Entangled Rigid Macromolecules under Continuous Startup Shear

Article pubs.acs.org/Macromolecules

Entangled Rigid Macromolecules under Continuous Startup Shear Deformation: Consequences of a Microscopically Anharmonic Confining Tube Daniel M. Sussman*,† and Kenneth S. Schweizer‡,§,∥ †

Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104, United States ‡ Department of Materials Science and Engineering, §Department of Chemical and Biomolecular Engineering, and ∥Frederick Seitz Materials Research Laboratory, University of Illinois, 1304 W. Green Street, Urbana, Illinois 61801, United States ABSTRACT: We build upon our recently developed microscopic theory for the tube confinement potential of rigid macromolecules to treat the relaxation of stress and orientation of fluids of topologically entangled needles during a continuous startup shear deformation. Two coupled and highly nonlinear evolution equations for stress and rod orientation are proposed. The novel feature is our ability to self-consistently relate the current stress- and orientationdependent state of the fluid with an effective instantaneous relaxation time that combines (perturbed) reptation and transverse activated barrier hopping. The effective relaxation time follows from a microscopic, time-dependent prediction of the tube confinement potential during the deformation. Results for the strain and Weissenberg number dependence of the confinement potential are presented for two degrees of entanglement. As a natural consequence of the predicted stress dependence of the confinement potential, our self-consistent single-rod theory emergently manifests features normally associated with many-polymer convective constraint release, such as a monotonic flow curve and an effective relaxation time that (nearly) scales with the inverse of the deformation rate. Comparisons with the original Doi−Edwards theory reveal qualitative differences as a consequence of stress- and orientation-induced softening of the tube constraints. Severe tube dilation often occurs, and the complete destruction of transverse localization is possible depending on system-specific conditions. and comb macromolecules.3 When applied to nonlinear rheology the reptation-tube approach must invoke additional strong assumptions and approximations that effectively correspond to informed guesses about the true nature of entanglements and tube confinement potentials. Direct testing of the fundamental assumptions of the tube theory under nonlinear deformation, as opposed to rheological consequences of specific ansatzes, remains a very difficult and largely open problem. For the continuous shear deformation of present interest, the original Doi−Edwards theory predicted an unphysically large degree of shear thinning at high deformation rates due to affinedriven polymer “overorientation”. To address this issue, qualitative modifications of the theory have been invoked, primarily the “convective constraint release” (CCR) idea. CCR postulates that fast flows can mechanically convect the many entangled polymers surrounding a tagged chain at a rate set by the imposed shear rate, effectively releasing constraints to its motion faster than quiescent reptation. Schematically, the essential idea is to modify the polymer relaxation rate under

I. INTRODUCTION Understanding the quiescent dynamics and rheological behavior of heavily entangled polymer melts and solutions has been an outstanding theoretical challenge for many decades. The difficulty lies in microscopically describing “topological entanglements”, the dynamical constraints that arise from polymer connectivity and uncrossability, which is generally viewed as an intractable problem. The primary framework for studying entangled polymer dynamics remains the phenomenological reptation-tube model of de Gennes,1 Doi and Edwards,2 and its many recent extensions.3 This singlechain approach is based on an effective mean field that assumes a harmonic confining tube around a tagged chain as a result of the interactions with the surrounding polymers. The diameter of the tube is not predicted, but rather treated as a fit parameter. The resulting long-time relaxation and mass transport of linear polymers can then only proceed via onedimensional curvilinear diffusion along the tube axis. For melts and solutions of chains in equilibrium the reptation-tube model has had many successes, particularly when competing relaxation processes (with a few new adjustable parameters), such as constraint release and contour length fluctuations,4 are included. Impressively, the tube approach can be generalized to treat many architectures, from rigid rods to branched star © 2013 American Chemical Society

