Alexei E. Likhtman

Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release

A refined version of the Doi and Edwards tube model for entangled polymer liquids is presented. The model is intended to cover linear chains in the full range of deformation rates from linear to strongly nonlinear flows. The effects of reptation, chain stretch, and convective constraint release are derived from a microscopic stochastic partial differential equation that describes the dynamics of the chain contour down to the length scale of the tube diameter. Contour length fluctuations are also included in an approximate manner. Predictions of mechanical stresses as well as the single chain structure factor under flow are shown. A comparison with experimental data is made in which all model parameters are fixed at universal values or are obtained from linear oscillatory shear measurements. With no parameter modification the model produces good agreement over a wide range of rheological data for entangled polymer solutions, including both nonlinear shear and extension.

Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-Poly equation

Abstract Recently we developed a theory for fast flows of entangled polymer melts which includes the processes of reptation, convective and reptation-driven constraint release, chain stretch and contour length fluctuations. The theory is derived from a stochastic microscopic equation of motion of the chain inside the tube and of the tube itself. As a result we obtain a partial differential equation for the tube tangent correlation function, the solution of which requires quite intensive calculations. At the same time the application of this theory to realistic flows (which is anything other than the laboratory rheometer) requires a simple and less computationally intensive set of equations for the stress tensor similar to the Giesekus, PTT, Larson or Pom–Pom equations. In particular, the last was derived from molecular theory for a generic type of branched polymer. In this paper we demonstrate that molecular tube theory can also provide a route to constructing a family of very simple differential constitutive equations for linear polymers. They capture the full model quite well and therefore can be used in flow solving software to model spatially inhomogeneous flows. We present a comparison of the proposed equations with our full model and with experimental data.

Definitions of entanglement spacing and time constants in the tube model

Numerous papers have recently appeared in the literature presenting quantitative comparisons of experimental linear viscoelastic data to the most recent versions of “tube” models for entangled polymer melts and solutions. Since these tube models are now being used for quantitative, rather than just qualitative, predictions, it has become important that numerical prefactors for the time constants that appear in these theories be evaluated correctly using literature data for the parameters (i.e., density, plateau modulus, etc.) that go into the theories. However, in the literature two definitions of the entanglement spacing in terms of plateau modulus have been presented, and confusion between these has produced numerous errors in the recent literature. In addition, two different definitions of the “equilibration time,” a fundamental time constant, have also appeared, creating additional potential for confusion. We therefore, carefully review the alternative definitions and clarify the values of the prefactors that must be used for the different definitions, in the hope of helping future authors to avoid such errors.

Linear and nonlinear shear flow behavior of monodisperse polyisoprene melts with a large range of molecular weights

Shear rheological data for a wide range of nearly monodisperse linear polyisoprene melts are reported herein using the results of linear and nonlinear measurements. The number of entanglements per chain ranges from about 0.5 to 230. In order to extend the range of available frequencies and shear rates the procedure of time-temperature superposition is improved and validated in both the linear and the nonlinear regime. In the linear flow regime a detailed comparison with the tube theory by Likhtman and McLeish [Macromolecules 35, 6332–6343 (2002)] was performed. The overall agreement was found to be satisfactory but several minor disagreements, e.g., at the crossover regime and for the steady-state compliance, were observed and have been discussed. Until now, most nonlinear models for entangled polymers were compared with shear data on solutions. However, the equivalence between solutions and melts is not well established. In this paper the results of a series of nonlinear start-up shear experiments are co...

Microscopic theory of convective constraint release

We develop a microscopic description of the contribution to stress relaxation in entangled polymer melts of convective constraint release, which is the release of entanglement constraints due to the effects of convective flow on chains surrounding a given chain. Our theory resolves three of the main shortcomings of the Doi–Edwards model in nonlinear rheology, in that it predicts (1) a monotonically increasing shear stress as a function of shear rate, (2) shear stress independent of molecular weight at sufficiently high shear rates, and (3) only modest anisotropies in the single chain scattering function, in agreement with experiment. In addition, our approach predicts that a stress maximum and resulting shear-banding instability would occur for living micelle solutions, as observed.

