Model potentials, molecular theories

The concept of corresponding states was based on kinetic molecular theory, which describes molecules as discrete, rapidly moving particles that together constitute a fluid or soHd. Therefore, the theory of corresponding states was a macroscopic concept based on empirical observations. In 1939, the theory of corresponding states was derived from an inverse sixth power molecular potential model (74). Four basic assumptions were made (/) classical statistical mechanics apply, (2) the molecules must be spherical either by actual shape or by virtue of rapid and free rotation, (3) the intramolecular vibrations are considered identical for molecules in either the gas or Hquid phases, and (4) the potential energy of a coUection of molecules is a function of only the various intermolecular distances. 

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. 

To go from experimental observations of solvent effects to an understanding of them requires a conceptual basis that, in one approach, is provided by physical models such as theories of molecular structure or of the liquid state. As a very simple example consider the electrostatic potential energy of a system consisting of two ions of charges Za and Zb in a medium of dielectric constant e. 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

Figure 2.5 Schematic representation of the Au/MPS/PAH-Os/solution interface modeled in Refs. [118-120] using the molecular theory for modified polyelectrolyte electrodes described in Section 2.5. The red arrows indicate the chemical equilibria considered by the theory. The redox polymer, PAH-Os (see Figure 2.4), is divided into the poly(allyl-amine) backbone (depicted as blue and light blue solid lines) and the pyridine-bipyridine osmium complexes. Each osmium complex is in redox equilibrium with the gold substrate and, dependingon its potential, can be in an oxidized Os(lll) (red spheres) or in a reduced Os(ll) (blue sphere) state. The allyl-amine units can be in a positively charged protonated state (plus signs on the polymer...

It remains to work out expressions for the system-dependent coefficients ao, uo, and Cq in terms of the molecular parameters of the RPM or other model potentials. Oq and o are determined by derivatives of the free energy of the homogeneous system. Moreover, theory shows that the correlation length of the fluctuations is given by... 

The molecular theory of Doi [63,166] has been successfully applied to the description of many nonlinear rheological phenomena in PLCs. This theory assumes an un-textured monodomain and describes the molecular scale orientation of rigid rod molecules subject to the combined influence of hydrodynamic and Brownian torques, along with a potential of interaction (a Maier-Saupe potential is used) to account for the tendency for nematic alignment of the molecules. This theory is able to predict shear thinning viscosity, as well as predictions of the Leslie viscosity coefficients used in the LE theory. The original calculations by Doi for this model employed a preaveraging approximation that was later... 

As expected, the total interaction energies depend strongly on the van der Waals radii (of both sorbate and sorbent atoms) and the surface densities. This is true for both HK type models (Saito and Foley, 1991 Cheng and Yang, 1994) and more detailed statistical thermodynamics (or molecular simulation) approaches (such as Monte Carlo and density functional theory). Knowing the interaction potential, molecular simulation techniques enable the calculation of adsorption isotherms (see, for example, Razmus and Hall, (1991) and Cracknell etal. (1995)). 

A key question about the use of any molecular theory or computer simulation is whether the intermolecular potential model is sufficiently accurate for the particular application of interest. For such simple fluids as argon or methane, we have accurate pair potentials with which we can calculate a wide variety of physical properties with good accuracy. For more complex polyatomic molecules, two approaches exist. The first is a full ab initio molecular orbital calculation based on a solution to the Schrddinger equation, and the second is the semiempirical method, in which a combination of approximate quantum mechanical results and experimental data (second virial coefficients, scattering, transport coefficients, solid properties, etc.) is used to arrive at an approximate and simple expression. 

Rullmann, J.A.C. and Duijnen P.Th. van, Potential energy models of biological macromolecules a case for ab initio quantum chemistry. CRC Reports in Molecular Theory (1990) 1 1—21. 

This point of interest is brought forward by the RISM approach to the structure of molecular liquids, and a RISM model with HNC closure supports a similar result for the excess chemical potential in terms of atom-atom correlations (Singer and Chandler, 1985 Hirata, 1998). RISM - reference interaction site model - is an acronym that refers to a class of theories for the joint two-atom distributions in molecular liquids. The most basic decision of RISM models is that theories of molecular liquids should focus first on the atom-atom distributions extracted from X-ray and neutron scattering data rather than more complex possibilities this highly practical point was not so obvious in an earlier epoch when models of molecular liquids were scarcely realistic on an atomic scale. That basic decision was encapsulated by invention of a site-site (or atom-atom) Ornstein-Zernike (SSOZ) (Cummings and Stell, 1982) equation that involved intramolecular atom-atom correlations. The original suggestions (Chandler and Andersen, 1972) were sufficiently successful as to support subsequent flamboyant developments, and to be substantially impervious to more fundamental improvements (Chandler et al, 1982). For these reasons a full discussion of the RISM models wouldn t fit here. Fortunately, a devoted exposition of current RISM work is already available (Hirata, 1998). 

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