Using a coarse-grained slip-spring model, the dynamics of rod-coil block copolymers is explored over a wide parameter space to fully capture the crossover between the short rod (activated reptation) and long rod (arm retraction) limits. An analytical, closed-form expression for curvilinear diffusion by activated reptation was derived by separating the drag into individual components for the rod and coil block. Curvilinear diffusion in the intermediate rod regime, where both mechanisms are important, was then found to be faster than predicted when both mechanisms are independently combined. The discrepancy in the crossover regime arises because the rod-coil copolymer’s exploration of space is not accurately described by either a coil homopolymer (assumed by activated reptation) or a rod homopolymer (assumed by arm retraction). This effect is explored by tracking the rod orientation as the polymer reptates, confirming that the polymer reptates along a path that becomes more rodlike as the rod fraction is increased. Thus, activated reptation under-predicts diffusion because the rod can choose reptation paths that are more extended than the coil homopolymer by renewal of the entanglement tube from the ends. Arm retraction under-predicts diffusion because minor rotations of the rod allow some motion before full retractions of the coil block. Finally, more familiar 3-dimensional center-of-mass diffusion measurements are related to the curvilinear diffusion analysis because the ratio of these two quantities varies smoothly between the coil and rod homopolymer limits as the reptation path becomes more extended.
Rod-coil block copolymers have attracted interest in the soft matter and polymer physics community for applications in functional nanostructured materials and as a comparatively simple example of a wider class of materials composed of polymers with complex chain shapes.1,2 The combination of these distinct shapes into hybrid molecules3,4 introduces new functionality such as electronic properties5,6 or biological activity7–9 into traditional polymers. These shapes also result in self-assembly behavior that depends on packing entropy10 and liquid crystalline interactions11 in new ways that can lead to interesting thermodynamic effects on material structure. In addition to thermodynamics, dynamic properties are important to understand for predicting material mechanics and designing manufacturing processes.12 For rod-coil block copolymers, entangled dynamics is particularly interesting because reptation theories and experimental measurements have shown that the nature of dynamic entanglement is quite different between rods and coils, leading to divergent scaling behaviors.13–18 A theory for the dynamics of these molecules is important for technological applications and must account for the new physics that directly arises from the motion of multiple domains of dissimilar geometries within the same molecule.
Recent studies by the authors have proposed a reptation theory of entangled rod-coil copolymers supported by simulation and experimental results. It has been shown that dual relaxation mechanisms govern the rod and coil homopolymer limits of copolymer composition due to the mismatch between the curvatures of the entanglement tubes of rod and coil polymers.19–21 In the small rod limit where the rod is a perturbation on coil motion, the randomly varying curvatures of the coil’s tube present entropic barriers to the reptation of the rod, modifying the unhindered motion of the coil along its tube to become an activated reptation process. In the large rod limit where the coil perturbs the rod motion, the long rod cannot rotate around the surrounding entanglements, so motion is only possible when the coil moves into a straightened entanglement tube in an arm retraction process. The predictions of these mechanisms for copolymer diffusivity have been verified using Kremer-Grest molecular dynamics (MD) simulations and experimental diffusion measurements by forced Rayleigh scattering. This reptation theory has been applied to tracer diffusion of coil-rod-coil triblock copolymers,19,20 tracer diffusion of rod-coil diblock copolymers,21 and self-diffusion of rod-coil diblocks in isotropic disordered melts.22
While there has been significant progress in understanding the physics of entangled rod-coil block copolymer relaxation mechanisms, further developments in the theory are required to provide quantitative predictions of diffusion and other dynamic properties. This is because the previous work has focused on the mechanisms of activated reptation and arm retraction individually at the limits of rod fraction equals zero or one, while a complete theory must quantitatively unify these two phenomena over the entire parameter space. Such a task has previously been challenging because experimentally, the range of molecules that can be precisely synthesized and measured is limited, and MD simulations are computationally expensive. An analytical closed-form prediction for diffusion remains elusive even for the activated reptation mechanism in isolation, because the amount of data has been insufficient and over a too narrow parameter space to determine empirical fits. Quantifying both the activated reptation and arm retraction mechanism in combination also requires more fundamental knowledge of how entanglement influences two domains that reptate very differently, such as how the shape of the entanglement tube varies with rod fraction or how the path that the polymer follows through space is related to this tube shape. Understanding these ideas would enable a more complete description of entangled rod-coil block copolymers and allow quantitative predictions of dynamics over the entire parameter space for this important class of polymers.
