Diffusion Mechanisms of Entangled Rod–Coil Diblock Copolymers

Jul 12, 2013 - ABSTRACT: Mechanisms of entangled rod−coil diblock copolymer diffusion are investigated using tracer diffusion simulations and experi...
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Diffusion Mechanisms of Entangled Rod−Coil Diblock Copolymers Muzhou Wang, Ksenia Timachova, and Bradley D. Olsen* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States S Supporting Information *

ABSTRACT: Mechanisms of entangled rod−coil diblock copolymer diffusion are investigated using tracer diffusion simulations and experiments in a matrix of entangled coil homopolymers, demonstrating that the diffusion mechanisms first identified in coil−rod−coil triblocks are universal for various molecular architectures. Diffusion measurements were performed using both Kremer− Grest molecular dynamics simulations and forced Rayleigh scattering experiments. In the large rod regime, diffusivity decreases exponentially with increasing coil size as predicted by an arm retraction mechanism. The ratio of diblock to rod homopolymer diffusivity was approximately equal to the ratio for triblocks squared, suggesting that the two coil blocks of the coil−rod−coil triblock relax independently. In the small rod regime, both experiments and simulation show that the slowing of diffusion with increasing rod length is the same for rod−coil diblock and coil−rod−coil triblock copolymers. This behavior occurs because both types of molecules reptate through tubes with Gaussian statistics, so diffusional slowing results from entropic barriers to rod reptation through curved sections of the entanglement tube.

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INTRODUCTION Rod−coil block copolymers have attracted extensive interest as functional nanostructured materials for organic electronics1,2 and biomaterials.2−4 The self-assembly of these copolymers is fundamentally different from coil−coil block copolymers due to the mismatch between rod and coil chain topology and anisotropic interactions between the rod blocks.2,5−7 This rigidflexible mismatch is a direct consequence of incorporating functional motifs such as extended π-conjugation8 and polypeptide α-helices9 into traditional block copolymers.10 Functional rodlike polymers have been incorporated into both rod−coil diblocks1,2 and coil−rod−coil triblocks,7,11,12 and the equilibrium thermodynamics of both types of molecules continues to be widely investigated. However, dynamic effects that are important for mechanics, processing, and self-assembly kinetics have only been explored in a few prior studies.13−16 Just as the rod’s rigidity leads to liquid crystalline and packing geometry effects that have profound implications for rod−coil self-assembly,5−7,17 the difference in rod and coil chain topologies leads to scientifically rich dynamic phenomena. For entangled homopolymers, this difference has been shown to create divergent scaling behaviors for entangled rods18−20 (isotropic self-diffusion D ∼ M−1, relaxation time τr ∼ M9) and coils19,21−23 (D ∼ M−2.3, τr ∼ M3.4). While scaling exponents in coil tracer diffusion experiments have been observed ranging from D ∼ M−2 to D ∼ M−3,24−26 the nature of dynamic entanglement is fundamentally different between rods and coils. Previous studies of rod−coil block copolymer dynamics have used rheology to empirically identify order−disorder transitions,13,14 compare molecular architectures,14 and measure intrinsic viscosities.15 Static and dynamic structure factors have also been analytically calculated for rod−coils in dilute solution.16 © XXXX American Chemical Society

Recently, we have contributed to the knowledge of rod−coil dynamics with a fundamental study combining theory, simulation, and experiment to investigate the dynamic relaxation mechanisms of entangled coil−rod−coil triblock copolymers.27,28 The entangled regime is particularly important for melt processing of copolymers with moderate to high molecular weight. Coil−rod−coil triblocks were chosen as a first system for investigation because they have a symmetric molecular structure, simplifying the theoretical treatment. Diffusion measurements of coil−rod−coil triblock copolymers in entangled coil homopolymer matrices showed that when rod and coil blocks are incorporated into the same molecule, the diffusion of the resulting copolymers is slower than either rod or coil homopolymers of the same total molecular weight.27 This effect was observed in diffusion measurements by molecular dynamics simulation and forced Rayleigh scattering experiments. The slowing is hypothesized to result from a mismatch between the curvatures of the entanglement tubes of the rod and coil blocks, because the rod’s tube is straight while the coil’s tube has a characteristic curvature of order a, the entanglement length. In the small rod limit where the rod length L is of order a, reptation of the coil block is hindered due to free energy barriers as the rod moves through curved sections of the entanglement tube. These barriers arise from the entropy decrease of deforming surrounding chains to accommodate the straight rod in a curved tube. Reptation slows as the characteristic free energy barrier height increases with increasing L/a, so diffusion of coil−rod−coils is slower than Received: March 28, 2013 Revised: June 23, 2013

