Letter Cite This: ACS Macro Lett. 2018, 7, 218−222
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Physics of the Stress Overshoot and Chain Stretch Dynamics of Entangled Polymer Liquids Under Continuous Startup Nonlinear Shear Kenneth S. Schweizer*,†,‡,§,∥ and Shi-Jie Xie†,∥ Departments of †Materials Science, ‡Chemistry, and §Chemical and Biomolecular Engineering, and ∥Frederick Seitz Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801, United States S Supporting Information *
ABSTRACT: We construct a new theory for transient aspects of the shear rheology of entangled chain liquids. Within an established tube model constitutive equation framework, four new physical features are introduced: a tension blob scaling derivation of the interchain grip force that generates chain stretch, a force imbalance condition for the termination of affine stretch deformation, a delayed chain retraction process that after loss of grip is accelerated for fast deformations, and a distribution of tube diameters. Nonclassical predictions are made for the stress−strain curve to just beyond the overshoot, the existence of a master curve, and fractional power law scaling of the overshoot strain and stress at high shear rates, all in good agreement with experiment and simulation. Testable new predictions are made for chain stretch dynamics.
T
heoretical understanding of entangled polymer dynamics is an extremely challenging problem in statistical mechanics, necessitating the development of phenomenological approaches based on physically motivated, but ad hoc, postulates.1−3 Equilibrium dynamics is controlled by thermal energy scale processes, and the reptation-tube model with many elaborations captures well the consequences of interchain forces on polymer dynamics based on the confining tube (diameter dT) concept.1−3 In contrast, nonlinear rheology involves forces far beyond thermal, chain deformation, and other nonequilibrium effects. Thus, additional strong guesses must be made about how deformation modifies entanglements, the tube, primitive path (PP) contour length, and so on. Beginning with Doi and Edwards (DE),1 tube models generally assume deformation does not modify the tube diameter and stretched chains can freely retract along the PP in a Rouse manner.3−5 Direct testing of such assumptions is difficult. A fundamental force level approach is desirable, as has been recently pursued in equilibrium6,7 and for rod rheology.8,9 Our interest here is the early stages of continuous startup shear rheology of entangled polymer melts with an emphasis on the stress overshoot (Figure 1a). All regimes of Rouse Weissenberg number, WiR ≡ γ̇τR, are considered where τR = N2τ0/3π2 is the Rouse time and τ0 = ζ0b2/kBT is the segmental relaxation time. In classic tube models,1 at WiR < 1 there is effectively no chain stretch and the overshoot arises from affine overorientation. At high WiR > 1, chains are driven to stretch, which competes with Rouse relaxation along the PP. The causality question10 of what is the microscopic interpolymer force induced by a macroscopically applied deformation that induces chain stretching is not addressed. To address this, the © 2018 American Chemical Society
Figure 1. (a) Shear stress as a function of strain with the overshoot and hashed region of present focus indicated. (b) Depiction of the interchain grip and intrachain retraction forces per ref 10. (c) Schematic of unentangled polymers under shear that stretch in onespatial direction beyond the Pincus tension blob size per ref 17.
