Hard spheres interaction between

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. 

In order to approach Bom s approximation more closely, let us consider the hard-sphere interaction between the ion and solvent molecules ... 

As discussed above, solvation free energy is t3q)ically divided into two contributions polar and nonpolar components. In one popular description, polar portion refers to electrostatic contributions while the nonpolar component includes all other effects. Scaled particle theory (SPT) is often used to describe the hard-sphere interactions between the solute and the solvent by including the surface free energy and mechanical work of creating a cavity of the solute size in the solvent [148,149]. 

Kinning DJ, Thomas EL. Hard-sphere interactions between spherical domains in diblock copolymers. Macromolecules 1984 17 1712-1718. 

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

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... 

Doll64 has applied classical S-matrix theory to the collinear A + BC collision where atoms A and B interact via a hard sphere collision this is the model studied quantum mechanically by Shuler and Zwanzig.65 Doll treats classically allowed and forbidden processes and finds good agreement between semiclassical and quantum mechanical transition probabilities. This is a remarkable achievement for the semiclassical theory, for the hard sphere interaction is far from the smooth potential that one normally assumes to be necessary for the dynamics to be classical-like. 

In this relation, N is the number density of the scattering microemulsion droplets and S(q) is the static structure factor. Equation (2.12) is only strictly valid for the case of monodisperse spheres. However, for the case of low polydispersities the occurring error is small [63, 64]. S(q) describes the interactions between and the spatial correlations of the droplets. These are in general well approximated by hard sphere interactions in microemulsion systems [65], The influence of inter-particle interactions as described by S(q) canbe estimated at least for S(0) using the Carnahan-Starling expression [52,64,66]... 

In summary, the major feature of the dynamic model just described is the approximation that solute-solvent and solvent-solvent collisions can be described by hard-sphere interactions. This greatly simplifies the calculations the formal calculations are not difficult to carry out in the more general case, but the algebra is tedious. We want to describe the effects of solute and solvent dynamics on the reactive process as simply as possible, and the model is ideal for this purpose. Specific reactive events among the solute molecules are governed by the interaction potentials that operate among these species. The particular reactive model described here allows us to examine certain features of the coupling between reaction and diffusion dynamics without recourse to heavy calculations. More realistic treatments must of course be handled via the introduction of species operators for the system under consideration. 

Although a full analysis of this result would be quite involved and has not yet been carried out, its physical content is easily appreciated. The form of the L(12 z) operator given above allows for the possibility of hard elastic collisions between the solute molecules. We omit these contributions below, although no new difficulties appear when they are included. Our considerations then apply to the case of solute molecules that interact with each other via soft forces and with the solvent via hard-sphere interactions. 

The different terms represent segment-segment hard spheres interactions, a, mean field contribution, permanent bond energy between segments in the chain, a "", and free energy of specific hydrogen bond interactions between associating sites, a ° , if any. 

Figure 6 shows the structure factor S(q) calculated for electrostatic repulsion (dotted line dashed line) and for hard core repulsion (solid line) for a system of spheres having a diameter of 80 nm and a volume fraction of 20% [46]. Again the abscissa has been scaled by the number-average diameter Dj. At low ionic strength there is a strong electrostatic repulsion between the spheres leading to a pronounced maximum in the structure factor (Fig. 6 dotted line). If the ionic strength in increased, however, the repulsive electrostatic interaction is screened and the variation of S(q) is much weaker (dashed line in Fig. 6). At high ionic strength the electrostatic repulsion of the latex particles is screened considerably and in first approximation the structure factor may be calculated in terms of effective hard sphere interaction (cf. Refs. [64] and [65]).

The former is obtained through a relation with the cavity pair distribution function, yAA Hs- between two non-polar solutes, assuming hard-sphere interaction among them and also between solute and water molecules. We discuss the evaluation of cavity distribution functions in Appendix 15.A. 

As would be determined 70 years later, orientationally averaged projections are exactly equal to ft (assuming hard-sphere interactions) for and only for contiguous bodies that lack concave surfaces and thus permit neither self-shadowing nor multiple scattering of gas molecules.For other bodies, ft always exceeds the projection. No polyatomic molecule is truly convex because of crevices between the atoms. However, the effect of such small locally concave areas on ft is only a few percent and PA is often passable for largely convex shapes. This is not true for objects... 

A + 2 —)Aads + 2e, where denotes an empty site. The interaction between adsorbates can be included in several ways. In the example in Fig. 3.57, a shell of purely hard sphere interactions is considered, in which the simultaneous bonding of two anions to neighboring sites is excluded. The isotherm can be calculated by including adsorption, desorption and surface diffusion steps and scanning potential E. The rate constants for adsorption and desorption are of the form... 

When the medium is a poor solvent for the attached polymers a rather different situation is encountered. The polymer chains then tend to assume collapsed configurations in order to minimize contact with solvent molecules and the polymer segments prefer to interact with each other. This results in (short-ranged) attraction between colloidal particles covered with polymer chains in a poor solvent (see Sect. 5.5 in [46]). The interaction of such sticky spheres (billiard balls with a thin layer of honey [47]) is often described in a simple manner using the adhesive hard sphere interaction (see right panel in Fig. 1.5)... 

It can be noticed that on figure 5.9 both choices of distance are in good agreement with experiment, due to the small contribution of hard sphere interactions at low concentrations. For a total concentration of 0.5 molal (fig. 5.10) and 0.75 molal (fig. 5.11), we observe a notable difference between the two sets of ionic radii, whereas the ideal and limiting law models are too different from the experimental data to be represented on the figures. 

Equation (3.30) formulates an interesting result. It reveals that the increase in the osmotic pressure over the ideal behavior, as described in lowest order by the second term on the right-hand side, may be understood as being caused by repulsive hard core interactions between the polymer chains which occupy volumes in the order of h z)R. To see it, just compare Eq. (3.30) with Eq. (2.73) valid for a van der Waal s gas. For a gas with hardcore interactions only, i.e. a = 0, the second virial coefficient equals the excluded volume per molecule 6/Nl. Therefore, we may attribute the same meaning to the equivalent coefficient in Eq. (3.30). Our result thus indicates that polymer chains in solution behave like hard spheres, with the radius of the sphere depending on Rf, and additionally on z, i.e. on the solvent quality. For good solvents, in the Kuhnian limit h z —> 00) = 0.353, the radius is similar to iJp-... 

The hard-sphere interaction has a clearly defined and energy-independent range, namely d, and so the colhsion cross-section has the exact value nd. Even when the interaction is not that of hard spheres, it is the case that the cross-section can be interpreted in such a form, where now is an effective range of the interaction. To do so we need to know more about realistic forces between molecules. 

Walz and Sharma [1439] calculated the depletion force between two charged spheres in a solution of charged spherical macromolecules. Compared to the case of hard-sphere interactions only, the presence of a long-range electrostatic repulsion increases greatly both the magnitude and the range of the depletion effect. Simulations and density functional calculations for polyelectrolytes between two planar surfaces extend these results [1440]. 

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