The recent development of slip-link models for entangled polymer dynamics can overcome the computational limitations and enable the study of rod-coil block copolymers over a much larger parameter space. In the original slip-link models applied to coil homopolymers,23 the reptation tube was discretized into a series of fixed points and a single tracer chain is allowed to move through them. Subsequent models contain additional features such as coarse-graining the chain positions between entanglements,24 accounting for the softness of entanglements using “slip-springs,”25 or simultaneously simulating multiple chains,26 while striving to match experimental and more detailed computational results.24,25,27–29 Recently, we have applied these models to study the tracer diffusion of coil-rod-coil triblock copolymers, allowing the activated reptation mechanism to be directly observed for the first time.30
In this study, a rod-coil slip-spring model is used to quantitatively investigate activated reptation and arm retraction both individually and in competition with each other. This model is applied to the tracer motion of coil-rod-coil triblock copolymers in a high molecular weight coil homopolymer matrix, as a model system suited for understanding the relative effects of these relaxation mechanisms. First, the model is reviewed and an algorithm for calculating curvilinear diffusion along the tube defined by the connected anchor points is described. The slip-spring model is effective for observing curvilinear diffusion because the entanglement tube can be rigorously defined by construction. This is in contrast to explicit MD, where many algorithms exist for defining the tube, each with its own advantages.31–36 Using data from this model, an empirical closed-form expression for curvilinear diffusion in the activated reptation mechanism is then derived. This analytical form is considered in combination with arm retraction in the intermediate rod regime, where both mechanisms play a significant role in dynamics. In this regime, the rod-coil moves in space along a path that becomes more extended as the rod length increases. This reptation path depends on a combination of the coil size and the rod length, and its relationship with other definitions of tube shape is explored. Finally, the analysis and discussion of curvilinear diffusion are related to center-of-mass diffusion, providing a connection to experimental measurements.
II. SLIP-SPRING MODEL OF ROD-COIL BLOCK COPOLYMERS
This study uses a single-chain slip-spring model of rod-coil block copolymers inspired by Likhtman’s work on coil homopolymers.25 The model was recently introduced30 for analysis of the tracer motion of coil-rod-coil triblock copolymers in a high molecular weight coil homopolymer matrix, which allows the effects of constraint release to be ignored. Coil blocks consist of Rouse chains connected to the ends of a rigid thin rod block, with entanglements simulated by slip-springs that connect fixed anchor points aj with Hookean springs to slip-links sj that reside on monomers. The total energy for a chain of N monomers is given by
where ri are the positions of the coil monomers and the rod ends, b is the coil statistical segment length, Nsb2 is the mean-square extension of a slip-spring, and Z = N/Ne is the number of slip-links per chain, with Ne as the average number of monomers between slip-links. Chains are simulated using Brownian dynamics with a coil monomer friction ζ. b serves as the characteristic unit length, and the unit time is τ = ζb2/(kBT). Slip-links move between monomers by a Monte Carlo scheme, with a trial move of one step between adjacent monomers occurring every 0.002τ with a Metropolis acceptance criterion. Only a single slip-link is allowed on a monomer at any time, preventing the crossing of entanglements. To simulate the tube renewal processes, slip-links that move off a chain end are destroyed. At least 20 chains are simulated simultaneously, so after each destruction event, a new slip-link is created at the end of a randomly selected chain to preserve the total number of entanglements. Constraint release events are neglected in order to isolate the effects of activated reptation and arm retraction.
The parameters of the slip-spring model were selected to quantitatively match observables from the Kremer-Grest MD simulation results,37 where tracer rod, coil, and coil-rod-coil polymers were observed in a matrix of coil homopolymers with 1000 monomers. This matching procedure was previously described in detail30 and is highlighted here for clarity. To match equilibrium distances and mobilities between the two models, scale factors for length and time are b = 1.32σMD and τSL = 41τMD, where σMD and τMD are the characteristic MD length and time units. These factors are similar to those used in other similar slip-spring models.27 The entanglement parameters for coil blocks are Ne,coil = 35 and Ns = 5, which were determined by matching monomer mean-squared displacement and end-to-end correlation functions of coil homopolymers.30 These parameters are consistent with previously observed entanglement spacings38 and tube radii.39
Rod blocks in the slip-spring model are perfectly rigid and thus fully determined by a center-of-mass position r, orientation u, and length Lrod. Since MD rods were represented by coil monomers with a stiff 3-bead bending potential, the rigid slip-spring rod was discretized into a series of monomers of size brod = b/1.32, such that the length Lrod = Nrodbrod matches the MD statistics. The strength of the slip-springs is the same as for coil monomers, Ns = 5, preventing discontinuities as the slip-links move smoothly between rod and coil monomers. This is consistent with both the rod and coil diffusing through the same coil homopolymer matrix in the tracer configuration, where the strength of entanglements is expected to be dominated by the common surroundings. Rod dynamics are governed by an anisotropy mobility tensor , with a parallel friction of and perpendicular friction of ζ⊥ = 6ζ||. These expressions were based on the algebraic form of hydrodynamic mobilities from slender-body theory.40 Entanglements move along the rod monomers in the same way as coil monomers, and Ne,rod = 35/4 is the average number of monomers between rod slip-links. The entanglement spacings and the prefactors in the mobility expressions were adjusted to empirically match the parallel and perpendicular mean-squared displacements between the MD and slip-spring models, and [Fig. 1(a)]. For example, even though slender-body theory predicts a ratio of perpendicular to parallel friction of 2, this value applies to motion within a Newtonian fluid, which is expected to differ significantly from the dynamics within a polymer melt. A larger number of entanglements per monomer are also expected for the rod, because rods are geometrically more extended in space than coils. Similarly, a rotational friction of was introduced to empirically match the orientational correlation function between the two models [Fig. 1(b)]. Together, these parameters matched the MD dynamics for coil and rod homopolymers as well as rod-coil block copolymers.30
Similar to the coil homopolymer model,25 equilibrated initial configurations can be generated using the simple Hamiltonian given by Equation (1), . To further confirm that the statistics are unchanged by the slow reptation processes studied here, 2400 triblock copolymers of N = 200 and Nrod = 100 were simulated for 106τ. After this time, all slip-links had been renewed, and the slip-link density and slip-spring extension at each monomer were averaged over the initial and final 1000τ. The results confirm a slip-link monomer density of 1/35 in the coil blocks, 4/35 in the rod blocks, and a mean-square extension of 5b2 that were all unchanged throughout the simulation [Fig. 2].
