Hydrodynamic interactions neglect

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. 

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. 

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 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]. 

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)... 

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. 

Under the simplifying assumption that hydrodynamic interactions may be neglected, the only new parameter that controls the dynamics is a monomeric friction coefficient (Rouse model). Then the prediction for the rate Pj is given by ... 

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]... 

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,... 

The dependence of the effective friction coefficient on the length of the macromolecule is of special interest. In a case when the hydrodynamic interaction of the particles of the macromolecule may be neglected, i.e. when the coil is, as it were, free-draining, the coefficient of resistance of the latter is proportional to the length of the macromolecule and the coefficient of friction of the particle associated with length M/N is proportional to this length... 

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... 

For the cases, when hydrodynamic interaction is neglected, that is l = 0, the equation for distribution function was found by Pokrovskii (1978) and was confirmed later by Schieber (1992)... 

The results for the case, when hydrodynamic interaction is taken into account (Altukhov 1986) are rather cumbersome. So, we consider here the more simple case, when hydrodynamic interaction is neglected but internal viscosity is retained. The results were obtained by Pokrovskii and Chuprinka (1973) (see also Pokrovskii 1978). 

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... 

This failure has been attributed to the fact that hydrodynamic interactions are neglected in the Rouse model. 

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 ... 

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