A viscoelastic constitutive model for thixotropic yield stress fluids: asymptotic and numerical studies of extension

Transcription

1 A viscoelastic constitutive model for thixotropic yield stress fluids: asymptotic and numerical studies of extension Holly Grant Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Yuriko Renardy, Chair Michael Renardy Shu-Ming Sun Pengtao Yue September 2017 Blacksburg, Virginia Keywords: Asymptotic Analysis, Viscoelasticity, Thixotropy, Yield Stress Fluid, Extension Copyright 2017, Holly Grant

2 A viscoelastic constitutive model for thixotropic yield stress fluids: asymptotic and numerical studies of extension Holly Grant ABSTRACT This dissertation establishes a mathematical framework for analyzing a viscoelastic model that displays thixotropic behavior as a model parameter gets very small. The model is the partially extending strand convection model, originally derived for polymeric melts that have long strands that get in the way of fully retracting. A Newtonian solvent is added. The uniaxial and equibiaxial extensional flows are studied using combined asymptotic analysis and numerical simulations. An initial value problem with a prescribed elongational stress is solved in the limit of large relaxation time. This gives rise to multiple time scales. If the initial stress is less than a critical value, the initial elastic elongation is followed by settling to an unyielded state at the slow time scale. If the initial stress is larger than the critical value, then yielding ensues. The extensional flows produce delayed yielding and hysteresis, both associated with thixotropy in complex fluids. This work is supported by the National Science Foundation under grant no. DMS

3 To my family. iii

4 Acknowledgments I would like to thank my advisor Yuriko Renardy, who is a wonderful mentor and friend, and my committee members, Michael Renardy, Shu-ming Sun, and Pengtao Yue, for their support and patience. I thank current and former faculty, staff, and graduate students at the Department of Mathematics at Virginia Tech for providing an excellent working environment. Last but not least, I would like to thank Phil Yecko from The Cooper Union, who encouraged me to study mathematics and to apply at Virginia Tech. iv

5 Contents 1 Introduction Yield stress and thixotropy Overview of main results in this dissertation Manuscripts contained in this dissertation Uniaxial Extension Introduction Governing equations Important Parameters Physical interpretation of parameters Uniaxial elongational flow Steady elongation Asymptotic analysis for small ɛ Fast dynamics Slow dynamics From fast dynamics to yielded dynamics Unloading the applied tensile stress Numerical results Unyielding Conclusion Appendix: Intersection of the initial fast curve with the slow curve v

6 2.7.1 τ < 1 has a root for 0 < C 22 < τ > 1 has no roots for α > Equibiaxial Extension Introduction Governing equations Steady equibiaxial extensional flow Transient dynamics Fast dynamics Slow dynamics From fast dynamics to yielded dynamics Unyielding by relieving stress Direct numerical simulations Comparison of uniaxial and biaxial extension for parameters that model wormlike micellar solutions The von Mises criterion for yielding Conclusion Appendix: Intersection of the fast curve from equilibrium with the slow curve 46 vi

7 List of Figures 2.1 Schematic for uniaxial elongational flow in the x 1 direction. The flow is of infinite extent (a) Steady state elongational stress τ v s. rescaled elongation rate κ = κ/ɛ. α = 2.9 ( ), -2.5 (- - -, non-monotone), -2.1 (, non-monotone), -1.9 (- -, monotone increasing). (b) Steady state τ vs. κ; α = 2, 2.2, 2.5, 2.8 in the direction of the arrow. ɛ = (a) The solid curve is the critical boundary obtained from the asymptotic analysis for ɛ << 1. F 1 is the stable fixed point below the boundary. Solutions from equilibrium, with applied tensile stress τ, yield for values of α above the boundary. This boundary begins at τ = 1, α = 2 and decreases in α as τ increases, approaching the line α = 3. (b) Direct numerical simulation from the original equations for elongation rate κ(t) vs. t; α = 2.5, ɛ = 10 5, τ = 0.5, 0.9, 1.1, 1.4, 1.45, 1.46, 1.5, 1.6, α vs. τ plane, with the contour curve s 1 = 3 ( ) for the singularity (2.31). (a) 0 < τ < 1. The region to the right of is the physically relevant regime, s 1 > 3. (b) 1 < τ. The region below is the physically relevant regime, s 1 > C 11 vs. τ, 0 < C 11 < 2, 0 < τ < 1, for the point of entry from the initial fast curve ( ), and stable fixed point (- -). (a) α = 2.8; (b) 2.5; (c) 2.2; (d) C 11 vs. τ, τ > 1. The point of entry to the slow manifold is ( ). Each solution increases C 11 upon entry, and may meet the singular point s 1 (... ), or a stable fixed point (- -). (a)α = 2.8; (b) 2.5; (c) 2.2; (d) (a)steady state τ vs. κ at ɛ = 10 5, α = 2.5, lifted from Fig. 2.2, together with arrows for the hysteresis loop for loading and unloading the applied stress. The first maximum in the curve is at τ The minimum is at τ 1. (b) Numerical simulation of apparent viscosity with τ = 2 initiated from equilibrium illustrates immediate yielding vii

