constant strain rate of dα/dt = 4 × 10−6 τ−1, and is such that the volume of the sample remains constant throughout. For networks made of triblock copolymer chains the strain rate used is dα/dt = 8 × 10−6 τ−1. For fully flexible LJ chains forming disordered (meltlike) chain-end-linked networks,43,44 the relevant relaxation time scale is the Rouse time τRouse ≈ 1.5N2τ (where N is chain length), and a period of 2τRouse was found to be sufficient to fully relax the stresses after large step changes in strain (∼10% or larger).43 Since in our case τRouse ∼ 600τ, strain changes of less than 1% occur over a 2τRouse period, suggesting that the stresses remain at quasi-equilibrium at all times during deformation. Our strain rates are also at the lower end of values that have been used to simulate similar deformations, even for polymer glasses.44 The comparisons above are only approximate given that our (entanglement-free) chains undergo polydomain transitions. Indeed, an appropriate strain rate was only chosen after analyzing the results from preliminary runs where selected networks were deformed at several strain rates

bonded beads in both homopolymer and block copolymer chains. Cross-links are represented as Lennard-Jones beads that are tetrafunctional; i.e., they form bonds with four (chain-end) LJ beads. Chain stiffness is modeled through the following bending potential between any two consecutive chain bonds,37 UBend /kBT = KBend(1 + cos ψ )

(4)

where ψ is the angle formed between the two consecutive bonds (see Figure 1d), kB is the Boltzmann constant, T is temperature and KBend is the chain stiffness constant. No bending potential is used if any of the two consecutive bonds involves cross-links, which hence renders a fully flexible crosslinking between chains. The persistence length of the chains at a given temperature T is monotonously dependent on KBend.38 KBend was varied from 0 to 5 to cover the range from fully flexible (free chain persistence length ∼1.3σLJ in a Θ solvent) to stiff/mesogenic chains (free chain persistence length ∼4.8σLJ in Θ solvent). Nonbonded beads interact via a cut and shifted LJ potential: ⎧ ⎡ 12 12 ⎛ σLJ ⎞6 ⎤ ⎛ σLJ ⎞6 ⎛ σLJ ⎞ ⎪ i , j⎢⎜⎛ σLJ ⎟⎞ ε − ⎜ ⎟ − ⎜ i , j ⎟ + ⎜ i , j ⎟ ⎥ if r /σLJ ≤ rci , j ⎪ i,j ⎝ r ⎠ ⎝ rc ⎠ ⎝ rc ⎠ ⎥⎦ (r )⎨ ⎣⎢⎝ r ⎠ ULJ ⎪ ⎪ if r /σLJ > rci , j ⎩0

(5)

The values used for the interaction energies, ε , and cutoff distances, rci,j, depend on the type of system studied. For networks made of homopolymer chains only (i.e., the first part of this study) we use εi,j = ε, and rci,j = 21/6σLJ; and for networks containing A(25%)−B(50%)−A(25%) triblock copolymer chains (second part of the study) we use εi,j = ε for all the interactions and rci,j=2.5σLJ if beads i and j are chemically identical (resulting in a potential with an attractive well), or rci,j = 21/6σLJ if beads i and j are chemically different (resulting in a purely repulsive potential). For 20-mer block copolymer chains (at T = 2.0ε/kB and ρ = 0.9/σLJ3) with the above-described interaction potentials, the resulting effective Flory−Huggins parameter is approximately χN ≈ 63.39 The cross-links in the systems containing triblock copolymers are similar to the chainend beads (A-blocks). Molecular dynamics (MD) in the canonical (NVT) ensemble is used to equilibrate and simulate the deformation of the model elastomer networks. In all cases, the system is kept at a i,j

