Hard-sphere interactions

Using the above relationships we readily obtain the hard-sphere second virial coefficient 

Note that because of the lack of temperature dependence of B we have the intriguing results 

In this case, the force acting between two particles is given by 

Note that the internal energy and entropy for this case can be obtained from Eqs. (4.82) and (4.86) using Eq. (4.97). 

Example 4.1 Thermodynamic Properties of Lennard-Jones (6-12) Gases 

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The interaction between particles belonging to the same species is a hard sphere interaction... 

The formation of a 3D lattice does not need any external forces. It is due to van der Waals attraction forces and to repulsive hard-sphere interactions. These forces are isotropic, and the particle arrangement is achieved by increasing the density of the pseudo-crystal, which tends to have a close-packed structure. This imposes the arrangement in a hexagonal network of the monolayer. The growth in 3D could follow either an HC or FCC struc-... 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

Early numerical estimates of ternary moments [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in the fundamental band, which we will consider in Chapter 6, and hard-sphere interaction potentials. While the main conclusions are at least qualitatively supported by more detailed calculations, significant quantitative differences are observed that are related to three improvements that have been possible in recent work [296] improved interaction potentials the quantum corrections of the distribution functions and new, accurate induced dipole functions. The force effect is by no means always positive, nor is it always stronger than the cancellation effect. 

Very little is known about the irreducible ternary dipole components. An early estimate based on classical electrodynamics, hard-sphere interaction and other simplifying assumptions suggests very small, negative contributions to the zeroth spectral moment [402], namely —0.13 x 10-10 cm-1 amagat-3. 

In the simplest approximations, molecules are assumed to be hard spheres. Interactions between molecules only occur instantaneously, with a hard repulsion, when the molecules centers come close enough to overlap. 

Evaluate the Chapman-Enskog expression for the thermal conductivity, Eq. 12.87, for the special case of a hard-sphere interaction. Show that for a pure-species, this gives the result cited earlier as Eq. 12.57. 

In addition, assumptions for the reaction probability of the hard spheres—which strictly speaking cannot react—are introduced. That is, the reaction probability is not calculated from the actual potentials or dynamics of the collisions but simply postulated based on physical intuition. Note that the assumption of a spherically symmetric (hard-sphere) interaction potential implies that the reaction probability P cannot depend on (j> (see Fig. 4.1.1), since there will be a cylindrical symmetry around the direction of the relative velocity. In addition, the assumption of structureless particles implies that the quantum numbers that specify the internal excitation cannot be defined within the present model. 

It is important to emphasize that the alcohols used as cosurfactants have high solubility in the oil phase because they are present there as both single molecules and aggregates. Therefore, in calculating X0o, one must consider the self-association of alcohol in the oil phase. The activity coefficients arise because of the hard-sphere interactions among the droplets. Therefore, in the dispersed phase, which is free of droplets, the activity coefficients are equal to unity, whereas, in the continuous phase, which contains the droplets, the activity coefficients differ from unity. 

As in the previous case,/ow is less than unity for a singlephase microemulsion but becomes equal to unity when an excess oil phase coexists. The activity coefficients ysw and ysA of the singly dispersed surfactant and alcohol molecules and the activity coefficient ygW of the aggregates differ from unity, since the water phase contains the droplets which contribute to solution nonideality through the hard-sphere interactions. For the same reason, the activity coefficient yoo of the oil in the dispersed oil phase is unity since no droplets are present in the oil phase. 

The first term on the right-hand side f(° is the standard chemical potential of component i, the second term is the ideal mixture contribution, and the third term is the nonideality caused by the hard-sphere interactions. The chemical potential of the hard-sphere droplet (component g) is obtained using the Gibbs—Duhem equation... 

The other kind of systems largely studied, consists of polymethylmethacrylate (PMMA) or silica spherical particles, suspended in organic solvents [23,24]. In these solvents Q 0 and uy(r) 0. The particles are coated by a layer of polymer adsorbed on their surface. This layer of polymer, usually of the order of 10-50 A, provides an entropic bumper that keeps the particles far from the van der Waals minimum, and therefore, from aggregating. Thus, for practical purposes uw(r) can be ignored. In this case the systems are said to be sterically stabilized and they are properly considered as suspensions of colloidal particles with hard-sphere interaction [the pair potential is of the form given by Eq. (5)]. 

The comparisons presented in Figures 7 and 8, show that the RY closure relation works better than HNC, PYand RMSA, for the two kinds of systems with repulsive interactions here considered. Before the introduction of the RY approximation, the picture was that HNC and PY were better approximations to describe the static structure of systems with repulsive long-range and hard-sphere interactions, respectively. The RMSA approximation, however, has been used extensively in the comparison with experimental data for the static structure of aqueous suspensions of polystyrene spheres, mainly because it has an analytical solution even for mixtures [36]. 

Anderko and Lencka find. Eng. Chem. Res. 37, 2878 (1998)] These authors present an analysis of self-diffusion in multicomponent aqueous electrolyte systems. Their model includes contributions of long-range (Coulombic) and short-range (hard-sphere) interactions. Their mixing rule was based on equations of nonequilibrium thermodynamics. The model accurately predicts self-diffusivities of ions and gases in aqueous solutions from dilute to about 30 mol/kg water. It makes it possible to take single-solute data and extend them to multicomponent mixtures. 

The hard sphere interaction energy is an accurate approximation for short-range interactions between particles. This occurs when we have steric stabilization [33,34] due to polymer adsorption and electrostatic stabilization with a thin double layer [35,36] (i.e., high ionic... 

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). 

Here, we propose a more realistic model of protein-electrolyte mixture. In the present case all the ionic species (macroions, co-ions and counterions) are modelled as charged hard spheres interacting by Coulomb potential as for the primitive model (Sec. 2), but the macroions are allowed to form dimers as a result of the short-range attractive interaction. Numerical evaluation of this multicomponent version of the dimerizing-macroion model has been carried out using PROZA formalism, supplemented by the MSA closure conditions (Sec. 3). 

Interactions in microemulsions have been studied using light scattering techniques. In water in oil systems, hard sphere interactions are dominant. The remaining interactions are usually attractive and are of the Van der Waals type. The case of oil in water microemulsion is less known. In those systems, interactions seem to be of a quite different kind. Entropic forces are thought to be important in those media, as will be shown by the study of a simplified system. 

B = 8 hard-sphere interaction B > 8 repulsive interaction B < 8 attractive interaction... 

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