8 2.8 (a) Evolution of apparent viscosity τ/κ for a relatively short time interval, 0 t 100. α = 2.5, ɛ = Each curve belongs to a different applied tensile stress, τ = 0.5, 0.9, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 2, increasing downwards. τ < 1.1 displays no yielding. τ = 2 shows immediate yielding from equilibrium. (b) Log-log plot over a longer time interval, to t = 10 9, of τ = 1.1, 1.2, 1.3, 1.4, 1.45, 1.46, 1.47, increasing downwards Evolution of apparent viscosity τ/κ for initially yielded elongation at α = 2.5, ɛ = The applied tensile stress is stepped down to τ = 1.8 (... ), 1.4 (- -), 1.1( ), 1 (-.), 0.9 (- -) (a) Evolution of elongation rate κ, corresponding to the step-down in applied stress to τ = 1 (-.) from τ = 2 in Fig α = 2.5, ɛ = There is retraction for 0 < t < 1.5, followed by elongation for t > 1.5. (b) Evolution of τ/κ on a linear scale, for step-down to τ = 1 (-.), shows negative values for 0 < t < 1.5. For t > 1.5, the viscosity comes down from infinity and achieves a minimum at t comparable to 100. At this t, the solution evolves on the slow manifold to unyield Schematic of r(c 22 ) vs. C 22, defined in (2.36). (a) τ < 1, α < 0. (b) τ < 1, α > 0. Two possibilities. In both, there is a transition from the fast curve to a slow curve at the nearest zero of r(c 22 ) Schematic of r(c 22 ) vs. C 22, defined in (2.36). (a) τ > 1, α > 0. The fast curve does not meet the slow curve, but proceeds to yielding. (b) τ > 1, α < 0. Two possibilities. There is a root if r(c 22e ) < 0 ( ), where C 22e is the local extremum, so the solution lands on a slow manifold. There is no root (- - -) if r(c 22e ) > 0 so the solution yields The curve r(c 22e ) = 0 in the τ vs. α plane, where C 22e is the local extremum of r(c 22 ) = 0. To the left of the curve, the fast curve intersects the slow curve. To the right of the curve, the fast curve does not intersect the slow curve. The vertical dashed line represents τ = Steady state τ vs. κ for τ, κ < 0. α = 0.2, 1.0, 1.6, 2.2, 2.8, and 2.95, decreasing in the direction of the arrow. ɛ = The parameter space α-τ is divided by the curve r(x) = 0 (solid line) into immediate yielding to the left of the curve, and slow dynamics to the right. Above α = 0, this critical curve continues as a vertical line τ = 1, so that if 2 < τ < 0, the slow manifold is reached for any α The α-τ pairs for which C 22f are complex (not physically relevant) and real (potentially accessible if positive) viii

9 3.4 The parameter space 3 < α < 0, τ < 1, and locations for immediate 2 yielding (left of the solid line), delayed yielding (denoted S for singular point ) and unyielding on the slow manifold (denoted F for fixed point ). (a) The solid line ( ) is from Fig The fixed points are real-valued in the region to the right of (- - -). For a fixed point to be accessible, the parameters must also lie on the right of (- - -). The region inside the ellipse is magnified in (b). (b) The solution lands on the slow manifold to the right of the solid curve. The fixed points are positive below 1 + (1 α)τ = 0 ( - -). Above ( - -), they are physically irrelevant. Above ( - -), and to the right of (- - -), the fixed points are not accessible. For 1 < τ < 0, there is no singular point, 2 and the solution settles to a fixed point for any α The α-τ pairs for which above ( ) singular points C 22s are encountered first and below ( ) the fixed points C 22f are encountered first Direct numerical simulations from the original equations for apparent biaxial elongational viscosity over a long time interval. α = 2.5 and ɛ = (a) From equilibrium. Each curve is for a prescribed τ, decreasing from the top curve: 0.5, 0.81, 0.82, 0.9, 1.2. (b) From yielded elongation at τ = 1.2, the prescribed stress is relieved to τ = 0.1 for unyielding Reproduction of Figs. 1 and 2 from [48]. (a) Reduced steady shear behavior for both solutions and all temperatures probed in this study. Data are plotted as reduced stress (σ/g 0 ) vs Weissenberg number ( γτ R ) for comparison to theory [46] and previous rheological studies [5]. (b) Difference in measured apparent elongation viscosity, η E,app, in uniaxial extension and biaxial extension ( ) experiments for 1.4% CPCl/Sal at 25 o C A comparison of uniaxial extension (- - -) and biaxial extension ( ) at (a) α = 5, ɛ = , uniaxial (- - -) and biaxial ( ); (b) α = 10, ɛ = , uniaxial (- - -) and biaxial ( ); α = 20, ɛ = , uniaxial ( ) and biaxial ( - -) α = 5, ɛ = Uniaxial and biaxial elongation, (a) The elongation rate κ vs applied stress τ for uniaxial elongation (- - -) and biaxial elongation ( ) have a linear dependence for larger shear rates. (b) Apparent viscosity τ/κ vs applied stress τ for uniaxial ( - - -) and biaxial ( ) elongation, showing maximal apparent viscosities Evolution of apparent viscosity τ/κ vs t to the same final value, for different values of applied stress. (a) Uniaxial elongation. α = 5, ɛ = τ = 0.35 ( ), τ = 0.65 (- - -); (b) Biaxial elongation. α = 20, ɛ = τ = 0.01 ( - -), 0.11 (- - -) and 0.4 ( ) ix

10 3.11 α = 5 and ɛ = Evolution of apparent biaxial elongational viscosity over a long time interval. (a) From equilibrium τ = 0.7. (b) From yielded elongation at τ = 0.7 to τ = 0.1 for ɛ = ( ) and ɛ = ( ) The (α, τ) pairs where the solution goes on the slow curve. Roots refers to the positive real roots of equation (3.26) between 0 and 1. At τ = 1 2 (dashed line) there are two roots, and one of them is (0, 0). Along the solid line there is one root and it is a double root. In summary, the solution goes on the slow manifold to the right of the solid curve (see also Fig. 3.2) x

11 Chapter 1 Introduction 1.1 Yield stress and thixotropy Yield stress fluids behave like solids unless a sufficient amount of stress is applied, after which they yield or visibly deform as liquids. The term thixotropic refers to a subclass of the yield stress fluids for which the value of yield stress depends on the amount of time that has passed since the last time the fluid was placed under stress. A familiar example of a complex yield stress fluid is ketchup, which requires a hard knock to flow out of a bottle, but once it flows, and the bottle is placed back on the table, it is easier for the next person to let it flow out. Equivalently, when a constant applied stress produces yielding and is then removed, the microstructure continues to evolve after the fluid appears to come to rest [45, 31, 24]. 1.2 Overview of main results in this dissertation This dissertation adopts the mathematical approach in [36] to describe several main features of a thixotropic yield stress fluid. In Chapter 2, we address a simplified model for homogeneous uniaxial elongation as a starting point for the investigation of more complex flows. The constitutive model for the microstructure combines the partially extending strand convection (PEC) model of [17] with a Newtonian solvent (N), and is abbreviated PEC-N. The PEC model was systematically derived from molecular theory for a polymer melt in which the polymers have long side branches that get in the way of full retraction. The side branches limit retraction. We show that the elongation flow solutions depend on an associated material parameter, called the elastic yielding parameter, α. A crucial ingredient in explaining thixotropic behavior is a separation of time scales, which allows a slow microstructural evolution in apparent absence of flow. This separation of time scales naturally 1

