Solution models Flory

Haeany Solution Model The initial model (37) considered the adsorbed phase to be a mixture of adsorbed molecules and vacancies (a vacancy solution) and assumed that nonideaUties of the solution can be described by the two-parameter Wilson activity coefficient equation. Subsequendy, it was found that the use of the three-parameter Flory-Huggins activity coefficient equation provided improved prediction of binary isotherms (38). 

So far two models have been employed to rationalize the solvation process the classical solution model, either the mole-fraction scale or any other concentration scale, and the Flory-Huggins model. The question is where to use which theoretical model to interpret the results of partitioning experiments, in which solute molecules distribute between two phases, a and ft. If the two phases are at equilibrium at the same temperature and the same pressure, /z = /xf. After rearrangement and applying Eq. (11-8), we can write... 

The Flory-Huggins model differs from the regular solution model in the inclusion of a nonideal entropy term and replacement of the enthalpy term in solubility parameters by one in an interaction parameter x- This parameter characterizes a pair of components whereas each S can be deduced from the properties of a single component. 

The sodium-lithium phase system has been studied by thermal analysis in the liquid and solid regions to temperatures in excess of 400°C. Two liquid phases separate at 170.6°C. with compositions of 3.4 and 91.6 atom % sodium. The critical solution temperature is 442° zt 10°C. at a composition of 40.3 atom % sodium. The freezing point of pure lithium is depressed from 180.5°C. to 170.6°C. by the addition of 3.4 atom % sodium, and the freezing point of pure sodium is depressed from 97.8° to 92.2°C. by the addition of 3.8 atom % lithium. From 170.6° to 92.2°C. one liquid phase exists in equilibrium with pure lithium. Regardless of the similarity in the properties of the pure liquid metals, the binary system deviates markedly from simple nonideal behavior even in the very dilute solutions. Correlation of the experimentally observed data with the Scatchard-Hildebrand regular solution model using the Flory-Huggins entropy correction is discussed. 

Corrales, L.R., Wheeler, J.C. Tetracritical and novel tricritical points in sulfur solutions A Flory model for polymerization of rings and chains in solvent. J. Chem. Phys. 1989, 90, 5030. 

The two VSM isotherm equations are given in eqs. (2.7-5) and (2.7-7) depending on whether the Wilson equation or the Flory-Huggin equation is used to calculate the activity coefficient. Observing the form of these equations, the vacancy solution model equation can be written in general form as follows ... 

To overcome some of the limitations of his early model, Flory, together with Krieg-baum (70), made a second assumption. They postulated the presence of the excluded volume, the volume occupied by a polymer chain that exhibits long-range intramolecular interactions. These interactions were innoduced as an enthalpic term (AT,) and an entropic term (Oi, as described later. The two terms are equal if AGm = 0. The temperature at which these conditions prevail in a given solvent is the 0 temperatme. At this temperature the effects of the excluded volume are eliminated and the polymer chain adopts its unperturbed conformation in dilute solution. In other words, the 0 temperature is the lowest temperature at which a polymer of infinite molecular weight is completely soluble in a given solvent. 

Limitations of the SCLF method include (1) electrostatic Coulombic interactions between the solute and surface moleucles are ignored, (2) the dependence of calculated adsorption results on the model parameters (such as the solute-solute, solute-surface and solute-solvent Flory-Huggins interaction parameters, the lattice site size, etc.) is difficult to determine, (3) adsorption must be determined by simultaneous solution of probability density equations derived from partition functions so that no single analytical adsorption equation is possible, (4) effects of pH on surface sites can only be considered implicitly through the Flory-Huggins interaction parameters, and (5) the Flory-Huggins interaction parameters do not allow explicit consideration of the molecular or chemical characteristics of the surface site molecules. 

The contribution of the elastic term in lightly cross-linked networks can be described by the Gaussian theory of rubber elasticity (2,3). In fully neutralized polyelectrolytes, in the presence of added salt, the ionic term is not expected to play an exphcit role. Ionic interactions, however, may modify the mixing free energy contribution. In neutral polymer solutions the Flory-Huggins theory (I), based on the lattice model of solutions, expresses the mixing pressure as... 

Figure A2.5.27. The effective coexistence curve exponent P jj = d In v/d In i for a simple mixture N= 1) as a fimction of the temperature parameter i = t / (1 - t) calculated from crossover theory and compared with the corresponding curve from mean-field theory (i.e. from figure A2.5.15). Reproduced from [30], Povodyrev A A, Anisimov M A and Sengers J V 1999 Crossover Flory model for phase separation in polymer solutions Physica A 264 358, figure 3, by pennission of Elsevier Science.

The constants K depend upon the volume of the solvent molecule (assumed to be spherica in slrape) and the number density of the solvent. ai2 is the average of the diameters of solvent molecule and a spherical solute molecule. This equation may be applied to solute of a more general shape by calculating the contribution of each atom and then scaling thi by the fraction of fhat atom s surface that is actually exposed to the solvent. The dispersioi contribution to the solvation free energy can be modelled as a continuous distributioi function that is integrated over the cavity surface [Floris and Tomasi 1989]. 

Figure 8.1 The entropy of mixing (in units of R) as a function of mole fraction solute for ideal mixing and for the Flory-Huggins lattice model with n = 50, 100, and 500. Values are calculated in Example 8.1.

Both Flory [143] and Huggins [144] in 1941 addressed themselves to this problem with the initial aim of describing solutions of linear polymers in low molecular weight solvents. Both used lattice models, and their initial derivations considered only polymer length (rather than shape, i.e. branching, etc.) The derivation given here will also limit itself to differences in molecular size, but will be based on an available volume approach. 

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

It is an arduous task to develop thermodynamic models or empirical equations that accurately predict solvent activities in polymer solutions. Even so, since Flory developed the well-known equation of state for polymer solutions, much work has been conducted in this area [50-52]. Consequently, extensive experimental data have been published in the literature by various researchers on different binary polymer-solvent sys-... 

The formation mechanism of structure of the crosslinked copolymer in the presence of solvents described on the basis of the Flory-Huggins theory of polymer solutions has been considered by Dusek [1,2]. In accordance with the proposed thermodynamic model [3], the main factors affecting phase separation in the course of heterophase crosslinking polymerization are the thermodynamic quality of the solvent determined by Huggins constant x for the polymer-solvent system and the quantity of the crosslinking agent introduced (polyvinyl comonomers). The theory makes it possible to determine the critical degree of copolymerization at which phase separation takes place. The study of this phenomenon is complex also because the comonomers act as diluents. 

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