The coarse-graining provided by the slip-spring model offers enormous computational speedups relative to the explicit MD simulation. Entanglements are replaced by slip-springs, so the stiff excluded volume Weeks-Chandler-Andersen (WCA) potentials in the Kremer-Grest model are not evaluated between every pair of monomers. This also allows the surrounding matrix chains which were explicitly simulated in MD to be replaced by the coarse-grained entanglements. This is particularly important for polymers with long rods, which would require enormous simulation boxes in explicit MD to maintain the dilute tracer condition. The overall speedups range from 104 for the smallest rod lengths to over 108 for the largest systems studied.
This study examines the tracer motion of symmetric coil-rod-coil triblocks to explore the relative effects of activated reptation and arm retraction on diffusion. 3-dimensional center-of-mass diffusion is measured from the mean-squared displacement of 2400 chains in the linear Fickian regime. Because both relaxation mechanisms directly affect the motion of the polymer along its entanglement tube, examining curvilinear diffusion is important for this study. To calculate curvilinear diffusion, the tube can be approximated as a series of segments connecting adjacent anchor points [Fig. 3(a)], and the position of the polymer along the tube in any configuration can be assigned to the projection of the center monomer on the segment connecting its adjacent anchors. This 1-dimensional position is well-defined whenever the center monomer is between two slip-links, which is almost always true for well-entangled chains. Curvilinear diffusion is then measured from the mean-squared displacement of this position.
This algorithm for measuring curvilinear diffusion is validated using coil homopolymers. At short times, 1D mean-squared displacement along the tube has a t1/2 dependence and no dependence on the total chain size because unentangled motion along the tube is expected to obey Rouse dynamics. At the Rouse time of τR = ζb2N2/(3π2kBT), there is a transition to a Fickian t1 regime. The curvilinear diffusion Dc determined from this regime obeys an N−1.2 power law while the 3D center-of-mass diffusion obeys D ∼ N−2.4, in close agreement with reptation predictions.13
III. CURVILINEAR DIFFUSION IN ACTIVATED REPTATION
While the slowed diffusion of rod-coils in the small rod regime has been directly attributed to entropic barriers to rod reptation,30 quantitative prediction of diffusivity has been elusive, especially when other variables such as coil size are varied. Previous studies have hypothesized a normalized diffusivity of the form D/D0 = f(Nrod),19 where D0 is the diffusivity of a coil homopolymer of the same size, and f is a monotonically decreasing function of unknown form. Though the underlying physical process in activated reptation has been studied in detail, the specific functional form of f remains undefined. Either an analytical or semi-empirical closed form expression for diffusion by activated reptation would allow quantitative estimates as well as comparison with other mechanisms such as arm retraction.
Normalizing the curvilinear diffusivity of rod-coil block copolymers with coil homopolymers reveals the familiar slowing of diffusion in the small rod regime [Fig. 4(a)]. Dc/Dc0 = 1 at zero rod length by construction, but as rod length increases, the magnitude of this slowing effect depends on the total polymer size. For the smallest polymers, normalized diffusivity decreases gradually with increasing rod length, but the decrease is more precipitous for the largest polymers. The N dependence of diffusivity was not apparent using MD simulations,19 which only considered N = 200 and 300 because of computational limitations. It was also difficult to notice in forced Rayleigh scattering measurements20 because of experimental uncertainty and the narrow range of experimentally accessible molecular weights. However, this phenomenon is readily observed in the slip-spring model where a much larger range of N is accessible.
