Solution lattice theory

The "solid-solution lattice theory" model of adsorption on zeolites has been shown to describe the experimental results in the literature with accuracy comparable to all existing theories, even though these theories are in many instances semi-empirical. Since the theory is related to actual physical phenomena, systematic studies can be made of the effect on adsorption of changing the zeolite. 

It can be concluded that the "solid-solution lattice theory" model shows great promise in describing adsorption on zeolites. 

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

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 The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

Aspler and Gray (65.69) used gas chromatography and static methods at 25 C to measure the activity of water vapor over concentrated solutions of HPC. Their results indicated that the entropy of mixing in dilute solutions is mven by the Flory-Huggins theory and by Flory s lattice theory for roddike molecules at very nigh concentrations. 

A ternary system consisting of two polymer species of the same kind having different molecular weights and a solvent is the simplest case of polydisperse polymer solutions. Therefore, it is a prototype for investigating polydispersity effects on polymer solution properties. In 1978, Abe and Flory [74] studied theoretically the phase behavior in ternary solutions of rodlike polymers using the Flory lattice theory [3], Subsequently, ternary phase diagrams have been measured for several stiff-chain polymer solution systems, and work [6,17] has been done to improve the Abe-Flory theory. 

Flory-Huggins /u. The Flory-Huggins n value measures the interaction between polymer and solvent (plasticizer). It derives from the so-called lattice theory, which represents a statistical approach to the behavior of polymer molecules in solution (10, 14, 75, 16, 22). The n value may be experimentally determined for any polvmer-plasticizer system (where the plasticizer can dissolve the polymer) by osmotic pressure measurements according to the relation ... 

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). 

Self-consistent field theory (SCFT, see Sections 2.3.3 and 3.4,2) has recently been applied to the phase behaviour of ordered micellar solutions. Noolandi et al. (1996) compared continuum SCFT to the lattice version of this theory for triblock copolymers such as the Pluronics in aqueous solution. From a different viewpoint, this work represents an extension of the SCFT employed by Hong and Noolandi (1981, 1983) and Matsen and Schick (1994) for the phase behaviour of block copolymer melts to block copolymers in solution. The approximations introduced by the adoption of a lattice model are found to lead to some significant differences in the solution phase behaviour compared with the continuum theory, as illustrated by Fig. 4.44. For example, the continuum theory predicts ordered phases for Pluronic L64 (PE013PP03oPEO 3), whereas the lattice theory (neglecting polydispersity) predicts none. 

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... 

A wide variety of theories have been developed for polymer solutions over the later half of the last century. Among them, lattice model is still a convenient starting point. The most widely used and best known is the Flory-Huggins lattice theory (Flory, 1941 Huggins, 1941) based on a mean-field approach. However, it is known that a mean-field approximation cannot correctly describe the coexistence curves near the critical point (Fisher, 1967 Heller, 1967 Sengers and Sengers, 1978). The lattice cluster theory (LCT) developed by Freed and coworkers (Freed, 1985 Pesci and Freed, 1989 Madden et al., 1990 Dudowicz and Freed, 1990 Dudowicz et al., 1990 Dudowicz and Freed, 1992) in 1990s was a landmark. 

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). 

On the other hand, the Flory-Huggins lattice theory (Flory, 1953) predicts for polymer solutions ... 

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. 

However, only limited experimental studies on the thermodynamic properties of polypeptide solutions have been carried out. The results of vapor sorption studies for PBLG and poly(P-benzyl L-aspartate) solutions at high polymer concentrations by Flory and Leonard could not be explained by the Flory model, but could be explained by assuming that mixing of solvent with flexible side chains dominates the thermodynamic behavior at high concentrations. Rai and Miller obtained similar results for the PBLG-dimethylformamide (DMF) system at high concentrations. They also showed that the results could be explained by the Wee-MiUer theory in which modification of Flory s lattice theory to allow for side chain... 

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