Repulsion potential, hard-sphere

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. 

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. 

The final scattering angle 0 is defined rising 0 = 0(t = oo). There will be a correspondence between b and 0 that will tend to look like what is shown in figure A3.11.5 for a repulsive potential (liere given for the special case of a hard sphere potential). 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

Figure C2.6.5. Examples of tire AO potential, equation (C2.6.12). The values of are indicated next to tire curves. The hard-sphere repulsion at r = 7 has not been drawn.

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. 

Stratification, as illustrated by the plots in Fig. 6, is due to eonstraints on the paeking of moleeules next to the wall and is therefore largely determined by the repulsive part of the intermoleeular potential [55]. It is observed even in the absenee of intermoleeular attraetions, sueh as in the ease of a hard-sphere fluid eonfined between planar hard walls [42,90-92]. For this system Evans et al. [93] demonstrated that, as a eonsequenee of the damped oseil-latory eharaeter of the loeal density in the vieinity of the walls, F-, is a damped oseillatory funetion of s., if is of the order of a few moleeular diameters, whieh is eonfirmed by Fig. 5. 

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... 

Another model which combined a model for the solvent with a jellium-type model for the metal electrons was given by Badiali et a/.83 The metal electrons were supposed to be in the potential of a jellium background, plus a repulsive pseudopotential averaged over the jellium profile. The solvent was modeled as a collection of equal-sized hard spheres, charged and dipolar. In this model, the distance of closest approach of ions and molecules to the metal surface at z = 0 is fixed in terms of the molecular and ionic radii. The effect of the metal on the solution is thus that of an infinitely smooth, infinitely high barrier, as well as charged surface. The solution species are also under the influence of the electronic tail of the metal, represented by an exponential profile. 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Equations (87)-(89) apply in aqueous solutions of two electrolytes in which the interaction potentials are conformal. For example, the assumptions utilized in the extensions of the Debye-Hiickel theory (e.g. water is considered as a continuous dielectric medium of dielectric constant D, that the cation-anion repulsive potential is that of hard spheres, and that all the... 

The term pair potential that contains only the attractive potential, because the repulsion effects have been allowed for by the effective volume fraction and hard sphere diameter. The new potential can be defined as... 

The molecular dynamics method is based on the time evolution of the path (p (t), for each particle to feel the attractions and repulsions from all other particles, following Newton s law of motion. The simplest case is a dilute gas following the hard sphere force field, where there is no interaction between molecules except during brief moments of collision. The particles move in straight lines at constant velocities, until collisions take place. For a more advanced model, the force fields between two particles may follow the Lennard-Jones 6-12 potential, or any other potential, which exerts forces between molecules even between collisions. 

One conclusion from this study is that although the hard-sphere fluid has been very successful as a reference fluid, for example, in developing analytical equations of state, it is unrealistic in representing the dynamical relaxation processes in real systems, even with very steeply repulsive potentials. Owing to the discontinuity in the hard-sphere potential, this fluid, in fact, is not a good reference fluid for the short time (fast or j9 ) viscoelastic relaxation aspects of rheology. 

In fact, the origin of the (8-N) rule resides in the delicate balance between the repulsive overlap forces and the attractive covalent bond forces. The bond lengths are not invariant as drawn in Fig. 8.1, since the atoms do not behave as hard spheres with fixed nearest-neighbour distances. Assuming a repulsive pair potential... 

The last entry in Table 10.1 is the least well defined of those listed. This is of little importance to us, however, since our interest is in attraction, and the final entry in Table 10.1 always corresponds to repulsion. The reader may recall that so-called hard-sphere models for molecules involve a potential energy of repulsion that sets in and rises vertically when the distance of closest approach of the centers equals the diameter of the spheres. A more realistic potential energy function would have a finite (though steep) slope. An inverse power law with an exponent in the range 9 to 15 meets this requirement. For reasons of mathematical convenience, an inverse 12th-power dependence on the separation is frequently postulated for the repulsion between molecules. 

The justification for using the combining rule for the a-parameter is that this parameter is related to the attractive forces, and from intermolecular potential theory the attractive parameter in the intermolecular potential for the interaction between an unlike pair of molecules is given by a relationship similar to eq. (42). Similarly, the excluded volume or repulsive parameter b for an unlike pair would be given by eq. (43) if molecules were hard spheres. Most of the molecules are non-spherical, and do not have only hard-body interactions. Also there is not a one-to-one relationship between the attractive part of the intermolecular potential and a parameter in an equation of state. Consequently, these combining rules do not have a rigorous basis, and others have been proposed. 

In order to derive a practical approximation for the repulsive contribution to vibrational frequency shifts the excess chemical potential, A ig, associated with the formation of a hard diatomic of bond length r from two hard spheres at infinite separation in a hard sphere reference fluid is assumed to have the following form. 

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