Solubility parameters Flory-Huggins model

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 solubility parameter is a very important property in science and has found widespread use in many fields and not just in the smdy of polymer-solvent thermodynamics. It is connected to the Flory-Huggins model as well, as explained in Section 16.3.3.2, but can also be used independent of it, as discussed in Sections 16.3.3.1 and 16.3.3.3. Several handbooks and reference books provide extensive lists of solubility parameters of numerous chemicals.The solubility parameter is defined as... 

Lindvig, Th., Michelsen, M.L., and Kontogeoigis, G.M., 2002. A Flory-Huggins model based on the Hansen solubility parameters. Fluid Phase Equilib., 203 247. 

These authors showed that for a number of polymer-solvent system with a = 0.6 this method performs similarly to group contribution methods using volume fractions to represent the segment fractions in the Flory-Huggins model. Values of solubility parameters are tabulated by Barton [36]. 

For the solubility of TPA in prepolymer, no data are available and the polymer-solvent interaction parameter X of the Flory-Huggins relationship is not accurately known. No experimental data are available for the vapour pressures of dimer or trimer. The published values for the diffusion coefficient of EG in solid and molten PET vary by orders of magnitude. For the diffusion of water, acetaldehyde and DEG in polymer, no reliable data are available. It is not even agreed upon if the mutual diffusion coefficients depend on the polymer molecular weight or on the melt viscosity, and if they are linear or exponential functions of temperature. Molecular modelling, accompanied by the rapid growth of computer performance, will hopefully help to solve this problem in the near future. The mass-transfer mechanisms for by-products in solid PET are not established, and the dependency of the solid-state polycondensation rate on crystallinity is still a matter of assumptions. 

In addition to the solubility parameter model to treat SEC adsorption effects, an approach based on Flory-Huggins interaction parameters has also been proposed (24-27). For an excellent review of both mechanisms, see reference 28.- A general treatment of polymer adsorption onto chromatographic packings can be found in Belenkii and Vilenchik s recent book (29). 

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... 

Equation (12-23) suffers from the same limitations as the simple solubilty parameter model, because the expression for Wm is derived by assuming that in-termolecular forces are only nondirectional van der Waals interactions. Specific interactions like ionic or hydrogen bonds arc implicitly eliminated from the model. The solubility parameter treatment described to this point cannot take such inler-actions into account because each species is assigned a solubility parameter that is independent of the nature of the other ingredients in the mixture. The x parameter, on the other hand, refers to a pair of components and can include specific interactions even if they are not explicitly mentioned in the basic Flory-Huggins theory. Solubility parameters are more convenient to use because they can be assigned a priori to the components of a mixture, x values are more realistic, but have less predictive use because they must be determined by experiments with the actual mixture. 

The Flory-Huggins theory has been modified and improved and other models for polymer solution behavior have been presented. Many of these theories are more satisfying intellectually than the solubility parameter model but the latter is still the simplest model for predictive uses. The following discussion will therefore focus mainly on solubility parameter concepts. 

Just a few years ago, the limitations of solubility parameter calculations and measurements discussed above were serious impediments to modeling the phasic and interfacial behaviors of polymeric systems. The coming of age of atomistic simulation methods over the last few years has improved this situation dramatically. As discussed in Section 5.A.3, whenever accuracy is important in calculating the phasic or the interfacial behavior of a system, it is nowadays strongly preferable to use atomistic simulations employing modem force fields of the highest available quality instead of solubility parameters in order to estimate the Flory-Huggins interaction parameters (%) between the system components as input for further calculations. 

For the description of such interactions as well as of polymer swelling, models based on the Flory-Huggins Theory (Flory, 1953 Mulder, 1991) and UNIQUAC are often applied for mixtures in general and, for binary mixtures, also the Solubility Parameter Theory if the feed components are hydrophobic (Hildebrand and Scott,... 

Prior to Harwood s work, the existence of a Bootstrap effect in copolymerization was considered but rejected after the failure of efforts to correlate polymer-solvent interaction parameters with observed solvent effects. Kamachi, for instance, estimated the interaction between polymer and solvent by calculating the difference between their solubility parameters. He found that while there was some correlation between polymer-solvent interaction parameters and observed solvent effects for methyl methacrylate, for vinyl acetate there was none. However, it should be noted that evidence for radical-solvent complexes in vinyl acetate systems is fairly strong (see Section 3), so a rejection of a generalized Bootstrap model on the basis of evidence from vinyl acetate polymerization is perhaps unwise. Kratochvil et al." investigated the possible influence of preferential solvation in copolymerizations and concluded that, for systems with weak non-specific interactions, such as STY-MMA, the effect of preferential solvation on kinetics was probably comparable to the experimental error in determining the rate of polymerization ( 5%). Later, Maxwell et al." also concluded that the origin of the Bootstrap effect was not likely to be bulk monomer-polymer thermodynamics since, for a variety of monomers, Flory-Huggins theory predicts that the monomer ratios in the monomer-polymer phase would be equal to that in the bulk phase. 

