Lattice theory of solution

A lattice theory of solutions has been proposed (.k) to describe the adsorption-desorption phenomena in zeolites. There are several reasons for this choice (a) forming a solid solution by two substances is analogous to the forming of an adsorbed phase in the cavities of a zeolite, (b) the theory of solutions is well understood and its mathematical techniques powerful, and (c) since the state-of-the-art in description of adsorption phenomena in... 

In the phenomenological characterization of small deviations from SI solutions, the concepts of regular and athermal solutions were introduced. Normally, the theoretical treatment of these two cases was discussed within the lattice theories of solutions. Here, we discuss only the very general conditions for these two deviations to occur. First, when Pt ab does not depend on temperature, we can differentiate (6.19) with respect to T to obtain... 

We noted in section 6.6 that care must be exercised when examining the conditions for stability, based on first-order deviations. The reason is simple -the first-order expansion is valid only for small values of PtAab- Clearly, one cannot use the first-order expansion to examine the behavior of these solutions at large values of PjAab To the best of the author s knowledge the molecular conditions for stability were never studied beyond the first-order correction to SI solutions. This is true also within the lattice theories of solutions where the conditions for stability were examined in terms of the exchange energy W, in equation (6.124). 

This expression, or the equivalent one in the lattice theory of solution, has been traditionally used to analyze deviations from SI solution and stability of the system. It is clear from (P.2) that when pAAB is positive (or negative), deviations from Raoult s law will be positive (or negative) and vice versa. 

The second hypothesis claims that the denaturants preferentially bind to the snr-face of the proteins (Timasheff 2002a) the larger the snrface area, the more denatur-ant molecules are bound to each protein the denatured state therefore becomes more stable than the native state. Both of these proposals have been founded upon primitive and antiquated models of solutions the lattice theory of solution is the foundation of the water structure breaker hypothesis (Frank and Franks 1968), whereas the stoichiometric binding model of solvation is the basis of the preferential binding hypothesis (Scheltman 1987 Timasheff 2002a). Consequently, the weak theoretical foundation had prompted much debate, not only over the validity of these hypotheses, but also over the true meaning of these hypotheses at a molecular level. 

This chapter is concerned with a few aspects of the theory of solutions which are either of fundamental character or are believed to be particularly useful in the study of aqueous solutions. With these limitations in mind, we have eliminated, for instance, discussion of lattice theories of solutions, which have played an important role in the development of the thermodynamics of mixing. We also confine ourselves to first-order expansion theories, which, for all practical purposes, are the most useful parts of the complete and general schemes. 

The quantity /if may be hypothetical, but the thermodynamic concept is very distinct, as is evidenced by Henry s law on solubility of gases in solution. Henry s law will be explained later by the lattice theory of solution. ... 

The lattice theory of solution is derived from several idealized assumptions. The assumptions that components A and B are the same size and have the same number of nearest neighbors, for example, are not applicable to real solutions. Ilie regular solution concept of Hildebrand is more versatile it takes into account a mixture of molecules of different sizes, where the principal idea is an ideal entropy of mixing at constant volume irrespective of heat. The activity coefficients in the form of (3.9) due to interaction between components A and B in a liquid mixture are derived by the following equations when the mixing term is expressed as a volume fraction ... 

The logic that leads us to this last result also limits the applicability of the ensuing derivation. Applying the fraction of total lattice sites vacant to the immediate vicinity of the first segment makes the model descriptive of a relatively concentrated solution. This is somewhat novel in itself, since theories of solutions more commonly assume dilute conditions. More to the point, the model is unrealistic for dilute solutions where the site occupancy within the domain of a dissolved polymer coil is greater than that for the solution as a whole. We shall return to a model more appropriate for dilute solutions below. For now we continue with the case of the more concentrated solution, realizing... 

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)...

Although the theory of solutions has been widely used in formulating problems of defects in solids the problems encountered differ in certain respects. The most obvious point is that defects are restricted to discrete lattice sites, whereas the ions in a solution can occupy any position in the fluid. Sometimes no allowance is made for this fact. For example, it has not been demonstrated that at very low concentrations, in the absence of ion-pair effects, the activity coefficients are identical with those of the Debye-Hiickel theory. It can be plausibly argued51 that at sufficiently low concentrations the effect of discreteness is likely to be negligible, but clearly in developing a theory for any but the lowest concentrations the effect should be investigated. A second point... 

It has been remarked in the preceding sections that the equilibrium concentration of monomer in solution of its living polymer is affected by the nature of the solvent and by the polymer concentration, because these factors influence the activities of the components. A quantitative treatment of these effects, based on Scott s modification of the standard lattice theory of polymer solutions (33), has been outlined recently by Bywater (34). 