12 Holly Grant Chapter 1. Introduction 2 makes the equations amenable to methods from singular perturbation theory and dynamical systems. In the limit of large relaxation time, ɛ 0, we distinguish three kinds of asymptotic regimes: fast, slow, and yielded dynamics. The dynamics resulting from the combination of these regimes explains both thixotropic behavior and delayed yielding; the latter is associated with a singularity in the slow dynamics and loss of stability of the slow manifold. The attractive feature is that these effects were derived from the model rather than phenomenologically built into it. As in the case of shear flow [19, 20], we perform time-dependent simulations of the governing equations, and also treat limiting cases with asymptotic expansions. The PEC-N model successfully predicts many of the salient phenomena of complex yield stress fluids [37, 19, 20]. In Chapter 3, the equibiaxial extension is considered as an initial value problem with a prescribed stress. The microstructure is modeled by the PECN. There are two free parameters that relate to the ratios of (i) yield stress to the stress modulus, called α, and (ii) yielded to unyielded viscosity, called ɛ. The dynamics is analyzed asymptotically for large relaxation time, and compared with direct numerical simulations of the original equations. The motion is initiated by an applied tensile stress, which is negative for equibiaxial extension. The results show a parameter regime where the time-dependent solution displays fast and slow time scales, which are characteristic of thixotropic yield stress fluids. In another regime, the fluid either yields or remains unyielded. The results complement prior work on uniaxial and equibiaxial extension. In particular, a comparison of uniaxial and biaxial extensions provides an interpretation for a published experimental observation of a wormlike micelle solution. In both Chapters 2 and 3, we also examined unloading the applied stress to a lower value from a yielded state, so that the initial condition is yielded flow at the old stress value, and the evolution occurs for the new value of stress. The solution initially goes on a fast curve or elastic deformation. If the new stress is close, then the solution settles to the fixed point of the yielded dynamics. If the new stress is further below, then the solution can retract, causing a negative elongation rate, and after some time, it reaches a slow manifold to unyield. We illustrate the separate cases with graphs of the elongational viscosity vs. a rescaled time. We analyzed a simple model, which does not need additional phenomenological assumptions. 1.3 Manuscripts contained in this dissertation 1. Renardy Y, Grant H (2013) Uniaxial extensional flow for a viscoelastic model that displays thixotropic yield stress behavior. Rheol. Acta 52: Grant H, Renardy Y (2015) Equibiaxial extension of a viscoelastic partially extending strand convection model with large relaxation time. Rheol. Acta 54: Not contained in this dissertation:

13 Holly Grant Chapter 1. Introduction 3 1. Renardy Y, Grant H (2016) Stretch and hold: The dynamics of a filament governed by a viscoelastic constitutive model with thixotropic yield stress behavior. Phys. Fluids 28:

14 Chapter 2 Uniaxial Extension This manuscript is published in Rheologica Acta. This work is authored with Yuriko Renardy. It is printed here with permission from the publisher. The copyright letter of permission is submitted in a separate document. Abstract Homogeneous uniaxial extensional flow of a viscoelastic fluid, namely the Partially Extending strand Convection model combined with a Newtonian solvent, is investigated for large relaxation time. Initial value problems are addressed, for prescribed constant tensile stress. The limit of large relaxation time introduces a slow time scale of evolution, in addition to a fast time scale for flow. Numerical solutions of the original equations show distinct stages of evolution, which are mathematically analyzed with asymptotic analyses for multiple time scales. We discuss the stages of evolution from equilibrium, as well as unloading the applied stress from a yielded solution. The overall picture which emerges captures a number of features which are usually associated with thixotropic yield stress fluids, such as delayed yielding, and hysteresis for up and down stress ramping. Even at large applied tensile stress, there is persistence of an interval of parameters where the deformation rate increases quickly, only after a delayed response. 2.1 Introduction A yield stress fluid is characterized by a microstructure which prevents its components from moving relative to each other until an applied stress exceeds a critical value known as yield stress, above which it flow [3, 10, 35, 30, 39]. There are a number of reasons for such a constraint such as entanglements of side branches of long chain molecules, networks of wormlike 4