Received: March 28, 2013 Revised: May 26, 2013 Published: July 8, 2013 5684

dx.doi.org/10.1021/ma400649k | Macromolecules 2013, 46, 5684−5693

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flow as τ−1 = τq−1 + γ̇, where τq is the quiescent relaxation time and γ̇ is the flow rate.5 Because of its phenomenological character, there are multiple CCR models; sophisticated changes to the relaxation time that depend on deformation time and history, contour length fluctuations, and other factors have been considered.3−5 The CCR idea can “cure” the shearthinning problem in the original reptation-tube theory, but at the cost of an additional fitting parameter that must be delicately tuned and is not fundamentally derived. Moreover, there remain troubling and puzzling issues. For instance, many of the ideas that work well in describing shear flow seem to fail to describe experiments in extensional flow.6 There is also some ambiguity about the physical meaning of the mathematical terms CCR adds to the tube model.7 Another set of poorly understood issues that we believe are crucial for nonlinear rheology concerns the strength of the physical entanglements that collectively define the confining tube potential. The standard assumption that the tube is a harmonic confinement field (i.e., acts like an unbreakable chemical cross-link) cannot be literally true, but the relevant question is whether it is a good approximation under deformation conditions of interest. There is a growing body of evidence that the barrier to transverse motion may be finite on a scale relevant to practical rheological testing conditions,8,9 which may or may not be related to the poorly understood microscopic physics of the CCR process. In this paper we demonstrate that CCR-like physics can emerge as a natural consequence of a self-consistently determined anharmonic tube confinement potential. The most spectacular consequence of tube breakability under stress would be “microscopic absolute yielding”, defined here as the deformation-induced destruction of the confining tube. Such an effect is well-known and accepted in the fields of nonlinear mechanics of polymer glasses and colloidal gels and glasses,8,10−13 where deformation can either greatly reduce the barrier to thermally activated hopping (“dynamic yielding”) or completely destroy the cage constraints and attendant transient localization (“absolute yielding”). Even in the absence of the more extreme form of yielding, recent simulations support the idea that the tube is strongly deformed under flow conditions corresponding to a weakening of the dynamic constraints on lateral motion. For instance, atomistic simulations and primitive chain network slip-link studies of shear and extensional flows have found power-law reductions of the steady-state number of entanglements and an increase in the effective tube diameter, dT.14−16 This may be a consequence of an effectively softerthan-harmonic tube, a feature observed in quiescent primitivepath simulation analysis of entangled chain melts17 and experiments on semiflexible biopolymers.18,19 Anharmonic softening may also be important for the microscopically unexplained power-law scaling of the “yield strain” (the value of strain at the stress overshoot), γy ∼ (γ̇τR)1/3, observed for Rouse−Weissenberg numbers in excess of unity, γ̇τR > 1 in both solutions8 and melts.20,21 Recent perspective articles7,22 have emphasized the need for conceptual advances under both quiescent and nonlinear driven conditionsincluding the microscopic foundation of CCR and the way entanglements and tubes are modified by large deformationsand this work continues our efforts in a new direction to address these fundamental issues. We initially choose to study a simpler systema solution of infinitely thin, nonrotating rods of length L. This is an attractive starting point, as it offers substantial simplifications from the full complexity of

entangled flexible chains. Moreover, the Doi−Edwards (DE) theory for entangled rod-like polymers2,23 shares many features with the tube rheology theory for flexible chains. This includes an excessive shear thinning behavior at high shear rates, but to our knowledge CCR elaborations have not been developed for rigid-rod polymers. Unfortunately, there is very little work on the rheological behavior of entangled rigid rods, and although experiments and simulations of such systems are feasible, there is currently insufficient data to incisively test the theory proposed. We hope the many testable predictions made in this article will stimulate such new efforts. By starting from a force-level microscopic perspective, we naturally arrive at an anharmonically softened transverse confining potential. If one accepts this feature of the tube (which has substantial experimental and simulation support as mentioned above), then a model-independent physical expectation is that the tube cannot be infinitely stable to externally applied forces. This scenario has been implemented in an intuitive but simplified manner in the context of our microscopic theory. We find that an external force results in polymer relaxation via competing parallel channels of (1) deformation-modified longitudinal reptative motion and (2) stress-assisted transverse “activated entropic barrier hopping”. In this article we show that using a simple physical idea, wellestablished in the context of colloid and polymer glasses,11,24 for how macroscopic stress results in an external force on a single polymer, an emergent CCR-like behavior in continuous shear flows is predicted without the need to explicitly include additional relaxation mechanisms by hand. The present article is a generalization of our recent work on instantaneous step-strain deformations of entangled needle systems,25 where we predicted an acceleration of terminal rotational relaxation due to tube dilation, and at sufficiently high deformations the emergence of a competitive transverse hopping relaxation process. Both of these are direct consequences of the predicted anharmonic form of the dynamic tube confinement potential. Our present treatment of continuous startup shear deformation involves new conceptual elements beyond what is in ref 25. Moreover, and very importantly, we believe the ability to capture CCR-like effectsincluding its consequences on the flow curve, rod orientational order, the terminal relaxation time, and tube dilation/possible collapsewithin a single self-consistent forcelevel theory is a significant advance containing new physics not in our prior work.25 The remainder of the article is organized as follows. Section II briefly reviews our previous work in microscopically constructing the transverse confinement potential from topological constraints under quiescent conditions and describes how we include applied stress and rod orientation. Section III introduces our new theory of dynamic stress and orientational relaxation during continuous shear deformations. It has mathematical similarities with the Doi−Edwards model, but there are qualitative physical differences associated with our ability to compute a time-, stress-, and strain-dependent effective relaxation time from the dynamic tube confinement potential. Section IV presents calculations for different degrees of entanglement and deformation rates. The paper concludes in section V with a brief summary and discussion. 5685