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Figure 1. (a) Synthesis scheme for dye-labeled rod−coil block copolymers. (b) SDS-PAGE showing the purity of the rod−coil polypeptide-b-PEO tracers synthesized for this study. Faint bands at low molecular weight that correspond to unreacted polypeptide are barely visible.

coil homopolymers. In the large rod limit where L ≫ a, rotation of the rod is severely hindered, so the coil block must adopt configurations parallel to the rod by an arm retraction mechanism in order for diffusion to occur. This causes diffusivity to decrease exponentially with increasing coil size; hence diffusion of coil−rod−coils is slower than rod homopolymers. Previous experiments and simulations have identified both diffusion regimes in coil−rod−coil triblock copolymers.27,28 Since both regimes involve the motion of one block to accommodate the characteristic curvature of the entanglement tube of the other block, we refer to this dual mechanism in rod−coil block copolymer diffusion as the curvature mismatch hypothesis. Varying the molecular architecture of rod−coil block copolymers may have a significant impact on these dynamic mechanisms. Coil−rod−coil triblocks were a convenient starting point to investigate curvature mismatch in the entangled state because the coil blocks at both ends of the molecule explore the surrounding space according to the motion of their Gaussian chain segments. For rod−coil diblocks, the rod and coil chain ends will explore space according to their individual dynamics, and it is unclear how this conformational and dynamical asymmetry will affect diffusion. Given the technological importance of rod−coil diblock copolymers and the questions that arise from molecular asymmetry, the detailed analysis of diffusion in these molecules is an interesting and important challenge. In this work, tracer diffusion of rod−coil diblock copolymers in entangled coil homopolymer solutions is studied by molecular dynamics simulation using the Kremer−Grest model29 and by experiment using forced Rayleigh scattering30−32 (FRS) of a model diblock composed of a poly(ethylene oxide) (PEO) coil and an alanine-rich α-helical polypeptide. Both the simulation and experimental data are compared to the previous coil−rod−coil triblock results, and extensions to the curvature mismatch hypothesis are proposed to explain the similarities and differences of the two molecular architectures.

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similarly to the previously described coil−rod−coil triblocks.28 The rodlike polypeptides are oligomers of the B repeat unit with length of 3.3 nm and molecular weight of 1982 Da and residue sequence AAAQAAQAQAAAEAAAQAAQAQ.33 The gene sequences originally used for synthesis of triblocks were modified by site-directed mutagenesis to replace the lysine residue at the C-terminus with glutamine. This resulted in only one primary amine for attachment of the coil block at the N-terminus, ensuring that only diblock copolymers are formed during bioconjugation. The full amino acid sequences of the polypeptides are shown in the Supporting Information. The genes were then expressed in Escherichia coli using the SG13009(pREP4) strain in Terrific Broth. After induction with isopropyl β-D-1-thiogalactopyranoside and harvesting by centrifugation, the hexahistidine-tagged polypeptides were purified from the cell lysate by metal affinity chromatography under denaturing and reducing conditions. The two cysteine residues near the N- and C-termini of the polypeptides were modified with a maleimide functionalized 4′-(N,Ndimethylamino)-2-nitrostilbene (ONS) photochromic dye for diffusion measurement by FRS. N-hydroxysuccinimide functionalized PEO (40 kDa) was then attached to the N-terminal amine in a buffer containing 8 M urea and 100 mM NaH2PO4 (pH = 8.0). Unreacted polypeptide and PEO were removed by anion exchange chromatography using buffers containing 6 M urea and 20 mM ethanolamine (pH = 10.0), eluting with a gradient of 0 mM to 200 mM NaCl. The diblock tracers are listed in Table 1 and their purity was verified by SDS-PAGE (Figure 1b).

