Mixtures Flory-Huggins equation

Note 1 The definition and the name of the term have been modified from that which appears in ref. [5] to reflect its broader use in the context of polymer blends. In its simplest form, the %. parameter is defined according to the Flory-Huggins equation for binary mixtures... 

Linking this molecular model to observed bulk fluid PVT-composition behavior requires a calculation of the number of possible configurations (microstructures) of a mixture. There is no exact method available to solve this combinatorial problem (28). ASOG assumes the athermal (no heat of mixing) Flory-Huggins equation for this purpose (118,170,171). UNIQUAC claims to have a formula that avoids this assumption, although some aspects of athermal mixing are still present in the model. 

In Section 7.1 we discussed the thermodynamic condition for a stable mixture given in the Flory-Huggins equation (Eq. 7.1-6), where AS denotes the increase of entropy due to mixing. This equation is based on Boltzmann s principle stating that the entropy of a... 

In polymer-polymer mixtures, both Na and Nb in the Flory-Huggins equation (2-33) are large hence phase separation is predicted to occur even for very small values of x For Na = Ab ] > 1 and u = ua = i>b, the spinodal point where phase separation first... 

The comparison of the experimental solubilities [4,5] of Ar, CH4, C2H6 and CsHg in the binary aqueous mixtures of PPG-400, PEG-200 and PEG-400 with the calculated ones is presented in Figs. 1-3 and Table 2. They show that Eq. (4) coupled with the Flory-Huggins equation, in which the interaction parameter x is used as an adjustable parameter, is very accurate. The Krichevsky equation (1) does not provide accurate predictions. While less accurate than Eq. (4), the simple Eq. (2) provides very satisfactory results without involving any adjustable parameters. It should be noted that Eq. (4) coupled with the Flory-Huggins equation with X (athermal solutions) does not involve any adjustable parameters and provides results comparable to those of Eq. (2). 

Flory—Huggins equation for real solutions will be used to derive an expression for 21 ( 23) for water (1) + protein (2) + cosolvent (3) mixtures. The activity coefficient of component 2 in the above ternary mixture can be written in the following form ... 

It is remarkable that no empirical mixture parameters and no experimental data are required to use the equation. The only parameters in the Flory-Huggins equation are the hard core volumes V, which are a pme-component property, and the atomic or group contribution values are found in standard compilations. Since the v/s are significant in the FH equation only in terms of their ratios, pure-liquid molar volumes are often used for V in place of hard core volumes. For solutions of polymers of the same chemical formula, molecular masses are legitimate substitutes for V , for the same reason. Thus the volume fractions ( ) can be substituted by mass fractions W . Either volume fraction or mass fraction is directly related to laboratory data. To avoid mole fractions, the activity tti from Equations (4.368) and (4.369) can be used to calculate by / = aj. ... 

The Wilson equation (suggested by Wilson72 in 1964) is suitable for a large variety of nonideal mixtures. This equation, a semiempirical generalization of the Flory-Huggins equation, is presented in Table 14-12. The term (A — An) is an empirically determined energy term which is closely related to the difference in cohesive energy between an i-j pair and an i-i pair. Over small temperature... 

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 U RE 10.4 Comparison between experimental (o) and calculated (solid lines) solubilities of phenacetin (S is the mole fraction of phenacetin) in the mixed solvent water/dioxane is the mole fraction of dioxane) at room temperature. The solubility was calculated using Equation 10.29. 1-activity coefficients expressed via the Flory-Huggins equation, 2-activity coefficients expressed via the Wilson equation. (From C. Bustamante, and P. Bustamante, 1996, Nonlinear Enthalpy-Entropy Compensation for the Solubility of Phenacetin in Dioxane-Water Solvent Mixtures, Journal of Pharmaceutical Sciences, 85, 1109. Reprinted from E. Ruckenstein, and I. L. Shulgin, 2003c, Solubility of Drugs in Aqueous Solutions. Part 2 Binary Nonideal Mixed Solvent, International Journal of Pharmaceutics, 260, 283, With permission from Elsevier.)... 

From the Flory-Huggins equation, the free energy of the homogeneous mixture (denoted as the disordered state) of binary blends in parallel to diblock copolymers is AFm W, while the free energy of microphase-separated state (denoted as the ordered state) is AF,n Here yr is normally called the segregation strength. 

For a mixture that is applicable to the FRRPP process, one can write the multi-component Flory-Huggins equation as... 

Fig. 16. (a) Phase diagram of poly(ethylene oxide) 3,500/100,000 mol-wt mixtures at different concentrations. The open circles represent the calculation of and, calculated by using the Flory-Huggins equations 46 and 47, respectively. The filled symbols represent the experimental data, (b) Phase diagram of polyethylene dissolved in 1,2,4,5-tetrachlorobenzene (TCB). The experimental data were obtained by melting after crystallization. The macromolecular crystals, 2, were not at equilibrium, but melted considerably lower than (see Table 1). 

Having established the Flory-Huggins equation, we now consider the consequences for polymer mixtures. Starting from A mix, the entropy of mixing. 

All these conclusions refer to the limit of large degrees of polymerization. It is important to see that the Flory-Huggins equation permits one to consider how the compatibility changes if the degrees of polymerization are reduced and become moderate or small. For the sake of simplicity, we choose for a discussion the case of a symmetric mixture with equal degrees of polymerization for both components, i.e. 

The Flory-Huggins equation (4.11) or (4.12) is famous and widely used. It sets the basis from which the majority of discussions of the properties of polymer mixtures emanates. 

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). 

Flory-Huggins Theory. The simplest quantitative model for AGmx that includes the most essential elements needed for polymer blends is the Flory-Huggins theory, originally developed for polymer solutions (3,4). It assumes the only contribution to the entropy of mixing is combinatorial in origin and is given by equation 3, for a unit volume of a mixture of polymers A. and B. Here, pt and... 

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