Tensor Oseen

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],... 

We can take the Rouse term l/ ke 02rm/0m2 (ke = 3kBT//2) entropic spring constant) into consideration formally, if we define the element Tnm of the Oseen tensor as Tnm = E/ . The equation of motion (13) thus becomes... 

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 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. 

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... 

When we use the Fourier representation of the Oseen tensor G, Ay is given by... 

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]. 

Equating this right-hand side to that of eqn. (208) and again inverting to use the Oseen tensor, T... 

The viscosity of the medium is t, and 1 is the unit tensor. (The Oseen tensor is the Green s function for the Navier-Stokes equation under the conditions that the fluid is incompressible, convective effects can be neglected, and inertial effects coming from the time derivative can be neglected.)... 

We note that Eq. (1.8) is still applicable to this case. Due to the asymmetry the formulas discussed in Section 4 are not valid for rod-like molecules. However, we observe that the Oseen tensor assumes the following form (41)... 

If we introduced the diagonalization approximation of the Oseen tensor we would have obtained... 

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. 

This conclusion is very natural cind important. Although we have considered in this section only rod-like molecules to reach this conclusion it seems that the diagonalization approximation of the Oseen tensor will cause similar errors even in the case of other types of polmer. 

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