Hydrodynamic interaction approximation

First approaches to approximating the relaxation time on the basis of molecular parameters can be traced back to Rouse [33]. The model is based on a number of boundary assumptions (1) the solution is ideally dilute, i.e. intermolecular interactions are negligible (2) hydrodynamic interactions due to disturbance of the medium velocity by segments of the same chain are negligible and (3) the connector tension F(r) obeys an ideal Hookean force law. 

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],... 

All measurements, of course, have to be made at a finite concentration. This implies that interparticle interactions cannot be fully neglected. However, in very dilute solutions we can safely assume that more than two particles have only an extremely small chance to meet [72]. Thus only the interaction between two particles has to be considered. There are two types of interaction between particles in solution. One results from thermodynamic interactions (repulsion or attraction), and the other is caused by the distortion of the laminar fiow due to the presence of the macromolecules. If the particles are isolated only the laminar flow field is perturbed, and this determines the intrinsic viscosity but when the particles come closer together the distorted flow fields start to overlap and cause a further increase of the viscosity. The latter is called the hydrodynamic interaction and was calculated by Oseen to various approximations [3,73]. Figure 7 elucidates the effect. 

Thus a simple power law behavior with an exponent of 1.5 would result if 0"= 1 [125]. Zimm and Kilb [128] made a first attempt to calculate g for star branched macromolecules on the basis of the Kirkwood-Riseman approximation for the hydrodynamic interaction. They came to the conclusion that... 

Since l is proportional to and q is proportional to 1/L, i is proportional to. Substitution of Eq. (67) into Eq.(62) gives the Langevin equation for the Rouse modes of the chain within the approximations of preaveraging for hydrodynamic interactions and mode-mode decoupling for intersegment potential interactions. Equation (62) yields the following results for relaxation times and various dynamical correlation functions. 

In Sect. 6.3, we have neglected the intermolecular hydrodynamic interaction in formulating the diffusion coefficients of stiff-chain polymers. Here we use the same approximation by neglecting the concentration dependence of qoV), and apply Eq. (73) even at finite concentrations. Then, the total zero-shear viscosity t 0 is represented by [19]... 

In 1959, Zimm and Kilb (34) made some calculations of the intrinsic viscosities of certain branched polymer molecules, taking into account the hydrodynamic interaction between portions of the polymer chain, using a modification of the Rouse procedure. They carried out these difficult calculations for a quite restricted range of models, obtaining numerical results for equalarmed stars with 3, 4, and 8 branches, and for one modified star with 2 long branches and 8 shorter branches. They found that their numerical results for this set of structures could be approximately represented by ... 

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

After approximately one minute the shock initiated by iron core collapse arrives at the outer edge of the helium core whose radius is typically 5 X 1010 cm. The hydrodynamic interaction with the envelope slows the helium core down, the deceleration propagating into the core as a reverse shock. Meanwhile the outgoing shock continues though the hydrogen. The time when the shock breaks through the surface of the envelope can be estimated (Shigeyama et a1. 1987 Woosley 1987),... 

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... 

Fig. 3. The dynamic scattering function for a pair of elements (j, k) after averaging over all orientations in the pre-average approximation of the hydrodynamic interaction, and without this approximation...

We will consider the cold-gas-convex surface of the flame front as a curved cell of the flame which had been formed after the plane flame lost its stability. The steady state of the convex flame is a result of the nonlinear hydrodynamic interaction with the gas flow field (see Zeldovich, 1966, 1979). In the linear approximation the flame perturbation amplitude grows in time in accordance with Landau theory, but this growth is restricted by nonlinear effects. 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

The form of the distribution function will depend on the approximations that have been incorporated into the model. In its simplest form, where finite extensibility, hydrodynamic interaction and excluded volume have been neglected, the following Gaussian function describes the distribution of conformations,... 

One may note that, in linear approximation with respect to the velocity of a particle (see, for example, equations (2.4) and (2.9)), the expression for forces are determined by small velocities of the particles and of the flow. The force, acting on a particle in the flow, does not depend on the specific choice of hydrodynamic interaction and can be written in the following general form... 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

Results (F.36) are valid with approximation to the term of second order with respect to the parameter of hydrodynamic interaction... 

