Preaveraging Approximation Research Articles

Dynamics of Regular Star Polymers: The Intrinsic Viscosity

The dynamics of a regular star polymer is investigated under good-solvent and Θ conditions. A self-consistent approach to the equilibrium and dynamics is proposed within the Gaussian approximation in terms of the polymer normal modes, i.e., suitable statistically-independent linear combinations of the bond vectors. The equilibrium normal modes used to study the star expansion through self-consistent free-energy minimization (see Allegra, G. ; Colombo, E. ; Ganazzoli, F. Macromolecules 1993, 26, 330) form a first-order approximation to the dynamical normal modes under partial-draining conditions. The latter modes are obtained by diagonalization of a matrix that depends on the intramolecular elasticity and on the hydrodynamic interaction in the preaveraging approximation, the effect of chain expansion on both being easily accounted for. If the equilibrium normal modes are used, the relaxation times, from which the intrinsic viscosity is calculated, are somewhat in error only for the collective modes describing the concerted motion of the arms, unlike those related with the independent motion of the arms ; on the other hand, they yield the intrinsic viscosity to a very good approximation, thus avoiding the lengthy diagonalization. Numerical results for regular stars and linear chains are summarized by analytical equations giving the mean-square radius of gyration , the hydrodynamic radius R H and the intrinsic viscosity [η]. The influence of the topology is expressed through the ratios g Q = Q star /Q lin , Q being any of the above quantities. In a good solvent, the star is only slightly more expanded than the linear chain, and so the g ratios are very close to the theoretical phantom-chain value both for (S 2 ) and for R H . Conversely, [η] increases less in the star than in the linear chain because of the different rate of change with expansion of the intramolecular elasticity and of the hydrodynamic interaction, and so the corresponding g ratio is lower than the phantom-chain value. In the Θ state, the residual three-body interactions give a finite expansion to the star, unlike the linear chain. Therefore, we have g Q ph ≤ g Q * and R H , in agreement with experiment (the asterisk refers to the good-solvent conditions) ; on the other hand, we get g η * < g η ph < g η Θ , whereas the experimental finding is g η * < g η Θ < g η ph . The reason for this discrepancy is attributed to the preaveraging approximation, which becomes questionable for stars in the Θ state because of their large density near the branch point.

The reptation model with segmental stretch

The Doi-Edwards model with segmental stretch and a non-linear finitely extensible spring law is described and examined in simple flow situations where analytic results are derivable; namely oscillatory flow and steady state flow at high deformation rates. The model is shown to be consistent with the Bueche-Ferry hypothesis in fast large strain unidirectional flows but to violate this rule in small strain reversing flows. The discrepancy is identified with a preaveraging approximation used to describe the relative tube-chain velocity. Experimentally verifiable scaling rule for the birefringence as a universal function of a planar flow-type parameter and deformation rate are identified. Sensitivity to the extensional flow character, absent in the original tube model, manifests itself with the introduction of segmental stretch. Although the model generates a non-separable memory function kernel the deformation dependence of the memory function is quantitatively shown to have negligible impact on the predicted theological properties relative to the original Doi-Edwards model. With this simplification, relatively uncomplicated approximations to the segmental stretch model can be deduced.

Cyclic polymers swollen in a good solvent dynamic scattering properties

AbstractA general formalism for the dynamic scattering properties of cyclic polymers in good solvent conditions is presented. Single‐chain dynamics are investigated in the framework of the Oseen tensor model with and without the preaveraging approximation. The chain swelling due to excluded volume interactions is introduced by a renormalization of the internal chain statistics of cyclic polymers. The relaxations of the concentration fluctuations are also described for both ring homopolymers and copolymers. © 1994 John Wiley &amp; Sons, Inc.

Static and dynamic structure factors for star polymers in .theta. conditions

The static and dynamic structure factors for star polymers in θ solutions are derived for semiflexible models and the partially stretched (PS) arm model. The PS model for the static structure factor is found to be in fairly good agreement with SANS experiments with exclusion of the high-a region, where only a qualitative agreement is obtained. The nonpreaveraged first cumulant for the PS model reproduces fairly well the peculiar effects found by neutron spin-echo experiments even though those experiments were carried out in good solvents. The effects of preaveraging and screening of the hydrodynamic interaction are considered. Results for the full dynamic structure factor are presented only in the preaveraging approximation

Gaussian approximation and Brownian dynamics simulations for Rouse chains with hydrodynamic interaction undergoing simple shear flow

A bead-spring-chain model including hydrodynamic-interaction effects is used to calculate the rheological properties of dilute polymer solutions undergoing steady shear flow. The hydrodynamic fluctuations are explicitly taken into account within the context of the recently developed Gaussian approximation for the hydrodynamic interaction. In this Gaussian approach expressions for the non-Newtonian stress tensor at arbitrary shear rate are calculated. For short chains, the shear-rate–dependent predictions for the viscometric functions are presented. Also, the Brownian dynamics simulation technique is applied to determine exact rheological behavior of the model. Comparing the simulation results to the corresponding predictions of the Gaussian, consistent-averaging, and preaveraging approximations the validity and the superiority of the Gaussian method is clearly demonstrated.