This dependence on total polymer size suggests that quantitative diffusion predictions in the small rod regime must account for the size of the coil blocks. To capture this behavior, it is hypothesized that the total curvilinear friction ζc = kBT/Dc should have independent components for the coil and rod, ζc = ζc,coil + ζc,rod. Since the activated reptation mechanism is only an interaction between the rod and the surrounding entanglements, the friction of the coil block should be unaffected. Therefore, the coil friction is given by ζc,coil = (1 − ϕ) ζc0, where ϕ = Nrod/N is the rod fraction, and ζc0 = kBT/Dc0 is the curvilinear friction of a coil homopolymer of the same total size. Since the friction of a coil of the same size as the rod block is ϕ ζc0, the effect of activated reptation on the rod’s curvilinear friction can be captured by a decreasing function f of Nrod, ζc,rod = ϕ ζc0/f(Nrod). The ratio Dc/Dc0 is expressed in terms of these individual friction components,
which can then be solved for f(Nrod),
Using the values of Dc/Dc0 from Fig. 4(a) and the values of ϕ = Nrod/N by construction, the function f(Nrod) can be determined empirically and the effect on curvilinear diffusion can be calculated [Fig. 4(b)]. Equation (3) approaches a singularity at Nrod = 0 because ϕ = 0 and Dc/Dc0 = 1, but the value of f(0) = 1 is consistent with the coil homopolymer limit at Nrod = 0. When ϕ is close to 0 and Dc/Dc0 is close to 1, the numerator and denominator of Equation (3) are both close to 0, so small statistical errors in Dc/Dc0 are magnified into large errors in f. This effect is most pronounced at Nrod = 4 and 8, where the calculated f has a large spread. At the longer rod lengths, f converges to an exponential dependence on Nrod as the total polymer size increases, as demonstrated by the good agreement with the fit of f(Nrod) = exp(−0.28Nrod) [Fig. 4(b)]. This exponential form of f is also in agreement with the physical picture in the activated reptation mechanism: the number of entanglements on the rod increases linearly with length, which suggests the free energy barriers to diffusion should also increase linearly in size, and so, the effect on rod friction should be exponential. Using this expression for f(Nrod), an empirical closed-form equation for curvilinear diffusion accounting for activated reptation can then be written as
As shown in Fig. 4(c), Equation (4) is in excellent agreement with the curvilinear diffusion data at the small rod lengths or the large polymer sizes. This closed-form prediction for diffusion in the small rod limit enables both quantitative comparison with the effects of arm retraction and analysis of the crossover between the two mechanisms.
The crossover between activated reptation and arm retraction is already apparent in this range of rod lengths. The f(Nrod) data depart from the exponential form at small N and large Nrod, suggesting that effects from tube renewal processes from the coil ends (i.e., arm retraction) compete with activated reptation. These effects should become more important as the coil blocks become smaller.
IV. COMBINING ACTIVATED REPTATION AND ARM RETRACTION
Using the slip-spring model, curvilinear diffusion of coil-rod-coil triblocks can be measured over the full range of rod fractions for many different total polymer sizes [Fig. 5(a)]. This data set confirms the slowed dynamics of rod-coils compared to both coil and rod homopolymers. Consistent with previous results, diffusion is slowed by activated reptation in the small rod regime, while in the large rod limit, the diffusion decreases exponentially with coil size because motion is only possible after arm retraction of the coil blocks.
The large amount of data from the slip-spring model enables a detailed closed-form expression for diffusion in the arm retraction regime. Curvilinear diffusion is given by the inverse addition of the rod and the two coils,
The diffusivity of the rod Drod is determined empirically from the simulation when N = Nrod, i.e., the rod fraction is unity [Fig. 5(a)]. These data are linearly proportional to the parallel friction as expected (see Section II). In the arm retraction regime, the diffusivity of a single coil block Dcoil is related to the diffusivity of a star polymer whose arm is the same size as the coil. 3-arm stars were simulated using the slip-spring model in order to determine Dcoil, which is given by
This expression is a linear combination of a Rouse term for unentangled stars and an exponential arm retraction term, with an empirical mixing rule capturing the smooth transition between the two regimes. Ncoil is the number of monomers in a single coil block, DA0 is an empirically determined exponential prefactor, and is an adjustable parameter corresponding to the average number of monomers between entanglements. Additional discussion of Equation (6) and simulation data on 3-arm stars are included in Appendix B. Excellent fits with the large rod regime data were achieved with parameters of DA0 = 0.0517kBT/ζ and . While is a parameter from the tube model that should be closely related to the slip-spring model parameter Ne because both are measures of the density of entanglements, these two definitions are distinct. It has been shown that because the slip-spring model has a distribution of monomers between entanglements, the tube model fit for should over-predict the slip-spring parameter of Ne = 35,27,41 as consistent with the result here.
For the first time, the closed-form diffusion predictions for both activated reptation and arm retraction allow analysis of both mechanisms in combination in the intermediate rod regime, where the diffusion of rod-coils at a fixed polymer size is near its minimum [Fig. 5(b)]. If the two mechanisms are assumed to be independent, the predicted diffusion in this regime is the sum of Equations (4) and (5). This assumption captures the fact that the rod length at the minimum diffusion increases as the total polymer size increases. However, it significantly under-predicts the simulated diffusion throughout this transitional regime.