15 Holly Grant Chapter 2. Uniaxial 5 micelles, or geometric interlocking of components in concentrated suspensions and emulsions [23, 11, 13]. Even though the microstructures are diverse, there are qualitative similarities in how they behave. Thixotropic refers to the dependence of yield stress on the time that has elapsed since the material was last stressed ( aging ) [24, 25]. The microstructure does not immediately return to a solid-like state when the applied stress is decreased to below the original yield stress, but instead, keeps flowing to much lower values. Yielding to flow is easier if the material has not rested long enough to return to equilibrium. This dependence of yield stress on aging gives the characteristic hysteresis curve for loading and unloading the applied stress. This is illustrated, for instance, by the experimental data for a 10% bentonite solution in Ref. [26]. The class of yield stress fluids called simple yield stress fluids does not display the dependence of yielding on the resting time. Popular constitutive models are the Bingham model [32] and Herschel-Bulkley model [42, 9, 21, 22]. The value of the yield stress is usually an input, although ideally, if the microsctructure is modeled well, then it should be an output. For thixotropic fluids, there are more elaborate models. A popular class of thixotropic models takes advantage of the non-monotonic response of steady solutions, together with an instability which switches the solution from a solid-like state to a flowing state and vice versa. Examples include the Johnson-Segalman and Giesekus models, combined with phenomenologically inspired additional terms to kill unphysical solutions [49]. A separate development is the phenomenological model based on structure parameters which fit the flow history for specific materials. The controversy about the concept of a yield stress compounds the difficulty of setting up a model [4, 3, 28, 44, 27, 26, 12]. For this paper, we abandon the a priori assumption of a yield stress [36], and seek a constitutive model that is systematically derived from molecular theory for a microstructure which has obstructions that get in the way of full extension, and add a solvent component. The conformation tensor C provides a direct measure of microstructural deformation through its trace. Since thixotropy is a time-dependent phenomenon, the time-dependent governing equations will be solved in terms of C. A viscoelastic model naturally contains a relatively long relaxation time and the solvent has a fast flow time scale. This gives rise to a small parameter ɛ, defined as the ratio of retardation time to relaxation time. The governing equations lead to a system of ordinary differential equations of the form dc = A(C) + ɛb(c), dt where A and B are operators on the conformation tensor, and 0 < ɛ << 1. A yield stress is recovered by letting ɛ tend toward 0. A creep experiment is addressed, meaning that the initial condition is the instantaneous application of a constant tensile stress which remains during the evolution. Since ɛ is small, the solutions for ɛ = 0 or dc = A(C), describe a part of the evolution. Since ɛ is absent, dt this system is referred to as fast dynamics. Depending on whether the conformation tensor grows or decays, these terms become comparable in magnitude to certain terms in ɛb(c). The system for which A(C) is of order ɛ is called slow dynamics. The system where C is large is called yielded dynamics. In order to calculate how these stages of evolution follow, one from the other, we perform direct numerical simulations of the original equations,

16 Holly Grant Chapter 2. Uniaxial 6 complemented by a separate perturbation analysis for multiple time scales. The work of [36] presents a discussion of classical viscoelastic constitutive models which do or do not make physical sense in the limit ɛ 0; for example, the Giesekus model leads to an infinite first normal stress in the yielded state, and the Johnson-Segalman model leads to unphysical oscillations for yielded states. Neither of these models are viable for our mathematical analysis in the asymptotic limit of large relaxation time. On the other hand, one model which behaves well is the Partially Extending strand Convection model [17], and for simplicity, this is combined with a Newtonian solvent (PEC-N model). This is the model we use in this paper. The investigation of a homogeneous unidirectional shear flow driven by a prescribed applied shear stress was conducted in Ref. [19, 20] for the PEC-N model and its more realistic generalization, PECR-N. The PECR-N model allows for an established flow to unyield at a non-zero stress, for instance, and has additional classes of solutions such as time-periodic aging-rejuvenation cycles with long aging phases. The mathematical analysis for large relaxation time finds solutions which are reminiscent of phenomena such as delayed yielding and hysteresis. The experimental data of Ref. [7] on Heinz ketchup is computationally obtained with the PECR-N model for a variety of prescribed shear stresses. The asymptotic stages of evolution begin with an initial fast response which is dominated by nonlinear elastic deformation, followed by a high rate of deformation (or immediate yielding ) if the applied shear stress is high enough, or a slow growth in deformation with eventual jump to a high deformation rate ( delayed yielding ), or if the applied stress is low, the deformation evolves slowly and remains small ( unyields ). If the initial condition is yielded flow, with a ramp down in applied stress, we calculate the unloading features of the hysteretic response. If the applied stress is decreased sufficiently, then the rate of deformation becomes of order ɛ and the material unyields. The perturbation analysis permits the calculation of an apparent viscosity during unyielding in a mathematically tractable form; for example, see Fig. 12 of [20]. Here, the trace of the conformation tensor, which is a measure of polymer elongation, is found to decrease inversely proportional to time. Hence, the apparent viscosity is found to increase linearly with time, and plateau to a value proportional to the large relaxation time. These features compare well with the experimental data of Fig. 9 from [27]. The multi-scale asymptotic analysis, while not simple, is an essential tool for the physical interpretation of the distinct stages in evolution. There is a need for experimental measurements and mathematical analysis for thixotropic yield stress fluids for up-and-down stress ramps, because this is part of a range of flows which can occur in industrial processes, such as fiber spinning. In contrast to elongation, there is a vast amount of work for shear flows, both for prescribed shear stress and prescribed shear rate. Recently, homogeneous unidirectional shear flow with prescribed shear stress is investigated with the mathematical methods of this paper [20]; direct comparisons with experimental measurements are described. The mathematical solutions represent the familiar phenomena for thixotropic yield stress fluids, such a delayed yielding and hysteresis. On the

17 Holly Grant Chapter 2. Uniaxial 7 other hand, unidirectional elongation driven by a prescribed tensile stress is far more difficult to set up in a laboratory. The case of fixed elongation rate is easier to achieve experimentally through boundary conditions, and the bulk of the literature focuses on aspects of rupture or inhomogeneities that occur with yielding. This is because fixing the elongation rate has the disadvantage that yielding often occurs with rupture and loss of homogeneity of the material. The case of constant tensile force is also called constant nominal tensile stress, but this is not a true tensile stress because stress is force divided by the cross-sectional area of the elongating material [2]. The case of constant force in the literature typically concerns a thinning filament. Hence, there needs to be a feedback mechanism for the time-dependent cross sectional area to calculate the stress. The advantage of keeping the true tensile stress constant is that filament breakage is avoided when the fluid yields, and it maintains a homogeneous microstructure, without neighboring patches of yielded and unyielded zones. The goal of maintaining a constant true tensile stress is becoming more practical with recent improvements in experimental design [1, 41]. There are still non-trivial experimental issues to be overcome, both for constant tensile stress and compressive stress (or squeeze). These include the effect of the end plates, friction with the sample, maintaining uniaxial elongation when deformation is large or if the plates approach each other for squeeze flow. Below, we address a simplified model for homogeneous uniaxial elongation as a starting point for the investigation of more complex flows. The mathematical formulation is defined in Sec Direct numerical simulations of the governing equations are performed to obtain an overall picture, but the stages in the evolution are distinct, and can be interpreted physically with tractable asymptotic solutions in the limit of large relaxation times. The Partially Extending strand Convection model was originally derived for a polymer melt, in which the polymers have long side branches which get in the way of full contraction [17]. Associated with this constraint comes a material parameter α, which will be defined in Sec This parameter was shown in [36] to scale out of the transient evolution for shear flow. However, we shall show that the elongational flow solutions do depend on α. Unlike prior models which modify existing models for simple yield stress fluids, our model is an inherently thixotropic model. 2.2 Governing equations Let v denote the velocity, p the pressure, ρ the density, η the Newtonian component of viscosity, C the conformation tensor, and T the extra stress tensor. The tensor C is positive definite, namely, x T Cx > 0 for any x R 3. In particular, if x = (1, 0, 0), (0, 1, 0) or (0, 0, 1), we see that the diagonal elements of C are strictly positive. It is normalized with the average equilibrium length of the polymer molecules. The trace measures molecular extension and