qualitatively new idea of a deformation-induced interchain “grip force” has been proposed by Wang and co-workers10−12 as the origin of affine stretching. The ability of a polymer to retract (onset of nonaffine behavior) is argued to be delayed until the elastic retraction force built by stretching exceeds the grip force, a condition called force imbalance or “loss of grip” (see Figure 1b). It is postulated this force imbalance10,11 triggers an immediate decrease of stress, thereby determining the overReceived: November 8, 2017 Accepted: January 29, 2018 Published: January 31, 2018 218
DOI: 10.1021/acsmacrolett.7b00882 ACS Macro Lett. 2018, 7, 218−222
Letter
ACS Macro Letters
ment” part due to tube confinement. Our hypothesis is the interchain drag or grip force for entangled polymers can be constructed using tension blob ideas suitably generalized to treat the transient response using reptation-tube theory results. Since the grip force controls stretching of the entire chain, the total (center-of-mass, CM) friction constant enters. In equilibrium, the CM mean square displacement (CM-MSD) law1,2 is Fickian beyond the Rouse time, MSD(t) = 6Dt, t ≥ τR, where D = D0Ne/(3N2), and non-Fickian on shorter times, MSD(t ) = 6D tτR , τe ≤ t ≤ τR . We introduce an effective time-dependent friction coefficient estimated using an Einsteinlike relation: ξCM(t) ≡ 6kBTt/MSD(t) or
shoot; retraction dynamics is not addressed. There is no theory for the grip force or force imbalance condition. The need for new theoretical developments seems clear from experiments performed in different laboratories in solutions and melts of different chemistries over a wide range of degree of entanglement Z = N/N e , 10−14 which find three key observations: (i) For WiR < 1, the overshoot strain is ∼2 and is (at most) weakly rate dependent; (ii) For WiR > 1, a fractional power law emerges over a remarkable two decades in rate,10,13,14 with the overshoot coordinates scaling as ∼WiR1/3; (iii) A universal stress−strain master curve can be constructed by nondimensionalizing the overshoot coordinates.13 A recent simulation observed point (ii).15 Although point (i) agrees with existing tube models,1,3−5 point (ii) is in strong disagreement; for example, the GLaMM model5 predicts a much stronger apparent power law (∼Wi0.7 R ) at high deformation rates (and ∼WiR asymptotically). Fundamental physics associated with chain stretching is apparently missing with regards to the stress overshoot behavior. This Letter proposes a new theoretical approach that addresses the above three points in a unified manner. For simplicity, and due to conflicting proposals and lack of understanding of the true behavior, we make the minimalist assumption that the tube diameter, Ne, and entanglement modulus, Ge, are deformation invariant. Physical processes important well beyond the overshoot, such as convective constraint release (CCR),3−5 are ignored. The theoretical advances fall into five categories: (1) A tension blob scaling construction of the interchain grip force and formulation of a mean force imbalance criterion; (2) Inclusion of tube diameter heterogeneity which introduces a distribution of force imbalance conditions; (3) A shear-rate-dependent acceleration of the contour length retraction rate relative to Rouse when WiR > 1; (4) A new evolution equation for stretch dynamics that combines (1)−(3); (5) Integration of (1)−(4) into a constitutive equation. Quantitative comparison to experiments and simulations are performed. We first recall the tension blob (size ξ) concept16 (Figure 1c) as used by Colby et al.17 to understand steady-state shear thinning (rate γ̇) in unentangled liquids. Here, ξ is the scale below which a polymer remains conformationally equilibrated and beyond which it stretches. It is determined by equating the total frictional drag force, fdrag, exerted on a tagged polymer by the surrounding chains (product of the total friction constant, ζCM = ζ0N, and velocity difference across a tension blob, γ̇ξ) to the corresponding tension blob force (kBT/ξ):17 fdrag = (ζ0N )(γξ̇ ) ≡
kBT ξ
⎛ k T ⎞1/2 ξ = ⎜ B ⎟ ∝ R eeWiR−1/2 ⇒ fdrag = ⎝ ζ0Nγ ̇ ⎠
t
−1 MSD(t ) = 6kBT ∫ dt ′ζCM (t ′). Since the MSD(t) is a power 0 law, the two approaches give the same result:
ζCM = ζ0
ζCM(t ) = ζ0
3N 2 Ne
(3)
t 3N 2 = ζ0 τR Ne
γ , τe ≤ t ≤ τR WiR (4)
where γ ≡ γ̇t is strain. Beyond the Rouse time, the effective friction is constant, but at earlier times it is time-dependent due to anomalous CM diffusion. We believe eq 4 remains valid even under fast deformation conditions because entanglement strands (to leading order) still obey random walk statistics for large Z due to back-folding, which is removed only on the (irrelevant for the overshoot) reptation time scale;18 see Supporting Information (SI) for additional discussion. Using these results in the second member of eq 2 yields our proposal for the entangled polymer “grip force”: fgrip = π
3kBT WiR , t > τR dT
fgrip = 3π
or
kBT (γWiR )1/4 , t < τR dT
γ > WiR or
γ < WiR
(5)
(6)
