Neglect of Hydrodynamic Interactions

Neglect of Hydrodynamic Interactions.—The coupling of hydrodynamic flow exerts a major influence on the dynamics of colloidal dispersions.In certain special cases, however, it has proved reasonable or expedient to neglect the hydrodynamic interactions. One such instance is the very dilute, electrostatically-stabilized dispersion in which particles interact via a screened Coulomb potential, that is, equation (2) with ku 1. 

The two parts of the van Hove correlation function have been extracted from the Brownian dynamics simulation of Gaylor et al. With increasing volume fraction, the van Hove self-correlation function deviates strongly from the free diffusion behaviour  

Evans and Watts have simulated an assembly of particles with repulsive pair potentials under the influence of homogeneous shear using a nonequilibrium molecular dynamics method.The shear viscosity is given by 

The derivation of equation (74) involves an amalgam of rheology and statistical mechanics. Simply stated, it is based on an integral equation of the same general form as equation (13), but involving a distribution function g(r), which now depends on the strain-rate tensor, and with the stress tensor replacing 

The viscosity of disordered, monodisperse polystyrene latices increases as the order-disorder transition is approached.This behaviour has been successfully interpreted in terms of the free-volume Cohen-Turnbull theory of molecular transport in a liquid. 

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The neglect of hydrodynamic interactions between rods in Eq. (4.35) was originally a matter of some concern. However, Allison subsequently demonstrated that their neglect introduces no significant error into the predicted correlation functions at times longer than 0.2 ns.(105, 106)... 

In the Rouse model, the relaxation time Ti of the first mode is proportional to (Eq. 3.136). The experimentally observed exponent for a polymer chain in a theta solvent is 3/2. The discrepancy also exists in the molecular weight dependence of Dq. Experimentally, we observe Dq (Section 3.2.7) in the theta solvent. In the Rouse model (Eq. 3.160), Dq The model fails to give the correct exponent. The shortcoming of the model is ascribed to the neglect of hydrodynamic interactions. In the following subsection, we take into account the hydrodynamic interactions. In Section 4.3, we will see an example in which the Rouse model can describe the motion of polymer chains correctly. 

In this section, we consider flow-induced aggregation without diffusion, i.e., when the Peclet number, Pe = VLID, where V and L are the characteristic velocity and length and D is the Brownian diffusion coefficient, is much greater than unity. For simplicity, we neglect the hydrodynamic interactions of the clusters and highlight the effects of advection on the evolution of the cluster size distribution and the formation of fractal structures. 

The system of entangled macromolecules becomes anisotropic when velocity gradients are applied, and one can assume that each Brownian particle of the chain moves in the anisotropic medium. The expressions for the discussed quantities (7.5) for case, when one can neglect the hydrodynamic interaction... 

We apply an alternative squared electric field in order to prevent the particles from reaching the electrodes, and the frequency is low enough to allow for a good tracking. The equation of motion along the vertical axis, neglecting the hydrodynamic interactions with the walls, can then be written as... 

To see what kind of result one expects from these equations, one may neglect the hydrodynamical interactions. Eq. (6.19) can be solved after diagonalyzing the matrix A and the complex viscosity is obtained in terms of the eigenvalues ZnofA ... 

In order to answer this question one has to find out what modifications are necessary in (a) the diffusion equation for the distribution function, and (b) the expression for the stress tensor. Kirkwood and coworkers (39,40,67) and Kotaka (42)w studied this problem for multibead dumbbells including complete hydrodynamic interaction. If one neglects the hydrodynamic interaction entirely, then from the articles cited above one concludes that all the results for rigid dumbbells can be taken over for the multibead dumbbells by replacing X — (,I / 2kT by XN — XN(N + l)/6(iV — 1) everywhere. For the case of complete hydro-dynamic interaction no such simple replacement is possible. 

Hess [13] neglected the hydrodynamic interactions among chain beads and treated the global motions of different chains as uncorrelated (this is to assume a small number of chain-chain contacts and thus to focus on the semi-dilute regime). He deduced that polymer self-diffusion consists of both lateral and longitudinal modes of chain motion until the entanglement parameter t/>(c, N) reaches unity, but it is dominated by the latter (i.e., chains move reptatively)... 

A crude estimation of W is done easily again by neglecting the hydrodynamic interaction in the shish-kebab model. Under the velocity gradient K, the rod rotates with the angular velocity eiu = X (ir ). Hence the velocity of the n-th bead relative to the fluid is... 

This discrepancy comes from neglecting the hydrodynamic interactions between the solvent molecules and the polymer chains. Because of this discrepancy, the Rouse model is not suitable for describing the dynamics of polymer chains in dilute solution, but it is still very useful for describing the dynamics of undiluted polymer with M < M,. 

Let us assume that sizes of drops are essentially various / , << R. Then, it is possible to consider that the small corpuscle moves simply in a hydrodynamic field big, and at definition of force of resistance of medium to its motion by inhomogeneity of this field it is possible with sufficient accuracy to neglect. If the distance between surfaces of drops several times is more R, it is possible to neglect also forces of hydrodynamic interacting of moving sphere with a motionless flat wall. These assumptions allow to present the equation of motion of a small corpuscle in an aspect ... 

In equation (51) the idea of hydrodynamic interaction was introduced, but this effect was neglected in subsequent equations. To include this effect it is necessary to insert an expression for, say, v l. A commonly used expression is vi = in which 12 is the Oseen-Burgers tensor,... 

These models are designed to reproduce the random movement of flexible polymer chains in a solvent or melt in a more or less realistic way. Simulational results which reproduce in simple cases the so-called Rouse [49] or Zimm [50] dynamics, depending on whether hydrodynamic interactions in the system are neglected or not, appear appropriate for studying diffusion, relaxation, and transport properties in general. In all dynamic models the monomers perform small displacements per unit time while the connectivity of the chains is preserved during the simulation. 

While mathematically attractive, this force law is of limited interest physically it represents only the interaction between permanent quadrupoles, and even this with neglect of angles of orientation. However, although the details of the dependence of viscosity upon temperature are affected by the force law used, the general form of the hydrodynamic equation in the Navier-Stokes approximation is not affected. 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

The dynamical properties of polymer molecules in solution have been investigated using MPC dynamics [75-77]. Polymer transport properties are strongly influenced by hydrodynamic interactions. These effects manifest themselves in both the center-of-mass diffusion coefficients and the dynamic structure factors of polymer molecules in solution. For example, if hydrodynamic interactions are neglected, the diffusion coefficient scales with the number of monomers as D Dq /Nb, where Do is the diffusion coefficient of a polymer bead and N), is the number of beads in the polymer. If hydrodynamic interactions are included, the diffusion coefficient adopts a Stokes-Einstein formD kltT/cnr NlJ2, where c is a factor that depends on the polymer chain model. This scaling has been confirmed in MPC simulations of the polymer dynamics [75]. 

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. 

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