Intermolecular potentials hard-sphere

In the first approach, we start with a molecular shape (spheres, dumbbells, etc.) and an expression for the intermolecular potential (hard-sphere, HS, Lennard-Jones, LJ, etc.) and proceed to calculate the macroscopic properties of this hypothetical fluid through numerical methods employing the appropriate - for the methodology used - equations. 

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. 

The program of calculating the BO-level potentials from Schroedinger level cannot often be carried through with the accuracy required for the intermolecular forces in solution theory. (9.) Fortunately a great deal can be learned through the study of BO-level models in which the N-body potential is pairwise additive (as in Eq. (3)) and in which the pair potentials have very simple forms. (2, 3, 6) Thus for the hard sphere fluid we have, with a=sphere diameter,... 

The macroscopic properties of the three states of matter can be modeled as ensembles of molecules, and their interactions are described by intermolecular potentials or force fields. These theories lead to the understanding of properties such as the thermodynamic and transport properties, vapor pressure, and critical constants. The ideal gas is characterized by a group of molecules that are hard spheres far apart, and they exert forces on each other only during brief periods of collisions. The real gases experience intermolecular forces, such as the van der Waals forces, so that molecules exert forces on each other even when they are not in collision. The liquids and solids are characterized by molecules that are constantly in contact and exerting forces on each other. 

All of the transport properties from the Chapman-Enskog theory depend on 2 collision integrals that describe the interactions between molecules. The values of the collision integrals themselves, discussed next, vary depending on the specified intermolecular potential (e.g., a hard-sphere potential or Lennard-Jones potential). However, the forms of the transport coefficients written in terms of the collision integrals, as in Eqs. 12.87 and 12.89, do not depend on the particular interaction potential function. 

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. 

Equations (32) and (34), which are based on a hard-sphere model, are in agreement in predicting no dependence of the diffusion constant on gas composition. However, for real molecules, a slight composition dependence should exist, which depends on the form of the intermolecular potential. ... 

The purpose of the present study is to apply the Tarazona method to the fluid systems of polyatomic molecules, thereby eliminating some disadvantages inherent to his method. We formulate a weighting function by the intermolecular potential alone ". Another improvement is with the use of the equation of state. A volume-corrected hard-sphere equation of state is used as the equation of state for polyatomic molecular fluids. 

A unique feature of H20 is the formation and sharing of hydrogen bonds with other molecules. Such bonds play a major role in determining the structure of both liquid and solid phases of H20. It is believed that for intermolecular spacings of less than 2 A, the two water molecules exert strong repulsive forces on each other. As such, there exists a hard sphere radius of little interpenetration of the molecules. Usually, the repulsive part of Lennard-Jones 6—12 potential can be considered appropriate to describe these repulsive characteristics. At distances of separation greater than 5 A, dipole-dipole interaction plays a dominant role. This is reasonable, because each H20 molecule has a large dipole moment, p = 1.84 D. 

There is only a qualitative justification for eqns, (3.3.6 and 3.3.7). The a parameter is related to attractive forces, and, from intermolecular potential theory, the parameter in the attractive part of the intermolecular potential for a mixed interaction is given by a relation like eqn. (3.3.6). Similarly, the excluded volume parameter b would be given by eqn. (3.3.7) if the molecules were hard spheres. However, there is no direct relation between the attractive part of the intermolecular potential and the a parameter in a cubic EOS, and real molecules are not hard spheres. 

The potential U(r ) is a sum over all intra- and intermolecular interactions in the fluid, and is assumed known. In most applications it is approximated as a sum of binary interactions, 17(r ) = IZ > w(rzj) where ry is the vector distance from particle i to particle j. Some generic models are often used. For atomic fluids the simplest of these is the hard sphere model, in which z/(r) = 0 for r > a and M(r) = c for r < a, where a is the hard sphere radius. A. more sophisticated model is the Lennard Jones potential... 

Fig. 2.2 Plot of the Lennard-Jones potential u r) in units of the attractive potential energy Clj against intermolecular distance, r, in units of the molecular diameter a. The vertical and horizontal straight lines at r = a show the potential energy for a hard-sphere representation of the system.

Stratification, as illustrated by the plots in Figs. 5.4-5.G, is due to constraints on the packing of molecules next to the substrate surface and is therefore largely determined by the repulsive part of the intermolecular potential [38). Stratification is observed even in the complete absence of intermolecular attractions, such as in the case of a hard-sphere fluid confined between planar hard walls [165-167]. For this system Evans et al. [168] demonstrated that, as a consequence of the damped oscillatory character of the local density in the vicinity of the walls, is itself a damped oscillatory function of s, if s is of the order of a few molecular diameters, which is confirmed by the plot in Fig. 5.3. 

The solid-fluid transition in the hard-sphere system has been the subject of great interest for about 50 years now. Although it was originally only a computer model, researchers in colloid science have come close to a complete experimental realization of the hard-sphere model [2,3]. The hard-sphere system is one with an intermolecular potential of the form... 

Since the intermolecular potential energy of a configuration of hard spheres is either zero or infinite, the Boltzman factor, exp(-pt/Af), is either one or zero and the configurational partition function is independent of temperature. Thus, the full behavior of this model is described by a single isotherm. 

Hansen and Verlet [156] observed an invariance of the intermediate-range (at and beyond two molecular diameters) form of the radial distribution function at freezing, and from this postulated that the first peak in the structure factor of the liquid is a constant on the freezing curve, and approximately equal to the hard-sphere value of 2.85. They demonstrated the rule by application to the Lennard-Jones system. Hansen and Schiff [157] subsequently examined g r) of soft spheres in some detail. They found that, although the location and magnitude of first peak of g r) at crystallization is quite sensitive to the intermolecular potential, beyond the first peak the form of g(r) is nearly invariant with softness. This observation is consistent with the Hansen-Verlet rule, and indeed Hansen and Schiff find that the first peak in the structure factor S k) at melting varies only between 2.85 n = 8) to 2.57 (at n= ), with a maximum of 3.05 at n = 12. 

Uij(r) = intermolecular pair potential for a pair of molecules of species i and /. Subscripts are dropped for pure fluid. u0(r) — hard-sphere intermolecular pair potential V = volume Vc = critical volume vc = V/No-3, reduced volume... 

A general method of predicting the effective molecular diameters and the thermodynamic properties for fluid mix-tures based on the hard-sphere expansion conformal solution theory is developed. The method of Verlet and Weis produces effective hard-sphere diameters for use with this method for those fluids whose intermolecular potentials are known. For fluids with unknown potentials, a new method has been developed for obtaining the effective diameters from isochoric behavior of pure fluids. These methods have been extended to polar fluids by adding a new polar excess function, to account for polar contributions in a mixture. A new set of pseudo parameters has been developed for this purpose. The calculation of thermodynamic properties for several fluid mixtures including CH —C02 has been carried out successfully. 

A theoretical basis for the computation of effective hard-sphere diameters is now well developed for the perturbation theory. Its numerical evaluation requires exact knowledge of the intermolecular potential so that the method is not immediately applicable to real molecules for which this potential is usually unknown. Despite this difficulty, the principles involved are important in designing a procedure for real molecules with the HSE theory. 

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