Oseen diffusion

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

Such a decomposition of the diffusion coefficient has previously been noted by Pattle et al.(l ) Now we must evaluate >. The time-integrated velocity correlation function Aj j is due to the hydrodynamic interaction and can be described by the Oseen tensor. The Oseen tensor is related to the velocity perturbation caused by the hydrodynamic force, F. By checking units, we see that A is the Oseen tensor times the energy term, k T, or... 

The components of represent stochastic displacements and are obtained using the multivariate Gaussian random number generator GGNSM from the IMSL subroutine library (30). p ° is the initial hydrodynamic interaction tensor between subunits iJand j. Although the exact form of D. is generally unknown, it is approximated here using the Oseen tensor with slip boundary conditions. This representation has been shown to provide a reasonable and simple point force description of the relative diffusion of finite spheres at small separations (31). In this case, one has... 

It is evident from a comparison of these equations with their counterparts in 3 that the only difference between the model with and without the equilibrium-averaged Oseen tensor is that l/ has been replaced by (1/0 (1 — f h). Therefore the diffusion equation and results derived from it (i.e., Eqs. (3.15) and (3.16)) are altered only by replacing l/( everywhere by (1/0(1 — f h). 

In conclusion we would like to mention the article by Altenberger and Dutch (1973) which discusses the contribution of the hydrodynamic interaction (Oseen interaction) between solute particles to the concentration dependence of the diffusion coefficient. This treatment yields Kb = 1 for small spheres if the m is introduced into the treatment. 

In the Zimm model (see Fig. 2A) the hydrodynamic interactions are included by employing the Oseen tensor Him the tensor describes how the mth bead affects the motion of the /th bead. This leads to equations of motion that are not Unear anymore and that require numerical methods for their solution. In order to simplify the picture, the Oseen tensor is often used in its preaveraged form, in which one replaces the operator by its equiUb-rium average value [5]. For chains in -solvents, this leads for the normal modes to equations similar to the Rouse ones, the only difference residing in the values of the relaxation times. An important change in behavior concerns the maximum relaxation time Tchain> which in the Zimm model depends on N as and implies a speed-up in relaxation compared to the Rouse model. Accordingly, the zero shear viscosity decreases in the Zimm model and scales as Also, in the Zimm model the diffusion coefficient... 

An intuitive approch to the problem is to assume an approximate form for the diffusion tensor Dj, and then see what happens in the computer simulation. Ermak and McCammon suggest the Oseen tensor ... 

In the Zimm theory, the flow perturbations and the co-operative hydrodynamic interactions between segments are treated using the Oseen tensor, pre-averaged for simplification. Pyun and Fixman (PF) avoided this approximation by a perturbation solution of the Kirkwood diffusion equation up to second order. One of the consequences was that [equation (3)] was re-evaluated (see Table 1). 

Extensive reviews of turbulent diffusion were provided by Levich andHinze. Tchen ° was the first investigator who modified the Basset-Boussinesq-Oseen (BBO) equation and applied it to study motions of small particles in a turbulent flow. Corrsin and Lumley pointed out some inconsistencies of Tchen s modifications. Csanady showed that the inertia effect on particle dispersion in the atmosphere is negligible, but the crossing trajectory effect is appreciable. Ahmadi and Ahmadi and Goldschmidt smdied the effect of the Basset term on the particle diffusivity. Maxey and Riley obtained a corrected version of the BBO equation, which includes the Faxen correction for unsteady spatially varying Stokes flows. 

A quite different model for nematic dif-fusivities, based on Oseen s hydrodynamic theory of isotropic liquids, was elaborated by Franklin [14] it describes the diffusivity components in terms of the five Leslie viscosities a, to 05, a scalar friction constant... 

Tests of the validity of the Kirkwood-Riseman picture, inquiring directly if diffusing objects actually have cross-diffusion tensors that match their supposed hydrodynamic interactions, have recently been accomplished Crocker used videomicroscopy and optical tweezers to study the correlated Brownian motions of a pair of 0.9 xm polystyrene spheres, thereby determining their cross-diffusion ten-sors(3). Crocker found that the diffusion tensors are accurately described by the hydrodynamic interaction tensors, exactly as Kirkwood and Riseman had assumed. An optical trap experiment by Meiners and Quake observed the motions of two Brownian particles, further confirming the validity of the Oseen approximation for hydrodynamic interactions(4). 

Hydrodynamic interactions on a long length scale can be measured with two-point rheology, in which fluorescent or other beads are mixed with a polymer solution, and videomicroscopy is used to measure the Brownian displacements AR, of pairs of beads. The cross-correlations AR, ARy) determine the cross-diffusion tensors as a function of the separation between beads. For beads a fraction of a micron in size in polymer solutions and interbead distances out to 100 xm, measurements of Crocker, et al.(25), Gardel, et a/.(26), and Chen, et al.(21) agree the cross-diffusion tensor falls off with distance as /R, and has at least approximately the magnitude expected for the Oseen interaction in these viscous polymer... 

Fundamental theories of diffusion for low-molecular weight liquid crystals in the nematic phase have been studied by Franklin [103-105] based on Oseen-Kirkwood hydrodynamic theory for isotropic liquids. Further theories of diffusion for low-molecular weight liquid crystals have been developed. These theories explained partially the experimental data on Dy and D. Chu and Moroi [106], and Leadbetter [107] have obtained that the anisotropy ratio of the diffusion coefficients, Dy/Dj, for low-molecular weight liquid crystals are expressed by [2y(l - S) + 2S -b 1]/[7(S + 2) -b 1 — S], where y = Ttd/Al in which I is the length and d of the diameter of the... 

This minor modification does not allow for the description of the shear-rate dependence of rj and and 4 2 is still equal to zero. It may be remarked in passing that the translational diffusivity for Hookean dumbbells with an Oseen-Burgers equilibrium-averaged hydrodynamic interaction is Ar = (/cr/20(i + CO). 

In the case that hydrodynamic interactions arising from the backflow by movements of segments (or particles) are dominant over friction of each segment with a viscous fluid medium, the diffusion coefficient is often evaluated theoretically as follows. According to the mode-mode coupling theory for fluids with the Oseen tensor of hydrodynamic interactions,the diffusion coefficient is expressed in terms of the static correlation function S([Pg.307]

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