Note the distinctive strain-dependence in eq 6 and the nondimensionalization of shear rate by the Rouse time. The grip force unit is set by the ratio of thermal energy to tube diameter. Our formulation of a grip force based on an effective friction differs from that advocated by Wang et al.10 It also is not contained in tube model1,4,5,19 discussions of “tension” and “drag” forces. The grip force is not used in our calculation of stress, but only to formulate the force imbalance condition (see SI). The intrachain retraction force is determined from the entanglement plateau modulus20,21 Ge = 0.0023kBTp−3, where p is the packing length and dT ≈ 17.7p. To convert elastic stress (Geγ) to a microscopic force requires the relevant area, which is naturally set by the tube diameter. Adopting a circular crosssection, A = πd2T/4, yields
(1)
kBTγζ̇ CM
3N 2 , t ≥ τR Ne
(2)
where Ree = √Nb is the chain equilibrium end-to-end distance. These ideas give the viscosity laws: η ∝ N, WiR < 1, and η ∝ Nξ/Ree ∝ (τ0γ̇)−1/2, WiR > 1, which agree with experiment.17 The origin of shear thinning is that Rouse modes of wavelength exceeding ξ do not dissipate energy in the steady state.17 For entangled liquids, the essence of the reptation model is that interchain forces and connectivity result in anisotropic motion beyond the tube diameter scale. In the lab frame, there is an effective friction coefficient composed of both a local segmental contribution plus a nonlocal “topological entangle-
kT π fretract = δ d T2(Geγ ) ≈ 5 B γ 4 dT
(7)
The prefactor δ involves quantitative issues such as the fraction of elastic stress due to stretch versus orientation. The value adopted assumes 50% of the elastic stress determines the stretching force; other reasonable choices do not affect our results (see SI). 219
DOI: 10.1021/acsmacrolett.7b00882 ACS Macro Lett. 2018, 7, 218−222
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ACS Macro Letters The force imbalance criterion follows by equating the interchain grip and intrachain retraction forces to obtain the mean “loss of grip” strain beyond which it becomes possible for chain retraction to commence. Importantly, the tube diameter cancels out in this criterion. Since the time scale at the overshoot is much larger (smaller) than the Rouse time for WiR < 1 (WiR > 1), only eq 5 (eq 6) is used for simplicity (see SI for details). For WiR < 1, we thereby find π
3kBT kT WiR = 5 B γg ⇒ γg ≈ 1.8 WiR dT dT
(8)
This strain is (well) below the DE theory value of ∼2.25. Thus, the grip force is predicted to be unimportant in determining the overshoot. In contrast, for WiR > 1 one has: 1
kT kT 3π B (γgWiR )1/4 = 5 B γg ⇒ γg ≈ 2.3WiR1/3 dT dT
Figure 2. Grip loss function vs strain at three shear rates. Inset: Corresponding grip strain probability distribution function.
(9)
The 1/3 exponent arises from a competition between the strain dependences of the grip and retraction forces and agrees well 15 with experiment12−14 (γg ≈ 2Wi1/3 R ) and simulation. Have we explained the stress overshoot? No, since the force imbalance criterion is a necessary, but not sufficient, condition for the initiation of chain retraction and stress decrease. The reason is that, even if grip is lost, polymers can still stretch under continuous deformation. The force imbalance condition only signals breakdown of affine deformation of the PP contour length. One must integrate this idea with a constitutive equation and address the following open questions. What is the nature of retraction dynamics after grip is lost and how does it modify stress? How does tube diameter heterogeneity impact force imbalance and rheology? Can one predict a stress−strain master curve? Does the force imbalance criterion really determine the overshoot and 1/3 power law? Simulations have found a broad distribution of the equilibrium tube diameter22 for which an accurate microscopic theory7 exists (see SI). This modifies the intrachain retraction force strength via the conversion step from elastic stress to microscopic force, which requires the cross-sectional area in eq 7, f retract = Ω25γ(kBT/dT,0), where dT,0 is the mean tube diameter and Ω ≡ dT/dT,0. The grip force is taken to be unaffected since it is set by the chain-scale CM friction and tube diameter fluctuations are expected to be averaged out to leading order. The tube diameter fluctuations7 imply a distribution of force imbalance conditions and loss of grip strains, P(γgrip). Strands with larger tube diameters undergo force imbalance “easier”, that is, at lower strains. The fraction of strands that have achieved force imbalance at a given strain is Θgrip(γ) = ∫ γ0P(γgrip)dγgrip. Results for these quantities are shown in Figure 2. A continuous crossover from solid-like to liquid-like response is expected, as found in our rheological calculations below (Figure 3). We adopt the Mead-Larson-Doi (MLD) model4 as a simple constitutive equation framework. The chain PP contour length stretch ratio is λ(t) ≡ lPP(t)/lPP,0. It describes in an average sense the behavior of any chain in the liquid which includes the mean consequences of a distribution of tube diameter per Figure 2. Orientational stress relaxes via reptation on a time1,4 τrep = a3ZτR, a = 1 or λ2, where a = λ2 corresponds to deformation-induced PP lengthening.4 Our results are insensitive to a since orientational relaxation is not important for determining the stress overshoot. The relevant MLD equations are then:4
