Theory of Liquids

In the previous chapters of this book we have dealt mainly with solvable problems. The models could be solved either because they lack interactions (e.g., ideal gases) or because the interactions took place between a small number of particles (Chapter 3) or because the system itself was simple (Chapter 4). From this chapter through the rest of the book we shall be dealing with nonsolvable systems. Liquids, even the simplest ones consisting of hard-sphere particles, pose very difficult theoretical problems. The main difficulty is contained in the configurational partition function, which for simple particles has the form 

Even when we assume pairwise additivity of the total potential energy, this integral cannot be solved for any reasonable pair interaction. Of course the problems become immensely more difficult for nonspherical molecules interacting through nonadditive potentials. 

This chapter will not deal with theories of liquids per se. Instead we shall present only general relations between thermodynamic quantities and molecular distribution functions. The latter are fundamental concepts which play a central role in the modern theoretical treatment of liquids and solutions. Acquiring familiarity with these concepts should be useful in the study of more complex systems such as aqueous solutions, treated in Chapters 7 and 8. As an exception, a brief outline of the scaled particle theory is presented in section 5.11. This theory, although originally aimed at studying hard-sphere systems, has been used in systems as complex as aqueous protein solutions. The main result that will concern us is the work required to create a cavity in a fluid. This quantity is fundamental in the study of solvation phenomena of simple solutes, as well as very complex ones such as proteins or nucleic acids. 

The notion of molecular distribution functions (MDF) command a central role in the theory of fluids. Of foremost importance among these are the singlet and the pair distribution functions. The following sections are devoted to describing and surveying the fundamental features of these two functions. We shall also briefly mention the general definitions of higher-order MDFs. These are rarely incorporated into actual applications, since very little is known about their properties. 

The pair correlation function, introduced in section 5.2.2, conveys information on the mode of packing of the molecules in the liquid. This information is often referred to 

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J. Frenkel, Kinetic Theory of Liquids, Clarendon Press, Oxford, 1946. 

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

Born M and Green H S 1949 A General Kinetic Theory of Liquids (Cambridge Cambridge University Press)... 

Barker J and Henderson D 1967 Perturbation theory and equation of state for a fluids II. A successful theory of liquids J. Chem. Phys. 47 4714... 

Wdom B 1965 Equation of state near the critical point J. Chem. Phys. 43 3898 Neece G A and Wdom B 1969 Theories of liquids Ann. Rev. Phys. Chem. 20 167... 

The fimctiong(ri is central to the modem theory of liquids, since it can be measured experimentally using neutron or x-ray diffraction and can be related to the interparticle potential energy. Experimental data [1] for two liquids, water and argon (iso-electronic with water) are shown in figure A2.4.1 plotted as a fiinction ofR = R /a, where a is the effective diameter of the species, and is roughly the position of the first maximum in g (R). For water, a = 2.82 A,... 

Frank F C 1958 On the theory of liquid orystals Discuss.Faraday Soc. 25 19-28... 

Oseen C W 1929 The theory of liquid crystals Trans. Faraday See. 29 883-99... 

Ericksen J L 1976 Equilibrium theory of liquid crystals Adv. Liq. Cryst. 2 233... 

Cao, J., Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties. J. Chem. Phys. 100 (1994) 5093-5105 II Dynamical properties. J. Chem. Phys. 100 (1994) 5106-5117 III. Phase space formalism and nalysis of centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6157-6167 IV. Algorithms for centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6168-6183 V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys. 101 (1994) 6184 6192. 

T.E. Faber. An Introduction to the Theory of Liquid Metals. Cambridge University Piess (1972). R.A. Swalin. Liquid metal diffusion theory, Acta. Met., 1, 736 (1959). 

The theory of liquids is based on statistical mechanics [2-5]. A fundamental result in statistical mechanics is that the probabihty of a state having an energy Ei is given by... 

Another theory of liquid-liquid explosion comes from Board et al. (1975). They noticed that when an initial disturbance, for example, at the vapor-liquid interface, causes a shock wave, some of the liquid is atomized, thus enhancing rapid heat transfer to the droplets. This action produces further expansion and atomization. When the droplets are heated to a temperature equal to the superheat temperature limit, rapid evaporation (flashing liquid) may cause an explosion. In fact, this theory resembles the theory of Reid (1979), except that only droplets, and not bulk liquid, have to be at the superheat temperature limit of atmospheric pressure (McDevitt et al. 1987). 

Hinchliffe, A. and Munn, R. W. (1985) Molecular Electromagnetism, Wiley, Chichester. Hirschfelder, J. O., Curtiss, C. F. and Bird, R. B. Molecular Theory of Liquids and Gases, Wiley, New York. 

Frenkel, J. Kinetic theory of liquids, New York Dover Publ. 1955... 

He is the author of two other books. Nonequilibrium Thermodynamics (1962) and Vector Analysis in Chemistry (1974), and has published research articles on the theory of optical rotation, statistical mechanical theory of transport processes, nonequilibrium thermodynamics, molecular quantum mechanics, theory of liquids, intermolecular forces, and surface phenomena. 

In the theory of liquids, such a phenomenon is well known under the term of de Gennes narrowing [152]. There the peak in S(Q) renormalizes the relaxation rate for density fluctuations Deff = D/S(Q). While in the liquid this is an effect involving different independent particles, here it occurs for the internal density fluctuations of one entity. 

In a contrary to the DFT studies of isolated molecules, where there is a strong link between applications to biological systems and general developments in the theory of density functionals, approaches used for modeling properties of chemical molecules embedded in the biological microscopic environment combine developments in many fields. These fields include DFT, statistical physics, dielectric theory, and the theory of liquids. 

The transitions of the coil-globule type were considered not only in the usual space, but also in the space of monomeric units orientation, where such transition is equivalent to the nematic liquid crystal ordering [60,61]. Such an approach using the formalism developed by I.M. Lifshitz has led to the creation of the theory of liquid crystal ordering in the solutions of semi-flexible macromolecules [62,63]. 

The definition of correlation functions in this book differs from the definition of the correlation coefficient in the theory of probability. The difference is essentially in the normalization, i.e., whereas g(, ) can be any positive number 0 S g the correlation coefficient varies within [-1,1]. We have chosen the definition of correlation as in Eq. (1.5.19) or (1.5.20) to conform with the definition used in the theory of liquids and solutions. 

The quantity W a, b) is the analogue of the potential of the mean force in the theory of liquids. We shall see in the following sections that this quantity sometimes behaves as an energy, but in most cases it is a/ree energy. In terms of W(a, b) we may also define noncooperative systems whenever W(a, b) = 0, and positive and negative cooperativity for W a, Z>) < 0 and W a, b) > 0, respectively. The reader should note the potentially confusing statement that a positive cooperativity involves negative values of W(a, b), and vice versa. 

This is sometimes referred to as the superposition approximation. It is not, however, the superposition approximation used in the theory of liquids, first because Eq. (7.2.34) is exact (in the limit m —> o°), and second because the superposition approximation [as introduced by Kirkwood (1935) and used extensively in the theory of liquids] has the form... 

Clearly, the excluded volume change in reaction (9.5.4) will be noted only when a solvent molecule can interact simultaneously with the two ligands occupying the two sites (Fig. 9.8). This is exactly the same condition for the solvent-induced correlation in the theory of liquids. ... 

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