Flory-Huggins solvent-polymer interaction parameter

Often, the Flory-Huggins solvent-polymer interaction parameter is applied instead of 1 P or G. There are some books (Refs. 1-3) giving details for such procedures as well as extensive tables of polymer solubility parameters from which the table below is extracted. Methods for calculating solubility parameters can be found in Refs. 4-7. 

Knowing the crosslink density, it is possible to calculate the Flory-Huggins solvent-polymer interaction parameter x (Flory 1955) using Eq. (4.10) as suggested by Rabek (1977). 

The Flory-Huggins solute-polymer interaction parameter may be also calculated from activity coefficients [16,39]. In the case of pure solvents the equation is ... 

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 ... 

As originally defined by the Flory-Huggins theory, the interaction parameter is independent of concentration and inversely dependent on temperature. This feature reproduces the UCS behavior experimentally observed in some systems. To account for the existence of a LCST, it is necessary to introduce a free-volume component (5-7) that accounts for the difference in size between polymer and solvent molecules. In this approach, a flexible expandable lattice is introduced, which can be envisaged to arise from the different rates of expansion of the polymer and the solvent, with increasing temperature. 

The solvent-polymer interaction parameters were calculated from vapor pressure data of aqueous homopolymer solutions [25], using the Flory-Huggins expression [26] x//=6 ln/)//) —ln(l — 0) —(1 — l/N)0, where p is the vapor pressure and 6 is the polymer volume fraction. The chain length N was determined using 13/3 (EO) or 30/9 (PO) monomers per bead. This gives for the interaction parameters X s=l-4, Xi>s=l-7 (here S denotes solvent). For the EO-PO interaction parameter from group contribution methods [27] we estimated xep = 3.0. 

Flory and Huggins developed an interaction parameter that may be used as a measure of the solvent power of solvents for amorphous polymers. Flory and Krigbaum introduced the idea of a theta temperature, which is the temperature at which an infinitely long polymer chain exists as a statistical coil in a solvent. 

Another well-developed mean-field model, known as the free association model, was developed by Dudowicz, Douglas and Freed and is based on the mean-field Flory-Huggins incompressible lattice model. ° Douglas and co-workers incorporated two temperature-independent parameters - polymerization enthalpy AHp and entropy ASp -along with parameters describing the flexibility of the polymer and a solvent-monomer interaction parameter (x). The model allows the calculation of various temperature-dependent properties, such as the number-average DP, constant volume specific heat (Q,), and osmotic pressure, and predias similar temperature-dependent behavior with the van der Schoot treatment. 

The binary interactirMi generally refers to the interactions between polymer-polymer and polymer-solvent The nature of solvent-polymer interaction plays an important role in the miscibility of blends. Many thermodynamic properties of polymer solutions such as solubility, swelling behavior, etc., depend on the polymer-solvent interaction parameter (y). The quantity was introduced by Flory and Huggins. Discussions of polymer miscibility usually start with Flory-Huggins equation for free energy of mixing of a blend (refer to Chap. 2, Thermodynamics of Polymer Blends ). 

Ivin and Leonard, starting from Flory-Huggins theory [54], derived an equation relating the volume fractions of monomer, polymer and solvent (Om. I p and Og), the monomer-solvent, polymer-solvent and monomer-polymer interaction parameters (%ms, Xps and x p) with the absolute (i.e. independent of the polymerization conditions) thermodynamic parameters [55] ... 

Although the emphasis in these last chapters is certainly on the polymeric solute, the experimental methods described herein also measure the interactions of these solutes with various solvents. Such interactions include the hydration of proteins at one extreme and the exclusion of poor solvents from random coils at the other. In between, good solvents are imbibed into the polymer domain to various degrees to expand coil dimensions. Such quantities as the Flory-Huggins interaction parameter, the 0 temperature, and the coil expansion factor are among the ways such interactions are quantified in the following chapters. 

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

According to Flory-Huggins theory, the heat of mixing of solvent and polymer is proportional to the binary interaction parameter x in equation (3). The parameter x should be inversely proportional to absolute temperature and independent of solution composition. 