Table 1. Rod−Coil Diblock Copolymer Tracers name

rod Mn (kg/mol)

rod length L (nm)

coil Mn (kg/mol)

coil PDI

total Mn (kg/mol)

PEO-B3 PEO-B4 PEO-B6 PEO-B9

8.1 10.1 14.4 19.9

9.9 13.2 19.8 29.7

40 40 40 40

1.02 1.02 1.02 1.02

48 50 54 60

Forced Rayleigh Scattering Measurements. Tracer diffusion measurements of the molecules listed in Table 1 in entangled PEO homopolymer solutions were performed using forced Rayleigh scattering. The PEO homopolymer (Mn = 387 kDa, PDI = 2.52) from Sigma (St. Louis, MO) initially contained insoluble stabilizing agents that were removed by dissolving it in water, centrifuging, and then lyophilizing the supernatant. The matrix molecular weight was chosen to be significantly higher than the tracer molar mass to minimize the effect of constraint release. Samples were prepared by dissolving the proper quantities of lyophilized tracers and the PEO

METHODS

Synthesis of Rod−Coil Diblock Copolymers. As summarized in Figure 1a, model rod−coil diblocks composed of rod-like, alanine-rich α-helical polypeptides and coil-like PEO chains were synthesized B

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Figure 2. (a) Diffusion of rod−coil diblock and coil−rod−coil triblock copolymers by molecular dynamics simulation in a cubic lattice, where monomers are prohibited from crossing the lattice edges. The total tracer size was N = 200, the lattice spacing was a = 10σ, and the length of the rod was varied between 0 and 200 monomers. (b) Data from the arm retraction regime show an exponential dependence on Nc for diblocks and triblocks was plotted against Nc/a2 for different lattice spacings (a = 6, 8, and 10σ), where Nc is the size of a single coil block. homopolymer in excess Milli-Q water. These mixtures were then lyophilized and redissolved in 10 mM NaH2PO4 (pH = 7.0) to matrix weight fractions ranging from 25 to 50%. The final tracer concentration was such that the number density of rods was less than one per (40 nm)3 volume. Since the longest rod investigated was L = 30 nm, the tracer concentrations were sufficiently dilute to ensure minimal rod−rod interactions. The entangled polymer solutions were pressed between two circular quartz crystal slides with a 1 mm thick Teflon spacer in a brass holder19 and allowed to equilibrate for at least 24 h before measurement. The forced Rayleigh scattering measurements were then performed as previously described.28 Molecular Dynamics Simulation. Tracer diffusion simulations of rod−coil diblock copolymers were performed by tracking meansquared displacement in the Kremer−Grest model29 as previously described.27 The characteristic distance σ and time τ arise from nondimensionalizing the equations of motion, and the average separation between adjacent monomers is 0.97σ. For measurements in the small rod regime, rod−coils of total size N = 200 were equilibrated in entangled coil polymer melts of N = 1000 monomers,34 with tracer density less than L−3 to minimize rod−rod interactions. For each rod length, 760 tracers were simulated by dividing them into smaller independent simulation boxes to reduce computational time. For measurements in the large rod regime, the surrounding entangled melts were replaced with a fixed infinite cubic lattice and each tracer was simulated individually.27 The excluded volume potential was enforced between all pairs of tracer monomers, as well as between each monomer and the 12 edges of its unit cell. Rods were equilibrated parallel to the lattice axes to preserve the expected tube diameters, which is defined as the lattice spacing a. Diffusion was measured based on an ensemble of 1600 tracers.

