Polymer Kinetic Theory Research Articles

Polymer kinetic theory temperature dependent configurational probability diffusion equations: Existence of positive solution results

Polymer kinetic theory temperature dependent configurational probability diffusion equations: Existence of positive solution results

How important are fluctuations in the treatment of internal friction in polymers?

The Rouse model with internal friction (RIF), a widely used theoretical framework to interpret the effects of internal friction on conformational transitions in biomolecules, is shown to be an approximate treatment that is based on preaveraging internal friction. By comparison with Brownian dynamics simulations of an exact coarse-grained model that incorporates fluctuations in internal friction, the accuracy of the preaveraged model predictions is examined both at and away from equilibrium. While the two models predict intrachain autocorrelations that approach each other for long enough chain segments, they differ in their predictions for shorter segments. Furthermore, the two models differ qualitatively in their predictions for the chain extension and viscosity in shear flow, which is taken to represent a prototypical out-of-equilibrium condition.

Numerical Investigation of the Effect of Viscoelasticity on Drop Retraction and the Evaluation of Interfacial Tension between Polymer Melts

The purpose of this paper is twofold. The first is to numerically investigate and reveal the effect of polymer viscoelasticity on the retraction of a deformed drop using the lattice Boltzmann (LB) method and polymer kinetic theory. More importantly, the second is to propose a novel method to evaluate the interfacial tension between polymer melts based on the numerical study. Compared with the conventional deformed drop retraction method (DDRM), the present method is designed to greatly reduce the impact of polymer viscoelasticity on measuring interfacial tension. To verify, the interfacial tension between molten PP and POE is evaluated using the proposed method and obviously closer result to the true value is shown.

Hydrodynamic interactions between two slender tori in a viscous fluid

As a gift to Professor Bird, hydrodynamic interactions between two slender toroidal particles immersed in a viscous fluid are derived to extend the Rotne–Prager–Yamakawa (RPY) tensor formulation of bead‐spring and bead‐rod models of polymer kinetic theory to toroidal beads. As a relatively simple example of a multiply connected domain, this model geometry exhibits the special “role of the hole” and features new tensorial constructs beyond the Oseen–Burgers and Stokes quadrupoles that are encountered in the classical RPY theory. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1517–1522, 2014

A microstructural flow-induced crystallization model for film blowing: validation with experimental data

The two-phase microstructural/constitutive model for film blowing of Doufas and McHugh (D-M) (J Rheol 45:1085–1104, 2001a) is validated against online film data of a linear low-density polyethylene (LLDPE) at a variety of processing conditions. The D-M model includes the effects of thermal and flow-induced (enhanced) crystallization (FIC) coupled with the rheological response of both the melt and semicrystalline phases under fabrication conditions. The model predictions of bubble radius, velocity, and crystallinity profiles are in quantitative agreement with available experimental data over a wide range of blow-up ratios (BUR), take-up ratios (TUR), and bubble cooling rates using the same set of material/model parameters. The model naturally predicts the location of the frost line as a consequence of system stiffening due to crystallization overcoming the pitfalls of traditional modeling approaches that impose it as an artificial boundary condition. For a wide range of processing conditions, it is found that key film mechanical properties including elongation to break, yield stress, tensile modulus, and tear strength correlate well with predicted locked-in extensional stresses and molecular orientation at the frost line enabling development of quantitative structure-process-properties relationships that are useful in product and process development. The D-M model for film blowing is physics-based including elements of molecular rheology (polymer kinetic theory), suspension, and nucleation theories as well as irreversible thermodynamics principles, yet being tractable for continuum-based numerical simulations with practical industrial applicability. The FIC enhancement factor of the model is shown to be proportional to $\exp \left (\lambda _{\text {eff},\textnormal {w}}^{2} -1\right )$ , where λ eff,w is a molecular chain stretch ratio of the whole chain and proportional to exp (λ 2 − 1), where λ is the stretch ratio of the remaining (uncrystallized) amorphous chain, consistent with fundamental kinetic Monte Carlo simulations of flow-induced nucleation of Graham and Olmsted (Phys Rev Lett 103:115702-1–115702-4, 2009).