Figure 3. Dimensionless stress as a function of strain for Z = 25 at WiR = 0.13, 0.40, 0.67, 1.33, 4.00, 6.67, 13.3, 40.0, and 66.7 from bottom to top. Dots indicate the overshoot which nearly fall on the dashed line. Inset: Corresponding doubly normalized stress−strain curves; green crosses indicate the experimental master curve.13
σ(t ) = 5Geλ(t )2 Sxy
(10)
t
Sxy =
∫−∞ dt′ dψ (dt t−′ t′) Q xy[E(t , t′)]
(11)
where Qxy is the shear component of the DE tensor and ψ(t − t′) is the tube survival function.1 Based on our physical picture, the evolution equation for stretch is postulated as dλ λ−1 = Sxyγλ̇ − Θgrip(γ ) dt τR,eff
(12)
The first term describes affine stretching in the standard manner,4,19 but the second term describing retraction has two qualitatively new elements. First, it continuously “turns on” with increasing strain capturing the degree of interchain grip loss as quantified by Θgrip(γ). Second, as grip is continuously lost, the PP contour length increasingly behaves in an unentangled manner (but not the transverse or orientational dynamics which remains tube-confined); retraction is accelerated when WiR > 1 for the reasons discussed above.17 The effective retraction time is reduced in a rate-dependent manner as,17 τR,eff(γ̇) ∝ τRξ/Ree. Quantitatively, we determine it from a Maxwell model or tension blob argument as 220
DOI: 10.1021/acsmacrolett.7b00882 ACS Macro Lett. 2018, 7, 218−222
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ACS Macro Letters τR,eff (γ )̇ =
τR η(γ )̇ = G Rouse [1 + WiR2 ]1/4
seems due to a near cancellation of competing effects: the overshoot strain grows, but losing grip becomes harder, with increasing WiR. Figure 5 presents predictions for the stretch ratio up to its peak value. The latter occurs beyond the stress overshoot since
(13)
where the final equality is an accurate interpolation formula.17 Equations 10−13 constitute closed coupled relations for the rheological response and stretch dynamics. We now present representative calculations for Z = 25, and have verified the independence of trends on Z. Some plots include data from both simulation and experiment. There is no fitting in the comparisons. Figure 3 shows nondimensionalized stress−strain curves for a wide range of WiR. Interestingly, the overshoot points fall on a line of slope ∼0.6. This behavior was found experimentally,13 and interpreted as a deformation-softened elastic modulus Geff ≈ 0.5Ge. Per Figure 2, its physical origin here is the predicted distributed loss of grip. The inset of Figure 3 shows a fully nondimensionalized plot based on the overshoot coordinates. An excellent master curve is obtained, in accord with experiment.13 Figure 4 plots the overshoot strain and stress as a function of rate. We predict the former is ∼2.25 for WiR < 1, in agreement
Figure 5. Stretch ratio as a function of strain; curves terminate at the maximum. Color code indicates the WiR value per Figure 3. Dots indicate the strain at the stress overshoot. Inset: Logarithmic plot of stretch ratio at the stress overshoot as a function of WiR in two formats with indicated apparent scaling exponents.
total stress involves an orientational component (see SI). The stretch ratio variation at the overshoot, and its difference compared to the unstretched case, are shown as a function of rate in the inset of Figure 5. At low WiR, there is almost no stretching (ala DE theory1 and simulation15), but apparent power law scaling is predicted for the deviation from unity. For WiR > 1, apparent power law growth is predicted with exponents of ∼0.3 (inset (a)) or 0.5 (inset (b)). The former value is close to 1/3 because the stretch dynamics of eq 12 incorporates the delayed onset of retraction and accelerated relaxation rate ideas. These apparent power law behaviors are very different than what is predicted by MLD,4 GLaMM,5 or other tube models, and the physical reason is the same as for the stress overshoot. In summary, we have theoretically formulated the concepts of interchain grip force, force imbalance criterion, delayed onset of chain retraction, and accelerated retraction after loss of grip under fast deformation for entangled polymers including consequences of a tube diameter distribution. Predictions for a stress−strain master curve and power law scaling of overshoot coordinates agree well with experiment and simulation. Testable new predictions have been made for PP contour length dynamics. The present advance allows the construction of a full rheological theory of startup shear deformation23 including postovershoot effects such as stress undershoots and flow curves for all WiR values.14,24 Extensional rheology can also be studied, including the possible breakdown of the entanglement network.25
Figure 4. Overshoot strain as a function of dimensionless rate for four SBR melts,13 from simulation (diamonds),15 and our prediction with apparent scaling exponents indicated. The dashed lines are polystyrene melt data.14 Inset: Corresponding normalized overshoot stress.