An examination of the individual diffusion mechanisms provides explanations for this phenomenon. In the small rod limit, activated reptation assumes that the rod is a perturbation to coil motion, so the rod diffuses along the coil’s entanglement tube and experiences barriers that are completely determined by the shape of the coil’s tube. However, as these barriers become prohibitively large (≫kBT) with increasing rod length, renewal of the entanglement tube from the ends competes with rod reptation, causing partial reorganization of the barriers as the rod passes through them, which speeds diffusion. In the large rod limit, arm retraction assumes that rotation of the rod is slow relative to relaxation of the coils, so motion is only possible once a coil block retracts in its tube and adopts a straightened conformation. However, once the rod becomes sufficiently short, minor rotations of the rod become possible, allowing some motion before full retractions of the coil block. Both of these effects suggest that the combination of activated reptation and arm retraction should result in dynamics that are faster than the sum of the mechanisms acting independently.
Since the intermediate rod regime corrections to both mechanisms are related to the rotation of the rod as it moves through the entanglement tube, quantifying this rotation can reveal deeper mechanistic insights into this regime. To measure orientational correlations as the polymer reptates, both the rod orientation vector u and the 1D tube position s along the tube defined by the connected anchor points [Fig. 3(a)] are recorded, and the simulation is run normally while the rod explores the entanglement tube. For coil homopolymers, u can be defined as RN/2+Δ − RN/2−Δ where Δ = 4, and the selection of Δ has little effect on the calculation as long as Δ ≪ Ne. As time progresses, the center monomer traces a unique path along the tube. Once the polymer moves into a renewed tube segment, the algorithm is terminated because the 1D position s is no longer defined on the original tube [Fig. 6]. After termination, the recorded u(t) and s(t) are converted into u(s) by discretizing s into bins of size Δs = b and averaging u in each bin. Thus, u(s) is the average orientation vector as the chain reptates along its tube. This algorithm can be easily demonstrated for coil homopolymers.
For rod-coil block copolymers, the reptation path defined by u(s) is distinct from other approaches to defining the tube because it characterizes how the polymer moves through space. This is in contrast to many algorithms such as the connected anchor points, primitive path analysis or Z1,33,34 isoconfigurational averaging,32 or the tube axis,31 which calculate the tube based on the average positions of the entire chain at a fixed moment in time. The distinction between these concepts is not important for coil or rod homopolymers because reptation postulates that the motion of the polymer through space is along the tube defined by the polymer.13,42 However, this distinction becomes important for rod-coil block copolymers because the rod and coil motifs with incompatible flexibilities explore space very differently, and it is not immediately apparent whether the polymer should move like a rod, a coil, or some combination of both. For example, when arm retraction is the dominant relaxation mechanism, the entanglement tube is extended near the long rod and Gaussian near the coil. However, through arm retractions of the coil block, the polymer can choose its reptation path that is distinct from the entanglement tube, and this path is expected to be extended because rotation of the long rod is hindered. In the small and large rod limits, activated reptation and arm retraction assume that the polymer moves through a tube defined completely by a coil or rod homopolymer, respectively. When both mechanisms act in combination, the reptation path offers a method to explore the copolymer motion.
The reptation path analysis can be directly compared to other methods of defining the entanglement tube, confirming the distinctions described above. Using u(s) as the tangent vector of the connected anchor points, the orientational correlation function for both definitions is shown for N = 600 chains with Nrod = 0 and 40, corresponding to the coil homopolymer limit and the intermediate rod regime [Fig. 7]. The connected anchor point definition contains minor artifacts such as anti-correlations near s ∼ 10b for coil homopolymers because the slip-springs protrude off underlying Gaussian coil blocks, so the anchor points are not Gaussian. Nevertheless, it is immediately apparent that the two definitions are relatively similar for coil homopolymers, but decays much more slowly for the reptation path than the connected anchors for Nrod = 40. The connected anchor definition is an average over an instantaneous snapshot of the entanglement tube around the rod and coil blocks. Thus, an initial fast decay in is similar to the coil homopolymer and corresponds to the shape of the coil blocks, and a slower decay corresponds to the extended shape of the tube around the rod block. However, the reptation path is much more extended because arm retractions of the coil block allow the rod to choose the straightened paths that it prefers. This comparison confirms that the reptation path is equivalent to classical tube definitions for coil homopolymers, but significant differences arise when considering rod-coil block copolymers.
The statistics of the reptation path varies smoothly between the coil and rod homopolymer limits, describing the motion in the intermediate rod regime. The decay in the orientational correlation function can be captured with an exponential fit , which approximates the reptation path using a worm-like chain model with persistence length lp [Fig. 9(a)]. While the fits are excellent for small rod lengths, minor deviations from the exponential dependence are observed at small s for longer rods. This effect occurs because rod rotations are suppressed for small motions along the tube; these rotations only occur after the entire rod has moved through the entanglements in a curved tube section. While these deviations are difficult to remove, the lp fitting parameter still provides a good measure for the decay of the orientational correlation function, allowing comparison of reptation path statistics for polymers of different N and Nrod.
When arm retraction is the sole mechanism of rod-coil motion, the rod waits for the coil block to retract before it moves, so the reptation path is fully determined by the rod block while the diffusivity is controlled by the coil. In the reptation path analysis, lp of rod-coils approaches that of rod homopolymers in the limit of large Nrod and small N, confirming that arm retraction is dominant mechanism in this regime. As Nrod decreases, arm retraction is no longer the main relaxation mechanism, and the rod-coil reptation path significantly deviates from rod homopolymers. Activated reptation should then become dominant for the smallest Nrod and largest N.