Lattice Boltzmann method for polymer kinetic theory

The purpose of this work is to extend the applicability of the lattice Boltzmann method (LBM) to the field of polymer kinetic theory or more generally suspensions that could be described in the Fokker–Planck formalism. This method has been, in a first time, used for gas kinetic theory, where the resolution space corresponds to the physical space coordinate. In a second time is has been generalized to be applied to fluid flow involving different behaviours: turbulence, porous media, multiphase flow, etc. However this powerful, parallel, and efficient algorithm has not been applied for solving Fokker–Planck equations widely used to describe suspension kinetic theory. In this scale, molecular models involve a high computational costs because of the multidimensionality of the fully coupled micro–macro complex flow. The originality of this work consists to apply the lattice Boltzmann technique for solving Fokker–Planck equation based on a discretization of the configuration space where the resolution coordinates correspond to the microscopic configuration space (and not the physical coordinates). The result of this work emphasizes the optimality of the used technique that, in addition to its parallel ability, gathers the simplicity of the stochastic simulation and the robustness of the traditional fixed mesh support (such as the finite element method). Accuracy and convergence of the LBM will be compared to the stochastic and the finite element techniques for homogeneous shear flow.

Legendre integrators, post-processing and quasiequilibrium

A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of this toolbox are: Legendre integrators, dynamical post-processing, and the thermodynamic projector. The thermodynamic projector is the tool to transform almost any anzatz to a thermodynamically consistent model. The post-processing is the cheapest way to improve the solution obtained by the Legendre integrators. Legendre integrators give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium (“maximum entropy”, MaxEnt) approximations. The essentially new element of this toolbox, the method of thermodynamic projector, is demonstrated on application to the FENE-P model of polymer kinetic theory. The multi-peak model of polymer dynamics is developed.

Transport phenomena

Transport phenomena

Variance reduced Brownian simulation of a bead-spring chain under steady shear flow considering hydrodynamic interaction effects

To obtain numerical estimates for the properties of a model for polymers in dilute theta solutions in its long-chain limit we follow a stochastic approach to polymer kinetic theory. The model takes into account configuration-dependent hydrodynamic interaction (HI) and simplifies to the Zimm bead-spring chain model in the case of preaveraged HI, for which parameter-free “universal ratios” such as the ratio between radius of gyration and hydrodynamic radius are known. The Chebyshev polynomial method and a variance reduction simulation technique are used to implement an efficient Brownian dynamics simulation. We resolve the full dependence of several characteristic ratios versus both chain length and hydrodynamic interaction parameter, we extrapolate their values to determine universal behaviors, and compare with analytical and experimental results.

Brownian dynamics of bead–rod–nugget–spring polymer chains with hydrodynamic interactions

With Brownian dynamics simulations of solutions containing complex biological macromolecules in mind we have formulated a polymer chain model consisting of a selectable sequence of spherical subunits (beads) and nonspherical subunits of arbitrary shapes (nuggets) connected by rigid rods, rigid ball–socket joints, or springs. Many physical properties of both subunits and connections are selectable. First a very general diffusion equation of polymer kinetic theory is employed. Next the equivalent Itô stochastic differential equation of motion is established and finally an effective Brownian dynamics simulation algorithm is presented. In all these steps it is assumed that the numerical values of the subunit steady state mobility tensor are obtainable. The important and previously unanswered question of how to include translational–rotational hydrodynamic interactions in Brownian dynamics simulations of polymer chains containing nonspherical subunits linked by rigid constraints is resolved rigorously here. This means that during the formal derivation of the presented Brownian dynamics simulation algorithm all subunit hydrodynamic interactions of the proposed bead–rod–nugget–spring polymer chain model have in principle been taken fully into account. In addition the model includes rigid constraints to fixed points in the laboratory coordinate system, solvent flow, external forces, excluded volume effects, and bending and torsional stiffness between polymer chain subunits. Bead–spring, bead–rod–spring, needle–spring, needle polymer chains and liquid crystals are all special cases of the bead–rod–nugget–spring polymer chain model. The validity of the algorithm presented is limited to the diffusion time domain.

Variance reduced simulations of polymer dynamics

Stochastic simulation techniques play an important role in investigating polymer dynamics. Such simulations can be made more efficient by applying variance reduction methods. We present and compare various variance reduction schemes, based either on importance sampling strategies or on the use of control variates. In order to demonstrate that variance reduced simulations are a powerful tool in polymer kinetic theory, we show that for Hookean dumbbells with hydrodynamic interaction up to two orders in computer time can be saved uniformly in time.

Brownian dynamics of rigid polymer chains with hydrodynamic interactions.

