Power-Law Stretching of Associating Polymers in Steady-State Extensional Flow

We present a tube model for the Brownian dynamics of associating polymers in extensional flow. In linear response, the model confirms the analytical predictions for the sticky diffusivity by Leibler- Rubinstein-Colby theory. Although a single-mode DEMG approximation accurately describes the transient stretching of the polymers above a 'sticky' Weissenberg number (product of the strain rate with the sticky-Rouse time), the pre-averaged model fails to capture a remarkable development of a power-law distribution of stretch in steady-state extensional flow: while the mean stretch is finite, the fluctuations in stretch may diverge. We present an analytical model that shows how strong stochastic forcing drive the long tail of the distribution, gives rise to rare events of reaching a threshold stretch and constitutes a framework within which nucleation rates of flow-induced crystallization may understood in systems of associating polymers under flow. The model also exemplifies a wide class of driven systems possessing strong, and scaling, fluctuations.

We present a tube model for the Brownian dynamics of associating polymers in extensional flow. In linear response, the model confirms the analytical predictions for the sticky diffusivity by Leibler-Rubinstein-Colby theory. Although a single-mode DEMG approximation accurately describes the transient stretching of the polymers above a 'sticky' Weissenberg number (product of the strain rate with the sticky-Rouse time), the pre-averaged model fails to capture a remarkable development of a power-law distribution of stretch in steady-state extensional flow: while the mean stretch is finite, the fluctuations in stretch may diverge. We present an analytical model that shows how strong stochastic forcing drive the long tail of the distribution, gives rise to rare events of reaching a threshold stretch and constitutes a framework within which nucleation rates of flow-induced crystallization may understood in systems of associating polymers under flow. The model also exemplifies a wide class of driven systems possessing strong, and scaling, fluctuations.
The natural or artificial production of highperformance polymeric materials requires precise control over flow-induced crystallization. This phenomenon involves in turn a highly non-trivial interdependence between the molecular level of bond-orientation-dependent nucleation, and the macroscopic level, where the temperature-dependent rheology generates stretch of entire chain segments [1][2][3][4][5]. Remarkably, nature has found a way to control robustly the flow-induced selfassembly of silk from an intrinsically disordered state (a solution of random-walk polymers) prior to forming high-performance fibers under flow at ambient conditions [6][7][8][9][10][11][12][13]). Key to achieving the final properties is that silk is processed in semi-dilute aqueous conditions [9], where nucleation can be induced through the stretch-induced disruption of the solvation layer [14]. How sufficient polymer stretch can be achieved in a limited time under modest flow conditions has so far remained unexplained. Recent work has shown that microscopic chain stretch and the consequent macroscopic strain hardening is triggered by a small number of calcium bridges [13,15] that act as 'sticky' reversible intermolecular crosslinks akin to those in synthetic 'sticky polymers' [16][17][18][19][20][21][22][23][24]. For this class of molecules, a molecular understanding of the non-linear rheology and crystallization of 'sticky polymers' has so far relied on computationally expensive (coarse-grained) molecular dynamics simulations [5,[25][26][27][28][29]. Simpler molecular models coarse-grained at the level of entanglements, but able to capture the vital slow processes, remain absent.
In the present work, we address this need by following the central idea by de Gennes of replacing the many-chain problem with a single chain in a tube-like confinement imposed by its environment of entanglements [30], and solve the Brownian dynamics of the chain in 1D [31,32]. This approach is simple yet powerful, and has led to the development of widely applied finite-element solvers ( [33][34][35]), a physical explanation for the (apparent) 3.4 power dependence of the relaxation time of polymer melts on the molecular-weight [36], and a comprehensive understanding of the rich non-linear rheology of (bimodal) polymer blends [37,38]. The ingredient that we add in this letter is a 'sticky-reptation model' description [39] for the temporary binding of associating monomers to the tubular environment developed for full non-linear flows. The model shares some structural similarities with early 'transient network' approaches to polymer melt and solution rheology, also demonstrating a hitherto unrecognised feature of those models The starting point of our contribution is to consider a chain consisting of N Kuhn segments with length b, and Z e entanglements (hence, with tube diameter a = b(N/Z e ) 1/2 ). The configuration of the chain is given by the spatial coordinates R i of monomers i = 1, . . . , N along the curvilinear direction along the tube, which evolve with time according to the Langevin equation [32,36,37] with ∂R/∂i = a at i = 1 and at i = N , ζ the monomeric friction, k B T the thermal energy, and f i a stochastic force given by the equipartition theorem In the absence of stickers, this equation predicts the Rouse diffusivity [31] and the variance of quiescent contour-length fluctuations |R N − R 1 | 2 = aZ e /3. R C.M. is the center of mass of the chain, and the strain rate,ε, is in one spatial dimension equivalent to the strain rate in the GLaMM model [37].