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II. STRESS-DEPENDENT DYNAMIC CONFINEMENT POTENTIAL AND RELAXATION TIMES A. Transverse Confinement Potential. A key input to our rheological theory is the microscopic calculation of the relaxation time of a tagged rod given the current state of macroscopic stress and orientational order of the sample. Starting with our detailed theory for the anharmonic confining potential, the reptative relaxation time is then related to the width of the tube. In the presence of applied stress, there is a finite barrier to lateral motion which results in a competing relaxation channel quantified by the mean first-passage time for barrier hopping. We have previously discussed these aspects of our theory in great depth,25 so here we simply state the main results and briefly comment on their physical origins. The underlying microscopic, force-level dynamic mean-field theory treats a Brownian suspension of infinitely thin rods of length L and number density ρ. The dynamics are completely controlled by the uncrossability of the needles, which is exactly enforced at the binary-collision level, with higher order dynamical processes taken into account via a self-consistent renormalization. A dynamic transverse localization length (related to the tube diameter), rl = dT/2, is then computed based on a Gaussian analysis of tagged polymer density fluctuations transverse to the rod axis under conditions where reptation (longitudinal motion) is quenched. This method of extracting a tube diameter is conceptually similar to primitivepath methods employed in chain melt simulations.26 The use of a small-amplitude Gaussian analysis of the dynamic localization to determine the tube diameter corresponds to the assumption of a harmonic confinement potential. For highly nonlinear deformations this may no longer be sufficient, and the longer-time dynamics may be quite sensitive to the finite strength of entanglement constraints. Heuristically, our procedure for quantifying the anharmonicity of the confinement potential can be thought of as reinterpreting, in a local equilibrium spirit, the Gaussian localization result in terms of a balance between an entropic delocalizing force and an entanglement-based restoring force which act on the instantaneous transverse displacement of a tagged rod. Technical details can be found in refs 25, 27, and 28; the resulting force is f (r⊥) =

2kBT πρLkBT g (L /r⊥) − 2r⊥ r⊥ 8 2

The confinement potential then follows from integrating this instantaneous force up to a specified displacement, Fdyn(r⊥) = −∫ rr⊥l f(r) dr. The qualitative and quantitative features of this tube confinement potential for needles have been discussed previously.25,27,28 A sample calculation under quiescent conditions for a relatively modest density (ρ/ρc = 10, where ρcL3 = 3√2 corresponds to the critical density where a barrier to transverse motion first emerges) is shown in Figure 1B. Our key result, the anharmonic softening of the tube and hence the finite strength of the transverse entanglement force, is apparent.

Figure 1. (A) Schematic cartoon of the theoretical approach. (i) Uncrossable rods diffusing with bare dynamics and undergoing impulsive hard-core collisions. (ii) Self-consistent renormalization of many-body effects into an effective diffusion tensor at the two-roddynamics level. (iii) Analysis of short-time structure of the effective diffusion tensor determines the transverse dynamic confinement potential. (B) Dynamic confinement potential (units of kBT) vs transverse displacement (r⊥/L). From top to bottom solid curves correspond to S = 0 and σ/σy = 0, 0.75, 1, and the dashed curve corresponds to σ/σy = 0, S = 0.5. For illustrative purposes, on the S = 0, σ = 0 curve the localization length rl ≡ dT/2, and location of the maximum restoring force r* are labeled; for the S = 0, σ = 0.75 curve the barrier location rb is indicated.

(1)

One expects that a continuous shear deformation will induce rod alignment, which we have previously shown can be easily incorporated provided the distribution of rod centers of mass (CMs) remains random.27 To study the simplest possible model with a single alignment axis, an Onsager-like distribution2 for the relative orientation of two rods is adopted: f(μ,α) = α cosh(αμ)/4π sinh α, where α parametrizes the degree of orientation relative to some axis and μ is the dot product of the two-rod orientation vectors. Physically, one expects that alignment reduces the probability of inter-rod collisions, thus decreasing the effective entanglement density and leading to both tube dilation and enhanced transverse motion. We have shown27 that to a very good approximation the effect of rod alignment in the theory can be thought of as modifying the effective density as ρ → ρ(1 − S)1/2, where S = ⟨(3 cos2 θ − 1)/2⟩ is the usual nematic order parameter. Finally, the effects of an applied shear deformation are included at the single-polymer level in the framework of the nonlinear Langevin equation (NLE) theory via a stress-based

The Gaussian-theory result for the tube diameter29,30 can be expressed as the mechanical equilibrium condition f(r⊥ ≡ dT/2) = 0. The function g(x) has a closed-form solution,31 but for convenience in the numerical calculations presented below a simple Pade approximation is employed that correctly recovers the asymptotic behavior of g and deviates by at most a few percent over the entire range of x:

(

r

)

r a + 48π L r 2 ⎛⎜ L ⎞⎟ g ≈ 2 ⎝ ⎠ r r2 r 2L 2L a + 48π L + 18π 2 2

(

⎧ r , r≪L ⎪ ⎪ 2L ≃⎨ ⎪ 4 − L, r≫L ⎪ ⎩ 3π 4r

L

) (2)

where a = 128 − 9π2. 5686


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τrot,q ≡ τrot(ρ,0,0) as a shorthand for the quiescent terminal rotational relaxation time. The second relaxation channel is related to the orientationand stress-induced reduction of the transverse entropic barrier. The characteristic rate of this process is computed from Kramers’ mean first-passage time formula:32

microrheology approach, which has been previously formulated for both finite-excluded-volume complex liquids that interact via conservative forces11,13 and zero-excluded-volume needles.25,28 Given that the dynamical NLE approach is constructed at the single particle level, the only option is to describe stress as an external force on a moving polymer. Hence, the macroscopic stress, σ, results in a constant effective force on the rod CM, analogous to previous work on glassy colloidal and polymer liquids.11,13 We adopt a scalar (not tensorial) description of the stress to develop as simple a model as possible in this initial study. The dynamic free energy thus acquires a mechanical-work-like contribution that is linear in both stress and instantaneous transverse displacement: Fdyn(r⊥ , σ ) = Fdyn(r⊥ , σ = 0) − Aσr⊥