With the idea of a broad investigation of the flow behavior of dilute polymer solutions in mind, the dynamics of polymer chains with rigid constraints and hydrodynamic interaction is formulated in various equivalent ways. Starting from a very general diffusion equation of polymer kinetic theory, equivalent stochastic differential equations of motion both in terms of generalized coordinates and in terms of constraint conditions are derived. Then an efficient Brownian dynamics simulation algorithm is constructed rigorously, and a convenient expression for evaluating stresses in simulations is suggested. Furthermore, a modified simulation algorithm, which is appropriate for infinitely stiff rather than rigid systems, is discussed.

Variance Reduced Brownian Dynamics Simulations

Brownian dynamics simulations can be made more efficient by incorporating the idea of importance sampling. By introducing and compensating a bias in favor of those configurations which mainly contribute to the average of a given quantity of interest, one can considerably reduce the variance of the stochastic simulation results. This idea can be applied to general stochastic differential quations of Motion and is hence not restricted to Brownian dynamics simulations. The construction of variance reduced Simulations requires an approximate understanding of the dynamics described by the underlying stochastic differential equation. The basic procedure is first developed in general and then illustrated for the example of a Hookean dumbbell solution in start up of steady shear flow. Particular emphasis is put on the development of a clearly structured formulation of the procedure which immediately allows for variance reduced simulations of nonlinear models. Possible applications in polymer kinetic theory and in the flow calculation of viscoelastic liquids are discussed.

Internal viscosity in polymer kinetic theory: shear flows

The Gaussian closure method and Brownian dynamics simulations have been used to calculate the shear material properties of a dilute solution of Hookean dumbbells with internal viscosity. Results for the zero-shear-rate material properties and small amplitude oscillatory shear material properties have been found analytically, and numerical results for the steady state shear material properties are also presented. Two interpretations of the stress tensor are investigated and results are compared. Brownian dynamics simulations are used to obtain material properties of the Hookean dumbbell with internal viscosity without approximations. These simulation results are compared with the perturbation solution of Booij and van Wiechen as well as with a new Gaussian closure solution. Also presented are the contracted distribution functions as derived from the Gaussian closure method and from Brownian dynamics simulations.

Renormalization-group calculation of viscometric functions based on conventional polymer kinetic theory

In order to obtain useful information from a perturbation expansion of hydrodynamic-interaction effects for long Rouse chains the basic equations of polymer kinetic theory are generalized to d-dimensional space. After solving the Kirkwood diffusion equation to first order in the strength of the hydrodynamic interactions, the Kramers expression for the stress tensor is used to calculate the corresponding first-order expansions of the zero-shear-rate viscometric functions for arbitrary d. Singularities occurring in these expansions near four dimensions indicate that local hydrodynamic interactions are not described in a proper way and, therefore, a renormalization of one of the model parameters, the bead friction coefficient, is necessary. With the renomalized perturbation expansions, various ratios of experimentally accessible quantities are calculated; in particular, ratios involving normal-stress coefficients are calculated for the first time by renormalization-group methods.

Freeze-etch TEM of aqueous polymer gel network morphology

The rheological and mechanical properties of crosslinked polymer gels arise from the structure of the gel network. In turn, the structure of the gel network results from: thermodynamically determined interactions between the polymer chain segments, the interactions of the crosslinking metal ion with the polymer, and the deformation history of the network. Interpretations of mechanical and rheological measurements on polymer gels invariably begin with a conceptual model of,the microstructure of the gel network derived from polymer kinetic theory. In the present work, we use freeze-etch replication TEM to image the polymer network morphology of titanium crosslinked hydroxypropyl guars in an attempt to directly relate macroscopic phenomena with network structure.

Brownian dynamics simulation of macromolecules in steady shear flow

A nonequilibrium Brownian dynamics technique is used to simulate a dilute solution of polymers in a steady shear flow. Results for the mean square end-to-end length of a Rouse chain agree very well with theoretical results from polymer kinetic theory.

Use of simple molecular models in the study of the mechanical behavior of solutions of flexible macromolecules

Six illustrations are given of the use of single-particle theories with elastic dumbbells and other simple models in polymer kinetic theory: (1) Marrucci's incorporation of thermal history in rheological equations of state; (2) Radial migration of polymers in a coaxial-cylinder viscometer (Uhlenhopp effect); (3) Crude incorporation of excluded volume effects; (4) Finite extensibility effects and the use of Peterlin's approximation; (5) Use of the Wang—Zimm theory to describe concentration effects in the non-Newtonian viscosity of solutions; and(6) Hydrodynamic interaction and the “effective velocity gradient” approximation. The dumbbell kinetic theory results have supplied useful guidance in the rheology and fluid dynamics of polymers.