Flory-Huggins relationship

Now let s discuss the pressure computations. The observed reactor pressure is a sum of the partial pressures of nitrogen and the styrene monomer vapor. The vapor pressure of the styrene vapor is an increasing function of temperature and decreasing function of conversion. This is explained by the Flory-Huggins relationship ( ). 

We next estimated the conversions by using the observed pressures and temperatures and the Flory-Huggins relationship. Since the Flory-Huggins relationship is less accurate at higher conversions, we can expect these estimates of conversions to be of decreasing accuracy in Tests 1, 2 and 3 respectively. 

Roult s law is known to fail for vapour-liquid equilibrium calculations in polymeric systems. The Flory-Huggins relationship is generally used for this purpose (for details, see mass-transfer models in Section 3.2.1). The polymer-solvent interaction parameter, xo of the Flory-Huggins equation is not known accurately for PET. Cheong and Choi used a value of 1.3 for the system PET/EG for modelling a rotating-disc reactor [113], For other polymer solvent systems, yj was found to be in the range between 0.3 and 0.5 [96],... 

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. 

The second virial coefficient B2 can be related to the interaction parameter Xi via the Flory-Huggins relationship... 

For polymers with a high degree of polymerization and relatively small concentrations of the volatile component A, the Flory-Huggins relationship can be simplified to ... 

This type of sorption isotherm arises when the interactions between permeant molecules are strong relative to permeant/polymer interactions, and the solubility coefficient increases continuously with increasing pressure. This sorption can be described for non-polar systems by the Flory-Huggins relationship... 

For more than two decades researchers have attempted to overcome the inadequacies of Flory s treatment in order to establish a model that will provide accurate predictions. Most of these research efforts can be grouped into two categories, i.e., attempts at corrections to the enthalpic or noncombinatorial part, and modifications to the entropic or combinatorial part of the Flory-Huggins theory. The more complex relationships derived by Huggins, Guggenheim, Stavermans, and others [53] required so many additional and poorly determined parameters that these approaches lack practical applications. A review of the more serious deficiencies... 

From a thermodynamic point of view, the heteropolymer globule in hand represents a subsystem which is composed of a macromolecule involving lu l2 units Mi, M2 and molecules of monomers Mi, M2 whose numbers are Mi,M2. Among these variables and volume fractions a in the framework of the simplest Flory-Huggins lattice model there are obvious stoichiometric relationships... 

Different models determine A in different ways. Nation exhibits a water-uptake isotherm as shown in Figure 7. The dashed line in the figure shows the effects of Schroeder s paradox, where there is a discontinuous jump in the value of A. Furthermore, the transport properties have different values and functional forms at that point. Most models used correlate A with the water-vapor activity, since it is an easily calculated quantity. An exception to this is the model of Siegel et al., ° which assumes a simple mass-transfer relationship. There are also models that model the isotherm either by Flory—Huggins theory" or equilibrium between water and hydrated protons in the membrane and water vapor... 

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 ... 

From the theoretical viewpoint, much of the phase behaviour of blends containing block copolymers has been anticipated or accounted for. The primary approaches consist of theories based on polymer brushes (in this case block copolymer chains segregated to an interface), Flory-Huggins or random phase approximation mean field theories and the self-consistent mean field theory. The latter has an unsurpassed predictive capability but requires intensive numerical computations, and does not lead itself to intuitive relationships such as scaling laws. 

Originally x was stated to be independent of polymer concentration. The X-parameters determined by many investigators using one or another of the methods for measuring colligative properties of polymer-liquid solutions (mentioned below) show that this is not the case (see Tables 3-22 of Reference 43) nor does x vary linearly with 1/T as stated in Eq. 7. Later [44] a quantity Aws representing an entropic contribution from contact interaction was added to the Flory-Huggins definition of x to produce a relationship linear in 1/T. 

Fig. 61. Correlation of the Flory-Huggins interaction parameter, %, for polystyrene-liquid systems at 25 °C with the volume fraction (v) of polymer in the system. The filled circles represent experimentally determined data recorded in Table XIX of Ref. 43. The empty circles represent estimations by interpolation or extrapolation of the linear relationships established on the basis of the experimental data shown. The value for x reported for acetone at v = 1 is placed in brackets to indicate that this point seems too high, and therefore it was not included in the data set used to establish by linear regression the equation shown for this linear relationship...

This equation is not particularly useful in practice, since it is difficult to quantify the relationship between concentration and activity. The Flory-Huggins theory does not work well with the cross-linked semi-crystalline polymers that comprise an important class of pervaporation membranes. Neel (in Noble and Stern, op. cit., pp. 169-176) reviews modifications of the Stefan-Maxwell approach and other equations of state appropriate for the process. 

Progressing to the polymer-solvent system (Figure 1.29), the ideal entropy term above must be replaced by a relationship that recognizes the features of interconnection of segments in the polymer chain. This is the Flory-Huggins relation ... 

Thus it is clear that the term "clustering" can be used in a variety of situations, and is not precisely defined. In the present case, it will be used in the context of the statistical thermodynamic treatment of binary solutions developed by Zlmm ( ) and Zimm and Lundberg ( ), which provides a calculation of a cluster integral, and which can be extended to specify a cluster size for each component. In describing the role of the "clustering" theory in relationship to previously developed solution theories, such as the widely used Flory-Huggins theory ( ), Zimm and Lundberg point out that "Our considerations are not intended as a replacement for the previous theories, but as... 

The subscript 1, referring to the solvent, is used here because )j,A ris just the difference in energy of a solvent molecule immersed in pure polymer compared with that in pure solvent. Note that Xi is pivotal parameter in Flory-Huggins theory. With these relationships, the enthalpy of mixing can be expressed as... 

The macroscopic thermodynamic approach Croucher and Hair (1978) have followed Eichinger (1970) in considering the temperature dependence of x in macroscopic thermodynamic terms. In the classical Flory-Huggins theory, the enthalpy and entropy of dilution parameters Ki and are constants related to the d-temperature by 0 = (ici/( i)r and to Xi by (i—Xi) = l i —ki. These relationships lead immediately to... 

Flory-Huggins x parameter, and the volume fraction of the A block, Z. The structure-property relationship for coil-coil diblock copolymer systems is relatively well understood both experimentally [14-19] and theoretically [20-23]. Chain stretching normal to the interface leads to a positive deviation of the scaling exponent [14]. For example, scaling behavior for a conven-... 

Flory-Huggins theory This theory calculates the free energy of mixing of pure amorphous polymers with pure solvent. The entropy and the enthalpy of mixing can be calculated separately, and the following relationship applies ... 

Bodi the partial pressure and Henry s law eonstant depend on temperature, pressure, and solvent properties. This relationship does not eonsider interaetion between solute and solvent In the ease of polymers, the Flory-Huggins theory gives a simplified relationship for low eoneenlrations of solvent ... 

FIGURE 10.5 Comparison between experimental (o) and calculated (solid lines) solubilities of paracetamol (S is the mole fraction of paracetamol) in the mixed solvent water/ethanol is the mole fraction of ethanol) 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 S. Romero, A. Reillo, B. Escalera, and P. Bustamante, 1996, The Behavior of Paracetamol in Mixtures of Amphiprotic and Amphiprotic-Aprotic Solvents. Relationship of Solubility Curves to Specific and Nonspecific Interactions, Chemical and Pharmaceutical Bulletin, 44, 1061. 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.)... 

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