Hydrodynamic tensor, Oseen

In the preaveraged approximation for the Oseen hydrodynamic tensor [19, 20], the linear Langevin equation may be written as... 

It should be emphasized that the name Oseen tensor , in chemical physics attributed commonly to the tensor expressed by eq 1.75 is somewhat misleading. In hydrodynamics, the Oseen approximation refers to a second-level approximation, see Eqs. 1.71 and subsequent remarks on the approximations. Oseen introduced his second-level approximation in 1910 to get a reliable description of the velocity field at distances greater than H/Rr, where R is the radius of the flow. However, since the Reynolds numbers for the systems investigated in solution chemistry generally are particularly low, it follows that the first-level description presented here is entirely satisfying. 

Since the hydrodynamic interaction decreases as the inverse distance between the beads (Eq. 27), it is expected that it should vary with the degree of polymer chain distortion. This is not considered in the Zimm model which assumes a constant hydrodynamic interaction given by the equilibrium averaging of the Oseen tensor (Eq. 34). 

If the Brownian particles were macroscopic in size, the solvent could be treated as a viscous continuum, and the particles would couple to the continuum solvent through appropriate boundary conditions. Then the two-particle friction may be calculated by solving the Navier-Stokes equations in the presence of the two fixed particles. The simplest approximation for hydrodynamic interactions is through the Oseen tensor [54],... 

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. 

A model that can take these findings into account is based on the idea that the screening of hydrodynamic interactions is incomplete and that a residual part is still active on distances r > H(c) [40,117]. As a consequence the solvent viscosity r s in the Oseen tensor is replaced by an effective... 

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 physical nature of this phenomenon is related to the presence of hydrodynamic interactions described by the Oseen tensor [22, 25]. The role of the finely porous medium in classical electroosmosis is played in this case by the gel which can be roughly considered as a collection of pores of size where is the mesh size of the gel [22]. 

Our theory may be imderstood better if compared with the KR theory. Their theory has been developed along the observations discussed in Section 1. We note that Ff of Eq. (1.4) which depends on all the segments is replaced in their theory by a one body force determined by the ordering number of a segment irrespective of its location. For this reason it was necessary to replace the Oseen hydrodynamical interaction tensor by its average. 

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... 

The Yamakawa-Fujii theory [2, 3] was developed by using the Kirkwood-Riseman formalism with the effect of chain thickness approximately taken into account. The following remarks may be in order. The Oseen interaction tensor was preaveraged. Force points were distributed along the centroid of the wormlike cylinder (not over the entire domain occupied by the cylinder). The no-slip hydrodynamic condition was approximated by equating the mean solvent velocity over each cross-section of the cylinder to the velocity of the cylinder at that cross-section (Burgers approximate boundary condition). 

Turrently, hydrodynamic interactions between suspended particles cannot be included in a DDFT. However, it is well known that, e.g., the rheology of suspensions cannot be explained without taking these into account. Hydrodynamic interactions in a simple approximation based on Oseen tensors have been included in the Fokker-Planck equation (Eq. 3), and the equivalent of Eq. 4 has been derived and discussed [15, 16]. However, this equation contains three-point and two-point correlations in a form such that the sum rule in Eq. 5 cannot be used. 

The hydrodynamic coupling tensor Xik given by the Oseen or the Navier-Stokes equations for Newton s law... 

Hydrodynamic Properties of Aggregates where J is the Oseen tensor given by... 

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... 

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). 

Now Yoshizaki and Yamakawa°° have extended the calculation to third-order terms, but with the Oseen tensor pre-averaged. In this way a precise lower bound for Og was obtained, close to that obtained by Auer and Gardner using Kirkwood-Riseman theory. The paper by Bixon and Zwanzig performs an infinite order calculation based upon the PF treatment. Using a numerical method, they obtain g 2.76 x 10 , between the Zimm and PF values. For flexible polymers (as for rigid rods) the pre-averaging of the hydrodynamic interaction tensor thus introduces only a small error the effect on the spectrum of relaxation times is more dramatic cf. columns 3 and 4 of Table 2), and the relaxation time of the slowest mode (proportional to 1/A/) is more than twice as slow. This difference should be detectable experimentally. 

The Zimm theory includes an alternative treatment in which frictional resistance to motion of the beads in the bead-spring chain is dominated by the viscous drag from other beads in the same chain (dominant hydrodynamic interaction. Fig. 9-5-11). The interaction is treated approximately as in the theory of Kirkwood and Riseman for the intrinsic viscosity of dilute polymer solutions, by use of the equilibrium-averaged Oseen tensor for the influence of the motion of one bead on another the average distances between pairs of beads are supposed to correspond to those in a 0-solvent. 

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