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Results from strain-driven deformations are consequently expected to encompass those obtained from stress-driven ensembles and provide a more complete picture of the stressextension ratio dependence and its underlying free-energy landscape. In a simulation at fixed stress value σ, the system will tend to settle at the lowest α where the same stress σ occurs in the strain-driven simulations. Hence, stress-driven runs will visit the α regions where the upturns of the stress “teeth” occur (first encountered when coming horizontally from the left with fixed σ). Figure 2b sketches how the previously reported24,25 stressdriven staircaselike curve can be mapped from the strain-driven sawtooth shaped curve. Similar mapping and qualitative discrepancies between thenonmonotonoustensile responses under strain-driven and stress-driven deformations have been experimentally observed46 for main-chain LC elastomers undergoing necking instabilities (during strain induced polydomain smectic transitions). Chain Stiffness Induces Isotropic−Nematic Transition at the Undeformed State. The strain-driven, stress−strain curves in Figure 2a show that the overall isotropic (monotonous trend)−mesomorphic (sawtooth profile) transition occurs for KBend near 3.75. Although the diamond network with KBend = 4 clearly shows the sawtooth shaped tensile behavior, it does not seem to exhibit the signature frequency split-peak in the 2 HNMR spectrum of a nematic phase in the undeformed state (α = 1.0), as shown in the simulated spectrum in Figure 3b, or

ranging from 10 times faster to 10 times slower than the values given above. We found that for that range of strain rates, the resulting stress−strain curves were essentially unchanged (including the location of the ordering transitions, with some small quantitative discrepancies that were within the error bars of the results). Hence, the chosen strain rates are such that local equilibration is attained at each elongation. We calculate and collect values of the quantities of interest such as stress tensor components, bending angle ψ, and chain segment orientation parameter S. This order parameter S is defined as (3/2⟨cos2(θ)⟩ − 1/2) where θ denotes the angle between a chain segment and the strain axis, and the angular brackets represent average over all chain segments. The individual components of the stress tensor are computed from the virial contributions of all the energetic interactions involved; those corresponding to the bending potential, a threebody force field, are calculated using the methodology described by Carpenter.45 Values of the aforementioned quantities are averaged over extension ratio intervals of width |αi+1 − αi| = 10−2, and reported at the midpoint of the intervals (αi+1 + αi)/2. Note that although we impose a “strain rate”, the stress is always evaluated for a system at a given strain, i.e., in an “iso-strain” ensemble. The effects of simulation finite size on tensile response were assessed for a 20-mer diamond network with KBend = 4.0 and T* = 2.0 by comparing strain−stress results from the base network with those of an 8-fold larger 20-mer diamond network; the resulting stress−strain curves were fully consistent.

3. RESULTS 3.1. Diamond Networks with Semiflexible Chains. 3.1.1. Mesoscopic Deformation Behavior and Mesomorphic Activity. Stress-Driven and Strain-Driven Deformations Give Different Stress−Strain Curves. The onset of mesomorphic behavior can be mapped as a function of either chain stiffness or temperature as these two parameters are tightly coupled for the force field used in this work. As shown in Figure 2a, the stress−elongation ratio curve for networks with KBend ≥4 exhibits a distinct oscillatory or sawtooth shape. Unlike these strain-driven results, the elastic response curve under stressdriven deformation was found to have a staircaselike shape for KBend > 4.24,25 The discrepancy between the shapes of the stress-extension ratio curves for the two deformation schemes is due to the different thermodynamic ensembles associated with them that do not always necessarily sample the same states of the system. Such difference notwithstanding, the microscopic structure of the stretched network (for comparable values of α) obtained via stress-driven or strain-driven is the same; i.e., for α >2.8 the network forms multiple ordered domains where chains and cross-links are segregated, and these chain domains are smectic phases because the mesogenic chains are not only aligned but also organized into layers. The nonmonotonous, sawtooth shaped, dependence of stress with extension ratio (e.g., Figure 2a for KBend = 5) indicates that multiple states of the system can have the same stress but with different α, and will not be all accessible to a stress-driven process. This is because in the latter, where the macrostate of the system is defined for a given stress value, the specific microstates visited will depend on deformation history in a system with a nonmonotonous tensile response. In contrast, several states with equal stress values and different α can be readily sampled in a strain-driven process as the state of the system is defined by imposing α.