The effect of activated reptation on the reptation path can be isolated using rod-coil rings, where no tube renewal processes are possible. Since activated reptation postulates that the rod moves along a tube completely determined by the coil, one may expect that lp should be the coil homopolymer value throughout the activated reptation regime. However, using rings of N = 800 total monomers that were carefully prepared as previously described,30 lp was shown to increase with Nrod as the number of rod entanglements Zrod = Nrod/Ne,rod approaches 1 or Nrod ∼ 8. This effect is due to the geometry of the rod [Fig. 8]. Because the rod is extended in space, its orientation vector samples a large section of the entanglement tube, so small fluctuations in the coil’s reptation path are averaged out. The rod no longer feels the short range curvature of the entanglement tube and follows a less tortuous path, reducing the decay of the orientational correlation function. Thus, the measured lp increases with Nrod even if the rod moves strictly along to the coil’s reptation path. The smoothing of local curvature also explains the minor deviations from an exponential dependence of at small s for longer rods [Fig. 9(a)], because the longer rods are less likely to rotate as they move small distances along the tube. Even though the lp is not identical to the coil homopolymer, the data on rod-coil rings provide a useful comparison benchmark for the reptation path from activated reptation alone. For example, the quantitative agreement with rings suggests that activated reptation is dominant for N = 800 and Nrod ≤ 16.
The reptation path statistics show a smooth transition between the rod homopolymer limit when Nrod approaches N and the ring limit when Nrod is small. The transition occurs in the intermediate rod regime, when both the rod size Nrod and coil size are sufficiently large such that both activated reptation and arm retraction are important. This confirms the proposed modifications to the two mechanisms that lead to faster motion than the sum of their predicted diffusivities. Departure of lp from the ring limit is due to renewal of the entanglement tube from the ends occurring before the rod reptates, allowing the rod to choose reptation paths that are more extended than the coil homopolymer. This speeds diffusion relative to activated reptation alone, and the effect is largest at small N where the coil blocks are shortest.
In the large rod limit, rotations of the rod can allow motion before full retractions of the coil block if the rod is sufficiently short. Knowing u(s) enables analysis of these rotations by monitoring the perpendicular motion of the ends of the rod as it reptates. If the angle θ that the rod rotates after moving s along the tube is given by , then the perpendicular distance of the rod end from the original tube centerline is . With the rough approximation , d⊥ can be expressed in terms of the distance that the rod travels along the tube s,
Using s = 5.5b as the average distance between coil anchor points, the deflection of the rod end can be calculated as it moves past one entanglement [Fig. 9(c)]. For rod homopolymers, this deflection is mostly less than the radius of the tube, , which confirms that the rod does not significantly rotate outside its tube radius until it moves past one entanglement. For rod homopolymers outside the activated reptation regime (Nrod > 20), the deflection increases with increasing N (coil size) and decreasing Nrod, which suggests that the large coil blocks force the rod to rotate as it moves past entanglements. At Nrod = 40, the deflection of all rod-coils is significantly above the tube radius, so the rods can rotate outside the tube radius, and arm retraction under-predicts diffusion as shown in Fig. 5(b). For larger Nrod, d⊥ approaches the tube radius as the diffusion approaches the arm retraction prediction, so the coil blocks must retract into a more straightened tube before rod motion is possible. The agreement between d⊥ and is better for smaller N and smaller coil blocks, consistent with the better agreement in the diffusion prediction.
V. 3-DIMENSIONAL CENTER-OF-MASS DIFFUSION
Similar to curvilinear diffusion, the behavior of D also displays the classical signs of activated reptation and arm retraction as rod length and polymer size are varied [Fig. 10(a)]. Although the exponential dependence on coil size is valid for D in arm retraction, the behavior of D in the small rod limit is notably different than Dc [Fig. 10(b)], suggesting that additional effects for D are present beyond the previous analysis for curvilinear diffusion (Equation (4)). These effects must be investigated to provide a connection between the curvilinear diffusion results and experimentally accessible center-of-mass diffusion measurements.
The behavior of D is more easily analyzed using the ratio Dc/D [Fig. 10(b)]. For rod homopolymers, diffusion along the tube is unhindered while perpendicular motion is prohibited, so the center-of-mass diffusion is D = (D|| + 2D⊥)/3 = D||/3. Because Dc in the slip-spring model is defined along the connected anchor points which extend perpendicularly from the rod by , the measured curvilinear diffusion over-predicts D|| since the anchor path is longer than the rod. From the model, the ratio of the average anchor path length to the rod length was determined to be 1.18, so Dc = 1.182D|| and thus Dc/D = 4.20 for rod homopolymers. This prediction quantitatively captures the behavior of rod-coils, as the simulated ratio approaches this value and becomes independent of rod length and total polymer size as Nrod increases.