Calculation of the intrinsic viscosity for dilute polymer solutions undergoing shear flow

Starting from a set of coupled Langevin equations describing the dynamics of flexible (Gaussian) polymer chains in the presence of hydrodynamic interactions and undergoing a weak shear flow, the stress tensor is determined using Kramers formula. With the aid of renormalization group techniques the steady state intrinsic viscosity is extracted toO(ɛ) (ɛ≡4−d,d being the spatial dimensionality) and to lowest nontrivial order in the flow strength. Within the preaveraging approximation we find a numerically small effect of shear thinning.

An analytical preaveraging approximation for the evaluation of matrix elements

A method is developed for the rapid numerical evaluation of matrix elements which may be averaged in order to approximately account for explicit uncertainties in the integrand caused by, for example, uncertainties in the interaction potential. The exact matrix element average is approximated by an analytically preaveraged integrand which selectively filters out high-frequency, long-range components of the original integrand. Results of this approximation are compared with the original matrix elements and their exact averages over these uncertainties. The analytical preaveraging approximation (APA) is applied first to a simple, analytically soluble integral and is then extended to treat inelastic matrix elements which utilize more complicated (and realistic) uniformized JWKB wavefunctions. APA matrix elements are computed for vibrationally inelastic transitions in Ar-N 2 scattering and are compared with exactly averaged results.

Langevin approach to polymers in flow.

A Gaussian polymer chain in simple shear flow is studied using Langevin equations. Incorporating hydrodynamic interactions to first order in \ensuremath{\epsilon}==4-d (d is the spatial dimensionality) with the aid of field-theoretic methods, we solve these nonlinearly coupled kinetic equations for polymer-solvent dynamics and analytically evaluate the second normal stress coefficient ${\ensuremath{\Psi}}_{2}$ for small shear rates. It is found that the mean-field (consistent preaveraging) approximation for hydrodynamic interactions (HI) produces an unphysical positive ${\ensuremath{\Psi}}_{2}$, while inclusion of fluctuations in HI leads to a negative value for ${\ensuremath{\Psi}}_{2}$, in agreement with experimental evidence.

Reptation-breathing theory of pulsed electrophoresis: dynamic regimes, antiresonance and symmetry breakdown effects.

We apply the concepts of tube and reptation to the pulsed electrophoresis of DNA, considering both biased reptation and "breathing" modes (internal modes of the chain). Using suitable preaveraging approximations, analytical expressions are derived which relate displacement in crossed field electrophoresis to molecular weight, field strength, field period, pore size of the gel, and the angle between the field. These expressions provide scaling laws for the change of mobility when one (or more) of the parameters is varied as well as "universal" velocity versus molecular weight versus pulse time curves. These results are quantitatively compared with experiments. At some point which depends on field angle, field strength and chain length, however, we predict a failure of this model due to symmetry breakdown and loss of ergodicity. Qualitatively, this should lead to considerable band spreading and/or splitting of the highest DNA bands into two bands migrating sideways from the diagonal. The case of field inversion is also investigated. It is shown that only breathing modes can explain the strong differences in mobility experienced by chains of different length when opposite fields of equal amplitude are applied: the "trapping" of chains in conformations of low mobility is associated with an antiresonance-like coupling between the external field and the internal modes.

A comparison between simulations and various approximations for Hookean dumbbells with hydrodynamic interaction

The Brownian dynamics simulation technique is used to calculate exact rheological properties of the Hookean-dumbbell model with hydrodynamic interaction. The simulation results for steady shear flow are compared to the corresponding predictions of the preaveraging approximation, the consistent-averaging approximation, and the Gaussian approximation developed in the preceding article. Our calculations of the viscometric functions of the Hookean-dumbbell model are performed for various hydrodynamic-interaction tensors.

On the dynamical viscosity of dilute polymer solutions in theta solvents

The path integral formulation of the Kirkwood diffusion equation is used to derive the perturbation expansion of the intrinsic viscosity in powers of the hydrodynamic interaction. Within the preaveraging approximation the perturbation expansion yields the Zimm formula for the intrinsic viscosity. The renormalization of the dynamical viscosity [η(ω)] has been explicitly carried out for both zero and large frequencies. At intermediate frequencies the renormalization of [η(ω)] has been performed in an approximate manner. A simple analytical expression for the dynamical viscosity is given. For low ω the real part of [η(ω)] has a plateau, while for large ω the Zimm behavior ([η(ω)]∼ω−1/3) is obtained.

First cumulant of the dynamic structure factor for rigid rings

The first cumulant of the dynamic structure factor is evaluated for rigid ring polymers. The results, together with those for flexible polymers, suggest that a careful experimental intercomparison of rings and open-chain polymers at low magnitudes of the scattering vector could serve to assess the degree of validity of the preaveraging approximation for the hydrodynamic interactions.