Figure 3. Results for 20-mer diamond networks at their undeformed state. (a and b) Simulated NMR spectra and (c and d) are representative snapshots; (a and c) KBend = 5; (c and d) KBend = 4. In all cases T* = 2. The nematic order parameter [=3/2⟨cos2(ϕ)⟩ − 1/2, where ϕ is the angle between a segment and the nematic director] is 0.488 (c) and 0.045 (d), respectively.

any evidence of chain alignment in its structure (Figure 3d) (details of such calculations are given in refs 47 and 48). On the other hand, the network with KBend = 5 (Figure 3, parts a and c) does exhibit those typical features of a nematic elastomer.49 In fact, the NMR spectrum of the diamond network with KBend = 4 seems closer to that of an isotropic elastomer than that of a nematic one. Unless otherwise stated, only results KBend = 5 networks are described henceforth. Diamond LC Network Exhibits Liquid-Like or Soft-Elastic Behavior at Low Strains. In addition to its distinct sawtooth shape, the tensile response of the KBend = 5 diamond network 843

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Figure 4. Simulated strain-driven tensile response for loading−unloading−loading runs of a 20-mer diamond network with KBend = 5. (a) Second loading run is performed immediately after the unloading run is finished. Marked microstructural differences are evidence of the material’s memory of its deformation history. (b) A heating (above isotropization temperature) and cooling process is carried out between the end of the unloading run and the beginning of the second loading run. (c) A second uniaxial extension is performed along a direction perpendicular to the first strain axis immediately after the unloading run is finished. Solid, dashed, and dotted arrows point the first loading, unloading, and second loading runs, respectively. (d) Snapshots at α = 2.0 for both the first loading (left) and unloading (right) runs. Stress σ* in kBT/σLJ3 units.

derivative of the free energy with respect to the strain is negative for strain values slightly below the hardening transition.7 In a uniaxial deformation setup that does not allow for sample buckling like ours, such instability is manifested through negative stress values for strains preceding the transition to the hard regime. Real nematic elastomers,17,18,58−64 however, do not exhibit a perfect sof t elasticity, and some65 may not even show signs of a strong coupling between nematic order and applied strain. Instead, they require the application of a small nonzero force for their deformation7,54,66 during the rotation of the nematic director. Such departures from ideality are due to topological and compositional heterogeneities which also preclude negative stresses.7 This experimentally observed tensile response is known as semisof t elasticity,4,7,63 and is theoretically described in terms of a succession of hard−semisoft−hard elastic regimes. More recently, nematic elastomers with much less topological and compositional heterogeneities have been created21−23 by cross-linking the mesogenic polymer chains in the high temperature isotropic state and then cooling the elastomer to an aligned state. Such elastomers exhibit a small amount of nonidealities and hence a nearly ideal sof t elastic tensile response which has been termed super(semi)sof t elasticity.22,23 Our model elastomer of mesogenic chains displays a completely ideal sof t elastic responseincluding the buckling instability before the network hardeningup to moderate deformations (α < 3.0), by virtue of the lack of frozen orientation defects (that would arise from both randomly placed cross-links and chain entanglements). Absence of such quenched orientation defects allows for a complete coupling of nematic order with chain deformation and the presence of Goldstone sof t modes. Diamond LC Network Exhibits Nonreversible Hysteretic Elastic Behavior. The KBend = 5 diamond network shows

displays other remarkable features (see Figure 2) such as a liquid-like behavior (i.e., null elastic modulus) at small deformations (α ≈ 1.0−1.4), very small negative stresses at moderate deformations (α ≈ 1.6−2.6), and pronounced strain hardening at larger α (≈2.7). Such nonlinear and atypical elastic features are characteristic of ideal LC elastomers and described by the so-called neo-classical rubber elasticity theory.4,7,50−54 This liquid-like behavior (the ability to undergo finite deformations at no energy cost) is usually referred to as sof t elasticity.4,7,50−54 According to the neo-classical rubber elasticity theory, sof t elasticity in ideal nematic elastomers originates from a rotational invariance51 of the network free energy, with respect to the orientation of its nematic director, that results from a strong coupling between nematic order and applied deformation. As a consequence, a continuous manifold of infinite macrostates with nonzero strains, known as soft (or Goldstone) modes,4,50 arises and allows for rotations of the chain nematic directorscoupled with chain deformationsto occur at constant (and maximal) configurational plus orientational entropy of the chains.4,7,54 The stress−strain curve of an ideal LC elastomer has then two different and well-defined regimes:55−57 first, a sof t one with null elastic modulus over a finite range of strain values (where nematic orientation and deformation are fully coupled); followed by a second hard regime with a finite elastic modulus when molecules are moderately aligned and further chain deformations can no longer take place at constant configurational plus orientational entropy. Another expected feature of ideal nematic elastomers during strain-driven uniaxial extension, for imposed strain values near hardening, is the bucklingor Euler strutinstability (i.e., spontaneous expansion of the clamped elastomeric sample).7 This instability occurs because, for a perfect LC elastomer, the 844