For Nrod < 40, the Dc/D ratio is poorly described by rod homopolymers as the polymer reptates on a path whose curvature approaches the coil homopolymer limit. In this limit, Dc/D ∼ N so the ratio has a strong dependence on both rod length and the total polymer size. Because the curvature of the reptation path is well-defined by the previous analysis [Fig. 9], the relationship between curvilinear and center-of-mass diffusion can be quantitatively predicted. Inspired by Doi and Edwards who derived an expression for the mean-squared displacement of a fully flexible reptating chain,13,14 we apply this analysis to reptation along a wormlike chain with persistence length lp. Using a combination of analytical derivation and simulation (see Appendix A), the ratio of diffusivities is given by
where Z = N/Ne is the number of entanglements and λ = exp(−a/lp). Using a = 5.5b as the average distance between anchor points along a coil homopolymer with lp from Fig. 9(b), the behavior of Dc/D is well-described over the entire parameter space. Combined with the curvilinear diffusion predictions in the activated reptation and arm retraction mechanisms, Equation (8) provides a quantitatively accurate understanding of center-of-mass diffusion in entangled rod-coil block copolymers.
A coarse-grained single-chain slip-spring model for rod-coil block copolymers allows quantitative predictions of diffusion and other dynamic properties over a large parameter space. Using the computational advantages of this model, sufficient data on the tracer motion of coil-rod-coil triblock copolymers were generated for a more detailed analysis of motion by activated reptation and arm retraction both individually and in combination. First, a closed-form expression for curvilinear diffusion by activated reptation was derived that accounts for effects from both the rod length and coil size by separating the drag into individual components for the rod and coil block. The effect of activated reptation on rod friction is exponential with rod length, which is consistent with the free energy barriers increasing linearly with length. This expression was then applied to the intermediate rod regime, where a combination of activated reptation and arm retraction under-predicts the observed diffusion. This discrepancy was explored by studying the reptation path, which is the path that the polymer follows in space as it reptates along the tube. For rod-coil block copolymers, this path was shown to be distinct from other tube definitions based on the average positions of the monomers because the rod and coil explore space differently. The statistics of this reptation path varies smoothly between the coil and rod homopolymer limits as the rod fraction was varied, implying that the polymer moves along a path in space that becomes more rodlike as the rod fraction is increased. This suggests that activated reptation under-predicts diffusion in the intermediate rod regime because renewal of the entanglement tube from the ends can occur before the rod reptates, allowing the rod to choose reptation paths that are more extended than the coil homopolymer and reducing the barriers to reptation. Likewise, arm retraction under-predicts diffusion because minor rotations of the rod allow some motion before full retractions of the coil block. Finally, these conclusions for curvilinear diffusion are also applied to 3-dimensional center-of-mass diffusion. The ratio of these two diffusivities varies smoothly between the coil and rod homopolymer limits as the reptation path becomes more extended, and this behavior is captured by a simple reptation model on a wormlike tube.
This paper is dedicated to our friend and collaborator Alexei Likhtman, who sadly passed during the publication of this manuscript. We are immensely grateful for all of his contributions to the scientific community, and his presence will be missed.
We gratefully acknowledge funding from NSF Award No. CMMI-1246740. Simulations were performed on the Darter cluster at the NICS through a generous XSEDE allocation No. TG-DMR110092. M.W. acknowledges support through a NSF Graduate Research Fellowship. A.E.L. acknowledges a funding from EPSRC Award No. EP/K017683.
APPENDIX A: RELATIONSHIP BETWEEN CURVILINEAR AND CENTER-OF-MASS DIFFUSION
To understand the effect of reptation path curvature on center-of-mass diffusion, a simple reptating tube model was simulated. The tube is approximated as a freely jointed chain with segment length a and a bending potential between segments (Fig. 11),
which results in wormlike chain statistics with persistence length lp over long length scales,
While this definition does not capture the wormlike behavior at small length scales, it is a necessary approximation because the tube is not well-defined for length scales smaller than the entanglement length. This model allows both the contour length and the end-to-end distance of a tube segment to be fixed at a. On the other hand, a continuous wormlike chain has a contour length that depends on lp if the end-to-end distance is fixed at a. In the fully flexible case, the Doi-Edwards model also ignores behavior at small length scales by assuming a for contour length and end-to-end distance of one tube segment, because the Gaussian chain is particularly pathological since the contour length is infinite.13,14
To generate the tube, chain segments of length a are placed in space to follow bending potential (A1) as follows. An initial segment is placed at a random position and orientation. In reference to the end segment, additional segments are placed with a uniformly distributed azimuthal angle, 0 ≤φ < 2π, and a polar angle following a probability density function derived from (A1),
After a tube of the proper length is generated, the simulation then proceeds by allowing the tube to reptate along itself. For the tube coordinates R1, …, RZ where 1 is at the head and Z = N/Ne is at the tail, a random variable is generated at each time step of size Δt. If ξ(t) = 1, then the chain moves one segment towards its head, and vice versa. This results in a curvilinear diffusion constant of
The evolution equations for the tube coordinates are given by
where v is a random vector representing a new segment created at the tube end. The length of this segment is a, and the angle between the new segment and the original end segment is determined by (A3). The center-of-mass diffusion is then determined from the slope of the 3-dimensional mean-squared displacement vs. time. These simulations were performed for 2000 tubes over 500 time steps, with Z from 3 to 40 and lp from 0.01a to 1000a. For each Z, the ratio D/Dc varies smoothly from the coil homopolymer limit of 1/3Z at low lp to the rod homopolymer limit of 1/3 at high lp [Fig. 12].