τhop(ρ , S , σ )

=

τ0

2π e βFB(ρ , S , σ ) K 0KB

(6)

where τ0 = L2/D∥,0, K0 and KB are the absolute magnitudes of the local curvatures at the confinement potential minimum (rl) and barrier location (rb), respectively, and FB is the height of the barrier. Since the lateral hopping times are comparable to the terminal rotational relaxation time only when the barrier is relatively small, the relevant part of our predicted quiescent potential for both the acceleration of (reptation-controlled) terminal relaxation and transverse barrier hopping is the quasilinear regime that falls in the intermediate displacement range rl < r < L (Figure 1). Physically, as the strain or stress approaches a critical (“absolute yield”) value, entanglements and tube localization are destroyed, FB → 0, and the transverse hopping time smoothly recovers the bare relaxation time τhop → τ0. In practice, the Kramers formula is a poor approximation for barrier heights comparable to the thermal energy. Thus, in numerical calculations when an activated hopping time is required for small barriers (rarely relevant in practice) a simple approach is adopted to estimate the hopping time: we fix S and linearly interpolate between the hopping time at the stress value that gives a barrier of 2 kBT and the bare time τ0 relevant at σ = σy(S). In our recent study of step-strain deformations, several examples were given of both the response of the dynamic confinement potential and the associated terminal relaxation rates for a variety of densities, stresses, and values of the orientational order parameter.25 We note that over the range of shear rates we have studied this issue only arises in the lower density calculations for fast flows. Where it arises the final results depend only modestly on the details of the approximation scheme as the lateral hopping channel starts to become competitive with the usual reptation channel. As seen in Figure 5B, though, for ρ/ρc = 10 the accelerated reptation time is at least an order of magnitude faster than lateral hopping for all but the fastest flow we study.

(3)

25,28

We previously argued that the prefactor converting macroscopic stress to microscopic force should be A = πL2/4. Allowing this prefactor to also depend on S may make some quantitative differences in our results below, but qualitatively we would expect little to change. Furthermore, we previously showed that assuming that the prefactor is orientationindependent leads to values of a yield strain consistent with recent experiments.25 Combining the quiescent confinement potential with the results above, the total effect of applied stress and orientation on the transverse confinement potential is described as Fdyn(r⊥ , ρ , σ , S) = Fdyn(r⊥ , ρ 1 − S , σ = 0) − πL2σr⊥/4 (4)

Although physically motivated, it is fair to call our treatment of external stress on the tube confinement potential and polymer dynamics phenomenological in that it has not been derived from rigorous nonequilibrium statistical mechanics. Our primary aim is to show that, in the presence of anharmonically softened tube confinement, certain qualitative features of the nonlinear dynamics can be properly captured in the context of a self-consistent single-polymer theory where the motion of a tagged needle is functionally coupled with the dynamic transverse confinement potential and the imposed flow. Qualitatively, one may think of the anharmonic tube as a physically clear implementation of a local constraint-release-like effect. B. Competing Relaxation Times. Knowledge of the tube confinement potential allows the two longest relaxation times to be computed as a function of density, stress, and orientational order. Having discussed the derivations of these in depth before, here we simply quote the results.25 The first relaxation channel is the stress- and orientationmodified reptation-controlled rotational relaxation

III. THEORY OF CONTINUOUS SHEAR DEFORMATIONS In our work on instantaneous step-strain deformations,25 a generalized Maxwell model was employed to describe the relaxation of stress and orientation after the deformation:

2 τrot(ρ , S , σ ) ⎛ rl(ρ , 0, 0) ⎞ =⎜ ⎟ ≈ ( 1 − S − σ ̃ /3(ρ /ρc ))2 τrot(ρ , 0, 0) ⎝ rl(ρ , S , σ ) ⎠

σ (t ) dσ =− dt τeff

(5)

S(t ) dS =− τeff dt

Equation 5 invokes the simple and well-known connection between the width of the tube (expressed in terms of our theory for the tube radius as a function of polymer density, orientation, and stress), the transverse diffusion constant, and the rotational relaxation time.25 The final approximate equality holds at high density and modest degrees of stress and orientation; σ̃ = σL3/kBT. For larger deformations or densities closer to the critical value the full numerical result for the localization length must be used. We will occasionally write

(7)

The richness of the model arises from the fact that the effective relaxation time, τeff−1 ≡ τrot−1 + τhop−1, is a function of the instantaneous stress and orientation. Thus, while the above equations appear simple, they are actually highly nonlinear and strongly coupled. Here we generalize this model to timedependent shear deformations. 5687


