Lattice theory of liquids

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. 

An indirect route has been developed mainly by Kirkwood, which involves molecular distribution functions (MDF) as an intermediate step. The molecular distribution function approach to liquids and liquid mixtures, founded in the early 1930s, gradually replaced the various lattice theories of liquids. Today, lattice theories have almost disappeared from the scene of the study of liquids and liquid mixtures. This new route can be symbolically written as... 

A variant of the lattice theory of liquids has been used to calculate liquid and gas densities and the surface tension [41]. 

J.A. Barker, Lattice theories of liquid state, Oxford Pergamon Press (1%3). 

Baker A.J. (1963) in The International Encyclopedia of Physical hemistry and Chemical Physics Topic 10, the fluid state, J.S. Rowlinson ed. Vol. I, Lattice Theories of Liquid State, Macmillan, New York. 

The zero point density was calculated to be 2.514 g/cm [13]. An equation to calculate the liquid and the vapor density based on a variant of the lattice theory of liquids was derived. Values of the vapor density dg (in mg/cm ) are given below [38] ... 

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

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

The mixing term can be expressed using the liquid-lattice theory of Flory and Huggins [4, 5] as ... 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

The question whether the dispersion bands observed with feebly damped waves by Colley, Obolensky, Romanoff, Potapenko, and others (Handbuch der Physiky 15, 514 et seq. (Berlin, 1927)) in the region of wave-lengths of a few decimetres correspond to intramolecular vibrations might be determined by comparison of the spectra of the vapours with those of the liquids. As is well known, the kinetic theory of liquids has recently exhibited a decided tendency to follow the theory of crystal lattices more closely in many respects for a comprehensive account of some of the most important papers on this subject see K. Jellinek, Lehrbuch der physikalischen ChemiCy 1, 824 et seq, especially pp. 828, 831 (Stuttgart, 1928). 

We shall not attempt to review and compare critically various theories of liquid crystallinity in this chapter. Inasmuch as theory based on a lattice model has proved most successful in the treatment of liquid crystallinity in polymeric systems, we shall present an abbreviated account of that theory confined to its essential aspects. The versatility of this theory has permitted its extension to polydisperse systems, to mixtures of rodlike polymers with random coils and to some of the many kinds of semirigid chains. These ramifications of the theory will be discussed in this chapter... 

The lattice model, as put forth by Flory [84, 85], has been proved successful in the treatments of the liquid crystallinity in polymeric systems, despite its artificiality. In our series of work, the lattice model has been extended to the treatment of biopolypeptide systems. The relationship between the polypeptide ordering nature and the LC phase structure is well established. Recently, by taking advantage of the lattice model, we formulated a lattice theory of polypeptide-based diblock copolymer in solution [86]. The polypeptide-based diblock copolymer exhibits lyotropic phases with lamellar, cylindrical, and spherical structures when the copolymer concentration is above a critical value. The tendency of the rodlike block (polypeptide block) to form orientational order plays an important role in the formation of lyotropic phases. This theory is applicable for examining the ordering nature of polypeptide blocks in polypeptide block copolymer solutions. More work on polypeptide ordering and microstructure based on the Flory lattice model is expected. 

This chapter summarizes the thermodynamics of multicomponent polymer systems, with special emphasis on polymer blends and mixtures. After a brief introduction of the relevant thermodynamic principles - laws of thermodynamics, definitions, and interrelations of thermodynamic variables and potentials - selected theories of liquid and polymer mixtures are provided Specifically, both lattice theories (such as the Hory-Huggins model. Equation of State theories, and the gas-lattice models) and ojf-lattice theories (such as the strong interaction model, heat of mixing approaches, and solubility parameter models) are discussed and compared. Model parameters are also tabulated for the each theory for common or representative polymer blends. In the second half of this chapter, the thermodynamics of phase separation are discussed, and experimental methods - for determining phase diagrams or for quantifying the theoretical model parameters - are mentioned. 

Theories of Liquid Mixtures 2.5.1 Lattice, Cell, and Hole Theories... 

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 most successful statistical theory of liquids is that derived by Simha and Somcynsky. The model considers liquids to be mixtures of voids dispersed in solid matter, i.e., a lattice of unoccupied and occupied sites. The occupied volume fraction, y (or its counterpart the free volume fraction f = 1 - y), is the principal variable y = P, T). From die configurational partition function the configurational contribution to the Helmholtz molar free energy of liquid i was expressed as [3] ... 

Ballauf extended the lattice theory of mixtures for blends of nematic liquid crystals and flexible polymers to include the effect of isotropic interactions between the components ... 

Cell or lattice models. Cell theories of liquids, such as the Lennard-Jones-Devonshire theory [177] have been applied to adsorption phenomena. For example, cell models including lateral interactions [178] permit the interpretation of experimental isosteric heats in multilayer adsorption [179,180]. 

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