David W. Mead

Theoretical derivation of molecular weight scaling for rheological parameters

Abstract Recently, a method has been established to determine moments (functionals) of the molecular weight distribution (MWD) of a given polymer directly from measurements of the linear viscoelastic relaxation modulus of that polymer. In part, the need to compute such quantities (functionals) is motivated by the experimentally observed scaling of rheological properties of polymers with respect to moments of their MWD. Although various authors have advanced different ad hoc arguments to derive various molecular weight scaling results for a variety of rheological parameters, such as the zero-shear viscosity, no formal procedure for deriving molecular weight scaling for rheological parameters has been proposed. In this paper, a natural parametric generalization of the reptation based mixing rules is introduced which includes single and double reptation as special cases. For this generalization, it is shown, by invoking the mean value theorem for integrals, how to formalize the derivation of molecular weight scaling for rheological parameters. In particular, from the point of view of choosing practical mixing rules, this paper establishes that when the relaxation function is characterized by a single time constant, the molecular weight scaling is independent of the standard linear mixing rules.

The response of entangled polymer solutions to step changes of shear rate: Signatures of segmental stretch?

Experiments measuring the orientation angle and birefringence in startup and double-step strain rate flows were conducted on a 3.0 wt % 8.42 × 106 molecular weight polystyrene solution in a Couette flow cell. A phase-modulated flow birefringence apparatus was used to noninvasively probe the sample. Upon startup from rest, the orientation angle undershoots its final steady-state value, as seen by earlier investigators. When the shear rate undergoes a step increase from one nonzero value to another, the amplitude of this undershoot is decreased. However, a more significant effect is a shorter time scale overshoot in the orientation angle that is highly counterintuitive in the sense that an increase of shear rate initially produces a rotation of chain segments away from the flow direction. Similarly, a step decrease in shear rate yields an initial transient rotation toward the flow direction. In both cases, the height of the peaks depends upon the magnitude of the shear rate jump, and the width of the peaks is a function of the final shear rate. The longer time transients in the startup and step increase experiments reflect an apparent change in the relaxation time for segment orientation, which we tentatively attribute to a combination of tube dilation and convective constraint release. The shorter time scale over- and undershoots in the orientation angle appear to be qualitatively explained by considering the differences in extension or contraction of segments along the polymer chain.

Stochastic simulation of entangled polymeric liquids in fast flows: Microstructure modification

We have modified the full-chain stochastic tube (XDS) model developed by Xu et al. [J. Rheol. 50, 477–494 (2006)] to simulate the rheology of entangled melts and solutions of linear monodisperse polymers. The XDS model, which has a single adjustable parameter that is equivalent to the Rouse time, successfully describes steady and transient shear and normal stress data at low to moderate rates, but the results deviate systematically from experimental data at high rates. The algorithm for re-entanglement was revised, and a configuration-dependent friction coefficient (CDFC), as originally proposed by Giesekus, was incorporated to account for microstructural change of the tube away from equilibrium. The simulation results from the modified model significantly reduce the deviation from the experimental data in shear, and they also agree well with extensional data for entangled solutions, including an initial −0.5-power dependence of the steady extensional viscosity on extension rate. We also applied the CDFC ...

On the recovery of molecular weight functionals from the double reptation model

Abstract Recently, Mead (J. Rheology, 38 (1994) 1797–1827) has shown, for the double reptation model with a monodisperse relaxation function, characterized by a single time constant, how moments for the molecular weight distribution (MWD) can be determined directly from measurements of the linear viscoelastic relaxation modulus. The purpose of this paper is to show that this result is a special case of a more general result which allows linear inner-product functionals defined on the MWD, such as its moments, to be determined directly from the measurements. The applicability of this strategy is verified and validated using synthetic data.

Experimental studies of an entangled polystyrene solution in steady state mixed type flows

Experimental measurements of birefringence and velocity gradient components are reported for steady mixed type flows of a 0.076 g/cm3 solution of 2.89×106 MW polystyrene in a mixed toluene/oligomer solvent. The flow field is produced in a co-rotating two-roll mill with a series of different ratios of the gap width to roller radius chosen so that the flow type at the stagnation point for a Newtonian fluid would range from 0.0196⩽λ⩽0.20, where ‖E‖/‖Ω‖=(1+λ)/(1−λ). Additional data are also reported, for comparison purposes, for a similar polystyrene solution in a simple Couette flow. Finally, the stress-optical relationship is used to obtain a generalized extensional viscosity as a function of strain rate. This viscosity shows a range of strain rate thinning as predicted by reptation theory, followed at a critical Weissenberg number of 0(1) based on the Rouse relaxation time by the initial stages of a region of strain rate thickening, as predicted by the Marrucci–Grizzuti extension of reptation theory that a...

