Perturbation Theories of Liquids

In this section, we present another demonstration of the application of the general theorem of Section 3.2 to obtain a first-order term in a perturbation expansion of the free energy. 

Consider a system in the T, F, N ensemble obeying the pairwise additivity assumption for the total potential energy  

suppose we can separate the pair potential into two parts 

C/ (X ) is referred to as the total potential energy of the unperturbed system, whereas C/ (X ) is considered to be the perturbation energy. The basic distribution function in the unperturbed system is 

The symbol stands for an average over the unperturbed system (here, in the T, V, N ensemble). Clearly, even when we assume the pairwise additivity (3.171), the average in (3.178) is not an average of a pairwise quantity. However, if the perturbation energy C7y (X ) is considered to be small compared with kT, we can expand the exponential function on the rhs of (3.178) to obtain 

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The eelebrated Pratt-Chandler (PC) theory is usually the starting point of any diseussion on the hydrophobie effeet. This theory ean be regarded as an application of the Weeks-Chandler-Andersen (WCA) perturbative theory of liquids to the solvation of one and a pair of non-polar solute moleeules. While Stillinger discussed the ehemieal potential involved in ereating a hard-sphere cavity in water using the scaled partiele theory, the Pratt-Chandler theory used an integral equation deserip-tion and showed how to properly discuss the effect within a general statistieal mechanical theory. 

Hermann(145) used a method based on the first order perturbation theory of liquid mixture in order to calculate the solubilities of hydrocarbons in water and the hydrophobic interaction. 

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... 

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. 

Micellar aggregates are considered in chapter 3 and a critical concentration is defined on the basis of a change in the shape of the size distribution of aggregates. This is followed by the examination, via a second order perturbation theory, of the phase behavior of a sterically stabilized non-aqueous colloidal dispersion containing free polymer molecules. This chapter is also concerned with the thermodynamic stability of microemulsions, which is treated via a new thermodynamic formalism. In addition, a molecular thermodynamics approach is suggested, which can predict the structural and compositional characteristics of microemulsions. Thermodynamic approaches similar to that used for microemulsions are applied to the phase transition in monolayers of insoluble surfactants and to lamellar liquid crystals. 

The following discussion is more technical but is useful in a subsequent section. We consider again perturbative interactions and Eq. (4.4). It may sometimes happen that the perturbative interactions I>q, are uncertainly known, but preliminary calculations can obtain conditional densities and density variances. Thus, more primitive available information might be [pj r) Ol ) and [8py r)8p r ) Ol ). In the traditional theory of liquids, attention is often directed to density functional aspects of the theory, and the perturbative interactions are perfectly known as a model. 

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. 

In view of the failure of the rigid sphere model to yield the correct isochoric temperature coefficient of the viscosity, the investigation of other less approximate models of the liquid state becomes desirable. In particular, a study making use of the Lennard-Jones and Devonshire cell theory of liquids would be of interest because it makes use of a realistic intermolecular potential function while retaining the essential simplicity of a single particle theory. The main task is to calculate the probability density of the molecule within its cell as perturbed by the steady-state transport process. 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

Relation (3.183) is useful whenever we know the free energy of the unperturbed system and when the perturbation energy is small compared with kT. It is clear that if we take more terms in the expansion (3.179), we end up with integrals involving higher-order molecular distribution functions. Therefore, such an expansion is useful only for the cases discussed in this section. For a recent review on the application of perturbation theories to liquids see Barker and Henderson (1972). 

In solutions in general, and in polymer solutions in particular, the most frequently employed treatments are based on different variants of lattice theory. Perturbation methods, modelling and special theories of liquids are also applied, and some elements are borrowed from models. 

The first two of these results (4.141) and (4.142) are of little use unless we know or can approximate the function c(ri. t2 ap) for all values of a. They were obtained first by Stillinger and Buff by a cluster expansion and by Lebowitz and Percus by using functional integration, but we owe to Saam and Ebner the comment that since F[p] is a unique functional of p(i) then the values of F and 0 calculated in this way are independent of the path in p-space (4.136). The third result (4.146), although restricted to pair potentials, is a useful starting point for the development of perturbation theories of both bulk liquids and of the gas-liquid surface. 

Barker, J.A, and Henderson, D., 1967, Perturbation Theory and Equation of State for Fluid. II. A Successful Theory of Liquids, J. Chem. Phys., 47 4714. 

In the small enlarged part of the jet, the dashed line shows the stretching of the jet because of acting forces. Additionally, due to the mentioned capillary instability, the actual jet surface is wavelike, which is illustrated by the solid fine. Assuming the flow of the particle-laden liquid to be rotational symmetric, the jet surface can be described by the two radii of curvature Ci and C2- According to the perturbation theory of small disturbances [ 1, 35], the flow is decomposed into a time steady basic flow and superimposed wavelike disturbances. Consequently, the jet radius is separated as it is shown in Fig. 5.1 and Eq. (5.1). 

More modem approaches borrow ideas from the liquid state theory of small molecule fluids to develop a theory for polymers. The most popular of these is the polymer reference interaction site model (PRISM) theory " which is based on the RISM theory of Chandler and Andersen. More recent studies include the Kirkwood hierarchy, the Bom-Green-Yvon hierarchy, and the perturbation density functional theory of Kierlik and Rosinbeig. The latter is based on the thermodynamic perturbation theory of Wertheim " where the polymeric system is composed of very sticky spheres that assemble to form chains. For polymer melts all these liquid state approaches are in quantitative agreement with simulations for the pair correlation functions in short chain fluids. With the exception of the PRISM theory, these liquid state theories are in their infancy, and have not been applied to realistic models of polymers. 

Most studies of disordered solids have been based on simple tight binding Hamiltonians of the kind described in Section 3.3. While this approach is of limited validity, it is at least susceptible to a certain amount of rigorous mathematical analysis. Other Hamiltonians, such as pseudopotential Hamiltonians, which might be more desirable in a given context, pose many more difficulties in a disordered system unless simple lowest-order perturbation theory happens to be adequate, as in the case of the Ziman theory of liquid metals, which is quite successful for the simple metals. 

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

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