18 Holly Grant Chapter 2. Uniaxial 8 is denoted s = trc = C 11 + C 22 + C 33. The governing equations are momentum conservation, ρ Dv = ( pi + S + T), incompressibility, v = 0, and the constitutive model for the microstructure, which combines the Dt Partially Extending strand Convection (PEC) model [17] together with a Newtonian solvent (N), and is abbreviated by PEC-N. The PEC model was systematically derived from molecular theory, for a polymer melt composed of entangled polymers with long side-branches which hinder complete extension; C + ɛ(φ(s)c χ(s)i) = 0, (2.1) ψ(s) = k 1 s + α, φ(s) = χ(s) = k 2(s + α), (2.2) where the upper convected derivative is denoted C = DC Dt ( v)c C( v)t, D Dt t +v, and ɛ is proportional to the inverse of the relaxation time [19, 20]. The total stress tensor is composed of the Newtonian contribution S = η( v + ( v) T ), and the viscoelastic contribution T = ψ(s)c. (2.3) We consider stress-controlled experiments where the prescribed elongational stress τ is the difference (T 11 + S 11 ) (T 22 + S 22 ), or τ = ψ(s)(c 11 C 22 ) + 3κη Important Parameters We rewrite the governing equations in dimensionless form. The dimensionless model parameters are α and k 2. The conformation tensor, and thus s =trc, are dimensionless. ɛ and the shear rate κ scale as the inverse of time. ψ and the stress modulus, k 1, have units of stress. k 1 appears in ψ which appears only in the term (τ ψc 12 )/η, or (τ/k 1 ψ/k 1 C 12 )/(η/k 1 ). Hence, we define τ = τ/k 1, ψ = ψ/k1, η = η/k 1. The viscosity η scales as mass/length/time. The parameter k 2 only occurs in the combination k 2 ɛ in the governing equations. This combination is the inverse of the relaxation time, which we denote by ɛ = k 2 ɛ.

19 Holly Grant Chapter 2. Uniaxial 9 Since k 1 is a stress modulus, and η is viscosity, the ratio η is the retardation time. We define the ratio of retardation time divided by the relaxation time as Time is rescaled with the retardation time as ɛ = ɛ η = ηk 2 ɛ/k 1. t = t/ η = tk 1 /η. A Weissenberg number W e = κ(s)/ɛ expresses a ratio of the relaxation time vs. flow time. A Bingham number Bi = k 1 /(ηκ(s)) expresses the yield stress vs. viscous stress. Both W e and Bi evolve with time. We drop the bars on ɛ, τ, ψ, and t. This is equivalent to setting k 1 = 1, k 2 = 1 and η = Physical interpretation of parameters Consider the homogeneous shear flow with a prescribed shear stress τ, with the notation of section 4.1 of [19]. The dimensional maximum elastic stress is T12 e = T12c e s where c s is the chosen scale for dimensional stress. We do not need to know what c s is because we form a ratio with another dimensional stress quantity, the stress modulus k1 = k 1 c s, where k 1 = α, T e 12 = α. (2.4) The value of T e 12 is the yield stress one observes during fast deformation; i.e., the waiting time is short compared with the relaxation time. The value of k 1 is the ratio stress/strain in the linearized regime. The rescaled stress modulus k 1 is the coefficient of the strain, denoted γ in equation (30) γ of [19], and T 12 =, linearized in γ. The ratio of the maximum elastic shear stress 3+α+γ 2 T12 e to the stress modulus k1, T12 both observable quantities, cancels out the c s : e k 1 = 3+α. 2 Therefore, α = ( e 2T12 ) 2 3. (2.5) k1 Thus, the model parameters have been related to observable quantities. We shall rescale the variables in order to minimize the number of free parameters. The rescaled applied shear stress τ denotes (dimensional applied shear stress)/(stress modulus). The rescaled time is (dimensional time)/(retardation time), where the retardation time equals (solvent viscosity)/(stress modulus).

20 Holly Grant Chapter 2. Uniaxial 10. x. 1 Figure 2.1: Schematic for uniaxial elongational flow in the x 1 direction. The flow is of infinite extent. The parameter k 2 appears only in the combination k 2 ɛ, which is the inverse of the relaxation time, and is absorbed into the rescaled ɛ, where now ɛ = retardation time relaxation time. (2.6) In effect, our rescaling is equivalent to setting k 1 = 1, k 2 = 1 and η = 1 in (3.1). Finally, the governing equations are (3.1), (3.3) and ψ(s) = 1, φ(s) = χ(s) = (s + α). (2.7) s + α The PEC parameter α is called the elastic yielding parameter, and α > 3 [20]. The strict inequality for the lower bound ensures that the stress components avoid the singularity in (s + α) Uniaxial elongational flow Fig. 2.1 shows a schematic of uniaxial elongational flow in the x 1 -direction. We investigate homogeneous uniaxial extension in the x 1 -direction, and of infinite extent in all three dimensions, and therefore Ö C è C = 0 C 22 0, (2.8) 0 0 C 22 where C 22 = C 33 because of axial symmetry.

