Jay D. Schieber

A full-chain, temporary network model with sliplinks, chain-length fluctuations, chain connectivity and chain stretching

A full-chain, temporary network model is proposed for nonlinear flows of linear, entangled polymeric liquids. The model is inspired by the success of a recent reptation model, but contains no beads or tubes. Instead, each chain uses a different (and smaller) set of dynamic variables: the location of each entanglement and the number of Kuhn steps in chain strands between entanglements. As before, the model requires only a single phenomenological parameter that is fit by linear viscoelasticity. The number of Kuhn steps varies stochastically from imbalances in chemical potential and Brownian forces. In the language of reptation, the model exhibits chain connectivity, chain-length fluctuations, chain stretching, and tube dilation. The current implementation in this framework does not include constraint release, although its addition is possible. The entanglements are assumed to move affinely. Because of the affinity assumption and lack of constraint release, the model should be expected to approximate well a ...

SEGMENT CONNECTIVITY, CHAIN-LENGTH BREATHING, SEGMENTAL STRETCH, AND CONSTRAINT RELEASE IN REPTATION MODELS. II. DOUBLE-STEP STRAIN PREDICTIONS

Predictions for double-step strain flows are presented using a newly proposed reptation theory that accounts for segment connectivity, chain-length breathing, segmental stretch and constraint release in a self-consistent, full-chain theory. In this part of the work emphasis is on double-step shear strains where the second step is reversed and the imposition time of the second strain is earlier than the estimated retraction time, for which the Doi–Edwards model and single-integral models have been found to be incapable of describing experimental trends. Transient stress relaxation properties of two types of reversing flows, types B and C, have been examined and compared to the predictions obtained from the Doi–Edwards model and a single-integral model. The simulations show excellent agreement with the experimental trends based on recent mechanical and optical measurements.

Segment connectivity, chain-length breathing, segmental stretch, and constraint release in reptation models. III. Shear flows

A previously proposed self-consistent reptation model that includes chain stretching, chain-length fluctuations, segment connectivity, and constraint release is used to predict transient and steady shearing flows. Quantitative comparisons are made with the concentrated solution data considered in the previous papers of the series. The model is able to capture quantitatively all features of experimental data considered, including overshoot in both shear and first normal stresses, the strain-rate dependence of the strain magnitude at maximum stress, the steady-state viscosity and first-normal-stress coefficient as functions of shear rate, the viscosity curves for different molecular weight, the transient and steady-state behavior of the extinction angle, and the stress relaxation in cessation of steady shear flow. The model can describe all aspects of the data very well except the magnitude of the overshoot in stress at high shear rates, where the model is somewhat over-predictive. A new method of analysis for shear stress decay following cessation of steady shear is proposed, based on the physics of the model.

Fluctuations in entanglements of polymer liquids

A method for incorporating reptation ideas in a temporary network model is used to predict the fluctuations of entanglement number and monomer density between entanglements in entangled polymeric liquids. The dynamic variables are chosen to be the number of Kuhn steps and the position of the entanglements. A chain of fixed number of Kuhn steps in a bath that fixes the chemical potential conjugate to the number of entanglements is considered. The static equilibrium statistics of such a model can be calculated analytically. Since the dynamics of the model may also be simulated, these analytic expressions may be used to check the algorithm. Also, the damping function is calculated analytically from these distributions, as well as normal stresses following a step shear.

Brownian dynamics simulation of reversible polymer networks under shear using a non-interacting dumbbell model

Abstract A reversible network of associative, telechelic chains is represented by a mean-field model similar to that proposed recently by Vaccaro and Marrucci [J. Non-Newtonian Fluid Mech. 92 (2000) 261]. The model contains neither the topology of the network nor explicit interactions between different chains. Instead, it consists of two separate ensembles of dumbbells. One of these ensembles represents the ‘active’ chains, which are connected by both ends to other chains and carry most of the stress on the system. The other ensemble represents ‘dangling’ chains, connected to the network by one end only. Association of dangling chains to the network, to become active, and dissociation, the reverse process, are simulated by appropriate transition rules. The stochastic differential equations for this model are solved numerically using a standard Brownian dynamics method. This circumvents the need for (questionable) closure approximations to solve analytically the equivalent Fokker–Planck equations. Under simple shear flow, this system shows the main characteristics of a reversible network of telechelic chains, e.g. a Newtonian plateau, shear thickening and shear thinning. The simulation results confirm some of the predictions of Vaccaro and Marrucci.

