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 MWD\u27s. 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 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. The MBP model is tested against experimental data in steady and transient extensional and shear flows. The MBP model captures the monotonic thinning of the extensional flow curve of entangled monodisperse polystyrene (PS) melts [A. Bach et al., Macromolecules 36, 5174-5179 (2003)] while simultaneously predicting the extension hardening found in PS semidilute solutions where CDFC is diluted out [P. K. Bhattacharjee et al., Macromolecules 35, 10131-10148 (2002)]. The simulation results also show that the rheological properties in nonlinear extensional flows of PS melts are sensitive to CDFC but not to convective constraint release (CCR) while those for shear flows are influenced more by CCR. The monodisperse MBP toy model is generalized to arbitrary polydispersity

The Effect of Weak Confinement on the Orientation of Nanorods under Shear Flows

We performed a numerical analysis to study the orientation distribution of a dilute suspension of thin, rigid, rod-like nanoparticles under shearing flow near a solid boundary of weak confinement. Brownian dynamics simulation of a rod was performed under various ratios of shear rate and rod diffusivity (Peclet number), as well as the center-of-mass position (wall confinement). We discuss the effects of Peclet number and wall confinement on the angle distributions, Jeffery orbit distribution and average orientation moments. The average orientation moments, obtained as a function of Peclet number and wall confinement, can be used to improve a previous shear-induced migration model. We demonstrate that the improved model can give excellent prediction of the orientation moment distributions in a microchannel flow

Shape-Based Separation of Micro-/Nanoparticles in Liquid Phases

The production of particles with shape-specific properties is reliant upon the separation of micro-/nanoparticles of particular shapes from particle mixtures of similar volumes. However, compared to a large number of size-based particle separation methods, shape-based separation methods have not been adequately explored. We review various up-to-date approaches to shape-based separation of rigid micro-/nanoparticles in liquid phases including size exclusion chromatography, field flow fractionation, deterministic lateral displacement, inertial focusing, electrophoresis, magnetophoresis, self-assembly precipitation, and centrifugation. We discuss separation mechanisms by classifying them as either changes in surface interactions or extensions of size-based separation. The latter includes geometric restrictions and shape-dependent transport properties

Modeling and Simulation of Biopolymer Networks: Classification of the Cytoskeleton Models According to Multiple Scales

We reviewed numerical/analytical models for describing rheological properties and mechanical behaviors of biopolymer networks with a focus on the cytoskeleton, a major component of a living cell. The cytoskeleton models are classified into three categories: the cell-scale continuum-based model, the structure-based model, and the polymer-based model, according to the length scales of the phenomena of interest. The criteria for classification of the models are modified and extended from those used by Mofrad [M. R. K. Mofrad, Annual Rev. Fluid Mech. 41, 433 (2009)]. The main principles and characteristics of each model are summarized and discussed by comparison with each other. Since the stress-deformation relation of cytoskeleton is dependent on the length scale of stress elements, our model classification helps systematic understanding of biopolymer network modeling

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 MWD\u27s 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 time scale of the longest chains in the MWD. This is the first molecular constitutive equation to predict the phenomenon of shear modification. We provide detailed insight into the molecular mechanism responsible for this previously enigmatic and important phenomenon. Additionally, the presence of shear modification is not necessarily associated with the presence of shear stress peak overshoot transients in interrupted shear flow. Specifically, we examine and analyze the interrupted shear experiments reported by Tsang and Dealy [J. Non-Newtonian Fluid Mech. 9, 203-222 (1981)] and demonstrate quantitatively their lack of a relationship to shear modification. We also demonstrate that the new MP model accurately predicts the Cox-Merz rule, Laun\u27s rule and Gleissele\u27s mirror relations in steady shear

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 to the molecular model developed by Mead et al. [Macromolecules 31, 7895-7914 (1998)] and obtained improved predictive performance at high deformation rates, reinforcing the idea that there is a structural change in the tube far from equilibrium that accelerates relaxation processes. Finally, noting that molecular models make fundamentally different assumptions about the effect of the deformation on the entanglement density but give essentially equivalent rheological predictions, we explored the effect of the dynamics of the entanglement density by changing the entanglement assumptions in the stochastic model

An Improved Model for the Steric-Entropic Effect on the Retention of Rod-like Particles in Field-Flow Fractionation: Discussion of Aspect Ratio-Based Separation

We developed an improved model for predicting the steric-entropic effect on the separation behaviors of rod-like particles in flow field-flow fractionation. Our new model incorporates the “pole-vault” rotation of a rod-like particle near a wall under shear flow into the original model developed by Beckett and Giddings which considered only Brownian rotation. We investigated the effect of the aspect ratio on the retention ratios and the cross-sectional concentration distribution in the separation of rods in field-flow fractionation (FFF). Our analyses involved comparing the results predicted using the original model and those from the new model under various rod geometries and flow conditions. We found that the new model can show the aspect ratio-enhanced elution trend in certain flow conditions for the assumption of non-constant cloud thickness (ratio between the cross flow rate and the rod diffusivity). We also deducted that the flow conditions allowing for the aspect ratio-enhanced elution are related to the interplay among the axial flow rate, cloud thickness, and rod geometry. The new model can be viewed as a prototype to qualitatively show the aspect ratio-enhanced trend since its quantitative agreement with the experimental data must be improved for our future work

Analysis of the Migration of Rigid Polymers and Nanorods in a Rotating Viscometric Flow

The dynamics and rheology of a rigid polymer or nanorod suspended in a Newtonian, viscous fluid under torsional flow have been studied. Our theoretical analysis predicts that rigid rods migrate in the radial direction according to their orientational configuration which is controlled by the competition between the shear flow, which tends to align the rods in the direction of flow, and Brownian motion, which tends to randomize the orientation. Steady and transient distributions of the center-of-mass in a dilute solution are derived from a kinetic theory and are confirmed by performing a Brownian dynamics simulation. The migration shifts the distribution toward the axis of rotation and enhances the shear-thinning behavior of the suspension

Inhomogeneous Distribution of a Rigid Fibre undergoing Rectilinear Flow between Parallel Walls at High Peclet Numbers

We use slender-body theory to simulate a rigid fibre within simple shear flow and parabolic flow at zero Reynolds number and high Péclet numbers (weak Brownian motion). Hydrodynamic interactions of bulk fibres with the bounding walls are included using previously developed methods (Harlen, Sundararajakumar & Koch, J. Fluid Mech., vol. 388, 1999, pp. 355-388; Butler & Shaqfeh, J. Fluid Mech., vol. 468, 2002, pp. 205-237). We also extend a previous analytic theory (Park, Bricker & Butler, Phys. Rev. E, vol. 76, 2007, 04081) predicting the centre-of-mass distribution of rigid fibre suspensions undergoing rectilinear flow near a wall to compare the steady and transient distributions. The distributions obtained by the simulation and theory are in good agreement at sufficiently high shear rates, validating approximations made in the theory which predicts a net migration of the rigid fibres away from the walls due to a hydrodynamic lift force. The effect of the inhomogeneous distribution on the effective stress is also investigated