with DE,1 indicating the unimportance of the grip force for the overshoot. However, for WiR > 1 an effective power law with exponent ∼0.3 is predicted, which is very close to the mean (no tube diameter distribution) result of eq 9. We find this connection requires the idea of accelerated retraction in fast deformation after losing grip and qualitatively supports the argument that force imbalance determines the overshoot. The predicted overshoot strains are within a factor ∼2 of data on different polymers with different Z values 13 and also simulation.15 The power law rate dependences of melt experiments, simulation and theory are all close. The analogous results for the overshoot stress are shown in the inset of Figure 4. As found experimentally,10,12,13 it scales the same as the overshoot strain. Significantly, the effective power law extends to WiR values well below unity, as observed experimentally.13 No existing tube model makes the above predictions. This suggests the key missing physics involves the interchain grip force concept of delayed onset of retraction and the idea that once grip is lost chains retract faster than Rouse. Additional insight follows from asking what fraction of entanglement strand grip is lost at the overshoot. We find roughly 75%, which is weakly dependent on WiR > 1. The latter
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.7b00882. Additional discussion of the physical ideas of our theory, technical simplifications adopted, and numerical calcu221
DOI: 10.1021/acsmacrolett.7b00882 ACS Macro Lett. 2018, 7, 218−222
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ACS Macro Letters
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(17) Colby, R. H.; Boris, D. C.; Krause, W. E.; Dou, S. Shear thinning of unentangled flexible polymer liquids. Rheol. Acta 2007, 46, 569− 575. (18) Desai, P. S.; Larson, R. G. Constitutive model that shows extension thickening for entangled solutions and extension thinning for melts. J. Rheol. 2014, 58, 255−279. (19) Pearson, D. S.; Kiss, A. D.; Fetters, L. J.; Doi, M. Flow-Induced Birefringence of Concentrated Polyisoprene Solutions. J. Rheol. 1989, 33, 517−535. (20) Fetters, L. J.; Lohse, D. J.; Richter, D.; Witten, T. A.; Zirkel, A. Connection between polymer molecular weight, density, chain dimensions, and melt viscoelastic properties. Macromolecules 1994, 27, 4639−4647. (21) Everaers, R.; Sukumaran, S. K.; Grest, G. S.; Svaneborg, C.; Sivasubramanian, A.; Kremer, K. Rheology and microscopic topology of entangled polymeric liquids. Science 2004, 303, 823−826. (22) Tzoumanekas, C.; Theodorou, D. N. Topological analysis of linear polymer melts: a statistical approach. Macromolecules 2006, 39, 4592−4604. (23) Xie, S.-J.; Schweizer, K. S. Manuscript in preparation. (24) Ravindranath, S.; Wang, S.-Q. Steady state measurements in stress plateau region of entangled polymer solutions: Controlled-rate and controlled-stress modes. J. Rheol. 2008, 52, 957−980. (25) Zhu, X.; Wang, S.-Q. Mechanisms for different failure modes in startup uniaxial extension: Tensile (rupture-like) failure and necking. J. Rheol. 2013, 57, 223−248.
lations of the distribution of tube diameters and orientational shear stress as a function of strain (PDF).
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Shi-Jie Xie: 0000-0003-3317-4720 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by DOE-BES via Oak Ridge National Labotatory. Helpful discussions with S. Q. Wang, Z. G. Wang, D. Vlassopoulos, Y. Wang, and H. Watanabe are gratefully acknowledged.
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REFERENCES
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DOI: 10.1021/acsmacrolett.7b00882 ACS Macro Lett. 2018, 7, 218−222