To model the binding and unbinding of monomers to the environment, we introduce a stochastic state variable p i (t), which takes values of either zero or unity for each monomer i. If it is zero, the monomer is free to diffuse and respond to the drag exerted by the flow field, as well as to respond to the stress within the polymer, including at least some relaxation of stress in segments it adjoins. However, if the value of the state variable is unity, the monomer is kinetically trapped by its environment and is unable to diffuse or to respond to the differences in chain tension exerted by the polymer. Hence, the closed sticker advects with the background flow. While the state variable of regular (non-sticky) monomers is always zero, the state variable for sticky monomers can either be zero ('the sticker is open'), or unity ('the sticker is closed'). In our simulations, the opening and closing of the stickers is simulated using a simple stochastic algorithm, where at every time step ∆t a sticker is opened with probability k i,open ∆t or closed with probability k i,close ∆t.
By defining the rules by which p i may switch between the 'open' and 'closed' states, copolymers with arbitrary monomer sequences may be modeled. Here, we consider chains with N − Z s monomers that are non-sticky (for these monomers k i,close = 0 and p i = 0 at all times) and Z s sticky monomers that may switch state using rates k close and k open . The opening rate is related to the rheological sticker lifetime, τ s = k −1 open [13,17,[24][25][26][27][28], and the closing rate is given by k close = k open p/(1 − p), with p the time-or ensemble-averaged fraction of closed stickers. Here, we ignore the underlying dissociation-association or bondswap mechanisms that determine the concentrationdependence of the opening and closing rates [40,41], and view p and τ s as free model parameters.
We have benchmarked our model in the absence of flow using the Likhtman-McLeish model for linear non-sticky polymers (results not shown here) and using the sticky-Rouse diffusivity, D SR = D SR (Z e , τ e , Z s , τ s , p) as calculated by Leibler et al. [17] (see the inset of Figure 1). For the non-linear dynamics of sticky polymers, so far no comparisons between analytical predictions with simulations or experiments have been reported. The first strategy to address this is to evaluate how well a DEMG-type single-mode approximation performs, with chain friction renormalized by averaging over the stochastic sticker dynamics: where the stretch ratio, λ ≡ (R N − R 1 )/Z e , is presumed to be uniform over the backbone of the chain. The extension rate is proportional to the stretch ratio itself. The retraction rate is determined by 1 − λ (in the absence of flow, λ = 1 at steady state) and by the sticky-Rouse time, In the main graph of Figure 1, we present comparison between this simple approximation and our simulations, (the approximations inherent in the DEMG require that the simulation time be divided by a factor 1.2 to result in the close agressment shown). This confirms that the intuitive 'sticky Weissenberg number' for the stretch transition is Wi =ετ SR . For Wi > 1 an exponential runaway stretch emerges as expected. In contrast to non-sticky polymers, however, we will argue that the stress and fluctuation in stretch may diverge below this stretch transition when the pre-averaging approximation inherent in DEMG is avoided.
While non-sticky polymers in steady state show a Gaussian stretch distribution with a width that is determined by the (effective) number of entanglements, we have observed rather large stretch fluctuations for the sticky polymer at extension rates of the order of, but below, the critical value. Indeed, the symbols in Figure 1 are averaged over five simulations for a chain with 10 stickers which are on average closed a fraction p = 0.95 of time. For simulations with p < 0.9 these fluctuations become much larger and difficult to distinguish graphically. Indeed, while the mean stretch is finite, the fluctuations in stretch diverge above a certain flow rate below the stretch transition.
For three flow-rates of Figure 1 we have plotted the stretch distribution, P (λ), in Figure 2. For small flow rates, the stretch distribution is Gaussian, ln P (λ) ∝ (1− λ) 2 (solid curves), as in the quiescent state. However, for increased flow rates deviations emerge in the high-λ tail of the distribution. Importantly, the polymer stretch may resemble the mean stretch for long times compared to the sticky-Rouse time, and only in 'rare events' the stickers may remain closed sufficiently long for the stretch to reach deep into the tail of the distribution (see inset). For an increasing flow rate, the high-stretch tail is no longer Gaussian but becomes a power law, P (λ) ∝ λ −ν . The inset shows the stretch ratio against time, and visualizes how this distribution includes 'rare events' of enormous chain stretch. For a sufficiently large flow rate, ν decreases. If ν > 2, the mean value of λ is finite (as it should in steady state); however, if also ν ≤ 3, the fluctuations in stretch, characterized by the expectation value of λ 2 , diverge.
