Liquid-state theory

Chandler D 1982 Equilibrium theory of polyatomic fluids The Liquid State of Matter Fluids, Simple and Complex ed E W Montroll and J L Lebowitz (Amsterdam North-Holland)... 

Stell G 1999 New results on some ionic fluid problems, new approaches to problems in liquid state theory Proc. NATO Advanced Study Institute (Patte Marina, Messina, Italy 1998) ed C Caccamo, J P Hansen and G Stell (Dordrecht Kluwer)... 

Stillinger F 1973 Structure in aqueous solutions from the standpoint of scaled particle theory J. Solution Chem. 2 141 Widom B 1967 Intermolecular forces and the nature of the liquid state Sc/e/ ce 375 157 Longuet-Higgins H C and Widom B 1964 A rigid sphere model for the melting of argon Mol. Phys. 8 549... 

Rasaiah J C 1987 Theories of electrolyte solutions The Liquid State and its Electrical Properties (NATO Advanced Science Institute Series Vol 193) ed E E Kunhardt, L G Christophous and L H Luessen (New York Plenum)... 

In the theory of the liquid state, the hard-sphere model plays an important role. For hard spheres, the pair interaction potential V r) = qo for r < J, where d is the particle diameter, whereas V(r) = 0 for r s d. The stmcture of a simple fluid, such as argon, is very similar to that of a hard-sphere fluid. Hard-sphere atoms do, of course, not exist. Certain model colloids, however, come very close to hard-sphere behaviour. These systems have been studied in much detail and some results will be quoted below. 

The theory of sublimation, t.e. the direct conversion from the vapour to the sohd state without the intermediate formation of the liquid state, has been discussed in Section 1,19. The number of compounds which can be purified by sublimation under normal pressure is comparatively small (these include naphthalene, anthracene, benzoic acid, hexachloroethane, camphor, and the quinones). The process does, in general, yield products of high purity, but considerable loss of product may occur. 

J. A. Barker, Lattice Theories of the Liquid State Pergamon, New York (1963). 

In the liquid state molecules are in intimate contact, so the energetics of molecular interactions generally make a contribution to the overall picture of the mixing process. There are several aspects of the situation that we should be aware of before attempting to formulate a theory for ... 

In this chapter, we have reviewed the RISM-SCF/MCSCF method, which combines electronic structure and liquid-state theories to deal with the chemistry of solutions. The ability... 

In addition to various analytic or semi-analytic methods, which are based on the theory of the liquid state and which are not the subject of this chapter, almost the entire toolbox of molecular computer simulation methods has been applied to the theoretical study of aqueous interfaces. They have usually been adapted and modified from schemes developed in a different context. 

The generic inhomogeneity in liquid state theory is the planar wall (Fig. 7). It is relatively simple. First, the position of the wall does not matter, as long... 

Our second goal is to introduce these simple phenomena in a statistical mechanical scheme such that the calculations keep a transparent significance at each step. Nowadays, the predictions of theoretical approaches depend on approximations of a high level of technicality in the domain of liquid state theory. These approximations seem to have a mathematical rather than... 

S. Dietrich. Fluids in contact with structured substrates. In C. Cacamo, ed. Proceedings of the NATO-ASI, New Approaches to Old and New Problems in Liquid State Theory, Messina, 1998. (Dordrecht Kluwer, 1999). 

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. 

Thus, from an investigation of the compressibility of a gas we can deduce the values of its critical constants. We observe that, according to van der Waals theory, liquid and gas are really two distant states on the same isotherm, and having therefore the same characteristic equation. Another theory supposes that each state has its own characteristic equation, with definite constants, which however vary with the temperature, so that both equations continuously coalesce at the critical point. The correlation of the liquid and gaseous states effected by van der Waals theory is, however, rightly regarded as one of the greatest achievements of molecular theory. 