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(dα/dt > 0), n → n + 1 transitions can be aided by the deformation energy being transferred to the system, but for unloading (dα/dt < 0), no such energy bias is available to aid n → n − 1 transitions. In attaining any particular layering state, the deformation history is hence much more important than the deformation rate (as long as it is slow enough that the system is allowed to locally relax). The importance of polydomain kinetic trapping is further evidenced in loading− unloading simulations of networks that did not form such domains (e.g., for KBend = 0): they exhibit completely reversible stress responses (without hysteresis). 3.1.2. Molecular Deformation Mechanism of a Diamond LC Network (KBend = 5). Although strain-induced smectic phases can occur frequently in certain LC elastomers,46,61,67,70−75 cross-link exclusion and layering due to the chains’ mesomorphic activity is rather atypical. This happens in our system due to the combination of three factors: regular topology of the network, strong smectic field, and cross-link flexibility. Regular connectivity and lack of trapped entanglements allow chains to suitably rearrange mostly through disinterspersion,24,25 and also to strengthen the smectic field due to the lack of frozen topological heterogeneities. In addition, lattice-like connectivity can favor the cross-link layering as lattice nodes are usually layered as well. A high mesogenic field (a function of chain stiffness, density, and strain) is also essential to align the chains, and drive the microsegregation between the mesogenic chains and the isotropic sections around the tetrafunctional fully flexible cross-links into separate alternating layers. Layering Mechanism Is Different for Stress-Driven and Strain-Driven Deformations. For the KBend = 5 diamond LC network, it was reported that the staircaselike stress−strain curve24,25 obtained under stress-driven deformation occurs due to the abrupt and discrete formation of ordered chain domains that exclude the cross-links, where each “step” involved an order−disorder−order transition. Specifically, upon a significant increase in the applied stress (enough to overcome the free-energy barrier between successive basins), the state with “n” ordered domains underwent first a destruction of its segregated structure (during which cross-links become uniformly distributed) followed by the formation of a new segregated structure with “n+1” ordered domains (where crosslinks are again excluded in layers).24,25 Although both stressdriven and our strain-driven deformations produce the same structural features of ordered chain domains and cross-links segregation (e.g., see Figure 5a), we find that order−disorder− order transitions do not occur in the strain-driven sawtooth response. Instead, the overall deformation mechanism under strain-driven elongation involves successive creation and distortion of the smectic chain domains. Indeed, such domain creation processes are now observed to occur more smoothly, reflecting the much slower “dynamics” imposed on the process (where the system is forced to take on a continuous set of strain values as opposed to the discrete set associated with the staircase response24,25). Details of this process are given below (see also video in Supporting Information76). A Nematic−Smectic Phase Transition Initiates Layering. For 1.0 < α < 1.4, where the elastic modulus is null, the deformation happens at constant entropy by a simultaneous rotation of the nematic director with minor chain deformations and cross-link rearrangements. Recall that this KBend = 5 diamond network exhibits a nematic phase at the undeformed state (α = 1.0), with most of the chains moderately oriented