21 Holly Grant Chapter 2. Uniaxial 11 The elongation rate is denoted κ. The velocity components for elongational flow in the x 1 - direction satisfies u v w x 1 = κ, x 2 = κ/2, x 3 = κ/2. This satisfies the incompressibility condition, v = 0, where tr( v) = v, and v = The evolution equations to be solved are, Ö κ κ κ 2 è. (2.9) dc 11 dt dc 22 dt = 2κC 11 + ɛ[χ(s) φ(s)c 11 ], = κc 22 + ɛ[χ(s) φ(s)c 22 ], κ = 1 3 (τ ψ(s)(c 11 C 22 )), s = C C 22, (2.10) for small ɛ. The variables C 11, C 22, κ and s depend on time. The tensile stress τ = (T 11 + S 11 ) (T 22 + S 22 ), τ = ψ(s)(c 11 C 22 ) + 3κ, (2.11) is a prescribed quantity which is fixed during the evolution. We see from (3.9c) that the viscous component of the stress evolves at the flow time scale, which is several orders of magnitude faster than the relaxation time of the microstructure. If the flow is turned off (v = 0), the nonlinear elastic response is C t which scales with ɛ. An alternative equation, without κ, will be useful in analyzing the slow dynamics of section The elongation rate κ can be eliminated from (3.9c): multiply the first equation by C 2 22, C22 2 dc 11 dt and the second by 2C 11 C 22, and add, to obtain = 2C 2 22κC 11 + C 2 22ɛ(χ(s) φ(s)c 11 ), 2C 11 C 22 dc 22 dt = 2C 11 C 22 κc C 11 C 22 ɛ(χ(s) φ(s)c 22 ), d dt (C 11C 2 22) = C 2 22ɛχ(s)(1 C 11 ) + 2C 11 C 22 ɛχ(s)(1 C 22 ) = ɛχ(s)c 22 (C 22 (1 3C 11 ) + 2C 11 ). (2.12) This will turn out to be a useful equation for the slow dynamics of Sec

22 Holly Grant Chapter 2. Uniaxial Steady elongation We present the steady solutions of (3.9c). From (3.9c), 0 = 2κC 11 + ɛ(χ(s) φ(s)c 11 ), and 0 = κc 22 + ɛ(χ(s) φ(s)c 22 ). For the PEC model, χ(s) = φ(s), so we have 0 = 2κC 11 + ɛχ(s)(1 C 11 ), 0 = κc 22 + ɛχ(s)(1 C 22 ). These become: C 11 = C 22 = ɛχ(s) 2κ ɛχ(s), (2.13) ɛχ(s) κ + ɛχ(s). (2.14) We eliminate C 11 and C 22 and arrive at an equation in terms of s, by using C 22 = (s C 11 )/2 in (3.15b) to obtain C 11 = s. This is substituted in (3.15a), to find an equation for κ, 2ɛχ(s) (ɛχ(s)+κ) 2κ 2 s + ɛχ(s)κ(s 3) + ɛ 2 χ 2 (s)(3 s) = 0. (2.15) This is a quadratic equation for κ but one solution is negative and physically unacceptable. Therefore, κ(s) = ɛχ(s) 4s (» ) 3(s 3)(3s 1) s + 3, (2.16) or, using the form of χ(s) = s + α, κ(s) = α + s (» ) 4s ɛ 3(s 3)(3s 1) s + 3. (2.17) From (2.11), we have the total tensile stress τ(s) = ψ(s)(c 11 (t) C 22 (t)) + 3κ(s), s = s(t). This is simplified by substituting (3.15a), (3.15b) and (2.16). Therefore, the steady tensile stress is τ(s) = 3κ(s) Substitution for the model functions yield an expression in terms of s: τ(s) = 3κ(s) ɛψ(s)χ(s)κ(s) (2κ(s) ɛχ(s))(κ(s) + ɛχ(s)). (2.18) s 3 + 3(s 3)(3s 1). (2.19) (α + s) Note that if s is sufficiently large, κ sɛ/2, and τ 3κ + 1. From this, it follows that in the range of small elongation rate κ, comparable to ɛ, the tensile stress τ approaches 1. We need to investigate whether τ as a function of κ is non-monotone, that is, achieves a maximum value for some κ = O(ɛ) before reaching 1 (The notation O denotes order of magnitude; i.e., κ is proportional to ɛ as ɛ 0.) Such a non-monotone curve naturally induces fast and slow dynamics, yielded and unyielded states, and a coupling between those states. We introduce a rescaled elongation rate, κ = κ/ɛ, and focus on the regime κ = O(1), with τ 1 small. First, the square root term in (3.19) is approximated for large s: a Taylor expansion in 1 s yields the dominant terms 3s(1 5 3s 8 9s ). Thus, τ 3 κɛ + s 2 2 α+s. Secondly, τ 1 3 κɛ 2+α s+α. In comparison with the first term, the second term dominates. Hence τ 1 > 0

23 Holly Grant Chapter 2. Uniaxial Τ steady Τ (a) Κ Ε 0.8 (b) Κ steady Figure 2.2: (a) Steady state elongational stress τ v s. rescaled elongation rate κ = κ/ɛ. α = 2.9 ( ), -2.5 (- - -, non-monotone), -2.1 (, non-monotone), -1.9 (- -, monotone increasing). (b) Steady state τ vs. κ; α = 2, 2.2, 2.5, 2.8 in the direction of the arrow. ɛ = when 2 + α < 0. Thus, we expect non-monotonicity for α < 2. The lower bound provided by the model is 3 < α. Fig. 2.2(a) shows the tensile stress τ vs. rescaled elongation rate κ. The horizontal axis is scaled to focus on the regime κ = O(ɛ). The figure illustrates the change in the non-monotonicity for 3 < α < 2, to monotonically increasing curves for 2 < α. In addition, the peak becomes sharper as α approaches -3. The non-monotonic regime is exemplified in the figure by α = 2.9 ( ), α = 2.5 (- - -, non-monotone) and -2.1 (, borderline non-monotone), all with a local minimum at τ = 1.0. The curve at α = 1.9 (- -) exemplifies the monotonic case. An unexpected result is that there is an interval of α between -2 and roughly -1 where there will be slow and fast time scales even though the steady curve is monotone increasing. This is displayed in Fig for a small interval of τ larger than 1; the details are calculated in the Appendix. The total stress as a function of κ is shown in Fig. 2.2(b), with each curve representing a different value of α = 2, 2.2, 2.5, 2.8. In the direction of the arrow, α decreases to -3. The peaks become sharper as α 3, and occur at smaller values of κ when ɛ 0. At ɛ = 10 5, a curve for α > 2 is monotone increasing, which is a property that is not associated with hysteresis upon loading, followed by unloading the applied stress. For a non-monotone curve, the negative slope is an unstable branch, so that the solution jumps at the first maximum to the second increasing branch; when unloading the stress, the solution moves down the second branch, and jumps at the minimum to the first increasing branch. We return to this issue in Sec. 2.5.