The normal substances, however, really exhibit small deviations which are all the greater the more complex is the molecule of the substance. The theory of van der Waals, or in fact any hypothesis from which a theorem of corresponding states could be derived, assumes however that the transition from the gaseous to the liquid state, as well as the changes of density in either state, result from alterations in the propinquity of molecules which otherwise remain unaltered. Any association or dissociation of the substance would therefore give rise to abnormalities, and in fact the substances which deviate most from the normal relations (e.g.l water, acetic acid) are those which appear, on other grounds, to be associated in the liquid state. In the case of acetic acid the commencement of polymerisation, even in the state of vapour, is evident from the abnormal densities. 

The vapour-pressure equation (22) has been verified for brom-and iodo-naphthalene by L. Rollo(7) (1909), and for toluene, naphthalene, and benzene by J. T. Barker (1910). The latter finds that the solid and liquid states give the same chemical constant, which is in agreement with Nemst s theory. 

Now calculations of Ta and Tk are plagued by the usual difficulties of liquid state structure theory and the accuracy of approximations, some of which are hard to control. Still, even in the face of such approximations, such microscopic considerations lead us to expect a universal value of y/Tg at Tg as we shall discuss next. 

In this section, rather than give a detailed account of theories of the liquid state, a more qualitative approach is adopted. What follows includes first a description of the structure of ice then from that starting-point, ideas concerning the structure of liquid water are explained. 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

Pratt, L. R., Hummer, G., and Garde, S. (1999). Theories of hydrophobic effects and the description of free volume in complex liquids. In New Approaches to Problems in Liquid State Theory (C. Caccamo, J.-P., Hansen, and G. Stell, eds.), vol. 529, pp. 407-420. Kluwer, Netherlands. NATO Science Series. 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

A quantity of central importance in the study of uniform liquids is the pair correlation function, g r), which is the probability (relative to an ideal gas) of finding a particle at position r given that there is a particle at the origin. All other structural and thermodynamic properties can be obtained from a knowledge of g r). The calculation of g r) for various fluids is one of the long-standing problems in liquid state theory, and several accurate approaches exist. These theories can also be used to obtain the density profile of a fluid at a surface. 

There have been extensive computer simulations and liquid state theories, and a good understanding of these systems is now available. The majority of work has focused on simple hard-chain systems, and the depletion and enhancement effects in these systems are well understood and there are several theories that are in quantitative agreement with computer simulations. In contrast, there has been relatively little attention focused on the effect of wall-fluid and fluid-fluid attractions on the behavior of confined polymers. Only the simplest of DFT approaches has been attempted, and the results are promising although the quantitative performance leaves a lot to be desired. 

The properties of the two helium isotopes in the liquid state are strongly influenced by quantum effects. In Fig. 2.8, the specific heat of 3He, calculated from the ideal gas Fermi model (Tp = 4.9 K) with the liquid 3He density, is compared with the experimental data. The inadequacy of this model is evident. A better fit, especially at the lower temperatures, is obtained by the Landau theory [25]. 

The theory of surface tension, in other words, the problem how certain known facts and certain assumptions about the liquid state can be made to account for the existence of a surface tension, has been treated exhaustively by Laplace, by Gauss, and more recently by Van der Waals. The mathematical apparatus employed is very considerable, and we must confine ourselves to a statement of the... 

The gas A must transfer from the gas phase to the liquid phase. Equation (1) describes the specific (per m2) molar flow (JA) of A through the gas-liquid interface. Considering only limitations in the liquid phase, this molar flow notably depends on the liquid molecular diffusion coefficient DAL (m2 s ). Based on the liquid state theories, DA L can be calculated using the Stokes-Einstein expression, and many correlations have been developed in order to estimate the liquid diffusion coefficients. The best-known example is the Wilke and Chang (W-C) relationship, but many others have been established and compared (Table 45.4) [28-33]. 

In the next sections we will focus on analyzing the dynamics of supercooled liquids in more detail and discuss our findings in terms of the modecoupling theory of the glass transition, which is a liquid state theory that predicts the dynamics from the structural properties of the liquid. 

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