evidence of significant irreversibility in its deformation mechanism. Figure 4a shows the stress−strain curve for three consecutive uniaxial deformation runs: a first loading, unloading, and a second loading. Smectic elastomers have been found67,68 to exhibit experimentally such irreversible elastic response (known as poor strain or shape recovery) that leads to shape-memory effects, which are distinguishing properties of multiple smart materials. The pronounced hysteresis observed during the combined loading−unloading− loading run (Figure 4a) arises from the ability of the stiff diamond network microstructure to retain information about its previous deformation history. For instance, upon unloading at α = 2.0, the smectic diamond network “remembers” its previous highly extended state and preserves the four smectic chain domains (formed at much higher extension ratios) although the network displayed only two smectic domains at the same α during the first loading (Figure 4d). Successive merging of ordered chain domains (reversing the first loading elongation mechanism) is not observed because the associated energetic barrier cannot be surmounted during unloading when no energy is transferred to the system. One common way to “erase” the shape memory of “smart” LC elastomers is by increasing the temperature above the isotropization transition.67,68 The stress−strain curve shown in Figure 4b corresponds to a combined loading−unloading− loading deformation run in which the network was heated (to T* ∼ 10, beyond the isotropization transition) and then cooled down back to the original temperature between the unloading and the second loading deformations. Because of this heating− cooling process, the segregated layers formed during the first loading are destroyed, and the system hence “forgets” its previous highly stretched state recovering the sawtooth shaped tensile response. Also consistent with experiments with smectic elastomers,61,64 we found (Figure 4c) that if the second loading extension is performed along any direction perpendicular to the original strain axis, the sawtooth stress−strain curve is also recovered without any heating and cooling. This occurs because the previously formed smectic chain domains need to be completely rearranged and reassembled in an orthogonal direction. The dependence on deformation history of the system’s microstructure can be further clarified as follows. Assume that the system has attained a segregation state with n smectic− cross-link bilayers and is at a particular α value. The transformation from the n to the n ± 1 domain structure can be considered as not only a phase transition, but also an activated process (in a kinetic sense, akin to polymer crystallization). For instance, by integrating over the first stress upturn (from α = 2.8 to α = 3) of the first load in the network of Figure 4a (replotted as force vs length) leads to a work of ∼60 kBT, a value that can be interpreted as the free-energy barrier that the system overcomes in going from 2 to 3 bilayers. While the free-energy barrier heights for surmounting different peaks while increasing or decreasing α (for the associated n → n ± 1 transition) are different, they are all in the order of tens of kBT, well beyond the reach of typical thermal fluctuations, hence resulting in the system being “trapped” in a free-energy well. If the simulations could be run for much longer times at a fixed α, the system could spontaneously move to a deeper well (if existing) undergoing a n → n ± 1 transition. However, estimating the time scale for such processes would require specialized rare-event simulation methods40,69 which are beyond the scope of this work. As indicated earlier, for loading 845

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For 1.6 < α < 2.9, the dominant deformation mechanism is the expansion of bellows of the bundled chains rather than chain-ends stretching or intrachain rearrangements. When α ≈ 2.7, the stress is still close to zero, signaling a stable structure of that specific (first) accordion-like Sm CA phase. Above this, around α ≈ 2.9, the bellows become fully extended with the chain domains forming now a Sm A phase, and S reaching a maximum (Figure 5c). A Stress Drop Signals the Birth of a New Smectic Domain from Adjacent Domains. Beyond the first maximum in S, the expansion of chain domain bellows stops being the dominant extension mechanism. This is because the Sm A phase, where most of the chains are extremely aligned to the strain axis, cannot be further stretched without affecting the internal configuration of the chains (e.g., Figure 5a at α = 2.95). Consequently, the main deformation mechanism becomes hairpin unfolding of chains. In strong nematic (or smectic) fields, main-chain mesogenic polymers randomly incorporate hairpins along the chain to increase their configurational entropy, and to compensate for the orientation entropy lost upon alignment.11 This has been observed experimentally77 and described theoretically.11 For α ≈ 2.9−3.45, a few hairpins unfold and take up the strain in a highly nonaffine way, some chains disintersperse and a few cross-links rearrange to minimize any misalignment with the strong smectic field. As a result, a new smectic chain domain is created in a continuous way from adjacent smectic domains (Figure 5a at α = 3.45). During the “birth” of the new smectic chain domain, the overall extent of order decreases not only because chain rearrangements but also from the frustration of this new domain as it packs between closely spaced cross-link domains. Such frustration, just as before, leads to the formation of accordion-like Sm CA phases between the newest domain and the adjacent “parent” domains. The occurrence of the Sm CA phase is associated with a drop in stress and in the overall orientational order, both arising from the increase of orientational entropy of the bundled chains (Figure 5, parts a and c, at α = 3.45). After the new Sm CA domains are formed, the principal deformation mechanism is again the expansion of bellows of the bundled chains, and this continues until all the chain domains become Sm A (α ≈ 3.45−3.90) and the overall order reaches another maximum. After that, the deformation cycle (Sm A−hairpin unfolding−Sm CA−bellows expansion− Sm A) keeps repeating until all possible chain Sm A domains are created and all chain hairpins have been unfolded. A simplified sketch of this strain-driven deformation cycle is shown in Figure 5b. Although no experimental counterpart exists for our semiflexible diamond network, model PDES networks undergoing an extension-induced mesophase formation (where domains of aligned mesophase and amorphous material coexist) share some key similarities.78 In particular, X-ray scattering experiments revealed that, like in our system, the increase in mesophase content with strain is caused by an increase in the number of mesophase domains rather than an increase in domain size.78 3.2. Triblock Copolymer Elastomer: Enhancing the Sawtooth Response through Enthalpic Microsegregation. A key feature of the sawtooth tensile behavior displayed by the LC diamond network is the strain-induced microsegregation between mesogenic chains and cross-links in alternating layers; indeed, new teeth are associated with the creation of new microsegregated domains. One way to