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where Sa(γ) is the orientation that results from an affine deformation of strain γ

The starting point for a time-continuous shear deformation in the xy-plane2,32 is ⎛

t

σ(t ) = Ge

⎞

∫−∞ dt′⎜⎝ dψ (dt t−′ t′) ⎟⎠Q [E(t , t′)]

Sa(γ ) =

(8)

The plateau modulus for rigid rods is Ge = 3ρkBT/5, a simple result that originates from the decay of single-rod orientational autocorrelations.2 A key point is that even though the tube diameter is a function of orientation and stress, the elastic modulus of rods is not. The quantity Q(x) ≈ 5x/(5 + x2) is the classic deformation “overorientation” factor, and E is the accumulated deformation that, for start-up shear deformations, is E(t , t ′) =

∫t′

t

⎧ γ(̇ t − t ′), t ′ > 0 dt ″γ(̇ t ″) = ⎨ t′ < 0 ⎩ γ(̇ t ),

⎛ dS − S(t ) dS = +⎜ a τeff (ρ , S(t ), σ(t )) ⎜ dγ dt ⎝

∫t′

t

⎞ dt ″ ⎟ τeff (ρ , S(t ″), σ(t ″)) ⎠

(9)

γeff (t ) = (10)

t

∫−∞ dt′ψ (t , t′) ddt′ Q [E(t , t′)]

⎧ ⎪ γ ̇ − γ ̇ dQ (x)/dx|x = γ ( ̇ t − t ′) , t ′ > 0 d Q (t , t ′) = ⎨ ⎪ dt ′ t′ < 0 ⎩ 0,

(11)

(12)

Combining these expressions, the final formula for the stress as a function of time is then σ(t ) = Geγ ̇

∫0

t

The evolution equation governing orientational order for time-dependent deformations must now be generalized. Again, we do not claim a full microscopic derivation of these evolution equations, and we employ the simplest possible extension of prior work that reduces to our step-strain formalism in the limit of a very rapid, interrupted continuous shear deformation (physically distinguished from a step strain as a delta-function deformation). The initial value of the orientation following a step-strain is the result of a purely geometric (affine deformation) argument. For an arbitrary γ̇(t) in the absence of relaxation (i.e., in the limit where deformation occurs much faster than any relaxation process), one expects the orientational order to have the same value after a total strain of γ = ∫ t0γ̇(t′) dt′ as it would have following a step-strain of equal total amplitude. Evidently, then, in the absence of relaxation one has ⎞ ⎟γ(̇ t ) ⎟ t γ =∫0 γ(̇ t ′)dt ′ ⎠

(16)

3S(t ) 1 + S(t ) − 2S2(t )

(17)

IV. RESULTS A. Approach to Steady State. We now present representative numerical calculations for two reduced densities which correspond to modestly (ρ/ρc = 10) and heavily entangled (ρ/ρc = 1000) rod systems, at a variety of deformation rates. These reduced densities are relevant, e.g., to entangled solutions of long synthetic PBLG rods and semiflexible f-actin or rigid microtubules, respectively.2,19 An important limitation is the range of deformation rates we expect to be able to describe given that our theory has no mechanism for single rod motion to proceed faster than the “bare” time scale, τ0. In light of this, the computations are restricted to a range of flow rates γ̇τrot,q ≲ τrot,q/τ0. Under quiescent conditions the separation in time scales between the terminal rotational relaxation time and the bare time is controlled by the ratio D⊥/D⊥,0, which is highly density-dependent: at ρ/ρc = 10, (D⊥/D⊥,0)−1 ≈ 30, and for ρ/ρc = 1000, (D⊥/D⊥,0)−1 ≈ 1.6 × 105. Thus, we study the lower density system at Weissenberg numbers up to Wi ≡ γ̇τrot,q = 50 and the higher density system up to Wi = 1000. Figure 2 shows the stress normalized by the shear modulus as a function of accumulated strain for different shear rates. As the Weissenberg number exceeds a value of order unity, a weak overshoot characteristic of entangled rod solutions emerges. Recall that in the absence of chain stretch the stress overshoot

⎛ ⎞ ⎡ d ⎛ 5x ⎞⎤ ⎟ ⎟⎥ dt ′⎜⎜ψ (t , t ′)⎢ ⎜ 2 ⎟ ⎝ ⎠ ⎣ ⎦ d x + x 5 ⎝ x = γ(̇ t − t ′)⎠ (13)

⎛ dS dS = ⎜⎜ a dt dγ ⎝

⎞ ⎟γ(̇ t ) ⎟ γ = γeff ⎠

Note that in the limit of τeff−1 ≪ ̇ γ one has γeff → γ = ∫ t0γ̇(t′) dt′, and thus eq 16 reduces to eq 14. Equations 13, 16, and 17 form a closed set of coupled nonlinear equations, functionally coupled to the dynamic tube confinement potential, which govern the evolution of an entangled needle system under a general time-dependent shear deformation. In the next section these equations are solved numerically for a constant deformation rate, γ̇(t) → γ̇ and strain γ = γ̇t. The extension to oscillatory shear deformations is straightforward. For continuous start-up shear the full transient response is obtained, during which stress, orientation, relaxation time, and tube confinement potential all evolve in a dynamically coupled and self-consistent manner.