rod−coil diblock copolymers will be presented in the framework of these dual mechanisms, with each regime analyzed individually. Large Rod Regime. Molecular dynamics simulations performed in a fixed cubic lattice of line obstacles of spacing a confirmed that rod−coil diblock copolymers diffuse more slowly than either rod or coil homopolymers (Figure 2a). At longer rod lengths the excluded volume interactions between the monomers and the edges of the fixed lattice prohibit rod rotation, which geometrically guarantees the dynamic criterion for the large rod regime for rod length L ≫ a by preventing rod rotation through the coil tube with Gaussian statistics. In this range, the simulations confirm that diffusion decreases exponentially with coil size, which is consistent with the arm retraction mechanism. This behavior was previously observed for triblocks in the long rod regime.27 As the rod length is decreased further, the rod is able to rotate within the lattice, and a minimum in the diffusivity is observed. This behavior indicates a crossover between the arm retraction and activated reptation (small rod regime) mechanisms. At rod lengths below this crossover the diffusion depends on the rod length rather than the coil size; this small rod regime behavior will be discussed in detail in the next section. A comparison of rod−coil diblocks with coil−rod−coil triblocks shows that the exponential slowing of diffusion is governed by the degree of polymerization of a single coil block rather than the total number of coil monomers. In the MD simulation, the total size of the tracers was held at a constant N = 200 and diffusion was measured for rods of various lengths at either the end or the center of the tracers. For tracers of the same rod length and same N = 200 total degree of polymerization, the ratio of diblock to rod homopolymer diffusivity was approximately the ratio for triblocks squared (Figure 2a). Because triblocks contained two coil blocks of half the size of the diblock’s single coil, this suggests that the exponential parameter Nc in the arm retraction mechanism is the size of a single coil block rather than the total coil molecular weight. This is shown below, where Dr0 is the diffusivity of a rod homopolymer of the same total degree of polymerization, Nc is the molecular weight of a single coil block, a is the entanglement length, and ν is an order unity prefactor. Because Nc for a diblock is twice Nc for a symmetric triblock of the same total size, Ndic = 2Ntri c implies

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RESULTS AND DISCUSSION Previous experimental, simulation, and theoretical studies of entangled rod−coil block copolymer dynamics have suggested two competing mechanisms of molecular motion,27,28 where the relative contributions of each mechanism to the overall dynamics depend on the molecular parameters of the tracer and the surrounding matrix. In the small rod regime, reptation of the rod block though the coil tube is fast compared to arm retraction of the coil block, and the effect of the rod may be conceptualized as a perturbation on coil reptation. In the large rod regime, rod reptation is slow compared to coil arm retraction, so diffusion must occur primarily parallel to the director of the rod. The effect of the coil is thus a perturbation on the motion of the rod. Both of these mechanisms result in slower diffusion compared to the respective coil and homopolymer cases. The results and theoretical analysis of C

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characteristic monomer length as defined by ⟨R2⟩ = Nb2, the triblock scaling simplifies to

⎛ Dtri ⎞2 D di di 2 tri 2 ∼ exp( −νNc /a ) ∼ exp(− 2νNc /a ) ∼ ⎜ r ⎟ D0r ⎝ D0 ⎠

D ∼ exp( −νNc /a 2) exp(−ν /b2)

which is observed in Figure 2a. This definition of Nc as the size of a single coil block implies that the coil blocks of a coil−rod− coil triblock relax independently. When the diffusivities from various values of a for both molecular architectures are plotted against Nc/a2 defined for a single coil block, the data from both triblocks and diblocks have the same slope which indicates that the prefactor ν is unaffected by the molecular architecture (Figure 2b). While the rescaled diblock and triblock data collapse quantitatively when Nc/a2 is small, the triblock diffuses slightly slower than the diblock at larger values of Nc/a2 (by approximately a factor of 2). Since the direct rescaling assumes that arm retraction of only one coil block at a time is necessary for diffusion of the triblock, the minor discrepancy may be explained by an additional relaxation of the second coil block. After one coil block retracts, motion of the rod past one entanglement length requires the unretracted coil block to straighten in the vacated space behind the rod (Figure 3). This