Figure 5. Strain-driven uniaxial deformation of 20-mer diamond network with KBend = 5. (a) Snapshots at elongations α = 2.95, 3.45, and 3.90 showing the different smectic phases, formed by the chain and cross-link domains. (b) Simplified mechanism of deformation cycle with chain/cross-link domain creation (for α > 2.9). (c) Average segment orientation order parameter “S” vs. “α”. In parts a and b, black beads are cross-links and red strands are chains.

perpendicular to the direction arbitrarily prechosen as the strain axis, and its average chain segment orientation parameter S is −0.22 (Figure 5c). In the 1.4 < α < 1.6 range, a nematic− smectic phase transition occurs, detectable in sequential snapshots and by the change in the slope of the order parameter S (Figure 5c). This newly formed smectic phase is somewhat frustrated because the distance between consecutive domains of layered cross-links is shorter than the characteristic length of the “packed bundle” of mesogenic chains. As a result, chains form an anticlinic smectic C (Sm CA) phase rather than further bend or reduce their characteristic length by increasing the number of hairpins. This accordion-like arrangement of the ordered chain domains is responsible for the negative stresses (buckling instability) observed for α ≈ 1.6−2.6 before network hardening. The elastic buckling instability arises because the accordion-like configurations entropically favor mildly folded bellows of chain domains over highly folded (or extended) ones. 846

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Figure 6. Strain-driven uniaxial deformation of 20-mer end-linked diamond triblock copolymer network with KBend = 4 and χN ∼ 63. (a) Stress− elongation ratio curve. (b) Average segment orientation order parameter “S” vs α. (c) Simulation snapshots at different α. Creation of newer smectic domains occurs continuously and from neighboring domains. Black beads, red, and green strands represent cross-links, B (central) block, and A (end) block chains, respectively. For clarity, snapshots without the A blocks are also shown in the left side of the panels.

pronounced sawtooth shape, and (3) the equilibrium elastic modulus of the network (α = 1) is finite (greater than zero). The deformation mechanism for this KBend = 4 triblock copolymer diamond network is the same as the one described for the KBend = 5 homopolymer diamond network (discussed in section 3.1.2 and video in Supporting Information76): successive creation and distortion of smectic chain domains (see Figure 6c). We attribute the remarkable enhancement of the elastic response of the regular triblock copolymer network to (1) the fewer (or even lack of) smectic defects, (2) the stabilization of smectic chain domains and stronger cross-link confinement, and (3) the increase of the energy barriers for new domain formation. The stabilization of the smectic phases and the occurrence of fewer smectic defects allow for a much more concerted strain-induced creation of new smectic domains (i.e., the “birth” of new smectic chain domains occurs more simultaneously and in a narrower range of strain values). This concertedness is promoted by the much higher energetic penalty that chains of one type would experience if they individually cross domains of the other block type. Therefore, a more collective (or cooperative) crossing motion takes place which leads to an overall deformation of the chemical domains involved in the formation of a new smectic domain. Deforming the existing domains and creating new smectic domains require an additional amount of work to overcome the energetic barriers associated with the increase of interfacial area between dissimilar chemical domains. We also attribute the complete disappearance of the sof t region (liquid-like behavior) to such chemical-disparity barrier that is active even at small deformation. Interestingly, we also found that a sawtooth elastic response can be attained in a diamond network consisting of triblock

