Hard spheres, hydrodynamic model

In Simha s early model (Simha and Zakin 1962), transition from a dilute to a concentrated polymer solution was envisioned as being due to interpenetration of polymer chains that occurs when concentration lies somewhere in the region 1 < [ryjc < 10. This transition is evident from the change in the concentration dependence of viscosity in polymer solutions. The quantity [r ]c, the Simha-Frisch parameter (Frisch and Simha 1956), also sometimes called the Berry number (Gupta et al. 2005), is therefore a reasonable measure of chain overiap in solution. As Shenoy et al. (2005b), however, correctly point out, the dependency, being ultimately based on the equivalent hard sphere hydrodynamic model, is strictly applicable only at low polymer concentrations. 

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. 

In Chapter 3 (Section 3.5.2) the viscosity of a hard sphere model system was developed as a function of concentration. It was developed using an exact hydrodynamic solution developed by Einstein for the viscosity of dilute colloidal hard spheres dispersed in a solvent with a viscosity rj0. By using a mean field argument it is possible to show that the viscosity of a dispersion of hard spheres is given by... 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

This functional form is derived from exact results in the dilute vapor and hydrodynamic solvent limits. The coefficients A, B, C and D used in modeling high density fluids are determined uniquely from the equation of state of the corresponding hard sphere reference system (33,34)- This hard sphere fluid chemical potential model has been shown to accurately reproduce computer simulation results for both homonuclear and heteronuclear hard sphere diatomics in hard sphere fluids up to the freezing point density (35) ... 

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 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). 

Interestingly, this slip behaviour of hard-sphere glasses is different in nature from that found earlier by Meeker et al. in jammed systems of emulsion droplets [60]. There, a non-linear elasto-hydrodynamic lubrication model, appropriate for deformable particles, could quantitatively account for their observations. It therefore appears that, while slip is ubiquitous for yield stress fluids flowing along smooth walls, the mechanism for its occurrence can be highly system dependent. 

In classical analysis, the concentration dependence of Dt has been described by Equations 35-37. For the simplest hydrodynamic model, that of impermeable neutral hard spheres, Pyun and Fixman (75) derived the result ... 

Recently, Hynes et al. [221, 222] have pointed out that continuum models of rotational relaxation become unreliable when the molecule of interest rotates in a solvent comprising molecules of similar size. To improve on the model, they considered a sphere to be surrounded by a first co-ordination shell of solvent molecules. All these were taken as rough spheres, that is hard spheres which reverse their relative velocity (normal and tangential components) on impulsive collision. Of specific interest are CCI4 and SF. The test sphere and its boundary layer is surrounded by a hydrodynamic continuum. To model this, Hynes et al. used linearised hydrodynamic equations for the solvent with a modified boundary condition between solvent and test molecule, which relates the rotational stress on the test sphere to the angular velocity of the sphere. A coefficient of proportionality, 3, is introduced as a slip coefficient (j3 0... 

In summary we see that the use of the triple relaxation time model gives the same thermodynamic properties and essentially the same hydrodynamic properties as the exact hard-sphere system. It therefore follows that the corresponding kinetic equation (135) should give the proper limit for S k, to) at long wavelengths and low frequencies. The hydrodynamic behavior of S k, to) can be calculated using the macroscopic equations of fluid dynamics. All the essential features are summarized in the expression ... 

The above emulsion system is fairly simple, since it is likely that the nonyl phenyl and propylene oxide chain is on the oil side of the interface, whereas the poly(ethylene oxide) chain is on the aqueous side of the interface. The hydrodynamic thickness of the surfactants, given above, is much smaller than the droplet radius and hence these sterically stabilized emulsions may approximate hard-sphere dispersions very closely (with an effective radius = R + 8 ). This can be tested by fitting the data to the hard-sphere model suggested by Dougherty and Krieger (40,41). 

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. 

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. 

Since experimental values of diffusion coefficients are known in relatively few instances, it is often convenient to express them in terms of the viscosity (rj) of the medium, which is known for many solvents and is easy to measure. Application of classical hydrodynamic theory to the hard-sphere model leads, as was shown in Section 1.3.2, to the Stokes-Einstein equation (1.3) for the relative diffusion coefficient Dab, which may be... 

The mechanical properties of suspensions containing a narrow size distribution of particles have been studied extensively because they offer the best chances for testing models for flow behavior. The most detailed studies can be found for hard spheres where particles experience only volume exclusion, thermal and hydrodynamic interactions. Based on the models developed for these systems, a great deal can be learned about the behavior of suspensions experiencing longer range repulsions and attractions. 

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