In the following, we will simplify the problem using a 'sticky dumbbell model' to explore and clarify the underlying causes of the power-law tail in the stretch distribution, and explore how it can be tuned by the flow rate. This minimal model that captures the essential physics is equivalent to a single polymer strand either attached to the bulk deformation at both ends (the 'closed state) or free to relax (the 'open state). The rate by which the polymer switches between the two states is given by the usual opening and closing rates. We can now address the development of stretch under extensional flow through a pair of coupled partial differential equations for the timedependent stretch distributions P o (t, λ) and P c (t, λ) for each state using the master equation We calculate the steady-state stretch distribution at strong stretch by setting the left-hand side of Eq. (5) to zero and taking λ ≫ 1. The result can be solved analytically since in these conditions the differential system becomes homogeneous. We therefore find the power-law relation with the exponent given in terms of the three dimensionless parameters of the system, p,ετ R , τ R /τ s by We compare this power-law to our sticky dumbbell simulations in Figure 3.
For sufficiently small flow rates, we find a reasonable agreement between our multibead simulations and the analytical approximation for the simple sticky dumbbell (under these conditions, ν > 3). When ν approaches a value 3 (this occurs at (1 − p)ετ R ≈ τ R /(2τ s )) the discrepancies become larger. This is not a coincidence: if ν = 3 the magnitude of the fluctuations diverge, λ 2 → ∞. Indeed, near this condition the variations in stretch are enormous and the single-mode approximation breaks down for multibead chains. We confirm this using simulations with a variable number of beads per chain: for fewer beads the higher-order Rouse modes are removed and the simulations show a better agreement with the single-mode approximation. Although the fluctuations diverge for ν = 3, the mean λ remains finite as long as ν ≤ 2 (the equality holds approximately when (1 − p)ετ R ≈ τ R /τ s ). For even larger flow rates, i.e., for ν ≤ 1 (at (1 − p)ε τ s = 1) the stretch distribution can no longer be normalized and true runaway stretch emerges. These various regimes are displayed in Figure 4 in terms of the dimensionless parameters of the system. Note that the stress is σ ∝ (1 − λ) 2 and the tail of the stress distribution is P (σ) ∝ λ −ν/2 : the mean stress diverges for ν ≤ 4 and its variance diverges for ν ≤ 6. The single-mode toy model clarifies the route through which the divergent fluctuations arise. Crucially, when a stretched strand is freed from the network, it may not relax entirely before reattachment (this effect is ignored in classical treatments of transient network models, which in consequence overlook the strong stochastic fluctuations they physically imply). Such continuous interchange between convecting and relaxing strands, together with the occurrence of longer-than-average attachment times for some segments, allow the exploration of very large chain stretches in steady-state.
To illustrate the potential consequences of this effect, we consider nucleation rates in steady-state extensional flow, assuming that polymer crystal phase may nucleate around chains beyond a critical stretch ratio λ * [1]. Assuming that the chain is relaxed prior to sticker closing at time t = 0, its stretch ratio develops as λ(t) = exp(εt) until it closes at a time τ open . This time is drawn from the probability distribution p(τ open ) = τ −1 s exp(−τ open /τ s ), so the probability that the critical stretch is reached is p * = λ −1/ε τs * . The probability that λ * is not reached after n attempts is (1 − p * ) n , and therefore the expected number of attempts needed is An attempt occurs, on average, after time intervals 1/k open + 1/k close = τ s /p. If the number density of chains is ρ, then combining these results gives as the extension-rate-dependent nucleation rate per volume.
In conclusion, we have numerically solved the stochastic Langevin equation of an aligned entangled sticky polymer in an effective medium and in extensional flow. We have found that the stretch transition is determined by the sticky-Rouse time and that the early stretch transients are well described using a single-mode approximation. Below the transition, we have identified a steady-state regime where the time-or ensembleaveraged stretch distribution has a power-law tail that renders large stretches much more likely than in the usual contour-length-fluctuation-dominated Gaussian distribution. We have shown that this behavior originates from a stochastic coupling between stretching and relaxation states of the polymer stretch distribution. We expect that these insights provide new means to calculate nucleation rates for polymer crystallization in extensional flow. It also provides an example of one of a family of driven, stochastic, systems in which a divergent and scaling structure of fluctuations arises, not just at a single critical point, but within a large region of state space, and with a universal critical exponent replaced by a family, dependent on the degree of forcing. This research was funded by the Engineering and Physical Sciences Research Council [grant number EPSRC (EP/N031431/1)]. Jorge Ramírez is thanked for sharing Alexei Likhtman's Brownian dynamics code and Richard Graham is thanked for sharing his GLaMM code; both codes helped us to benchmark our simulation software. Chris Holland and Pete Laity are thanked for useful and encouraging discussions.