A closed-form expression for center-of-mass diffusion can be determined by examining this model analytically. Evolution equations (A5) are taken to the continuous limit,
where s is the position along the tube contour length varying from 0 to Lc, and the random variable ξ(Δt) is Gaussian with moments,
The mean-squared displacement of a chain segment s, , can be calculated using the correlation function,
The initial value of this correlation function, , is simply the static mean-squared end-to-end distance of a polymer between s and s′. Using the definition , this initial value can be calculated as
where λ is defined in Equation (A2). While this expression is strictly valid only when Z′ is an integer, it is an excellent smooth approximation in the continuous limit because length scales smaller than a are ignored as previously discussed.
The center-of-mass diffusion of the chain is related to the slope of mean-squared displacement at long times,
This diffusion should be the same for all chain segments, and thus independent of s and s′. Thus,
The last step above is possible from the symmetry relationship . This implies that the derivatives are anti-symmetric,
We then proceed using the definition below for convenience,
so (A12) implies
The first term above can be simplified by noting that at t = 0, initial value (A10) depends only on the absolute difference . By applying the chain rule through w, it can be shown that
where u(s, t) is a unit vector along the chain at segment s and time t. Thus, the center-of-mass diffusion of the tube is
The integral depends on the orientational correlation time of the tube ends. In the case of a fully flexible chain, the orientations decorrelate instantly, so the integral is zero. For rigid rods, both correlation functions are always one, so the total integral is also zero. In the intermediate case, it is expected that the integral should be nonzero and negative. However, it can be shown empirically that in this range of parameters, the first term is sufficient to capture the diffusion behavior. This is confirmed by plotting Equation (A20) against the simulation data, with excellent agreement (Fig. 13).
The final expression for the diffusion ratio has an intuitive interpretation. In the first term of (A19), the expression is the mean-squared end-to-end distance of the chain. Thus, the ratio of center-of-mass to curvilinear diffusion is the squared ratio of the 3-dimensional end-to-end distance to the curvilinear contour length of the tube. This is expected since the tube moves one end-to-end distance in 3-dimensional space in the time that it moves one contour length in curvilinear space. The additional factor of 3 accounts for the difference in dimensionality between the two diffusion processes. This simple explanation suggests that even when neglecting the higher order effects from the orientational correlation of the tube ends, this analytical solution (A20) is a close approximation to the actual diffusion of a semi-flexible reptating tube.
APPENDIX B: DIFFUSIVITY OF COIL BLOCKS IN ARM RETRACTION REGIME
The arm retraction mechanism of coil blocks in the long rod regime is analogous to the motion of star polymers. Thus, we simulated 3-arm stars using the slip-spring model in order to estimate the diffusivity of coil blocks in the expression for coil-rod-coil triblock diffusion. This slip-spring model for star polymers has been explored previously43 and is briefly described here. The model is identical to the model described in Section II for coil homopolymers, except one of the monomers is the branch point of the star. 3 monomers are attached to this center monomer by Hookean springs, corresponding to the 3 arms of the star. Thus, the total number of monomers on the star is N = 3Narm + 1, where Narm is the number of monomers on each arm. Slip-links are not allowed to move from one arm to another, which is enforced by rejecting all slip-link moves attempting to occupy the branch point.
The center-of-mass diffusion as a function of arm size is shown in Fig. 14. For arms that are significantly smaller than Ne = 35, the stars are not entangled and thus follow Rouse predictions,
For Narm ≫ Ne, the dominant diffusion mechanism is arm retraction, and thus, diffusivity decreases exponentially with arm size,16
where is an adjustable parameter corresponding to the average number of monomers between entanglements, and DAR0 is a proportionality constant. With an exponential fit to data for Narm ≥ 70, these constants were determined to be DAR0 = 0.0172 and . The parameter over-predicts the slip-spring model Ne parameter as discussed in the main text. The overall 3-arm star diffusion over both the Rouse and entangled regimes can be described by a combination of both expressions with an empirically determined mixing rule,
This expression is an excellent fit to the star polymer data [Fig. 14].
Because the stars have 3 arms, the contribution of a coil block to the curvilinear diffusion of coil-rod-coils should be equal to Dcoil = 3Dstar, where the size of the coil block is the same as a single arm on the star, Ncoil = Narm. To summarize, the contribution to coil block diffusivity is given by
where DA0 = 3DAR0 = 0.0517 kBT/ζ, , Ncoil is the size of a single coil block, and the contribution of the star polymer’s center monomer to Rouse diffusion term was dropped. Using Equation (5) in the main text, the curvilinear diffusion of coil-rod-coils can be accurately described for sufficiently short coils using this expression for Dcoil [Fig. 15].