Thermodynamic model Flory-Huggins equation

Traditionally, the thermodynamics of polymer mixtures was developed in terms of a lattice model, with each monomer unit of the polymer chains occupying a single lattice site. The free energy of mixing of polymers in solution can be described by the Flory-Huggins equation ... 

Fig. IS. Cloud-point conversion as a function of the volume fraction of CO in a DGEBA-EDA system for two different reaction temperatures. Full lines are theoretical predictions from the Flory Huggins equation using x(T) from Eq. (35) (Reprinted from Polymer International, 30, R.A. Ruseckaite, R.JJ. Williams, Castor-oil-modified epoxy resins as model systems of rubber-modified thermosets. 1 Thermodynamic analysis of the phase separation, 11-16, Copyright (1993), with kind permission from the Society of Chemical Industry, London, UK)...

Statistical thermodynamic theories provide a powerful tool to bridge between the microscopic chemical structures and the macroscopic properties. Lattice models have been widely used to describe the solution systems (Prigogine 1957). Chang (1939) and Meyer (1939) reported the earliest work related with the lattice model of polymer solution. The lattice model was then successfully established by Flory (1941, 1942) and Huggins (1942) to deal with the solutions of flexible polymers by using a mean-field approximation, and to derive the well-known Flory-Huggins equation. 

Current thermodynamic theories for polymer systems are combinations of the Flory -Huggins, Guggenheim, and Equations-of-State approaches. All of these theories make use of empirical parameters and are based on assumptions about the underlying molecular model. 

The thermodynamic definition of the spinodal, binodal and critical point were given earlier by Eqs. (9), (7) and (8) respectively. The variation of AG with temperature and composition and the resulting phase diagram for a UCST behaviour were illustrated in Fig. 1. It is well known that the classical Flory-Huggins theory is incapable of predicting an LCST phase boundary. If has, however, been used by several authors to deal with ternary phase diagrams Other workers have extensively used a modified version of the classical model to explain binary UCST or ternary phase boundaries The more advanced equation-of-state theories, such as the theory... 

For ideal systems (usually as in elastomers), the solubility wiU be independent of concentration and the sorption curve will follow Henry s law (Equation 4.6), i.e., gas concentration within the polymer is proportional to the applied pressure. For nonideal systems (usually as in glassy polymers), the sorption isotherm is generally curved and highly nonlinear. Such behavior can be described by free-volume models and Flory-Huggins thermodynamics—comprehensive discussions on this may be found elsewhere [1,25,26]. 

Thermodynamic modehng of the above-mentioned phase diagrams requires a model that is able to account for the polymer chain-like structure, the polymer/ solvent interactions, and the influence of pressure on the phase behavior. Whereas the first two issues can be at least qualitatively covered by using a lattice theory of the weU-known Flory-Huggins type, such an approach is in general not able to describe the influence of pressure. Fulfillment of the third requirement requires a thermodynamic equation of state. Such a model naturally accounts for density effects in a system. 

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 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. 

One of the key results from application of the equation of state approach to predicting phase behavior is the observation that lest behavior can be predicted based upon a non-combinatorial contribution to entropy inherent with this formalism. The Flory-Huggins lattice model theory is an incompressible model that does not allow for the compressibility effects on the system thermodynamics. For equation of state approaches that allow for compressibility effects, the miscibility condition expressed by Eq. 2.2 [18] becomes ... 

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