Translational friction tensor

The tensor R,-, termed the translation tensor, for a rigid body depends solely on the size and shape of the body. The translation tensor has the dimensions of length and may be interpreted as an equivalent radius. In the polymer literature the force is usually expressed in terms of a translational friction tensor T. The components are called translational friction coefficients... 

Here, U, is the terminal velocity and f is the translational friction tensor that is, the force depends on the orientation for a particle of arbitrary shape. 

The error concerned an explicit formula for the translational diffusion coefficient. Kirkwood calculated the diffusion tensor as the projection onto chain space of the inverse of the complete friction tensor he should have projected the friction tensor first, and then taken the inverse. This was pointed out by Y. Ikeda, Kobayashi Rigaku Kenkyushu Hokoku, 6, 44 (1956) and also by J. J. Erpenbeck and J. G. Kirkwood, J. Chem. Phys., 38, 1023 (1963). An example of the effects of the error was given by R. Zwanzig, J. Chem. Phys., 45, 1858 (1966). In the present article this question does not come up because we use the complete configuration space. 

Zimin, each frictional element is assumed to be a point and the hydro-dynamic interactions between these elements and the solvent are described by the Oseen tensor (23,35). This method is derived from solution of the Navier-Stokes equation assuming the existence of point resistances (34). Although frictional elements of finite size were used in the calculation of translational friction coefficients by Edwards and Oliver (35,36), they have not been applied to the intrinsic viscosity or to dynamic mechanical properties to date. 

The tensor Rij is called the translational tensor, or the resistance tensor. Its components depend on particle s size and shape and have the dimensionality of length. They can be interpreted as equivalent radii of the body. The tensor is called the friction tensor, and the values Vy are known as mobilities. They are similar to mobilities introduced in Section 4.5. Therefore the tensor with components Vij is called the mobility tensor. 

Substituting the preaveraged result of the Oseen tensor and performing the normal mode analysis of the Zimm equation for the various Rouse modes, we can calculate the mean square displacement of the center-of-mass of the chain, mean square displacement of a labeled monomer, translational friction coefficient of the chain, and the relaxation times of the various Rouse modes with the Zimm dynamics (Doi and Edwards 1986). The main results of these calculations are the following. 

Since the expressions (8.5) and (8.15) are identical, they can be combined, by introducing so-called global tensors of friction f and mobility V, including translational and rotational components. In the Stokes flow, these tensors have some universal properties [2], of which the most important are dependence on instant configuration and independence of velocity, as well as symmetry and positive definiteness of matrixes fj and V. ... 

The friction coefficient f has been determined either from the hydrodynamic diffusion tensor using Monte Carlo simulation [67] or by using one of several simple analytical expressions for a sphere translating through a right cylindrical pore [31, 68-70]. The most commonly used is Eq. (11) [71], where X is the characteristic ratio of solute size to pore size... 

In the case under investigations, which includes nematic (anisotropic) phase environments, we shall assume the usual approximation of considering isotropic local friction, and the macroscopic local viscosity is taken equal to half of the fourth Leslie-Ericksen coefficient 1/4 [92-95]. The diffusion tensor of the system is obtained, neglecting translational contributions, as a 4 x 4 matrix, that is. 

---