Constriction flows of monodisperse linear entangled polymers: Multiscale modeling and flow visualization

We explore both the rheology and complex flow behavior of monodisperse polymer melts. Adequate quantities of monodisperse polymer were synthesized in order that both the materials rheology and microprocessing behavior could be established. In parallel, we employ a molecular theory for the polymer rheology that is suitable for comparison with experimental rheometric data and numerical simulation for microprocessing flows. The model is capable of matching both shearand extensional data with minimal parameter fitting. Experimental data for the processing behavior of monodisperse polymers are presented for the first time as flow birefringence and pressure difference data obtained using a Multipass Rheometer with an 11:1 constriction entry and exit flow. Matching of experimental processing data was obtained using the constitutive equation with the Lagrangian numerical solver, FLOWSOLVE. The results show the direct coupling between molecular constitutive response and macroscopic processing behavior, and differentiate flow effects that arise separately from orientation and stretch.

Microphase separation in block copolymer/homopolymer blends: Theory and experiment

The process of microphase separation in diblock copolymer/homopolymer blends is studied both theoretically and experimentally for an asymmetric diblock copolymer and for homopolymer concentration less than 25%. The degree of polymerization of the added homopolymer (Nh) covered all possible cases; N≈Nh, N>Nh, and N<Nh. SAXS and rheology are employed and provide the order–disorder transition temperature through the discontinuous changes of the structure factor and the storage modulus. The minority phase can solubilize only a small amount of added homopolymer; addition of a higher amount results in the formation of nonequilibrium structures. Theoretical calculations performed in the strong segregation limit provide the period and the critical value of χN for the stability of the disordered phase. The theory predicts that (χN)c always increases with the addition of the majority phase. When the minority phase is added, (χN)c can increase (N≫Nh and N⩾Nh) or decrease (N≪Nh). The experimental results are in good ...

Efficient on the fly calculation of time correlation functions in computer simulations

Time correlation functions yield profound information about the dynamics of a physical system and hence are frequently calculated in computer simulations. For systems whose dynamics span a wide range of time, currently used methods require significant computer time and memory. In this paper, we discuss the multiple-tau correlator method for the efficient calculation of accurate time correlation functions on the fly during computer simulations. The multiple-tau correlator is efficacious in terms of computational requirements and can be tuned to the desired level of accuracy. Further, we derive estimates for the error arising from the use of the multiple-tau correlator and extend it for use in the calculation of mean-square particle displacements and dynamic structure factors. The method described here, in hardware implementation, is routinely used in light scattering experiments but has not yet found widespread use in computer simulations.

Viscoelasticity and Molecular Rheology

This chapter contains a review of available models for describing unentangled and entangled polymer liquids such as polymer melts and solutions. These simple single-chain models are investigated by numerical solution of underlying stochastic differential equations, and then compared with the many chain molecular dynamics (MD) simulations. The MD model contains only simple bead-spring potentials, which are supposed to contain all essential information about entanglements but not the chemical details. The tube model is criticized for containing many hidden assumptions and ambiguities and several alternative models are analyzed.

Linear rheology of multiarm star polymers diluted with short linear chains

We present experimental results on the linear rheology of multiarm star/linear polymer mixtures, the latter having molecular weight much smaller than the star arm molecular weight. In such a case the linear chains act as ideal macromolecular solvents, which dilute entanglements of the arms. Using different star polymers we show that it is possible to account for this dilution and describe the linear rheology of the mixtures using the Milner–McLeish theory for arm relaxation, complemented by the longitudinal modes of stress relaxation and high frequency Rouse modes. A universal description of the isofrictional arm relaxation time as a function of the number of entanglements is obtained for stars of any functionality and degree of dilution. The slow structural mode, related to the diluted star’s colloidal core, also depends on the number of entanglements, but in a more complex way.