Hard spheres soft repulsions

We now proceed to more realistic models of adsorption systems. As a preliminary step, comparative simulations of various gases on various model surfaces should be mentioned. These include hard spheres at a soft repulsive wall [73] and hard spheres, soft repulsive spheres, and Lennard-Jones atoms between hard, soft repulsive, and soft attractive walls [741. For coverages greater than one monolayer, these simulations show that the local density n z) is relatively insensitive to the detailed nature of the interactions [74]. It is the repulsive cores of the adsorbed atoms that are the determining factor. This point is illustrated in Fig. 9. 

Tadros (1986) describes four types of interparticle forces hard sphere, soft (electrostatic), van der Waals, and steric. Hard-sphere interactions, which are repulsive, become significant only when particles approach each other at distances slightly less than twice the hard-sphere radius. They are not commonly encountered. 

Whittle and Dickinson used a more elaborate model, incorporating harmonic bonds, short-ranged repulsive forces, and a shifted-center Leimard-Jones potential to study aggregates of soft particles typical of food colloids [70]. These authors also studied the failure of a colloidal gel under strain [71]. Similar studies, using either hard spheres or repulsive-core particles, have also been performed by other groups [72-75]. 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

In the previous section, non-equilibrium behaviour was discussed, which is observed for particles with a deep minimum in the particle interactions at contact. In this final section, some examples of equilibrium phase behaviour in concentrated colloidal suspensions will be presented. Here we are concerned with purely repulsive particles (hard or soft spheres), or with particles with attractions of moderate strength and range (colloid-polymer and colloid-colloid mixtures). Although we shall focus mainly on equilibrium aspects, a few comments will be made about the associated kinetics as well [69, 70]. 

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

Note that, for = 0, the potential given above does not reduce to the Lennard-Jones (12-6) function, because the soft Lennard-Jones repulsive branch is replaced by a hard-sphere potential, located at r = cr. The results for the nonassociating Lennard-Jones fluid can be found in Ref. 159. 

Instead of the hard-sphere model, the Lennard-Jones (LJ) interaction pair potential can be used to describe soft-core repulsion and dispersion forces. The LJ interaction potential is... 

Equation 20.5 describes exactly the ideal gas equation, where the individual particles do not interact, that is, do not show any attractive and repulsive forces. At low distances between the particles, attractive and repulsive forces start to become visible. Thus, the third and the higher terms in Nb/V take into account the hardness respectively softness of the individual particles. In a Lrst approximation, the atom, molecule is considered as a sphere with a hard core oftvdHiBie behaviorcan be better approximated by taking into accountalso higherterms ofthe Taylor expansion. It is impressive that the van der V feals equation permits to calculate the critical parapgpflq s andVc, where the gas condenses and becomes a liquid ... 

Several models have been developed to describe these phenomena quantitatively, the main difference being the interaction potential between the particles. There are two major approaches the hard sphere and the soft sphere. The hard sphere assumes that the only interaction between particles is a strong repulsion at the point of contact. The soft sphere is more realistic and assumes a potential with a barrier and a primary minimum like in DLVO theory (Figure 11.8). 

Fig. 18 Schematic representation of crowded soft systems (a) entangled polymers, (b) repulsive colloidal hard spheres, (c) colloidal star polymers, and (d) attractive hard spheres. The former are described by the tube model for entanglements, whereas the latter three by the general cage model for colloidal glasses...

As an alternative to adding attraction, the hard-sphere model may be made more realistic by making the repulsion softer. The soft-sphere or inverse-... 

The BH diameter is designed to include temperature-dependent, soft-repulsion contributions in the hard-sphere equation of state. The WCA procedure also does this and further modifies the diameter to be the proper value when using a hard-sphere distribution function in perturbation terms due to attraction. The VW procedure corrects the WCA result for some of the limitations of the Percus-Vevick (PY) approximation and expresses the result in a simple algebraic form. A table of dB and 8Vw values is given so no integration is required. 

The column headed HSE uses an approximation made originally by Mansoori and Leland (3) that the diameter used in the hard sphere equations of state is c0o-, the LJ a parameter for each molecule multiplied by a universal constant for conformal fluids. This approximation then requires that be replaced by equations defining the HSE pseudo parameters, Equations 10 and 11. The results in the HSE column use c0 = 0.98, the value for LJ fluids obtained empirically by Mansoori and Leland. This procedure is correct only for a Kihara-type potential and it is not consistent with the LJ fluids in Table I. Furthermore, this causes only the high temperature limit of the repulsion effects to be included in the hard-sphere calculation. Soft repulsions are predicted by the reference fluid. 

For nonpolar fluids and symmetric reference fluids for polar substances we will assume that the unknown potential function for each may be modeled with a symmetrical potential consisting of a hard-sphere repulsion potential for spheres of diameter d plus an excess which depends on (r/d) and a single energy parameter, e, in the form e i(r/d). If the fluids are nonspherical, e is an average which may depend on temperature and to some extent on density. If the unknown true potential involves a soft repulsion, d may depend on both temperature and density. 

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