Liquid-lattice theory

Thermodynamic predictions based on the liquid lattice theory do not fit osmotic experimental data [5],... 

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

Ottewill and Walker derived an equation for the energy change for the overlap of adsorbed layers by using the Flory s liquid lattice theory for polymer solutions ... 

Insofar as the liquid lattice theory may be viewed as a random walk process, where n. steps can be taken in each of i=l,2,...0" directions, each with a probability W., we see that the total number of different configurations VJ untethered is... 

B. Liquid Lattice Theory (Flory-Huggins Theory)... 

B. LIQUID LATTICE THEORY (FLORY-HUGGINS THEORY)... 

Flory and Huggins used a liquid lattice theory to express p — p as a function of the volume fraction Vj of polymer in the mixture, the polymer-solvent interaction parameter % and the ratio x of molar volumes of polymer Vj and solvent V. ... 

Of particular interest has been the study of the polymer configurations at the solid-liquid interface. Beginning with lattice theories, early models of polymer adsorption captured most of the features of adsorption such as the loop, train, and tail structures and the influence of the surface interaction parameter (see Refs. 57, 58, 62 for reviews of older theories). These lattice models have been expanded on in recent years using modem computational methods [63,64] and have allowed the calculation of equilibrium partitioning between a poly-... 

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

By using the liquid lattice approach to treat the random mixing of a disoriented polymer and a solvent, the so-called Flory-Huggins theory is often used to correlate the penetrant activity and the composition of the solution ... 

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 The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

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. 

Taking into account the modes in which the water can be sorbed in the resin, different models should be considered to describe the overall process. First, the ordinary dissolution of a substance in the polymer may be described by the Flory-Huggins theory which treats the random mixing of an unoriented polymer and a solvent by using the liquid lattice approach. If as is the penetrant external activity, vp the polymer volume fraction and the solvent-polymer interaction parameter, the relationship relating these variables in the case of polymer of infinite molecular weight is as follows ... 

Later Hildebrand [10] obtained the same result assuming that free volume available to the molecules per unit volume of liquid is the same for the polymer as for the solvent. The heat of mixing is defined as the difference between the total interaction energy in the mixture compared with that of pure components. Based on their lattice theory model, Flory [7,8,9] and Huggins [11,12] obtained the following expression for the heat of mixing ... 

The diamter of the PBG a-helix has been estimated variously to be between 15-25 A as noted above (25). Values of d used in the determination of persistence len s are listed in the solvent categories of Table I. Values of Vp calculated from Equation 1 using x = L/d are invariably too low for the solvents of this study. Better agreement with the lattice theory has been reported for critical volume fractions of PBG in dimethylformamide and m-cresol (32). Experimental volume fractions for liquid crystal formation of PM in dioxane are lower, however, than those calculated from Flory s theory (33). PBG is known to undergo extensive... 

To calculate AWm (the enthalpy of mixing) the polymer solution is approximated by a mixture of solvent molecules and polymer segments, and AW is estimated from the number of 1,2 contacts, as in Section 12.2.1. The terminology is somewhat different in the Flory-Huggins theory, however. A site in the liquid lattice is assumed to have z nearest neighbors and a line of reasoning similar to that developed above for the solubility parameter model leads to the expression... 

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. 

Figure 5.14 (a) Temperature-volume fraction phase diagram for PBLG Ma, = 310,000) in DMF, where I denotes an isotropic phase, LC denotes a chiral nematic liquid-crystalline phase, and I + LC is a gel that is presumed to be two coexisting phases that are unable to separate macroscopically. (b) The x-volume fraction phase diagram predicted by the Flory lattice theory for rigid rods of axial ratio (length/diameter) = 150. (From Miller et al. 1974, with permission.)... 

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