Renormalization group theory of the Rouse–Zimm model of polymer dynamics to second order in ε. II. Dynamic intrinsic viscosity of Gaussian chains

The dynamical intrinsic viscosity [η(ω)] of Gaussian chains is studied within the Rouse–Zimm bead-spring model of polymer dynamics using conformational space renormalization group methods. Calculations are performed through order ε2 (ε=4−d, where d is the spatial dimensionality) both with and without the preaveraging approximation. We find an 8% preaveraging error in the steady state limit. Preaveraging corrections to [η(ω)] involve couplings between all Rouse modes, leading to more complicated expressions for the Rouse mode relaxation rates and to weight factors for mode contributions to [η(ω)].

Intrinsic viscosity from the Green–Kubo formula

The Green–Kubo formalism and the conventional Kirkwood–Riseman formalism (with and without hydrodynamic preaveraging approximation) are compared in the calculation of the intrinsic viscosity of a Gaussian chain. It is discussed that the Green–Kubo-type formula for the intrinsic viscosity is correct to order ε(=4−d, d being the spatial dimension). Then it is shown that without self-avoiding interaction the two formalisms give identical results to order ε; i.e., the Kirkwood–Riseman formalism is as reliable as the Green–Kubo formalism without self-avoiding interactions. With self-avoiding interaction it is argued that the validity of the Kirkwood–Riseman formalism cannot be expected.

Renormalization group study of Rouse–Zimm model of polymer dynamics through second order in ε

Gell–Mann–Low style renormalization group methods are employed in conjunction with dimensional regularizations and ε expansions to study the bead-spring model of polymers with excluded volume in dilute solutions. Perturbation calculations are carried out through order ε2 for the translational diffusion coefficient D (ε=4−d, d being the spatial dimension) both with and without the use of the preaveraging approximation. Polymer excluded volume interactions are systematically incorporated into the Rouse–Zimm model by using the well-known two parameter model along with the ε expansion. The nondraining limit hydrodynamic interaction fixed point is found to be unaffected by preaveraging through order ε2 and is identical to that obtained previously from the order ε2 preaveraged Kirkwood–Riseman formalism. Our work indicates that the nonpreaveraged renormalization group perturbation expansions appear more well behaved (controllable) in the Rouse–Zimm model than in the Kirkwood–Riseman rigid body model.

Corrections to preaveraging approximation within the Kirkwood–Riseman model for flexible polymers: Calculations to second order in ε with both hydrodynamic and excluded volume interactions

The polymer hydrodynamic radius RH is calculated to second order in the ε=4−d (d is the dimension) perturbation theory using the Kirkwood–Riseman model in conjunction with the renormalization group. Excluded volume effects on RH are modeled by the two-parameter theory, and the hydrodynamic radius is evaluated both with and without the preaveraging approximation. These second order in ε calculations are then used to evaluate the leading corrections in ε to the preaveraging approximation for RH. Gaussian and self-avoiding chains in three dimensions yield nondraining limit preaveraging corrections of 26% and 16%, respectively, whereas simulations by Zimm and others using the same rigid body model of dynamics produce only about a 12% correction for Gaussian chains. It thus appears that the dynamical renormalization group theory only provides a qualitative prediction of the preaveraging error to this order in the ε-perturbation theory.

Transport coefficients of weakly bending rods. Effects of the preaveraging approximation

Influence sur le coefficient de diffusion moyen translationnel et sur la viscosite intrinseque d'apres un modele de tige en arc

Renormalization group theory of transport properties of polymer solutions. II. Rotatory friction coefficient vs intrinsic viscosity

The rotatory friction coefficient fr is calculated with the aid of the renormalization group theory. The present calculation shows that the ratio of fr to the intrinsic viscosity [η] depends on the excluded volume and the draining effects. The calculated ratio fr/[η] is about ∼5%–7% less than the conventional value which is obtained by using the preaveraging approximation of the Oseen tensor. The conventional value is independent of the excluded volume and draining effects.

Concentration dependent friction coefficients of polymer molecules in dilute solutions. II

Using a multiple scattering cluster expansion approach, an explicit calculation of the leading terms of the concentration dependent translational and rotational frictional coefficients is presented. Within the preaveraging approximation, a value of 0.75 [η] is obtained for the leading factor for the translational friction coefficient where [η] is the intrinsic viscosity. This is in excellent agreement with the recent experimental results of Mulderije. Similarly, a value of 0.43 [η] is obtained for the same term for the rotational friction coefficient.

Multiple scattering theory calculation of the concentration dependence of the tracer and cooperative friction coefficients for Gaussian polymer chains

The cooperative and tracer translational friction coefficients of Gaussian polymer chains are evaluated through order c in concentration using the full dynamic bead-to-bead space time correlation functions within the framework of a recent reformulation involving a discrete bead-type model of the multiple scattering theory for the concentration dependence of the viscoelastic properties of polymer solutions. It is shown that, using preaveraging approximation of the Oseen tensor, the order c cooperative translational friction coefficient differs from the tracer friction coefficient due to a higher order contribution in the multiple scattering expansion, one varying as the inverse seventh power of the separations between chains. The leading term for the cooperative translational friction coefficient of order c, equal to the tracer one, is calculated to be 0.62, about 10%–25% smaller than recent experimental results obtained in sedimentation experiments. The difference may be due, in part, to the higher terms in the multiple scattering expansion which turn out to be very difficult to compute even in a very rough approximation.