24 Holly Grant Chapter 2. Uniaxial Asymptotic analysis for small ɛ Fast dynamics Suppose the initial condition is equilibrium: C 11 = C 22 = 1. An instantaneous tensile stress τ is applied. It is evident that the solution is close to equilibrium for some time, and we can neglect terms of O(ɛ) in (3.9c). At this stage, t = O(1) which is a fast time scale: dc 11 dt = 2κC 11, dc 22 dt = κc 22. (2.20) The elimination of κ from (2.20) gives d dt ln C 11 = 2 d dt ln C 22. Consequently, we find the relationship C 11 C 2 22 = C 11 (t 0 ) C 2 22(t 0 ), where t 0 is the initial time. In particular, since we have equilibrium at t = 0, we substitute C 11 = C 22 = 1 to obtain C 11 C 2 22 = 1 (2.21) as the equation for the solution trajectory; this is called a fast curve. The question now is: Does the fast dynamics continue until yielding occurs, or does the solution switch to a slow time scale? At t = 0, we use C 11 = C 22 = 1 in (2.11) to obtain κ = τ 3 > 0. Since κ is positive, C 11 grows and C 22 diminishes. This behavior changes if κ becomes 0 on the fast curve, or specifically O(ɛ) in the original equations. When the elongation rate decreases to an order of magnitude ɛ, the time scale switches to the slow time scale, of order 1 ɛ, and the fast dynamics analysis is no longer valid. In (2.11), τ = ψ(s(t))(c 11(t) C 22 (t)) and C 11 = C 2 22, then τ = which leads to the roots of the cubic polynomial, = = 1 C C 22 + α (C 11(t) C 22 (t)) C22 2 C 22 C C 22 + α 1 C C (2.22) αc2 22 C 3 22(1 + 2τ) + ταc τ 1. (2.23) The question is whether there is a root between 0 and 1, given τ > 0 and α > 3. If so, then C 22 decreases from 1 on the fast curve, and at the first root it meets, lands on the slow curve. The details of the roots of (2.23) are given in the Appendix. In summary, the fast curve meets the slow curve if τ < 1. If τ > 1, then this occurs only for α in the region to the left of the curve shown in Fig When τ becomes larger, then α needs to be closer to -3 for this to occur. Otherwise, the fast curve transitions to yielded dynamics. We remind the reader that our model is not a modification of a simple yield stress model, and therefore there is no limiting case that corresponds to that physical interpretation. In order for a material to be a simple yield stress fluid, there would not be any small parameter ɛ, so that the material either yields or does not depending on the applied stress. Moreover, the case ɛ = 0, which we call fast dynamics is a nonlinear regime, and bears no parallel to linear elasticity with its concept of plastic yielding at a von Mises criterion. In the limit of fast dynamics, yielding in our model occurs because the nonlinear elastic response allows only a finite maximum stress.

25 Holly Grant Chapter 2. Uniaxial Slow dynamics In this section, we address the dynamics on the slow manifold, after the initial fast dynamics. We have seen in the previous section that this occurs if τ < 1 and for any α. If τ > 1, then this occurs only for a restricted region of α. Once on the slow manifold, κ O(ɛ), and t ɛ 1. Hence, we define a slow time variable, t = ɛt. We express C 11, C 22 in terms of s, together with C 11 = s 2C 22 and (2.11); The slow manifold is therefore a line in the C 11 -C 22 plane, The evolution equation (2.12) becomes C 11 = 1 3 (s + 2τ ψ ) = 1 3 (s(1 + 2τ) + 2τα), (2.24) C 22 = 1 2 (s C 11) = 1 3 (s(1 τ) τα). (2.25) C 22 = 1 τ 1 + 2τ C 11 τα 1 + 2τ. (2.26) d d t (C 11C 2 22) = χ(s)c 22 (C 22 (1 3C 11 ) + 2C 11 ). (2.27) The right hand side of (2.27) is now a function of only s. Fixed points of the slow manifold The fixed points are defined by ds ds d = 0. The left hand side of (2.27) is d t d t ds (C 11C22), 2 and therefore vanishes at the fixed points. Thus, C 22 (1 3C 11 ) + 2C 11 = 0. Upon substitution of (2.24)-(2.25), we find 1 3 (s τ ψ )(1 (s + 2τ ψ )) (s + 2τ ψ ) = 0, (s τ ψ )(1 (s + 2τ ψ )) + 2(s + 2τ ψ ) = 0, (s τ(s + α))(1 (s + 2τ(s + α))) + 2(s + 2τ(s + α)) = 0. The fixed points are the roots F 1 and F 2 : Ä 3 3τ + ατ 4ατ τ + 2ατ + 3τ ατ 2 + 3α 2 τ 2ä F 1 = 2 (τ 1) (2τ + 1), (2.28) and F 2 is the same except for a positive sign in front of the square root. Recall that physically realizable states must satisfy s 3. Through numerical evaluation, it is found that F 2 is not viable; hence, we focus the discussion on F 1. We find that F 1 is viable and stable for 0 < τ < 1. For τ > 1, the solid curve in Fig. 2.3(a) denotes the boundary, above which the fixed points are either complex conjugates or real and unstable or physically irrelevant (s < 3), and below which F 1 is stable. Therefore, the solution remains on the slow manifold, attracted to a stable fixed point F 1, only for the region below the solid curve. This region narrows as τ increases above 1, and α approaches 3 from above.