The value of the additional term can be directly calculated for the molecular dynamics simulation. Linear regression on the triblock data from Figure 2b for Nc/a2 > 0.5σ−2 (to reduce end effects near zero coil block size) provides ν = 3.20 ± 0.31σ2, and b = 1.99σ is determined from the end-to-end distance of an N = 200 coil homopolymer with a = 8σ. This gives a value of the additional coil block term of exp(v/b2) = 2.24, which approximates the discrepancy between diblocks and triblocks at large values of Nc/a2 where end effects are minimized. As this straightening effect only adds an extra order 1 prefactor to the triblock diffusion, the underlying molecular mechanism of independent coil block relaxations by arm retractions should be unaffected. Because the arm retraction mechanism is analogous to the motion of star polymers, with the rod block’s hindered rotation replacing the fixed position of the star polymer’s branch point, the comparison of diblock and triblock diffusion is related to the dependence of diffusion on the number of arms in star polymers. Experimental measurements have shown that the diffusion of many-armed star polymers do not require the simultaneous retraction of multiple arms.35 Hypotheses explaining this effect allow the branch point to move within a local area of constraints upon retraction of a single arm,35 or when all arms pass through two gates in the branch point’s cell.36 These theories are analogous to this analysis of the rod− coil diblock and coil−rod−coil triblock diffusion, allowing the rod block to move within a small area before the second coil block relaxes in the triblock case. The arm retraction diffusion mechanism predicted from simulation is experimentally observed by forced Rayleigh scattering measurements on the PEO−polypeptide diblocks. From the raw tracer diffusion data at various rod lengths and matrix concentrations (Supporting Information), the dependence of diffusion on a was determined using a = 3.5 nm for the melt state from linear rheology37,38 and the known scaling of a ∼ c−2/3 for semidilute Θ solutions.19,39 For rod lengths L ≥ 20 nm, the relationship between log D and a−2 is linear for both diblocks and triblocks,28 confirming diffusion by an arm retraction mechanism (Figure 4a). Because the size of individual coil blocks is equal for diblocks and triblocks in

Figure 3. Unlike diblock copolymers which have only one coil block, the second coil block of coil−rod−coil triblock copolymers contributes an additional relaxation to a typical diffusion process. Starting at an equilibrium configuration (1), one coil block retracts to form a straight line parallel to the rod (2). The rod then moves toward the retracted coil block, causing the other coil block to stretch in the region highlighted in light blue (3). Both coil blocks then relax in the new rod position (4).

straightening contributes an additional term to the scaling relationship for triblocks compared to diblocks, D ∼ exp(−vNc/ a2) exp(−vNa/a2), where Na is the number of monomers in a chain of end-to-end distance a. Since a2 = Nab2 where b is the

Figure 4. (a) Diffusivity of the L ≥ 20 nm tracers for both diblocks and triblocks vs 1/a2. The slopes of all data sets are equal to within experimental error, consistent with diffusion by arm retraction for both diblocks and triblocks. (b) Triblock data shifted by a factor of (Mtri/Mdi)1.8 to reflect the expected scaling from the literature shows collapse of diblock and triblock data onto a common curve, suggesting universal diffusion behavior for the rod−coil and coil−rod−coil architectures. D