Note that ψ does not in general decay exponentially in time due to the orientation and stress dependence of the effective relaxation time, a crucial difference between our theory and the DE model. Undoing an integration-by-parts used to get to the above equation for σ gives σ(t ) = −Ge

(15)

Note that the derivative of Sa is computed at an ef fective value of the strain. This reflects the physical intuition that as the orientation relaxes, the amount by which the deformation reorients the rods depends on the current state of the system, not on some hypothetical state that exists in the absence of relaxation. The theory is then closed by introducing an effective strain, γeff, which satisfies Sa(γeff) = S(t). Explicitly inverting eq 15 gives

The normalized “tube survival” relaxation function, ψ, is taken to be the accumulated exponential relaxations associated with the two parallel channels: ⎛ ψ (t , t ′) = exp⎜ − ⎝

γ − 3 γ2 + 4

To reintroduce orientational relaxation we posit the following evolution equation:

⎪ ⎪

−2γ

(14) 5688


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Figure 2. Shear stress normalized by the equilibrium shear modulus as a function of strain at increasing deformation rates (bottom to top) for (A) ρ/ρc = 1000 and (B) ρ/ρc = 10. Inset of (A): value of stress at the yield peak normalized by the entanglement modulus as a function of deformation rate. Curves are the calculation according to Doi− Edwards,2 orientation-modified Doi−Edwards, and the present work at high and low densities (top to bottom).

Figure 3. Orientational order parameter as a function of strain at increasing deformation rates (bottom to top) for (A) ρ/ρc = 1000 and (B) ρ/ρc = 10.

of linear flexible polymers. Although the orientation does not go through a maximum, there does seem to be a qualitative change in its rate of growth at an accumulated strain close to the stress maximum. The absolute magnitude of S at high strains is also significantly smaller than predicted by Doi− Edwards theory (not shown)a trend that follows from our prediction that tube confinement weakens with growing stress and rod alignment. One can also study the microscopic dynamical state of the sheared fluid. In Figure 4, calculations of the effective relaxation time normalized by its quiescent, reptation-controlled value are shown (note that for the fastest rates in this and later figures, small amounts of noise are evident associated with numerical

is expected to be smaller for rods than for flexible chains.23 The modulus-normalized stress curves in Figure 2A,B at the two densities are very similar: at the lowest rate they overlap, and at higher rates (subject to γ̇τrot,q ≲ τrot,q/τ0) there are only relatively minor differences in the region of crossover to steady state. In the inset to Figure 2A the magnitude of the “stress overshoot” peak is plotted as a function of Weissenberg number. Qualitatively, a strong increase is predicted at low shear rates, followed by a slower approach to saturation at high Wi numbers. Overall, the behavior is very similar to the classic Doi−Edwards result,2 which is also shown in the inset, although with a lower magnitude due to our prediction that the tube softens with increasing deformation. The corresponding strain at the stress peak, γy, is a weakly nonmonotonic (concave down) function of imposed shear rate (not shown). For the high-density system we find γy ≈ 1−1.1, quite close to the “absolute yield strain” corresponding to the complete destruction of the transverse entropic barrier for step-strain deformations.25 For the lower density fluid γy ≈ 0.8−0.9, slightly higher than the corresponding step-strain yield strain.25 In all cases, the strain at the overshoot is smaller than the DE result of γy ≈ √5. Figure 3 shows the growth of the orientational order parameter during the deformation; in contrast to the stress curves, there is no overshoot. This raises an important point: in our prior step-strain work the form of the generalized Maxwell model guaranteed a type of stress-optical law relation between stress and orientation,25 but here our use of γeff suggests, in general, that no such relationship will hold. The degree to which this relationship is violated can be intuited from eq 15 by looking at the magnitude of the second term, which governs deviations of the orientation that are controlled by the effective rather than the true strain. Thus, in slow enough flows or in the time regime t ≪ 1/γ̇ where the system has not relaxed any orientation and γeff ≈ γ, the stress-optical law will still approximately hold, just as in the continuous shear deformation

Figure 4. Effective relaxation time normalized by the quiescent terminal relaxation time as a function of strain at increasing deformation rates (top to bottom) for (A) ρ/ρc = 1000 and (B) ρ/ρc = 10. 5689


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integration of the evolution equations). Strong acceleration of relaxation is generically predicted, which is more pronounced as the Weissenberg number increases. One also sees a weak undershoot at a strain close to the stress maximum for high Wi number. Figure 5 shows the relative importance of (accel-

Figure 6. Entropic barrier height (in units of thermal energy) as a function of strain at increasing deformation rates (top to bottom) for (A) ρ/ρc = 1000 and (B) ρ/ρc = 10.