Fig. 16.9 shows the low frequency slopes of 2 and 1, respectively, as expected for viscoelastic liquids and the high frequency slopes Vi and 2/3 for Rouse s and Zimm s models, respectively. Experimentally it appears that in general Zimm s model is in agreement with very dilute polymer solutions, and Rouse s model at moderately concentrated polymer solutions to polymer melts. An example is presented in Fig. 16.10. The solution of the high molecular weight polystyrene (III) behaves Rouse-like (free-draining), whereas the low molecular weight polystyrene with approximately the same concentration behaves Zimm-like (non-draining). The higher concentrated solution of this polymer illustrates a transition from Zimm-like to Rouse-like behaviour (non-draining nor free-draining, hence with intermediate hydrodynamic interaction).

Estimates of ut based on pairwise additivity by Glendinning and Russel (1982) may also be cited. These authors find, however, that the approximate, pairwise-additive treatment of hydrodynamic interactions fails to be adequate for all but very dilute suspensions. Their theoretical formalism can nonetheless be systematically improved by including three- (or more) particle interaction effects. 

Monte Carlo techniques were first applied to colloidal dispersions by van Megen and Snook (1975). Included in their analysis was Brownian motion as well as van der Waals and double-layer forces, although hydrodynamic interactions were not incorporated in this first study. Order-disorder transitions, arising from the existence of these forces, were calculated. Approximate methods, such as first-order perturbation theory for the disordered state and the so-called cell model for the ordered state, were used to calculate the latter transition, exhibiting relatively good agreement with the exact Monte Carlo computations. Other quantities of interest, such as the radial distribution function and the excess pressure, were also calculated. This type of approach appears attractive for future studies of suspension properties. 

Analysis of the hydrodynamic interactions of many particles in a laminar flow, carried out by Saito [56] showed that in view of the complexity of the physical picture of interactions in many body systems introduction of Stokes approximations in a theoretical consideration of the flow of dispersions can lead to incorrect results. For the laminar flow Saito proposed a formula containing a power series ... 

The basis for comparing the ratios of the free diffusion coefficients and permeability coefficients was the assumption that hindrance considerations could be ignored. In the instance that this assumption is valid (i.e., the case of large pore dimensions relative to solute radii), the free diffusion coefficients are a reasonable approximation to the diffusion coefficients of the solutes in the membrane. In the instance that hindrance considerations are not negligible, due to pore dimensions that lead to diffusion-restricting hydrodynamic interactions between the solute and the membrane, the diffusion coefficient of the solute in the membrane is a function of both the solute parameters and the properties of the membrane. In this case, the effective diffusion coefficient can be approximated by the product of the free diffusion coefficient and a diffusional hindrance factor, HQC) (Deen, 1987) ... 

For dense suspensions of spherieal particles, an espeeially aecurate method ealled Stokesian dynamics has been developed by Bossis and Brady (1989). In Stokesian dynamics, one solves a generalized form of Eq. (1-40), in which the simple Stokes law for the drag force on sphere i, = — (x/ — v ), is replaced by a more accurate tensor expression that accounts for the hydrodynamic interactions—that is, the disturbances to the solvent velocity field produced by the relative motions of the other spheres. The Stokesian dynamics method accounts for hydrodynamic interactions among widely separated spheres by a multipole expansion, as well as for closely spaced ones by a lubrication approximation. Results from this method appear in Figs. 6-8 and 8-8. 

The intrinsic viscosity can be related to the overlap concentration, c, by assuming that each coil in the dilute solution contributes to the zero-shear viscosity as would a hard sphere of radius equal to the radius of gyration of the coil. This rough approximation is reasonable as a scaling law because of the effects of hydrodynamic interactions which suppress the flow of the solvent through the coil, as we shall see in Section 3.6.1.2. The Einstein formula for the contribution of suspended spheres to the viscosity is... 

In a real situation, the motion of the segments of a chain relative to the molecules of the solvent environment will exert a force in the liquid, and as a consequence the velocity distribution of the liquid medium in the vicinity of the moving segments will be altered. This effect, in turn, will affect the motion of the segments of the chain. To simplify the problem, the so-called free-draining approximation is often used. This approximation assumes that hydrodynamic interactions are negligible so that the velocity of the liquid medium is unaffected by the moving polymer molecules. This assumption was used in the model developed by Rouse (5) to describe the dynamics of polymers in dilute solutions. 

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