Here % is the Flory-Huggins interaction parameter and ( ), is the penetrant volume fraction. In order to use Eqs. (26)—(28) for the prediction of D, one needs a great deal of data. However, much of it is readily available. For example, Vf and Vf can be estimated by equating them to equilibrium liquid volume at 0 K, and Ku/y and K22 - Tg2 can be computed from WLF constants which are available for a large number of polymers [31]. Kn/y and A n - Tg can be evaluated by using solvent viscosity-temperature data [28], The interaction parameters, %, can be determined experimentally and, for many polymer-penetrant systems, are available in the literature. 

The role of the solvent in polymer adsorption has been the subject of much discussion. For example, theories have made predictions about the effect of the polymer/solvent interaction (i.e. Flory Huggins x parameter) on adsorption. For many systems, x parameters had already been tabulated so that a number of adsorption studies focused attention on this parameter. In spite of much effort, available data are ambiguous, sometimes verifying and sometimes contradicting the trends predicted by theory. 

Further development of the Flory-Huggins method in direction of taking into account the effects of far interaction, swelling of polymeric ball in good solvents [4, 5], difference of free volumes of polymer and solvent [6, 7] leaded to complication of expression for virial coefficient A and to growth of number of parameters needed for its numerical estimation, but weakly reflected on the possibility of equation (1) to describe the osmotic pressure of polymeric solutions in a wide range of concentrations. 

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. 

Several authors have attempted to corrolate the degradation rate with such solvent parameters as osmotic coefficient [35], viscosity [36-38] and the Flory Huggins interaction parameter, % [39,40] - a low % value indicates a good solvent in which the polymer is expected to exhibit an open conformation (as opposed to coiled) and therefore is more susceptible to degradation (Fig. 5.15). 

There have been many attempts to describe the process of mixing and solubility of polymer molecules in thermodynamic terms. By assuming that the sizes of polymer segments are similar to those of solvent molecules, Flory and Huggins derived an expression for the partial molar Gibbs free energy of dilution that included the dimensionless Flory Higgins interaction parameter X = ZAH/RT, where Z is the lattice coordination number. It is now... 

The deterioration of the solvent qnality, that is, the weakening of the attractive interactions between the polymer segments and solvent molecules, brings about the reduction in the coil size down to the state when the interaction between polymer segments and solvent molecules is the same as the mutual interaction between the polymer segments. This situation is called the theta state. Under theta conditions, the Flory-Huggins parameter % assumes a value of 0.5, the virial coefficient A2 is 0, and exponent a in the viscosity law is 0.5. Further deterioration of solvent quality leads to the collapse of coiled structure of macromolecules, to their aggregation and eventually to their precipitation, the phase separation. 

Flory-Huggins Approach. One explanation of blend behavior lies in the thermodynamics of the preceding section, where instead of a polymer-solvent mixture, we now have a polymer-polymer mixture. In these instances, the heat of mixing for polymer pairs (labeled 1 and 2) tends to be endothermic and can be approximated using the solubility parameter. The interaction parameter for a polymer-polymer mixture, Xi2, can be approximated by... 

Another important application of experimentally determined values of the osmotic second virial coefficient is in the estimation of the corresponding values of the Flory-Huggins interaction parameters x 12, X14 and X24. In practice, these parameters are commonly used within the framework of the Flory-Huggins lattice model approach to the thermodynamic description of solutions of polymer + solvent or polymer] + polymer2 + solvent (Flory, 1942 Huggins, 1942 Tanford, 1961 Zeman and Patterson, 1972 Hsu and Prausnitz, 1974 Johansson et al., 2000) ... 

Here Vi and v are the partial specific volumes of the polymer (/ = 2,4) and the solvent, respectively M is the molar weight of the solvent and Xu and 724 are the Flory-Huggins interaction parameters, quantifying the energy of interaction between unlike lattice-based polymer segments (%24) or between polymer segments and solvent molecules (%u). 

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