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the experimental model system, the factor νNc should also be equal, meaning log D vs a−2 should have the same slope. The diffusion measurements for diblocks have slopes of −28.6 ± 0.7 nm2 for L = 20 nm and −28.1 ± 0.7 nm2 for L = 30 nm, which is within experimental error of −29.0 ± 0.5 nm2 and −28.6 ± 0.5 nm2 for triblocks. While the slopes of the diblock and triblock data are equivalent, the diblock diffusivities are consistently higher because of their lower total moleculear weight. This difference can be corrected using the diffusivity prediction in the long rod regime of D ∼ Dr0exp(−vMc/a2), where Dr0 is the diffusivity of a rod homopolymer of the same total degree of polymerization, and Mc is the molecular weight of a single coil block. Quantitative estimation requires the scaling exponent for rod homopolymer diffusion vs molecular weight. However, diffusion measurements of the rod homopolymers in this study were outside the range of the experimental apparatus. Instead, the triblock data was vertically shifted to compare with the diblock results using the experimentally observed Dr0 ∼ M−1.8 exponent for the self-diffusion of the structurally similar poly(γ-benzyl glutamate) in isotropic pyridine solutions (Figure 4b).40 Although the exponent was obtained from a different experimental system, remarkably good collapse of the data was observed at the largest L = 30 nm rod length. This supports the predicted D ∼ Dr0 exp(−vMc/a2) scaling in the long rod regime. The inherent asymmetry of diblocks requires careful analysis of the proposed diffusion mechanisms in both the short and long rod regimes to guarantee that diffusion is in fact symmetric in the rod and coil directions, as required by the second law of thermodynamics. The rod encounters the coil’s curved entanglement tube only as it moves in the coil direction, but it can diffuse in the rod direction without rod rotation or arm retraction. This asymmetry implies that the molecule is more likely to diffuse in the rod direction than in the coil direction, and the rod−coil can avoid curvature mismatch in this manner. In the long rod regime, the lack of rod rotation guarantees that diffusion can only occur through arm retractions or coil extensions as shown by the simulation and experimental data on diblocks. Therefore, while the coil must maintain its equilibrium ensemble of Gaussian configurations, only those configurations that are confined in tubes parallel to the rod have sufficient mobility to diffuse. These straightened configurations are equally likely to diffuse in both the rod and coil directions, confirming that diffusion is directionally symmetric even though the molecular architecture is highly asymmetric. While diffusion is directionally symmetric in the long rod regime when the coil is confined to a straightened tube, an equilibrium ensemble of Gaussian coil configurations should also be equally likely to diffuse in the rod and coil directions when diffusion is considered on the submolecular length scale of the reptation tube diameter, a. Considering the classic reptation model of diffusion through a cubic lattice with no contour length flucutations and no rod rotation (Figure 5), the probability of diffusing in either the rod and coil direction is proportional to the product of the number of ways to form a new tube segment at the free end of the chain multiplied by the fraction of coil configurations in which the rod can enter the coil’s tube, or vice versa. For diffusion in the direction of the coil, there are 5 possible configurations for new tube in the cubic lattice, but the rod can only diffuse into the coil tube in 1 /5th of the coil configurations where the tube segment immediately adjacent to the rod is oriented parallel to the

Figure 5. Reptation of a rod−coil block copolymer on a lattice illustrates that motion is symmetric in both directions. In the pictured configurations a and b, the molecule can only reptate to the right because the coil tube segment adjacent to the rod is not parallel to the rod. However, in configuration c, the molecule is more likely to reptate to the left because the coil end has more ways to explore space than the rod end. These two effects balance to guarantee that diffusion is directionally symmetric even on the length scale of the tube diameter.

direction of rod reptation. For diffusion in the direction of the rod, the rod can only generate new tube in a single direction, and the coil can enter the rod tube in all configurations. In both cases these effects multiply to an equal probability of diffusion in the rod and coil direction. While the physical origin of the 1 /5 term is the hindered motion of a rod in the coil’s tube associated with reptation barriers from curvature mismatch, the 5 term arises from the larger number of new tube segments available at the coil end. This effect is equivalent to an effective equilibrium end “tension” that appeared in the original Doi− Edwards treatment coil homopolymer reptation, and arises from the projection of a 3-dimensional random walk onto a 1dimensional tube.19,22,41 This end tension that favors motion in the coil direction is exactly balanced by the reptation barriers that favor motion in the rod direction. This argument easily generalizes to off-lattice diffusion by replacing the number of lattice directions with an accessible solid angle for the possible directions of rod diffusion given its rotational constraints. Finally, we note that the slowing of diffusion is the same between diblocks and triblocks at the shortest rod lengths in the simulation data (Figure 2a). While the fixed lattice condition is a poor approximation to an entangled matrix in this limit, this result provides an important clue to the behavior to be examined in more detailed simulations and experiments in the short rod regime. Small Rod Regime. Tracer diffusion was simulated using the Kremer−Grest model and measured in experiment by forced Rayleigh scattering in the small rod regime, where reptation of the rod block is fast compared to arm retractions of the coil block. To compare theory and simulation, the raw data E

