Average Oseen tensor

In the case of dynamic mechanical relaxation the Zimm model leads to a specific frequency ( ) dependence of the storage [G ( )] and loss [G"(cd)] part of the intrinsic shear modulus [G ( )] [1]. The smallest relaxation rate l/xz [see Eq. (80)], which determines the position of the log G (oi) and log G"(o>) curves on the logarithmic -scale relates to 2Z(Q), if R3/xz is compared with Q(Q)/Q3. The experimental results from dilute PDMS and PS solutions under -conditions [113,114] fit perfectly to the theoretically predicted line shape of the components of the modulus. In addition l/xz is in complete agreement with the theoretical prediction based on the pre-averaged Oseen tensor. 

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in 3 that the only difference between the model with and without the equilibrium-averaged Oseen tensor is that l/ has been replaced by (1/0 (1 — f h).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). 

The calculation of the average of the product of the Oseen tensor and the force exerted by the y th subchain is exceedingly difficult. The standard approximation is to carry out the average over each quantity separately and multiply the results. The average Oseen tensor is ... 

The average Oseen tensor is a scalar function of the intersubchain distance Rij. The average over the intersubchain tensor function gives a numerical constant of 4/3, due to the spherically symmetric nature of j. The force exerted by a subchain can then be expressed as ... 

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. 

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). 

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. 

In comparison, we note that our results involve a function F(2Ao). We can conclude that the diagonalization approximation of the Oseen tensor amounts to introducing an average interaction parameter. Our rigorous results show that the approximation gives results with a smaller characteristic parameter. 

Calculation of Complex Modulus. In the Zimm theory, the Oseen tensor is approximated by its average value over the equilibrium configuration ... 

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. 

Usually, an Oseen tensor is used for H . Its magnitude is reciprocally proportional to r - r and is therefore a fnnction of the chain conformation that changes with time according to Eq. 3.161. Zimm decoupled H from the rest of the equation and replaced it with its average at equilibrium (preaveraging approximation) ... 

The first term Sij/C is the local Rouse term the Oseen tensor describes the hydrodynamic interactions. A frequently used approximation, introduced first by Kirkwood and Riseman, averages the Oseen tensor over the configurations of the polymer chain in its equilibrium Gaussian state. This replaces the Oseen tensor by the preaveraged Oseen tensor... 

The inclusion of SI in equation (76) complicates the theory considerably. In most of the polymer literature it has been commonplace to replace the Oseen tensor by its equilibrium averaged value eq = jShj/ dQ = Sid, which is isotropic. Then all results for the Hookean dumbbell have to be modified only by replacing everywhere by and = C/4H by = 1/[4H(C — O)]. 

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. 

Another procedure is to assume that the Oseen-Burgers tensor is the equilibrium-averaged value, multiplied by a truncated Taylor series in the polymer contribution to the stress tensor such an assumption leads, for Hookean dumbbells, directly to the Giesekus constitutive equation ... 

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