The Flory-Huggins Equation

Flory defined the following interaction parameter, x which he made dimensionless by dividing by kT (Equation 11-25)  

Each segment (except the end groups) must have two neighbors that are also polymer segments. 

FIGURE 11-13 Schematic diagram showing the reduction in the probability that a B polymer segment is next to a solvent molecule A. 

Adding Flory s expressions for the entropy and enthalpy we arrive at Equation 11-27  

rearranging and converting to a molar basis we obtain Equation 11-28  

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The symbol Xs stands for the interaction energy per solvent molecule divided by kT. Combining equations (5.7) and (5.8) gives the Flory-Huggins equation for the free energy of mixing of a polymer solution ... 

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],... 

The Flory-Huggins equation (Eq. 10) allows one to reconstruct schematic phase diagrams to express the phase separation behavior as discussed below. 

One of the most common techniques for determining x parameters for polymer-solvent systems is the vapor pressure method.(10) In this approach, the uncrosslinked polymer is exposed to solvent vapor of known pressure, p. The polymer absorbs solvent until equilibrium is established, x is related to p and V2, the volume fraction of polymer at equilibrium, by the Flory-Huggins equation (ll)... 

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... 

Tanaka and collaborators [31] further investigated the polyacrylamide behavior. Based on the Flory-Huggins equation for osmotic pressure, they... 

The interesting features of the photo-heating phase transition are that irradiation causes the originally continuous transition to become discontinuous and the transition temperature to be lowered. These can be explained using the Flory-Huggins equation of state ... 

FIGURE I Predicted and observed DOM binding constants using the Flory-Huggins equation and methylsalicylic acid as a DOM analog. 

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... 

Fig. 14 Binary phase diagram for C246H494 in octacosane. The top curve shows the equilibrium liquidus for extended-chain crystals, and the bottom line the metastable liquidus for once-folded crystals. Experimental dissolution temperatures are fitted to the Flory-Huggins equation with / = 0.15 (solid lines). Vertical dotted lines (a) and (b) indicate the concentrations at which the growth rates were determined as a function of Tc in [29]. Horizontal dotted lines indicate the temperatures at which the rates were determined in [45] as a function of concentration. G(c) at Tc = 106.3 °C, measured along line (c), is shown in Fig. 12. The shading indicates schematically the crystal growth rate (black = fast), and the dashed line the position of the growth rate minimum...

The Flory-Huggins equation-of-state (22) provides simple expressions for the SCFs (Helfand, 1975c Helfand and Sapse, 1975, 1976 Scheutjens and Fleer, 1979 Hong and Noolandi, 1981 Ploehn et al, 1988),... 

Although the Wilson activity coefficient model has proven to be useful for solutions of small molecules, it has seen very limited use for polymer solutions most likely because of its increased complexity relative to the Flory-Huggins equation. 

The Flory-Huggins equation for the activity of a solvent in a polymer solution is ... 

Flory - a parameter in the Flory-Huggins equation to account for intermolecular interactions. X values less than 0.5 are indicative of a solvent which has relatively good solubility with a polymer. 

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 fourth factor determining polymerization rate is the monomer concentration in the particles. For some monomers the ratio of monomer to polymer in the particles is about constant during part of the polymerization. Smith (57) suggested that this results from a balance between the eflFect on the monomer activity of the dissolved polymer and the eflFect of interfacial tension of the very small particles. This equilibrium was put in a quantitative form by Morton, Kaizerman, and Altier (44), who derived the following equation by combining an expression for the interfacial free energy of the particle with the Flory-Huggins equation for the activity of the solvent (monomer) in the monomer-polymer particle. 

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). 

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