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Figure 6. (a) Simulation and experimental diffusion measurements for diblock copolymers normalized against coil homopolymer diffusion and entanglement length. The experimental data is derived from various concentrations of the matrix from Figure S1b (Supporting Information). (b) Simulation and experimental data for rod−coil diblocks show similar trends and quantitative values to the corresponding data for coil−rod−coil triblocks.

for the smaller rod lengths of L ≤ 20 nm at various matrix concentrations (Supporting Information) were nondimensionalized using a characteristic diffusivity D0 from previous data and scaling (D0 ∼ M−2.4c−3.5) of a coil homopolymer of the same molecular weight28 and a characteristic length a from the entanglement length of the surrounding matrix. For the simulation data, a = 10.7σ was estimated from primitive path analysis.42,43 For the experimental data, entanglement lengths were estimated from linear rheology and scaling as described for the large rod regime. The agreement between diblock copolymer simulation and experiment from various matrix concentrations is reasonable in the small rod regime where L/a ∼ 1 (Figure 6a), with minor discrepancies from possible errors in estimating entanglement length and the confining potential of the entanglement tube.28 For the L = 20 nm tracer corresponding to L/a > 2.5, the data also follows the arm retraction scaling as previously shown, and thus begins to cross over into the long rod regime. When compared with the corresponding measured and analyzed data for coil−rod−coil triblock copolymers, the simulations and experiments show similar normalized diffusivities for both copolymer structures (Figure 6b). Simulations of diblocks and triblocks are in quantitative agreement, while the experimental data for diblocks and triblocks follow the same qualitative trend and have similar values. This surprising agreement between diblocks and triblocks suggests the key predictions of the curvature mismatch hypothesis originally developed for coil−rod−coil triblocks apply equally to diblock copolymers. The agreement between diblocks and triblocks is explained by examining the time scales and mechanisms of diffusion in both cases. In the triblock copolymer case, arm retraction of the coil block is much slower than rotation of the rod block in the small rod regime, so the rod reptates along an activated path defined by the coil. Rotation of the rod in the triblock copolymer is distinct from the rotation of entangled free rods. In the reptation treatment of entangled isotropic rod homopolymers,19,20,44 small rotations within the tube occur by fast unentangled dynamics, while larger rotations are constrained by the surrounding obstacles. Thus, rotation and translation time scales were intimately linked because the rod is only able to rotate an angle of a/L upon diffusing a length of L/ 2. However, conjugation to coil blocks requires the rod to rotate as it reptates through the coil’s entanglement tube, so a triblock rod must orientationally decorrelate on the same length scale as the coil’s tube, a. For sufficiently long rods (L > a), a is smaller than the length scale required for the rotation of

a free rod, so rotation of the rod block requires deformation of the surrounding entanglements. The entropic penalty associated with these deformations, whose magnitude depends upon the varying local curvature of the coil blocks’ entanglement tube, leads to the previously identified activated reptation surface. Since the orientation of the entanglement tube decorrelates over a length scale a, this mechanism controls both the rotational relaxation time of the block copolymer rod and the time required to reptate a length a along the tube. The MD simulation on a triblock tracer confirms that the rod block rotation and reptation time scales are equivalent in the small rod regime (Figure 7).