Figure 5. Ratio of the terminal rotational relaxation time to the mean transverse activated barrier hopping time, τhop/τrot, as a function of strain at increasing deformation rates (bottom to top) for (A) ρ/ρc = 1000 and (B) ρ/ρc = 10.

erated) reptative relaxation and transverse barrier hopping. For the high-density fluid, even up to very large rates there is still a substantial entropic barrier, and only at the highest rate is transverse activated hopping competitive with reptation. In contrast, the entanglement constraints are much weaker for the lower density fluid, and as a consequence at the fastest deformation rate studied we find the barrier to transverse motion (the tube) is completely destroyed. This is akin to a microscopic “absolute yielding” event, seemingly in the spirit of recent experimentally motivated discussions for entangled chain polymers.8 The above behavior is more explicitly documented in Figures 6 and 7 in the context of key features of the tube confinement potential. Figure 6 shows the strain evolution of the barrier height for a wide range of shear rates. The entropic barrier generically drops by orders of magnitude with increasing deformation rate. However, at the shear rates studied the highdensity system always exhibits a transverse barrier large compared to thermal energy, in contrast to the more weakly entangled fluid. Figure 7 shows the concomitant widening (and breaking at high enough Wi for the lower density system) of the effective tube diameter during the deformation. Again, the long-time steady-state value is achieved at a strain roughly equal to, or just beyond, the stress overshoot. Quantitatively, the tube diameter can widen by a nonperturbative factor depending on rod density and Wi number. B. Steady-State Properties. We now turn to the longtime, nonequilibrium steady-state properties. Perhaps our most dramatic finding is the flow curve shown in Figure 8. In contrast to the Doi−Edwards model which displays an unphysical nonmonotonicity at high deformation rate,2,23 we predict a plateau-like behavior of the stress at both rod densities studied;

Figure 7. Ratio of the tube or localization length at fixed strain to its undeformed value, rl(γ)/rl(0), as a function of strain at increasing deformation rates (bottom to top) for (A) ρ/ρc = 1000 and (B) ρ/ρc = 10.

the modulus-normalized limiting stress is only weakly dependent on rod density or quiescent degree of entanglement. This stress-plateau behavior is an emergent aspect of the theory that is a consequence of our self-consistent construction of a deformation-dependent tube confinement field. Hence, it appears to be quite different than the phenomenological CCR approach where an ansatz concerning many chain physics in strong flows is effectively inserted by hand into the singlechain DE model. A Doi−Edwards-like calculation is also shown in Figure 8 where the tube is allowed to dilate with orientation but remains insensitive to stress. This delays the non5690


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Figure 10. Effective Weissenberg number as a function of strain rate. Curves are the calculation according to Doi−Edwards2 (dash-dotted), orientation-modified Doi−Edwards (black solid), and the present work at high (solid) and low densities (dashed).

Figure 8. Normalized steady-state shear stress vs strain rate. Curves are the calculation according to Doi−Edwards2 (dash-dotted), orientation-modified Doi−Edwards (black solid), and the present work at high (solid) and low densities (dashed).

monotonicity to higher deformation rates but does not remove the unphysical behavior. Thus, we conclude that the most dramatic predictions of our theory are primarily due to the selfconsistent coupling of the macroscopic stress to the microscopic dynamical state of the tube confinement potential. Figure 9 shows the corresponding long-time orientational order

Figure 11. Effective Weissenberg number vs accumulated strain for ρ/ρc = 1000. From bottom to top curves correspond to Wi = 0.1, 1, 5, 10, 50, 500, and 1000, and the stars correspond to the value of strain at the stress maximum at each of those rates.

stress overshoot peak. Note that the full flow-induced modification of the relaxation time is not complete until a strain of order γy ≈ 1. Figure 12 examines the steady-state tube diameter as a function of deformation rate. After an initial complicated Figure 9. Steady-state orientational order parameter as a function of strain rate. Curves are the calculation according Doi−Edwards2 (dashdotted), orientation-modified Doi−Edwards (black solid), and the present work at high (solid) and low densities (dashed).

parameter, which monotonically grows with strain rate. Though qualitatively similar, we note that we predict much less steadystate alignment under shear than the Doi−Edwards model, consistent with expectations based on Figure 3. The form of the stress plateau predicted by our theory is qualitatively similar to the modification of the tube-model that includes CCR effects in systems of flexible chains. For example, in Figure 10 the long-time value of the ef fective Weissenberg number, defined by Wieff = γ̇τeff, is plotted, and a plateau-like behavior is evident at Wieff of order unity. In detail, as seen in the figure, the plateau value is (weakly) density-dependent, and true saturation is not obtained over the Wi number ranged studied. Nevertheless, the long-time value of the effective relaxation time does scale roughly as τeff,∞ ∼ γ̇−1. Such a relation is essentially assumed in models of CCR,3 but here it is, again, an emergent feature of our self-consistent treatment which allows the tube confinement field to evolve dynamically. To get a sense of when this CCR behavior becomes important, in Figure 11 the effective Weissenberg number is plotted as a function of strain for the high-density system. In all cases it is clear that the qualitative change in the behavior of Wieff from the initial behavior to the steady-state behavior occurs at a strain comparable in magnitude to the location of the yield or

