Hydrodynamic interaction tensor

We note that the values of the hydrodynamic interaction tensor (2.6) averaged beforehand with the aid of some kind of distribution function, are frequently used to estimate the influence of the hydrodynamic interaction, as was suggested by Kirkwood and Riseman (1948).4 For example, after averaging with respect to the equilibrium distribution function for the ideal coil and taking the relation (1.23) into account, the hydrodynamic interaction tensor (2.6) assumes the following form... 

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... 

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. 

HO. is known as the Oseen hydrodynamic interaction tensor . In a common... 

For the more microscopic approach of an MD simulation of a chain in solvent particles, it is useful to also look at the theory from a more microscopic point of view, in particular in order to assess its limitations. The derivation of equations of motion of the Smoluchowski type and the discussion of the involved errors is a standard problem in modem transport theory. In the present case, the form of the hydrodynamic interaction tensor has to be derived from the microscopies, However, analytical... 

The superscript identifies the conformation at the beginnin of the time-step. For small timesteps At this should be reasonable to do. Fj in the above equation is the force exerted on particle j. The so-called spurious drift , i.e., the third term in the r.h.s. of eq. (3.20) usually vanishes, since most diffusion tensors which have been used in the literature have zero divergence (this is directly related to the assumption of incompressible flow). p] At) is the random displacement by the coupling to the heat bath. The crucial difficulty comes from the connection of the displacement by the heat bath and the hydrodynamic interaction tensor Dy via the fluctuation dissipation theorem. This fixes the first two moments to be... 

A rather efficient method to calculate the root of the hydrodynamic interaction tensor is Cholesky decomposition. The random displacements are then obtained via multiplying the root matrix with a vector of random numbers. The root is usually not unique, i.e., there are several matrices whose square is the diffusion tensor, but since any of these matrices yields random displacements which satisfy the condition eq. (3.22), this nonuniqueness averages out in the course of the simulation. These matrix operations become numerically rather intensive if the number of monomers becomes large (the number of operations is proportional to the third power of the number of monomers). The numerical algorithms for Langevin equations are well established, however, some details are still discussed today. ... 

Here is the drag coefficient of a single colloid particle in pure solvent, I is the identity tensor, and further terms in each sum involve the relative positions of several particles. The above considerations apply equally to colloidal spheres and to segments of polymer chains. If the diffusing objects are not spheres the ijlij and the F, all depend implicitly on the orientations of the objects. For interacting spherical colloids, the case under consideration in this chapter, the extended b, and Tij are the hydrodynamic interaction tensors obtained by Kynch(32), whose lead terms are... 

What physical forces affect colloid dynamics Three forces acting on neutral colloids are readily identified, namely random thermal forces, hydrodynamic interactions, and direct interactions. The random thermal forces are created by fluctuations in the surrounding medium they cause polymers and colloids to perform Brownian motion. As shown by fluctuation-dissipation theorems, the random forces on different colloid particles are not independent they have cross-correlations. The cross-correlations are described by the hydrodynamic interaction tensors, which determine how the Brownian displacements of nearby colloidal particles are correlated. The hydrodynamic drag experienced by a moving particle, as modified by hydrodynamic interactions with other nearby particles, is also described by a hydrodynamic interaction tensor. 

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). 

Several paths exist for improving the original renormalization group calculation. Merriam and Phillies(7) have since extended the author s original calculation(6) to determine the five-point chain-chain-chain-chain-chain hydrodynamic interaction tensor. The deviation of the observed stretched-exponential behavior from simple calculations yielding pure-exponential behavior was predicted to arise from the concentration dependence of the chain radius. Dielectric relaxation measures both a relaxation time and a chain radius. Analysis demonstrated that chain contraction accounts quantitatively for the form of the stretched-exponential concentration dependence of the dielectric relaxation time(8). 

HOs is known as the Oseen hydrodynamic interaction tensor. In a common situation there exists a certain flow field, vo r) and a suspended particle moves at first with the liquid. Application of an external force, f, on the par-... 

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],... 

These results show that hydrodynamic interactions and the spatial dependence of the friction tensor can be investigated in regimes where continuum descriptions are questionable. One of the main advantages of MPC dynamics studies of hydrodynamic interactions is that the spatial dependence of the friction tensor need not be specified a priori as in Langevin dynamics. Instead, these interactions automatically enter the dynamics from the mesoscopic particle-based description of the bath molecules. 

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... 

Hydrodynamic and frictional effects may be described by a Cartesian mobility tensor which is generally a function of all of the system coordinates. In models of systems of beads (i.e., localized centers of hydrodynamic resistance) with hydrodynamic interactions, is normally taken to be of the form... 

In models of beads with full hydrodynamic interactions, for which the mobility tensor is represented by a dense matrix, the Cholesky decomposition of H requires 3N) /6 operations. Eor large N, this appears to be the most expensive operation in the entire algorithm. The only other unavoidable 0 N ) operation is the LU decomposition of the K x K matrix W that is required to solve for the K constraint forces, which requires /3 operations, or roughly... 

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]. 

Suppose for example that there is no unperturbed velocity field, V°(R) =0. Then the frictional force on the, /th bead is determined by not only its own velocity ft , but also by the velocities ft, of all the other beads. The only exception can be when the friction tensor , is strictly diagonal and this may be expected to be true only in the complete absence of hydrodynamic interaction. 

Djk is the division tensor which describes the hydrodynamic interaction between the various segments, arising from the incompressibility of the solvent and the back flow of solvent molecules when a segment fluctuates around its equilibrium position. In the first approximation by Oseen76 this tensor is given for a homopolymer as... 

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