reinforce such a behavior is by replacing the homopolymer chains in the diamond network with, e.g., triblock linear copolymers A−B−A where the A and B blocks chemically “dislike” each other so that they would tend to microsegregate. We only consider the case when the volume fractions of A and B blocks are symmetric, i.e., for A(25%)−B(50%)−A(25%) triblock 20-mer chains (Figure 1b), so that these chains (at temperatures below their order disorder transition) will have a tendency to microphase separate into a symmetric lamellar phase even as free (un-cross-linked) chains in a melt. In a network, this tendency should help segregate and bury the cross-links within the A domains. Figure 1e displays a fully extended (for clarity) diamond network made of A(25%)−B(50%)−A(25%) triblock copolymer chains. After equilibration, the diamond network with moderately stiff (KBend = 4) triblock copolymer chains effectively microphase separate into a lamellar phase (see Figure 1g) that stabilizes a smectic phase formed by chain domains at the undeformed state (α = 1.0). Recall that equilibrium LC phases at the undeformed state were not observed for the KBend = 4 homopolymer network (see Figure 1f). Figures 6a and 6b show the stress−strain curve and the segment orientation order parameter vs extension ratio, respectively, for the KBend = 4 diamond network with triblock copolymer chains. The tensile response of the equivalent homopolymer diamond network is changed in at least three ways: (1) The stress values and hence the toughness (area under the stress−strain curve), are nearly 1 order of magnitude larger than for the KBend = 4 network with homopolymer chains (see Figure 2a); (2) the stress−strain curve has a more 847

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chains which are fully flexible (KBend =0); i.e., driven by enthalpic segregation only (recall from Figure 2 that KBend > 3.75 is required to have the sawtooth curve driven by the entropic alignment of semiflexible chains). The fully flexible 20mer diamond triblock copolymer network after equilibration displayed microphase segregation (a lamellar phase similar to the one shown in Figure 1g), and the stress−strain curve also exhibits a sawtooth profile (albeit less pronounced), as shown in Figure 7.

Pronounced elastic hysteresis is observed due to the irreversibility of the deformation mechanism wherein smectic domains do not merge upon unloading. After a loading− unloading cycle these networks have highly anisotropic configurations and, similarly to elastomers with shape-memory effects, can be reset to their “initial” states by increasing the temperature above the isotropization transition. It is also shown that the discontinuous staircase-like stress−strain curve previously reported for stress-driven deformations of these networks24,25 can be mapped from the sawtooth iso-strain stress−strain curve. The elastic sawtooth pattern arises from a distinct, highly nonaffine, deformation mechanism involving two processes: (1) the successive creation of smectic (Sm CA) chain domains, which (due to the regular network architecture) exclude crosslinks in parallel layers; (2) the accordion-like distortion of such chain domains. Contrary to the discontinuous transitions observed24,25 during stress-driven deformations, under straindriven elongation new Sm CA chain domains are created in a continuous and concerted fashion by hairpin unfolding and disinterspersion of mesogenic chains. The enthalpic forces that drive the microphase segregation in block copolymers were harnessed to increase the height of the free-energy barriers associated with a sawtooth elastic response. This exploits the fact that each stress−strain “tooth” is the manifestation of a free-energy barrier which can have (designable) entropic and enthalpic contributors. Specifically, the observed mechanical enhancement can be attributed to (1) the further stabilization of the smectic chain domains and increase in the cross-links confinement brought about by the microphase separation of the different blocks and (2) the higher deformation energy barriers caused by the enthalpic penalty associated with the transient increase of the interblock interfacial area during the creation of new smectic domains. In addition, we also found that the stabilization of the smectic chain domains due to the chemical disparity between the

Figure 7. Strain-driven stress−elongation ratio curves for a fully flexible (KBend = 0) 20-mer end-linked diamond triblock copolymer network. T* = 2.0, χN ∼ 63, and σ * is in kBT/σLJ3 units.

4. CONCLUSIONS AND DISCUSSION A distinctive sawtooth-shaped elastic response was found for a model elastomer network of semiflexible chains with diamond connectivity undergoing strain-driven uniaxial deformations at nanoscales, or scales comparable to the characteristic length of liquid crystalline smectic domains found in such networks. The chain mesomorphic activity (associated with chains with sufficient stiffness) and the network regular topology lead to a perfect sof t elastic behavior, including a buckling instability preceding hardening, up to moderate extension ratios (α < 2.9), followed by the oscillatory sawtooth pattern for larger strains.

