Abstract:
Linear viscoelastic properties for a dilute polymer solution are predicted by modeling the solution as a suspension of non-interacting bead-spring chains. The present model, unlike the Rouse model, can describe the solution's rheological behavior even when the solvent quality is good, since excluded volume effects are explicitly taken into account through a narrow Gaussian repulsive potential between pairs of beads in a bead-spring chain. The use of the narrow Gaussian potential, which tends to the more commonly used delta-function repulsive potential in the limit of a width parameter "d" going to zero, enables the performance of Brownian dynamics simulations. The simulations results, which describe the exact behavior of the model, indicate that for chains of arbitrary but finite length, a delta-function potential leads to equilibrium and zero shear rate properties which are identical to the predictions of the Rouse model. On the other hand, a non-zero value of "d" gives rise to a prediction of swelling at equilibrium, and an increase in zero shear rate properties relative to their Rouse model values. The use of a delta-function potential appears to be justified in the limit of infinite chain length. The exact simulation results are compared with those obtained with an approximate solution which is based on the assumption that the non-equilibrium configurational distribution function is Gaussian. The Gaussian approximation is shown to be exact to first order in the strength of excluded volume interaction, and is found to be accurate above a threshold value of "d", for given values of chain length and strength of excluded volume interaction.

Abstract:
The Rouse model has recently been modified to take into account the excluded volume interactions that exist between various parts of a polymer chain by incorporating a narrow Gaussian repulsive potential between pairs of beads on the Rouse chain (cond-mat/0002448). The narrow Gaussian potential is characterized by two parameters: z* - which accounts for the strength of the interaction, and d* - which accounts for the extent of the interaction. In the limit of d* going to zero, the narrow Gaussian potential tends to the more commonly used delta-function repulsive potential. The influence of the parameter d*, in the limit of infinite chain length, on equilibrium and linear viscoelastic properties, and on universal ratios involving these properties, is examined here. A renormalization group calculation of the end-to-end vector suggests that the value chosen for the variable d* will not affect critical exponents, or universal ratios. A similar trend is also observed for results obtained with an approximate solution, which is based on the assumption that the non-equilibrium configurational distribution function is Gaussian.

Abstract:
The development of a coherent conceptual basis for the treatment of non-linear microscopic phenomena, such as, hydrodynamic interaction, finite extensibility, excluded volume and internal viscosity, in molecular theories of dilute polymer solutions, is discussed. In particular, recent advances in the treatment of hydrodynamic interaction are reviewed, and the successive refinements which have ultimately led to the prediction of universal viscometric functions in theta solvents are highlighted.

Abstract:
The hydrodynamic radius of a polymer chain, obtained using Brownian dynamics simulations of the continuum Edwards model, is found to obey a crossover in the excluded volume parameter z, which is significantly different from that observed for the radius of gyration. It is shown that this difference arises from contributions due to dynamic correlations to the diffusivity, which are ignored in the commonly used definition of hydrodynamic radius based on the Kirkwood expression. The swelling of the hydrodynamic radius from the theta-state, obtained from simulations, shows remarkable agreement with experimental measurements.

Abstract:
The Gaussian Approximation, proposed originally by Ottinger [J. Chem. Phys., 90 (1) : 463-473, 1989] to account for the influence of fluctuations in hydrodynamic interactions in Rouse chains, is adapted here to derive a new mean-field approximation for the FENE spring force. This "FENE-PG" force law approximately accounts for spring-force fluctuations, which are neglected in the widely used FENE-P approximation. The Gaussian Approximation for hydrodynamic interactions is combined with the FENE-P and FENE-PG spring force approximations to obtain approximate models for finitely-extensible bead-spring chains with hydrodynamic interactions. The closed set of ODE's governing the evolution of the second-moments of the configurational probability distribution in the approximate models are used to generate predictions of rheological properties in steady and unsteady shear and uniaxial extensional flows, which are found to be in good agreement with the exact results obtained with Brownian dynamics simulations. In particular, predictions of coil-stretch hysteresis are in quantitative agreement with simulations' results. Additional simplifying diagonalization-of-normal-modes assumptions are found to lead to considerable savings in computation time, without significant loss in accuracy.