Figure 7. Rotational correlation function ⟨u(t)·u(0)⟩, where u is the orientation vector of the rod of length L = 16σ in a rod homopolymer, diblock, or triblock of N = 200 monomers. The arrow indicates the time required for the diblock and triblock tracers to diffuse a distance of a, which corresponds to the rotational relaxation time of the triblock (where ⟨u(t)·u(0)⟩ ≈ 1/e). The diblock rod relaxes faster because of its position at the end of the molecule.

While the equivalence of rotation and reptation is readily observed for coil−rod−coil triblocks, the analogous MD simulation suggests that the rod block in rod−coil diblocks rotates much faster than the rate of reptation (Figure 7). This is consistent with the idea that disengagement of a reptating polymer begins at the chain ends. Thus, diblock rods rotate faster because the polymer ends disengage from the entanglement tube before the center of the polymer reptates, resuling in a loss of the rod’s orientational correlation. While a triblock rod must reptate into the tube defined by coil blocks on both ends, the diblock rod explores many possible tubes along its free end. In the time required for the molecule to reptate a distance a, many rotational relaxation times and thus F

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Author Contributions

the diblock rod’s orientation is completely decorrelated. Therefore, the rod reptates into new tubes according to Gaussian statistics (new tube segments are directionally isotropic). This conclusion is consistent with the equilibrium Gaussian statistics of the coil block, which are maintained only if the molecule reptates along a Gaussian tube in both directions. Furthermore, the slowing of diffusion is a direct result of reptation of the rod block along the coils’ entanglement tube according to Gaussian statistics. Because the tubes are hypothesized to be Gaussian in both diblocks and triblocks, the slowing in diffusion should be quantitatively equal in both cases, as observed by the simulations and experiments. Finally, it is important to note that the directional symmetry of diffusion is preserved in the small rod regime despite the molecular asymmetry of the diblock copolymers. Because of the rapid orientational decorrelation of the rod, both polymer ends explore space according to the same directional statistics, meaning there is no effective end tension. Furthermore, diffusion within a Gaussian tube implies that the entropic barriers presented by the surrounding entanglements are equivalent for reptation in both the rod and coil directions, meaning there is no directionality from the barriers. Thus, the slowed reptation minimizes both effects and is consistent with molecular diffusion at equal rates in the rod and coil directions, as required by equilibrium thermodynamics.

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the National Science Foundation. We thank K. D. Wittrup and J. Mata-Fink for assistance with protein chromatography, K. J. Prather for equipment for DNA cloning, and A. Alexander-Katz for helpful discussions. Simulations were performed on the Kraken cluster at NICS through a generous XSEDE allocation TGDMR110092 as well as a local cluster administered by the T. A. Hatton group. M.W. acknowledges funding through a NDSEG Fellowship, and K.T. acknowledges funding through the MIT UROP Office.

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CONCLUSIONS Molecular dynamics simulation and forced Rayleigh scattering measurements showed that the tracer diffusion of entangled rod−coil diblock copolymers occurs by an arm retraction mechanism in the long rod regime (arm retraction time much shorter than rod rotational time) and by an activated reptation mechanism in the short rod regime (arm retraction time much longer than rod rotational time). These results were compared to previous data on coil−rod−coil triblock copolymers, showing that both molecular architectures follow the same diffusion mechanisms. In the large rod regime, diffusivity decreases exponentially with increasing coil size, and each coil block relaxes independently to within a minor correction factor. In the small rod regime, both experiments and simulation showed that the slowing of diffusion is quantitatively the same between diblocks and triblocks, meaning the curvature mismatch hypothesis originally developed for triblock copolymers directly applies to the diblock case. Finally, it was shown that the entropic barriers resulting from the rod being in the coil’s tube are balanced by an effective tension at the coil end, resulting diffusion that is directionally symmetric even though the molecular architecture of the rod−coil diblock is asymmetric. This balance is preserved in the mechanisms of both the small and large rod regimes.

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ASSOCIATED CONTENT

S Supporting Information *

Polypeptide amino acid sequences, and raw diffusion data from molecular dynamics simulations and forced Rayleigh scattering measurements. This material is available free of charge via the Internet at http://pubs.acs.org.

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