26 Holly Grant Chapter 2. Uniaxial 16 (a) (b) Figure 2.3: (a) The solid curve is the critical boundary obtained from the asymptotic analysis for ɛ << 1. F 1 is the stable fixed point below the boundary. Solutions from equilibrium, with applied tensile stress τ, yield for values of α above the boundary. This boundary begins at τ = 1, α = 2 and decreases in α as τ increases, approaching the line α = 3. (b) Direct numerical simulation from the original equations for elongation rate κ(t) vs. t; α = 2.5, ɛ = 10 5, τ = 0.5, 0.9, 1.1, 1.4, 1.45, 1.46, 1.5, 1.6, 2. In terms of the rescaled t = O(1) on the slow manifold, we denote ṡ = ds. The chain rule applied to the left d t hand side of (2.12) gives d dt (C 11C22) 2 = ds d dt ds (C 11C22). 2 Cancellation of ɛ yields the evolution equation 1 ṡ = d ds (C 11C22 2 )χ(s)c 22(C 22 (1 3C 11 ) + 2C 11 ). (2.29) The fixed point s = F 1 satisfies ṡ = 0. To examine whether this is linearly stable, let F 11 (t) be a small perturbation, and substitute s(t) = F 1 + F 11 (t) in (2.29) to obtain F 1 11 = d )χ(s)c 22(C 22 (1 3C 11 ) + ds (C11C2 22 2C 11 ). We verified numerically that the growth rate is negative for 0 < τ < 1, meaning that solutions approach s = F 1 and unyield. For τ > 1, the fixed point F 1 is stable in the region below the boundary in Fig. 2.3(a); this stability region becomes narrower in α as τ increases further from 1, and squeezes down to α 3. This narrow parameter regime corresponds to the first increasing portion of the non-monotone steady state curve in Fig. 2.2(b), above the non-zero minimum. Here, the solution is attracted to the stable fixed point. For τ larger than the first maximum of the non-monotone curve, the fixed point loses stability. Without a stable fixed point, the solution slowly approaches a singular point on the slow manifold, which is studied next in Sec Figure 2.3(b) shows the computed elongation rate κ vs. t for ɛ = 10 5, α = 2.5. We see in Fig. 2.3(a) that at this value of α, the fixed point loses stability at τ = 1.1. This agrees with the computational results in Fig. 2.3(b), where the evolution curves for τ 1.1 approach the fixed point on the slow manifold and κ remains of the order of ɛ. The evolution for τ > 1.1 eventually lead to yielded elongation.

27 Holly Grant Chapter 2. Uniaxial 17 Singular points The condition equation for s, d ds (C 11C 2 22) = 0 in (2.27) gives the values of s where ds d t is singular. This is a quadratic 1 3 (1 + 2τ)1 9 (s τ(s + α)) (s + 2τ(s + α)) 1 3 (s τ(s + α)) 1 3 (1 τ) = 0. (2.30) If a solution reaches such a point, it leaves the slow manifold. The roots are s 1 = s 2 = ατ(2τ 1) (1 + 2τ)(1 τ), ατ (1 τ). (2.31) That is, s 1 = s 2 ( 2τ 1 2τ+1 ). The singular point s 2, when substituted into (2.25), produces C 22 = 0. We require s 3 to be physically realizable. First, examine s 2. If τ < 1, then This is the condition for s 2 > 3. ατ 3(1 τ) (1 τ) > 3 when α > τ. Next,examine s 1. To have s 1 > 3, we require α > 3(1 τ)(2τ+1) τ(2τ 1). For any τ > 0, this expression satisfies α > 3. So to be physically realizable, we require C 22 > 0. However, one of the singular points always belongs to d C 22 = 0 because we found these from the equation, ds (C 11C22) 2 = C 22 ( dc11 ds C dc C ds ) = 0. The physically relevant singular point is s = s 1 provided C 11 > 0, C 22 > 0. Fig. 2.4 shows the regions in the τ vs. α plane where s 1 3, and are thus physically realizable. The curves are s 1 = 3 ( ) defined by α(τ) = 3(1 τ)(2τ+1) τ(2τ 1). Fig. 2.4(a) addresses the interval 0 < τ < 1, and Fig. 2.4(b) the interval τ > 1. In Fig. 2.4(a), we have s 1 > 3 to the right of ( ). In Fig. 2.4(b), the region below the curve represents s 1 > 3. Summary of slow dynamics Slow dynamics is associated with a microstructural evolution which is imperceptible, and occurs over a time scale of O(1/ɛ). For values of τ less than the first maximum of the non-monotone hysteresis curve (Fig. 2.7(a)), the slow dynamics is associated with the persistence of the unyielded state. The initial fast curve intersects the slow manifold under the conditions described in Sec , and detailed in the Appendix with positive dc11 dt. From the point of intersection onwards, Figs. 2.5(a-d) and 2.6(a-d) summarize the behavior; C 11 is given in terms of τ and α, which are fixed for a specific thought experiment. The α varies from -2.8 to -2 because the steady τ(κ) curve loses non-monotonicity for α > 2. The interval 0 < τ < 1 is given in Fig. 2.5(a-d), and τ > 1 in Fig. 2.6(a-d). Observe that for a chosen α, and τ = τ 0, the initial fast curve arrives ( ) with C 11 increasing along the vertical line τ = τ 0. A solution unyields if it reaches a stable fixed point (- -). If the singularity s 1 (...) is crossed, then the solution blows up and leaves the slow manifold. We shall see in Sec that when a yielded flow is stepped down in applied stress to below the minimum of the hysteresis curve, unyielding occurs, and it always occurs on the slow manifold.