Also a thermodynamic model based on the coupled Equation of State model and Flory-Huggins theory for polymer solutions was developed. The model parameters such as solubility-parameter of asphaltenes, molecular weight of asphaltenes, and molar volume of asphaltenes were obtained by fitting the model to experimental data. 

A thermodynamic model based on Flory-Huggins polymer-solution theory was developed and coupled with Equation of State model to predict the amount of asphaltene precipitation. The model prediction shows close agreement with the experimental data after regression of asphaltene properties such as molar volume, solubility parameter and molecular weight. The model, however, fails to account for the effect of large changes in the solubility parameters of the oil-solvent mixtures. 

The simple Flory-Huggins %-function, combined with the solubility parameter approach may be used for a first rough guess about solvent activities of polymer solutions, if no experimental data are available. Nothing more should be expected. This also holds true for any calculations with the UNIFAC-fv or other group-contribution models. For a quantitative representation of solvent activities of polymer solutions, more sophisticated models have to be applied. The choice of a dedicated model, however, may depend, even today, on the nature of the polymer-solvent system and its physical properties (polar or non-polar, association or donor-acceptor interactions, subcritical or supercritical solvents, etc.), on the ranges of temperature, pressure and concentration one is interested in, on the question whether a special solution, special mixture, special application is to be handled or a more universal application is to be foxmd or a software tool is to be developed, on munerical simplicity or, on the other hand, on numerical stability and physically meaningftd roots of the non-linear equation systems to be solved. Finally, it may depend on the experience of the user (and sometimes it still seems to be a matter of taste). 

The infortnation provided in this chapter can be divided into four parts 1. introduction, 2. thermodynamic theories of polymer blends, 3. characteristic thermodynamic parameters for polymer blends, and 4. experimental methods. The introduction presents the basic principles of the classical equilibrium thermodynamics, describes behavior of the single-component materials, and then focuses on the two-component systems solutions and polymer blends. The main focus of the second part is on the theories (and experimental parameters related to them) for the thermodynamic behavior of polymer blends. Several theoretical approaches are presented, starting with the classical Flory-Huggins lattice theory and, those evolving from it, solubility parameter and analog calorimetry approaches. Also, equation of state (EoS) types of theories were summarized. Finally, descriptions based on the atomistic considerations, in particular the polymer reference interaction site model (PRISM), were briefly outlined. 

In addition to numerous experiments, there have been attempts at modelings using solubility parameters and Flory-Huggins interaction pa-rametersS to predict separation characteristics and permeability of polysiloxane membranes. " Simulations indicate that at least the asymmetrically substituted polysilanes [-SiRR -] have gas permeabilities comparable to that of PDMS. ° The permeability, P, is the product of the solubility, S, of the gas in the polymer and its diffusivity, D. Values of P for the... 

The Flory—Huggins theory, which is based upon statistical thermodynamic models, has been used to assess the miscibility of polymer blends and was developed by Flory (1941, 1942) and Huggins (1941,1942) in the 1940s. Unlike the Hildebrand solubility parameter, it provides a fundamental understanding backed with classical thermodynamic theories. 

In order to explain the experimental phase diagrams of water-soluble polymers, a number of semiempirical approaches that assume the concentrational dependence of the Flory-Huggins solubility parameter were developed. " " Hie two-state models, which involve equilibtium coexistence of two interconvertible (solvophilic and solvophobic) states of the monomer units, as well as the n-duster model, which assumes temperature-dependent inversion of the higher order virial coeffident, allow to rationalize the apparent concentrational dependence of the solubility parameter in aqueous solutions of water-soluble polymers. 

In real systems, nonrandom mixing effects, potentially caused by local polymer architecture and interchain forces, can have profound consequences on how intermolecular attractive potentials influence miscibility. Such nonideal effects can lead to large corrections, of both excess entropic and enthalpic origin, to the mean-field Flory-Huggins theory. As discussed in Section IV, for flexible chain blends of prime experimental interest the excess entropic contribution seems very small. Thus, attractive interactions, or enthalpy of mixing effects, are expected to often play a dominant role in determining blend miscibility. In this section we examine these enthalpic effects within the context of thermodynamic pertubation theory for atomistic, semiflexible, and Gaussian thread models. In addition, the validity of a Hildebrand-like molecular solubility parameter approach based on pure component properties is examined. 

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