Abstract:
The steady flow of three viscoelastic fluids (Oldroyd-B, FENE-P, and Owens model for blood) in a two-dimensional channel, partly bound by a deformable, finite thickness neo-Hookean solid, is computed. The limiting Weissenberg number beyond which computations fail to converge is found to increase with increasing dimensionless solid elasticity parameter {\Gamma}, following the trend Owens > FENE- P > Oldroyd-B. The highly shear thinning nature of Owens model leads to the elastic solid always collapsing into the channel, for the wide range of values of {\Gamma} considered here. In the case of the FENE-P and Oldroyd-B models, however, the fluid-solid interface can be either within the channel, or bulge outwards, depending on the value of {\Gamma}. This behaviour differs considerably from predictions of earlier models that treat the deformable solid as a zero-thickness membrane, in which case the membrane always lies within the channel. The capacity of the solid wall to support both pressure and shear stress, in contrast to the zero-thickness membrane that only responds to pressure, is responsible for the observed difference. Compar- ison of the stress and velocity fields in the channel for the three viscoelastic fluids, with the predictions for a Newtonian fluid, reveals that shear thinning rather than elasticity is the key source of the observed differences in behaviour.

Abstract:
The role of solvent quality in determining the universal material properties of dilute polymer solutions undergoing steady simple shear flow is examined. A bead-spring chain representation of the polymer molecule is used, and the influence of solvent molecules on polymer conformations is modelled by a narrow Gaussian excluded volume potential that acts pair-wise between the beads of the chain. Brownian dynamics simulations data, acquired for chains of finite length, and extrapolated to the limit of infinite chain length, are shown to be model independent. This feature of the narrow Gaussian potential, which leads to results identical to a $\delta$-function repulsive potential, enables the prediction of both universal crossover scaling functions and asymptotic behavior in the excluded volume limit. Universal viscometric functions, obtained by this procedure, are found to exhibit increased shear thinning with increasing solvent quality. In the excluded volume limit, they are found to obey power law scaling with the characteristic shear rate $\beta$, in close agreement with previously obtained renormalization group results. The presence of excluded volume interactions is also shown to lead to a weakening of the alignment of the polymer chain with the flow direction.

Abstract:
The crossover region in the phase diagram of polymer solutions, in the regime above the overlap concentration, is explored by Brownian Dynamics simulations, to map out the universal crossover scaling functions for the gyration radius and the single-chain diffusion constant. Scaling considerations, our simulation results, and recently reported data on the polymer contribution to the viscosity obtained from rheological measurements on DNA systems, support the assumption that there are simple relations between these functions, such that they can be inferred from one another.

Abstract:
A narrow Gaussian excluded volume potential, which tends to a delta-function repulsive potential in the limit of a width parameter d* going to zero, has been used to examine the universal consequences of excluded volume interactions on the equilibrium and linear viscoelastic properties of dilute polymer solutions. Brownian dynamics simulations data, acquired for chains of finite length, has been extrapolated to the limit of infinite chain length to obtain model independent predictions. The success of the method in predicting well known aspects of static solution properties suggests that it can be used as a systematic means by which the influence of solvent quality on both equilibrium and non-equilibrium properties can be studied.

Abstract:
A dilute polymer solution is modeled as a suspension of non-interacting Hookean dumbbells and the effect of excluded volume is taken into account by incorporating a narrow Gaussian repulsive potential between the beads of each dumbbell. The narrow Gaussian potential is a means of regularising a delta-function potential---it tends to the delta-function potential in the limit of the width parameter going to zero. Exact predictions of viscometric functions in simple shear flow are obtained with the help of a retarded motion expansion and by Brownian dynamics simulations. It is shown that for relatively small non-zero values of the width parameter, the presence of excluded volume causes a swelling of the dumbbell at equilibrium, and shear thinning in simple shear flow. On the other hand, a delta function excluded volume potential does not lead to either swelling or to shear thinning. Approximate viscometric functions, obtained by assuming that the bead-connector vector is described by a Gaussian non-equilibrium distribution function, are found to be accurate above a threshold value of the width parameter, for a given value of the strength of excluded volume interaction. A first order perturbation expansion reveals that the Gaussian approximation is exact to first order in the strength of excluded volume interaction. The predictions of an alternative quadratic excluded volume potential suggested earlier by Fixman (J. Chem. Phys., 1966, 45, 785; 793) are also compared with those of the narrow Gaussian potential.