Figure 8. Similarities between tensile responses and deformation mechanisms of organic polymer adhesives in nacre76 (left) and LC triblock elastomers with diamond connectivity (right). Each stress “tooth” is associated with the loss of a folded domain in nacre (left) but with the creation of a new ordered domain in the elastomer (right). 848

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different blocks is sufficient to produce a sawtooth elastic behavior in fully flexible diamond networks. Sawtooth elastic responses have been observed at nanoscales, through atomic force microscopy, in natural materials with extraordinary mechanical strength such as the muscle protein titin79 and organic polymer adhesives80 found between carbonate plates in abalone shells. The outstanding toughnesstotal energy per unit volume that can be absorbed via deformation before breakage (i.e., the area under the stress− strain curve) of such natural materials has its origin in deformation mechanisms that involve discrete and sequential processes. For instance, the deformation mechanism of the polymer adhesive between nacre tablets80 consists of successive and concerted unraveling of discrete “modules” of folded adhesive polymer (protein) structures; each unraveling event is set off when the force reaches a value that is lower than that which would cause breakage of chemical bonds. A rough comparison between the discrete and sequential deformation mechanisms of both adhesive polymers in nacre and our LC crystal elastomers with regular topology is displayed in Figure 8. The oscillatory nature of these stress−strain curves allows to maintain a relatively high value of average stress over a large range of strains, resulting in considerably high toughness (much larger area under the stress−strain curve than that of typical elastomers). Similar toughening mechanisms14,81 are also found in spider and worm silk fibers where successive splitting of betasheet crystallites by breaking dense regions of hydrogen bonds (after the semiamorphous protein domains have been highly extended) plays an important role on their outstanding tensile strength. Note that the sawtooth tensile response observed in the networks discussed here, is expected to occur only at nanoscales. Indeed, the saw teeth will smear out as the size of the sample increases, a consequence of averaging the resulting stress over larger spatial regions, and leading to a smoother stress−strain curve and, e.g., to such phenomena as LC polydomain−monodomain transitions.26,46 Of course that this “averaging” also happens if one were to test numerous chains of the nacre adhesive. Despite such smoothing out, however, the toughening effect will still be active. This study can be extended in different directions which may change the sawtooth tensile response in quantitative or qualitative ways. For example, increasing the molecular weight of the semiflexible network chains would be expected to preserve the sawtooth response but to reduce the stresses at any given elongation, as a reflection to a lower concentration of cross-links (preliminary results82 for 50-mer KBend = 5 diamond networks confirm such an expectation). It would also be informative to undertake simulations to quantify the time scales associated with local relaxation at different strains, the transition between polydomain states (at fixed strain), and with the loading−unloading hysteretic behavior. While the combination of enthalpy-driven microphase separation, regular connectivity, and main-chain mesomorphic activity gives the best defined and enhanced sawtooth tensile response and toughness (i.e., compare Figure 6 to Figures 2 and 7), deviations from those conditions are expected to weaken such a behavior. Some preliminary results82 (not shown) on altering the chain-end connectivity from a simple diamond network into a double diamond or a double gyroid network also reveal that the amount of trapped entanglements introduced that way is small enough to preserve the sawtooth behavior reported here. One general area of exploration consists in making the model

network more realistic by adding imperfections, such as chainlength polydispersity, dangling chains, or creating networks with tetrafunctional connectivities that arise from a realistic chain-end cross-linking process.47,48 It is unclear to what extent the sawtooth type of elastic responses may be preserved under such conditions. In terms of block copolymers, one could explore the effect of different A:B compositions (away from the 50:50 explored here) and hence of nonlamellar morphologies on elastic behavior. It would also be of interest to simulate the shock response of block copolymer networks (with coarsegrained potentials calibrated to mimic known materials) to assess their shock-energy absorption ability.83 Research along these lines is under way.

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ASSOCIATED CONTENT

S Supporting Information *

Movie files corresponding to the simulated deformations of networks described in Figures 5 and 6. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

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

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Professor R.C. Hedden for insightful discussions. This work was supported by the National Science Foundation Polymers Program under Grant DMR-0705565 and by the American Chemical Society Petroleum Research Fund.

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REFERENCES

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