Figure 12. Inverse normalized steady-state localization length as a function of strain rate. Curves are the high-density (solid) and lowdensity (dashed) calculations, plus a power-law guide to the eye with an exponent of −0.4 Inset: effective entanglement density vs strain rate. Curves are the high-density (solid) and low-density (dashed) calculations, plus a power-law guide to the eye with an exponent of −0.8.

regime of tube widening, the transverse localization length continues to dilate in a power-law fashion, rl ∼ Wi−0.4. Computing the tube diameter using the full steady-state degree of orientational order, but neglecting the effect of stress, one can probe the relative importance of flow-induced orientation versus direct stress effects on tube dilation. We find that the dominant effect comes from the external stress−using only orientational order results in a power-law tube dilation with 5691


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much smaller exponent, rl ∼ Wi−0.05. This, together with the results of Figure 8, emphasizes that our results are robust to the particular choice of evolution equation we use for the orientational order. Taking standard tube-model relations between the tube diameter and the number of entanglements,2 the predicted tube dilation could also be interpreted as a powerlaw reduction in the number of effective entanglements, shown in the inset to Figure 12. Intriguingly, such power-law entanglement-reduction behaviors have been observed in simulations of flexible and DNA-like chains under both shear and extensional deformation, although for these flexible-chain systems the observed power laws have smaller exponents than those we predict for rigid rods.14−16 From both the above discussion and the various figures showing transient and steady-state properties, it is clear that there is a regime of flow rates for which the tube is no longer present: the stresses induced by the flow are sufficient to destroy it. This is seen explicitly for the upper range of rates at ρ/ρc = 10 and would also be predicted for the high-density fluid at higher rates (not shown). Interestingly, the behavior of the stress overshoot implies that there would be a (very narrow) range of flow rates at which the tube is only transiently destroyed in the vicinity of the stress peak but then re-forms at a very weak level in steady state. However, we emphasize that the possibility of transient or permanent tube destruction are of only modest practical importance for the quantities we study. Over the range of rates prescribed above, the main finding is that the tube is sufficiently softened for the transverse activated barrier hopping process to become competitive with (stress- and orientation-accelerated) longitudinal reptative relaxation. Moreover, since transverse hopping is exponentially accelerated by stress relative to reptation,25 any reasonable model for the flowinduced orientational order will lead to qualitatively similar results. Even without tube destruction, it is this feature that underlies the qualitatively sensible CCR-like behavior seen in Figure 8.

solutions34,35). Similarly there is little simulation data for the nonlinear rheology of Brownian solutions of entangled rigid needles. This is undoubtedly a tractable problem, and we suggest that new computer simulations be performed to critically probe the key features of our quiescent tube confinement potential, its response to prescribed applied stress and rod orientation, and other predicted consequences of deformation on entanglement dynamics. The next natural step in the development of our theory is to consider the relaxation of rigid-rod solutions under different deformation protocols. In particular, given the mysteries surrounding the observed differences between shear and extensional deformation experiments and simulations of entangled chain solutions and melts,2,6,14,16 formulating the theory to treat extensional deformation is a worthwhile endeavor. However, we believe the task of highest impact is to extend our approach to treat entangled flexible chain liquids. We have begun work in this direction based on a microscopic version of the classic DE ansatz of treating ideal coils as random walks of primitive path steps under equilibrium conditions.36 Generalization to nonlinear rheological conditions will at a minimum require introducing ideas about how chain stretching enters the problem. We anticipate it may also require a critical evaluation of how stress is stored, both at the intra- and interpolymer level.37,38

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (D.M.S.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Nanoscale Science and Engineering Initiative of the National Science Foundation under NSF Award DMR-0642573.

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V. CONCLUSION We have built on our microscopic theory for the confinement potential of rigid macromolecules interacting via topological constraints under quiescent and stressed conditions to construct a predictive, self-consistent theory for nonlinear response under continuous shear deformation. Our microscopic prediction of an anharmonic tube confinement potential immediately implies (i) a nonlinear coupling between stress, orientation, and effective relaxation rate and (ii) the introduction of a “transverse tube hopping” relaxation channel that competes with the usual reptation-based relaxation channel. A variety of novel phenomena are predicted, including (a) the strain and Weissenberg number dependent dilation of the tube in shear flows, (b) a purely monotonic growth, and ultimate apparent saturation, of the steady-state stress with shear rate, (c) an emergent CCR-like phenomena in which the terminal relaxation time is (nearly) inversely related to the shear rate, and (d) a decrease in the effective entanglement density under flow. Qualitatively, these effects all seem consistent with recent experiments and simulations of entangled flexible chain polymer liquids.8,14,15 Unfortunately, there is relatively little experimental data characterizing the response of high-aspect-ratio rod solutions in the isotropic state to continuous shear deformations. This would seem to be an experimentally accessible problem based on synthetic or biological rigid rod polymers (e.g., microtubule

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