An application of the generalized Fokker–Planck equation to the dynamics of dilute polymer solutions (original) (raw)

Dynamics of dilute polymer solutions

Fluid Dynamics, 1988

Taking volume effects, hydrodynamic no-flow conditions, and internal viscosity into account in the molecular dynamics has made it possible to formulate an equation of flow for a dilute polymer solution which in the region of comparatively slowly varying motions is described by the available experimental [acts .

Extended kinetic theory of polymeric fluids

Macromolecular Theory and Simulations, 1996

The setting of classical configuration space kinetic theory of polymeric fluids is extended by adopting the field of internal momentum into the set of independent state variables. The internal momentum predicted by the extended theory compares well with the internal momentum seen in molecular simulations. The extended theory is also shown to provide an intrinsically consistent formulation of the time evolution of macromolecules with internal friction.

The Rouse-Zimm-Brinkman theory of the dynamics of polymers in dilute solutions

arXiv (Cornell University), 2005

We propose a theory of the dynamics of polymers in dilute solution, in which the popular Zimm and Rouse models are just limiting cases of infinitely large and small draining parameter. The equation of motion for the polymer segments (beads) is solved together with Brinkman's equation for the solvent velocity that takes into account the presence of other polymer coils in the solution. The equation for the polymer normal modes is obtained and the relevant time correlation functions are found. A tendency to the time-dependent hydrodynamic screening is demonstrated on the diffusion of the polymers as well as on the relaxation of their internal modes. With the growing concentration of the coils in solution they both show a transition to the exactly Rouse behavior. The shear viscosity of the solution, the Huggins coefficient and other quantities are calculated and shown to be notably different from the known results.

GENERALIZED LANGEVIN THEORY OF THE BROWNIAN MOTION AND THE DYNAMICS OF POLYMERS IN SOLUTION

The review deals with a generalization of the Rouse and Zimm bead-spring models of the dynamics of flexible polymers in dilute solutions. As distinct from these popular theories, the memory in the polymer motion is taken into account. The memory naturally arises as a consequence of the fluid and bead inertia within the linearized Navier-Stokes hydrodynamics. We begin with a generalization of the classical theory of the Brownian motion, which forms the basis of any theory of the polymer dynamics. The random force driving the Brownian particles is not the white one as in the Langevin theory, but " colored " , i.e., statistically correlated in time, and the friction force on the particles depends on the history of their motion. An efficient method of solving the resulting generalized Langevin equations is presented and applied to the solution of the equations of motion of polymer beads. The memory effects lead to several peculiarities in the time correlation functions used to describe the dynamics of polymer chains. So, the mean square displacement of the polymer coils contains algebraic long-time tails and at short times it is ballistic. It is shown how these features reveal in the experimentally observable quantities, such as the dynamic structure factors of the scattering or the viscosity of polymer solutions. A phenomenological theory is also presented that describes the dependence of these quantities on the polymer concentration in solution.

Lattice Fokker Planck for dilute polymer dynamics

We show that the actual diffusive dynamics, governing the momentum relaxation of a polymer molecule, and described by a Fokker-Planck equation, may be replaced by a BGK-type relaxation dynamics without affecting the slow (Smoluchowski) dynamics in configuration space. Based on the BGK-type description, we present a lattice-Boltzmann (LB) based direct discretization approach for the phase-space description of inertial polymer dynamics. We benchmark this formulation by determining the bulk rheological properties for both steady and time-dependent shear and extensional flows at moderate to large Weissenberg numbers. Finally, we compare the usefulness of the different discrete velocity models, typically used in the LB framework, for solving diffusive dynamics based on the Fokker-Planck equation.

Mesoscopic constitutive relations for dilute polymer solutions

Physica A: Statistical Mechanics and its Applications, 2006

A novel approach to the dynamics of dilute solutions of polymer molecules under flow conditions is proposed by applying the rules of mesoscopic nonequilibrium thermodynamics (MNET). The probability density describing the state of the system is taken to be a function of the position and velocity of the molecules, and on a local vector parameter accounting for its deformation. This function obeys a generalized Fokker-Planck equation, obtained by calculating the entropy production of the system, and identifying the corresponding probability currents in terms of generalized forces. In simple form, this coarse-grained description allows one to derive hydrodynamic equations where molecular deformation and diffusion effects are coupled. A class of non-linear constitutive relations for the pressure tensor are obtained. Particular models are considered and compared with experiments.

