Hydrodynamically interacting hard spheres

For the volume fraction range 0.01 < c )< 0.2, Batchelor [22] derived the following expression for a dispersion of hydrodynamically interacting hard spheres ... 

The relative viscosity of a suspension of hydrodynamically interacting hard spheres can be expressed in the form... 

First, in this section, the influence of volume fraction of particles is discussed in the case where there are no surface forces between particles. Only hydrodynamic forces and Brownian motion are considered in this case, which is known as the non-interacting hard sphere model. The influence of surface forces is considered in the following section. 

At finite concentration, tire settling rate is influenced by hydrodynamic interactions between tire particles. For purely repulsive particle interactions, settling is hindered. Attractive interactions encourage particles to settle as a group, which increases tire settling rate. For hard spheres, tire first-order correction to tire Stokes settling rate is given by [33]... 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

This is our starting point and the infinite dilution case was analysed by Einstein in the early years of this century.12 This analysis was based on the dilation of the flow field because the liquid has to move around the flowing particle. The particles were assumed to be hard spheres so that they were rigid, uncharged and without attractive forces small compared to any measuring apparatus so that the dilatational perturbation of the flow would be unbounded and would be able to decay to zero (the hydrodynamic disturbances decay slowly with distance, i.e. r 1) and at such dilution that the disturbance around one particle would not interact with the disturbance around another. The flow field is sketched in Figure 3.10. The coordinates are centred on the particle so that the symmetry is clear. The result of the analysis for slow flows (i.e. at low Reynolds number) was ... 

The theory reflects the solvent properties through the thermody-namic/hydrodynamic input parameters obtained from the accurate equations of state for the two solvents. However, the theory employs a hard sphere solute-solvent direct correlation function (C12), which is a measure of the spatial distribution of the particles. Therefore, the agreement between theory and experiment does not depend on a solute-solvent spatial distribution determined by attractive solute-solvent interactions. In particular, it is not necessary to invoke local density augmentation (solute-solvent clustering) (31,112,113) in the vicinity of the critical point arising from significant attractive solute-solvent interactions to theoretically replicate the data. 

For liquid-phase diffusion of large adsorbate molecules, when the ratio = r /r of the molecule radius r to the pore radius is significantly greater than zero, the pore diffusivity is reduced by steric interactions with the pore wall and hydrodynamic resistance. When < 0.2, the following expressions derived by Brenner and Gaydos [/. Coll. Int Sci, 58,312 ( 1977)] for a hard sphere molecule (a particle) diffusing in a long cylindrical pore, can be used... 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

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... 

For monodisperse or unimodal dispersion systems (emulsions or suspensions), some literature (28-30) indicates that the relative viscosity is independent of the particle size. These results are applicable as long as the hydrodynamic forces are dominant. In other words, forces due to the presence of an electrical double layer or a steric barrier (due to the adsorption of macromolecules onto the surface of the particles) are negligible. In general the hydrodynamic forces are dominant (hard-sphere interaction) when the solid particles are relatively large (diameter >10 (xm). For particles with diameters less than 1 (xm, the colloidal surface forces and Brownian motion can be dominant, and the viscosity of a unimodal dispersion is no longer a unique function of the solids volume fraction (30). 

The model was also checked by evaluating the center-of-mass friction. It was shown that hydrodynamic interactions are important for solvent-separated atoms, 8 A, but not for the diatomic with 2.66 A. The mass dep>endences of the isolated iodine and argon frictions were not consistent with hydrodynamics estimates of the Stokes-Einstein theory (Section III E). Rather, they are in agreement with the Enskog theory corrected for caging by the Herman-Adler results for hard spheres. Further studies are required which avoid the use of Eq. (5.8). 

Fig. 14 Normalized Dcoii/Bo of PDMS coated silica suspension with = 0.3 in a symmetric mixture of toluene and heptane (solid circles) along with hard sphere suspension (open squares) at similar volume fraction. The hydrodynamic interactions expressed in H(q) for the two systems (solid squares for the hard sphere suspension) are shown in the inset [101]. This system is crystallized by sedimentation as seen in the photograph...

When 0 is larger than typically a few percent, the above description no longer applies. In the case of hard spheres, the osmotic pressure is well represented by the Carnaham and Starling formula up to large volume fractions [21]. The friction coefficient is more difficult to evaluate because the hydrodynamic interactions are long-range. Microemulsion droplets behave as hard spheres in many circumstances. However, in W/O microemulsions, droplets frequently exhibit supplementary attractive interactions. It has been proposed that the osmotic pressure... 

Adopting a different point of view, we tried to fit the variation of D versus (p at low (j) with existing theories taking into account the role of hydrodynamic interactions and interparticle forces. This has been done for microemulsions A and B, using Felderhof theory (11) with an interaction potential sum of hard sphere repulsion and W = A(2R/r), A = B. The agreement with experimental a values is quite satisfactory. 

Hard sphere colloidal systems do not experience interparticle inta-actions until they come into contact, at which point the interaction is infinitely repulsive. Such systems represent the simplest case, where the flow is affected only by hydrodynamic (viscous) interactions and Brownian motion. Hard spha-e systems are not often encountered in practice, but model systems consisting of Si02 spheres stabilized by adsorbed stearyl alcohol layers in cyclohexane (56,57) and polymer latices (58,59) have been shown to approach this behavior. They serve as a useful starting point for considering the more complicated effects when interparticle forces are present. 

Dynamic light scattering from dilute solutions provides the value of the diffusion coefficient, which can be converted to hydrodynamic radius J h,star of the star polymer. The ratio Rh/(Rg) characterizes the compactness of the macromolecule for the uniform hard sphere impenettable for the flow, it is Rb/Rg= (5/3) 1.29, whereas for the Gaussian coil, Rh/(Rg) = 3a- / /8 0.66. For ideal stars (without excluded-volume interactions), the ratio Rh/(Rg) can be derived within the Kirkwood-Riseman approximation, which gives the value of Rh/(Rg) 0.93. Reported experimental values of the Rh/(Rg) ratio for star polymers and starlike block copolymer micelles are usually found close... 

In addition, the droplets have a hydrodynamic radius, th, which is obtained from the diffusion coefficient extrapolated to infinite dilution. As will be shown below, the droplet interactions can, to a very good approximation, be described in terms of hard spheres. A third characteristic radius, the hard-sphere radius, ths, then enters to describe the interactions. Associated with the three radii, there are three different characteristic droplet volumes, and therefore three different characteristic droplet volume fractions. If 0 = 0 -t- 0o denotes the total volume fraction of surfactant and oil, the hard sphere volume fraction, 0hs, can be written as follows ... 

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