Spherical intermolecular potential

Using the effective interaction diameter approximation, one may write, for a spherical intermolecular potential at any density [see Eq. (49)],... 

The study of fluid mixtures allows the determination of the eflects of additional variables, such as molecular parameters and concentration. Mixtures, though more complex, can be dealt with relatively simply using, for spherical intermolecular potentials, the IBI hypothesis and the localization approximation. ... 

In the IBI model, for near-spherical intermolecular potential, the density dependence of the relaxation time is given by [Eq. (56)]... 

We may, for example, see what conclusion might be drawn for the liquid-solid phase transition using the IBI approach to energy relaxation to write, for approximately spherical intermolecular potentials. 

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

The empirical potentials for the molecules were obtained on the assumption of single attraction centers. This assumption is probably good for H2, fair for CH4 and N2, and very poor for Cl2. Even for molecules such as CH4 which are relatively spherical in shape, the fact that some atoms are near the outer surface rather than the center has an important effect. The closest interatomic distances are emphasized by the i 6 dependence of the potential. This point has been considered by several authors who worked out examples showing the net intermolecular potential for several models. 

Spherical nonpolar molecules obey an interaction potential which has the characteristic shape shown in Fig. 2. At large values of the separation r it is known that the potential curve has the shape — r 6, and at short distances the potential curve rises exponentially the exact shape of the bottom part of the curve is not very well known. Numerous empirical equations of the form of Eq. (78) have been suggested for describing the molecular interaction given pictorially in Fig. 2. The discussion here is restricted to the two most important empirical functions. A rather complete summary of the contributions to intermolecular potential energy and empirical intermolecular potential energy functions used in applied statistical mechanics may be found in (Hll, Sec. 1.3) ... 

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. 

One source of information on intermolecular potentials is gas phase virial coefficient and viscosity data. The usual procedure is to postulate some two-body potential involving 2 or 3 parameters and then to determine these parameters by fitting the experimental data. Unfortunately, this data for carbon monoxide and nitrogen can be adequately represented by spherically symmetric potentials such as the Lennard-Jones (6-12) potential.48 That is, this data is not very sensitive to the orientational-dependent forces between two carbon monoxide or nitrogen molecules. These forces actually exist, however, and are responsible for the behavior of the correlation functions and - In the gas phase, where orientational forces are relatively unimportant, these functions resemble those in Figure 6. On the other hand, in the liquid these functions behave quite differently and resemble those in Figures 7 and 8. 

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

In this model the interactions of the guest with the nearest neighboring z, water molecules of a spherical cage are summed in a pair-wise manner. The model obtains a function m (r) describing the resulting field, averaged over all positions of the molecules within the cavity. The fundamental intermolecular potential between a water molecule of the cavity wall and a solute molecule may be described by a number of intermolecular potentials. 

Recent refinements on the atom-atom potential method include the development of accurate anisotropic model intermolecular potentials from ab initio electron distributions of the molecules. The non-spherical features in these charge distributions reflect features of real molecules such as lone pair and 7t-electron density, and therefore are much more effective at representing key interactions such as hydrogen bonding. 

The realisation that lattice theories of liquids were getting nowhere came only slowly from about 1950 onwards. A key paper for chemists was that of Longuet-Higgins on what he called conformal solutions in 1951. In this he avoided the assumption that a liquid had a lattice (or any other particular) structure but treated the different strengths of the intermolecular potentials in a mixture as a first-order perturbation of the physical properties of one of the components. In practice, if not formally in principle, his treatment was restricted to molecules that could be assumed to be spherical, but it was so successful for many mixtures of non-polar liquids that this and later derivatives drove lattice theories of liquid mixtures from the field. 

Figure 16.1 is a sketch of the intermolecular potential energy U for an isolated pair of spherically symmetric neutral mo lecules,forwhichW depends only on the distance between the molecular centers, i.e., on the intermolecularseparationr. (More generally, ti is also afunction of the relative orientations of the two molecules.) The intermolecularforce F is proportional to the r-derivative of IA ... 

All intermolecular potentials are anisotropic to a certain extent. We need to consider anisotropy theoretically, first to define numerically the limits of the previously employed spherical approximation and second in an attempt to extend our simple theoretical tools to more complex potentials. 

Anisotropy of the repulsive core or of the attractive well, or both, can be treated in a similar manner, and as a practical example we will consider V-T transfer for an intermolecular potential composed of a spherical repulsive core surrounded by an anisotropic attractive well (such as dipole-dijjole interaction). We will further invoke an approximate localization of the relaxation interaction on the repulsive core, implying that the multipolar long-range forces do not directly influence the relaxation probability. This assumption is... 

The orientations of linear molecules, relative to the global frame, can be specified by two Euler angles (oP = 8P, /> the symmetry-adapted functions G icop) that occur in the intermolecular potential [Eq. (15)] reduce to Racah spherical harmonics C (0/>, />). If the molecules possess a center of inversion such as N2 (when we disregard the occurrence of mixed isotopes l4Nl5N, the natural abundance of l5N being only 0.37%),... 

The intermolecular potential in gases is usually assumed to be additive. It has been pointed out, however, that the effect of potential nonadditivity on the equation of state of gases does not seem to be negligible (Kihara9). The simplest system for which the nonadditivity of the intermolecular potential plays a role is the system composed of three spherically symmetric atoms, which will be treated in Sectiqn I. The aim of Sections I. A and I. B is to investigate quantum-mechanically the van der Waals interaction between three distant atoms. By use of the results, a model of nonadditive potential is introduced in Section I. C, which model will be applied, in Part II, to the equation of state of gases. 

---