Comprehensive comparisons with nonlinear flow data of a consistently unconstrained Brownian slip-link model

A consistently unconstrained Brownian slip-link model (CUBS) with constant chain friction is used to predict the nonlinear rheological behavior of linear, entangled, polymeric liquids. The model naturally incorporates primitive-path-length fluctuations, segment connectivity, monomer density fluctuations, entanglement fluctuations, and constraint release without making any closure approximations. Constraint release is imposed on the level of the dynamics of the chain, and the relaxation modulus follows from these rigorously. The model is a mean-field, single-chain slip-link model, or temporary network model, with a single phenomenological time constant, τe, fit by linear viscoelasticity. The nonlinear flow predictions are made without adjusting any additional parameters. We find that the addition of constant chain friction noticeably improves the model predictions in all the flows considered. In contradiction with tube models, the results suggest that the additional physics of constraint release and convec...

Viscoelastic flow through fibrous media using the CONNFFESSIT approach

A combined finite element and Brownian dynamics technique (CONNFFESSIT) is used to predict the steady-state flow field around an infinite array of square-arranged cylinders using kinetic theory models. A finitely extensible elastic (FENE) dumbbell model and a modified reptation model are considered. Since Brownian dynamics simulations are used to predict the stresses in the flow field, no closure approximations are necessary in the models, such as the Peterlin approximation, or independent alignment. Comparisons are made with analogous models that have closed-form constitutive equations, namely the FENE-P dumbbell, and Doi and Edwards reptation model with independent alignment using the same numerical technique. The modified reptation model contains information about the entire chain configuration instead of just single segment orientations. In this way, the problems involved with reversing flows for reptation with independent alignment can be avoided. Since the flow field presents alternately converging ...

The effects of bead inertia on the Rouse model

The Rouse model for dilute polymer solutions undergoing homogeneous flows has been generalized to include the inertia of the beads in the equations of motion. To obtain the correct ‘‘diffusion equation’’ for the probability density distribution function in phase space, we generalize the diffusion equation derived by Murphy and Aguirre [J. Chem. Phys. 57, 2098 (1972)] from Hamilton’s equations of motion for an arbitrary number of interacting Brownian particles at equilibrium. Material functions are found, and the noninertial case is seen to be obtained as the zero mass limit in all steps of the development. In particular, the steady‐state shear results are unaffected by the inclusion of inertia. It is also shown how two assumptions, ‘‘equilibration in momentum space,’’ and the neglect of acceleration, made independently by Curtiss, Bird, and Hassager in their phase‐space kinetic theory, are actually the result of assuming zero mass.

Approximations of the discrete slip-link model and their effect on nonlinear rheology predictions

The discrete slip-link model (DSM) was developed to describe the dynamics of flexible polymer melts. The model is able to predict linear viscoelasticity of monodisperse linear, polydisperse linear, and branched systems. The model also shows good agreement with dielectric relaxation experiments, except for the single data set available for bidisperse linear systems with a small volume fraction of long chains. In this work, both shear and elongational flow predictions obtained using the DSM without parameter adjustment are shown. Model predictions for shear flow agree very well with experimental results. The DSM is able to capture the transient response as well as the steady-state viscosity. However, for elongational flow, agreement is unsatisfactory at large strains. The DSM captures the onset of strain hardening, but after a Hencky strain between 2 and 3, it predicts transient strain softening, whereas experiments show only monotonic growth. We explore a number of assumptions and approximations of the model and their effect on flow predictions. The approximations are related to the neglect of these phenomena, which are expected to be more sensitive in elongational flow: finite extensibility, convective constraint

Entangled Polymer Dynamics in Equilibrium and Flow Modeled Through Slip Links

The idea that the dynamics of concentrated, high-molecular weight polymers are largely governed by entanglements is now widely accepted and typically understood through the tube model. Here we review alternative approaches, slip-link models, that share some similarities to and offer some advantages over tube models. Although slip links were proposed at the same time as tubes, only recently have detailed, quantitative mathematical models arisen based on this picture. In this review, we focus on these models, with most discussion limited to mathematically well-defined objects that conform to state-of-the-art beyond-equilibrium thermodynamics. These models are connected to each other through successive coarse graining, using nonequilibrium thermodynamics along the way, and with a minimal parameter set. In particular, the most detailed level of description has four parameters, three of which can be determined directly from atomistic simulations. Once the remaining parameter is determined for any system, all parameters for all members of the hierarchy are determined. We show how, using this hierarchy of slip-link models combined with atomistic simulations, we can make predictions about the nonlinear rheology of monodisperse homopolymer melts, polydisperse melts, or blends of different architectures. Mathematical details are given elsewhere, so these are limited here, and physical ideas are emphasized. We conclude with an outlook on remaining challenges that might be tackled successfully using this approach, including complex flow fields and polymer blends.