Two-fluid kinetic theory for dilute polymer solutions

2022

We provide a Boltzmann-type kinetic description for dilute polymer solutions based on two-fluid theory. This Boltzmann-type description uses a quasi-equilibrium based relaxation mechanism to model collisions between a polymer dumbbell and a solvent molecule. The model reproduces the desired macroscopic equations for the polymer-solvent mixture. The proposed kinetic scheme leads to a numerical algorithm which is along the lines of the lattice Boltzmann method. Finally, the algorithm is applied to describe the evolution of a perturbed Kolmogorov flow profile, whereby we recover the major elastic effect exhibited by a polymer solution, specifically, the suppression of the original inertial instability.

Correlation function formula for the intrinsic viscosity of dilute polymer solutions

The Journal of Chemical Physics, 1975

The correlation function formalism for the intrinsic viscosity of polymers is studied. A controversy concerning the correct force to use in the momentum flux is resolved. It is shown that when the diffusion equation is used in the full configuration space of polymer segments the forces entering the momentum flux are purely mechanical and there is no entropic contribution. A comparison is made with Kirkwood's theory of viscoelastic behavior. The correlation function expression we advocate is shown to yield the correct high frequency limiting behavior for the case of elastic dumbbells.

The dynamics of polymers in solution with hydrodynamic memory

arXiv (Cornell University), 2005

The dynamics of individual polymers in solution is fundamental for understanding the properties of polymeric systems. Consequently, considerable work has been devoted towards understanding polymer dynamics. In spite of the long-standing investigations, a number of problems remains between the theory and experiments, such as the dynamic light or neutron scattering. For example, the value of the first cumulant of the dynamic structure factor (DSF) is lower than the theoretical prediction for flexible polymers. The origin of these discrepancies is a matter of continuous discussion. The development of the theory of polymer dynamics is thus still of interest. This work represents an attempt of such a development. The main idea of the proposed generalization comes from the theory of the Brownian motion, which lies in the basis of the bead-spring models of polymer dynamics. In the classical (Einstein) description the resistance force on the bead moving in a liquid is the Stokes force, which is valid only for the steady motion. In a more general nonstationary case the hydrodynamic memory, which is a consequence of fluid inertia, should be taken into account. In the Brownian motion it displays in the famous "long-time tails" in the velocity autocorrelation function. The time dependence of the mean square displacement (MSD) of the particle changes from the "ballistic" regime at short times to the Einstein diffusion at long times. We have found similar effects in the dynamics of polymers. We give the corresponding generalization of the Rouse-Zimm (RZ) theory. It is shown that the time correlation functions describing the polymer motion essentially differ from those in the RZ models. The MSD of the polymer coil is at short times proportional to t 2 (instead of t). At long times it contains additional (to the Einstein term) contributions, the leading of which is ~ t 1/2. The relaxation of the internal normal modes of the polymer differs from the traditional exponential decay. This is displayed in the tails of their correlation functions, the longest-lived being ~ t-3/2 in the Rouse limit and t-5/2 in the Zimm case when the hydrodynamic interaction is taken into account. It is discussed that the found peculiarities, in particular a slower diffusion of the coil, should be observable in dynamic scattering experiments. The DSF and the first cumulant of the polymer coil are calculated. Finally, we extend the theory to the situation when the dynamics of the studied polymer is influenced by the presence of other polymers in dilute solution.

Canonical distribution functions in polymer dynamics. (I). Dilute solutions of flexible polymers

Physica A: Statistical Mechanics and its Applications, 2002

The quasi-equilibrium or maximum entropy approximation is applied in order to derive constitutive equations from kinetic models of polymer dynamics. It is shown in general and illustrated for an example how canonical distribution functions are obtained from the maximum entropy principle, how macroscopic and constitutive equations are derived therefrom and how these constitutive equations can be implemented numerically. In addition, a measure for the accuracy of the quasi-equilibrium approximation is proposed that can be evaluated while integrating the constitutive equations. In the example considered, it is confirmed that the accuracy of the approximation is increased by including more macroscopic variables. In steady elongational flow, it is found that more macroscopic variables need to be included above the coil-stretch transition to achieve the same accuracy as below.