A thermodynamic approach was put forward by one of us (10), based on the Flory-Huggins lattice theory of a polymer solution the chemical potentials of each monomer must be equal in each phase copolymerization increment causes a little change in the chemical potential in the particles diffusion of monomers from the water phase will reequilibrate the system and in turn diffusion from droplets to water phase takes place. For instance, expression from monomer 1 in the particles is ... 

With increase in salt concentration the approximations involved in the Debye-Hiickel theory become less acceptable. Indeed it is noteworthy that before this theory was published a quasi-lattice theory of salt solutions had been proposed and rejected (Ghosh, 1918). However, as the concentration of salt increases so log7 ,7 being the mean ionic activity coefficient, appears as a linear function of c1/3 (the requirement of a quasi-lattice theory) rather than c1/2, the DHLL prediction (Robinson and Stokes, 1959). Consequently, a quasi-lattice theory of salt solutions has attracted continuing interest (Lietzke et al., 1968 Desnoyers and Conway, 1964 Frank and Thompson, 1959 Bahe, 1972 Bennetto, 1973) and has recently received some experimental support (Neilson et al., 1975). 

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. 

Summarizing, one can say that the lattice theories need improvement and compact macromolecules need more refined treatment. We shall develop in this paper a refined and unified theory of macromolecular solutions with special emphasis on dilute solutions. We shall put our standpoint on the general theory of solutions developed by McMillan and Mayer in 1945 and Kirkwood and Buff in 1951 (9). TTiese theories do not use the lattice model and are more natural for application especially to dilute solutions. The theories extend statistical theories on gases and this is the reason why we used the name gas theories (70) in the beginning of this Introduction. 

The lattice theories have introduced very naturally the volume concentration of polymer molecules and the entire polymer solution properties have been described in terms of this volume concentration. It is convenient to use this quantity because expansions in the volume concentration converge rather rapidly in the case of pol3oner solutions as compared with expansions in other concentrations. In the general theory of solution, however, the volume of a polymer or of a solvent molecule is given only thror h the molecular potentials. Nevertheless, without using the potentials we shall define by Eq. (4.1) a quantity

[Pg.247]

The present calculations are in agreement with the conclusion of ref 59 (which employed both a lattice and the McMillan— Mayer theories of solution" ) that the solute—solute interactions in the systems investigated increase in the sequence MeOH < EtOH < 2-PrOH < 1-PrOH t-BuOH. There are, however, essential differences between the lower alcohols (MeOH and EtOH) and the higher ones. 

The term regular solutions was first coined by Hildebrand (1929). It was characterized phe-nomenoligically in terms of the excess entropy of mixing. It was later used in the context of lattice theory of mixtures mainly by Guggenheim (1952). It should be stressed that in both the phenomenological and the lattice theory approaches, the regular solution concept applies to deviations from SI solutions, (see also Appendix M). 

In this appendix, we present a very brief outline of the lattice theory of ideal and regular solutions, developed mainly by Guggenheim (1952). The main reason for doing so is to emphasize the first-order character of the deviations from SI, equation (M.12) below. 

The micelle has too small an aggregation number to be considered as a phase in the usual sense, and yet normally contains too many surfactant molecules to be considered as a chemical species. It is this dichotomy that makes an exact theory of solubilization by micelles difficult. The primary theoretical approaches to the problem are based on either a pseudophase model, mass action model, multiple equilibrium model, or the thermodynamics of small systems [191-196]. Technically, bulk thermodynamics should not apply to solute partitioning into small aggregates, since these solvents are interfacial phases with large surface-to-volume ratios. In contrast to a bulk phase, whose properties are invariant with position, the properties of small aggregates are expected to vary with distance from the interface [195]. The lattice model of solute partitioning concludes that virtually all types of solutes should favor the interface over the interior of a spherical micelle. While for cylindrical micelles, the internal distribution of solutes... 

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. 

Some novel statistical theories of solutions of polymers use the %i parameter, too. They predict the dependence of the %i parameter on temperature and pressure. According to the Prigogine theory of deformable quasi-lattice, a mixture of a polymer with solvents of different chain length is described by the equation ... 

The theory of solutions assumes that molecules are spherical. For long-chain polymer molecules, consisting of segments, the number of modes of arrangement in a solution lattice differs from those of spherical molecules. This results in deviations from the ideal entropy of mixing. The polymer-plasticizer interaction differs qualitatively because of the presence of segments coimected into chains. 

Hory-Huggins Lattice Theory of Polymer Solutions... 

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