A constitutive model for entangled polymers incorporating binary entanglement pair dynamics and a configuration dependent friction coefficient

Following recent work [e.g., J. Park et al., J. Rheol. 56, 1057–1082 (2012); T. Yaoita et al., Macromolecules 45, 2773–2782 (2012); and G. Ianniruberto et al., Macromolecules 45, 8058–8066 (2012)], we introduce the idea of a configuration dependent friction coefficient (CDFC) based on the relative orientation of Kuhn bonds of the test and surrounding matrix chains. We incorporate CDFC into the “toy” model of Mead et al. [Macromolecules 31, 7895–7914 (1998)] in a manner akin to Yaoita et al. [Nihon Reoroji Gakkaishi 42, 207–213 (2014)]. Additionally, we incorporate entanglement dynamics (ED) of discrete entanglement pairs into the new Mead–Banerjee–Park (MBP) model in a way similar to Ianniruberto and Marrucci [J. Rheol. 58, 89–102 (2014)]. The MBP model predicts a deformation dependent entanglement microstructure which is physically reflected in a reduced modulus that heals slowly following cessation of deformation. Incorporating ED into the model allows “shear modification” to be qualitatively captured. ...

A constitutive model for entangled polydisperse linear flexible polymers with entanglement dynamics and a configuration dependent friction coefficient. Part I: Model derivation

A new polydisperse “toy” constitutive model is derived and developed from fundamental principles and ideas governing the nonlinear rheology of linear flexible polymers [Mead et al., J. Rheol. 59, 335–363 (2015)]. Specifically, the new model is comprised of four fundamental pieces. First, the model contains a simple differential description of the entanglement dynamics of discrete entanglement pairs. Second, the model contains a differential description of the ij entanglement pair orientation tensor dynamics. Third, following a similar development by Mead and Mishler [J. Non-Newtonian Fluid Mech. 197, 61–79 and 80–90 (2013).], a diluted stretch tube is constructed to describe the relative stretch of each component in the molecular weight distribution (MWD). Fourth, a description of configuration dependent friction coefficients is generated by generalizing the monodisperse formulation of Ianniruberto et al. [Macromolecules 45, 8058–8066 (2012)]. The polydisperse stress calculator is developed from the orientation, stretch and entanglement density and is fundamentally different from other molecular models that assume a constant entanglement density. The resulting model is comprised of three differential evolution equations and is simple to code and fast to execute. The model can simulate arbitrary fast nonlinear flows of arbitrary MWDs. In the slow flow linear viscoelastic limit, the model collapses to the double reptation model. This welcome result has positive implications with respect to our model parameter determination [Ye et al., J. Rheol. 47, 443–468 (2003); Ye and Sridhar, Macromolecules 38, 3442–3449 (2005)] for making quantitative calculations.A new polydisperse “toy” constitutive model is derived and developed from fundamental principles and ideas governing the nonlinear rheology of linear flexible polymers [Mead et al., J. Rheol. 59, 335–363 (2015)]. Specifically, the new model is comprised of four fundamental pieces. First, the model contains a simple differential description of the entanglement dynamics of discrete entanglement pairs. Second, the model contains a differential description of the ij entanglement pair orientation tensor dynamics. Third, following a similar development by Mead and Mishler [J. Non-Newtonian Fluid Mech. 197, 61–79 and 80–90 (2013).], a diluted stretch tube is constructed to describe the relative stretch of each component in the molecular weight distribution (MWD). Fourth, a description of configuration dependent friction coefficients is generated by generalizing the monodisperse formulation of Ianniruberto et al. [Macromolecules 45, 8058–8066 (2012)]. The polydisperse stress calculator is developed from the orien...

A constitutive model for entangled polydisperse linear flexible polymers with entanglement dynamics and a configuration dependent friction coefficient. Part II. Modeling “shear modification” following cessation of fast shear flows

The polydisperse Mead–Park (MP) “toy” molecular constitutive model developed in Paper I [Mead et al., J. Rheol. 62, 121–134 (2017)] as well as our previously published work [e.g., J. Rheol. 59, 335–363 (2015)] is used in the “forward” direction to study model polydisperse melts of entangled linear flexible polymers in severe, fast shear flows. The properties of our new model are elucidated by way of numerical simulation of a representative model polydisperse polymer melt in step shear rate and interrupted shear flow. In particular, we demonstrate how the MP model simulates the individual molecular weight distribution (MWD) component dynamics as well as the bulk material properties. Additionally, we demonstrate that the polydisperse MP model predicts the phenomenon of “shear modification” for model MWDs with a long, high molecular weight tail. Specifically, the terminal dynamic moduli following cessation of severe, disentangling deformation, are shown to slowly heal/recover on the orientational relaxation...