Molecular distribution functions

In this chapter, we introduce the concepts of molecular distribution function (MDF), in one- and multicomponent systems. The MDFs are the fundamental ingredients in the modern molecular theories of liquids and liquid mixtures. As we shall see, these quantities convey local information on the densities, correlation between densities at two points (or more) in the system, etc. 

We start with detailed definitions of the singlet and the pair distribution functions. We then introduce the pair correlation function, a function which is the cornerstone in any molecular theory of liquids. Some of the salient features of these functions are illustrated both for one- and for multicomponent systems. Also, we introduce the concepts of the generalized molecular distribution functions. These were found useful in the application of the mixture model approach to liquid water and aqueous solutions. 

In this chapter, we shall not discuss the methods of obtaining information on molecular distribution functions. There are essentially three sources of information analyzing and interpreting x-ray and neutron diffraction patterns solving integral equations and simulation of the behavior of liquids on a computer. Most of the illustrations for this chapter were done by solving the Percus-Yevick equation. This method, along with some comments on the numerical solution, are described in Appendices B—F. 

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Born M and Green H S 1946 A general kinetic theory of liquids I. The molecular distribution functions Proc. R. Soc. A 188 10... 

Molecular Distribution Functions for the Entropy Density in an Infinite System. 

Recent developments in the theory of polymer solutions have been reviewed by Berry and Casassa (32), and by Casassa (71). Casassa, who has contributed very largely to these developments, has adopted a statistical mechanical approach using molecular distribution functions, as first outlined by Zimm (72), rather than using a lattice model like that used by Flory, Huggins, and many later workers. 

A similar situation exists in the molecular-distribution function theory of liquids and one usually resorts to a superposition approximation. This amounts to assuming that, e.g., = 2 or something similar. It will be seen shortly that, contrary to unimolecular reactions, for bi-molecular reactions the stochastic mean is not the same as the classical kinetic expression for the concentration. 

Cochran, H.D., Lee, L.L., "Solvation Structure in Supercritical Fluid Mixtures based on Molecular Distribution Functions," ACS Symp. Ser, 1989,406,28. 

This statistical mechanical expression for surface tension depends explicitly on the potentials of inteimolecular force and molecular distribution functions. Upon recognition that the two-phase system under consideration is thermodynamically open, it follows that the distribution functions must be represented in the grand canonical ensemble. Thus, the dependence of y on temperature, T, and chemical potentials, enters through the implicit dependence of the distribution func-... 

Prigogine expanded the molecular distribution function in an infinite series around the equilibrium molecular distribution function f0... 

Molecular Distribution Functions Ideal Gas in a Force Field... 

Kirkwood, J. G. and Salsburg, Z. W., The statistical mechanical theory of molecular distribution functions in liquids. Disc. Faraday Soc. 15, 28-34 (1953). 

Statistical Molecular Distribution Functions in a Static and in an Alter-natii Electric Field.— We now shall apply the statistical distribution function expansion (100) to a molecular gas (or dilute solution of polar molecules in a non-polar solvent) immersed in an external electric field E. Not taking into consideration the mutual correlations of molecules but solely their interaction with the field E, we are justified by equation (81) in writing the potential energy to within the square of the field as follows ... 

With the development of the non-Newtonian viscosity theories it is now possible to compare the rotary diffusion coefficient and thereby the calculated length (or diameter) of the rigid particles as obtained from this technique with that from the commonly used flow birefringence method. Since both measurements depend upon the same molecular distribution function (Section III) they should give an identical measure of the rotary diffusion coefficient. Differences, however, will arise if the system under study is heterogeneous. The mean intrinsic viscosity is calculated from Eq. (7) whereas the mean extinction angle, x, for flow birefringence is defined by the Sadron equation (1938) ... 

Tj aim UQ ruesciiueu ueiuw , me luiiiuiiiig paiameiei a, me secoim aim louiiii mume S2 and iS4 of the molecular distribution function and a parameter R(p) that depends the effective aspect ratio p of the rigid molecules or particles, namely. 

We propose the study of Lennard-Jones (LJ) mixtures that simulate the carbon dioxide-naphthalene system. The LJ fluid is used only as a model, as real CO2 and CioHg are far from LJ particles. The rationale is that supercritical solubility enhancement is common to all fluids exhibiting critical behavior, irrespective of their specific intermolecular forces. Study of simpler models will bring out the salient features without the complications of details. The accurate HMSA integral equation (Ifl) is employed to calculate the pair correlation functions at various conditions characteristic of supercritical solutions. In closely related work reported elsewhere (Pfund, D. M. Lee, L. L. Cochran, H. D. Int. J. Thermophvs. in press and Fluid Phase Equilib. in preparation) we have explored methods of determining chemical potentials in solutions from molecular distribution functions. 

Fluid microstructure may be characterized in terms of molecular distribution functions. The local number of molecules of type a at a distance between r and r-l-dr from a molecule of type P is Pa T 9afi(r)dr where Pa/j(r) is the spatial pair correlation function. In principle, flr (r) may be determined experimentally by scattering experiments however, results to date are limited to either pure fluids of small molecules or binary mixtures of monatomic species, and no mixture studies have been conducted near a CP. The molecular distribution functions may also be obtained, for molecules interacting by idealized potentials, from molecular simulations and from integral equation theories. 

We now introduce the singlet molecular distribution function, which is obtained from N(S) in the limit of a very small region S. First we note that Ai(Ri, S) can also be written as... 

We now observe that relation (2.45) has the same structure as relation (2.23) but with two differences. First, (2.45) refers to a system of N— 1 instead of N particles. Second, the system of AT— 1 particles is in an external field. Hence, (2.45) is interpreted as the local density at X" of a system of N— 1 particles placed in the external field Bx. This is an example of a conditional singlet molecular distribution function which is not constant everywhere. 

Molecular distribution functions in the grand canonical ensemble... 

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