Flory-Huggins interaction parameter polymer solution thermodynamics

It should also be mentioned that polymer-solvent interactions can be characterized by the second virial coefficients that appear in equations (8) and (13) and by the free energy of interaction parameter Z1 that appears in the Flory-Huggins theory of polymer solution thermodynamics.1,61... 

The classical thermodynamics of binary polymer-solvent systems was developed independently by P. J. Flory (1-3) and M. L. Huggins (4-6). It is based on the well-known lattice model qualitatively formulated by K. H. Meyer (7), who pointed out the effect of the differences in molecular size of polymer and solvent molecules on the entropy of mixing. The quantitative calculation of the entropy of mixing led to the introduction of a dimensionless quantity, the so-called Flory-Huggins interaction parameter for the thermodynamic description of polymer solutions. 

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) ... 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

The usefulness of inverse gas chromatography for determining polymer-small molecule interactions is well established (1,2). This method provides a fast and convenient way of obtaining thermodynamic data for concentrated polymer systems. However, this technique can also be used to measure polymer-polymer interaction parameters via a ternary solution approach Q). Measurements of specific retention volumes of two binary (volatile probe-polymer) and one ternary (volatile probe-polymer blend) system are sufficient to calculate xp3 > the Flory-Huggins interaction parameter, which is a measure of the thermodynamic... 

Classical polymer solution thermodynamics often did not consider solvent activities or solvent activity coefficients but usually a dimensionless quantity, the so-called Flory-Huggins interaction parameter % is not only a function of temperature (and pressure), as was evident from its foundation, but it is also a function of composition and polymer molecular mass. As pointed out in many papers, it is more precise to call it %-function (what is in principle a residual solvent chemical potential function). Because of its widespread use and its possible sources of mistakes and misinterpretations, the necessary relations must be included here. Starting from Equation [4.4.1b], the difference between the chemical potentials of the solvent in the mixture and in the standard state belongs to the first... 

The second contribution influencing polymer adsorption from solution is the Flory-Huggins interaction parameter between poljnner and solvent. Such an enthalpy of mixing term adds a contribution of X (t)2 z) — 4> ) to the interaction energy contribution, where 4) is the bulk solution concentration of the polymer (2) and, when approaches thermodynamically poor values, then the adsorbed amoimt of pol3mier increases significantly. Figure 5.13 shows experi-... 

Wolf BA (2003) Chain connectivity and conformational variability of polymers clues to an adequate thermodynamic description of their solutions II Composition dependence of Flory-Huggins interaction parameters. Macromol Chem Phys 204 1381... 

One further difficulty not touched upon in the foregoing discussion is the absence of a truly quantitative theory describing polymer solution thermodynamics. Even a second generation theory, such as the equation-of-state theory, probably only represents a qualitative or, at best, a semi-quantitative theory of polymer solution thermodynamics (Casassa, 1976). In the absence of a fully quantitative theory, it seems justifiable to make do with the classical Flory-Huggins theory, provided that the cracks that have appeared in its superstructure are papered over. These include using the concentration dependent interaction parameter % that is determined experimentally. Most of the theories of steric stabilization that have been developed to-date have unfortunately been based upon a concentration independent interaction parameter (see Table 10.1), although there are some exceptions (see, e.g. Evans and Napper, 1977). 

The thermodynamic data of polymer solutions were first derived from the approach of regular solutions of small molecules by Flory [29] and by Huggins [30]. In the original quasi-lattice model of Flory, the interaction parameter, x, was representative of the enthalpic difference of polymer/polymer, sol-vent/solvent and polymer/solvent contacts. Progressively % lost this simple signification and represented the shift with an ideal behavior (including entropic effects) but nevertheless the model was established for... 

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

In the early 1940s, Flory and Huggins proposed, separately, a lattice model to describe polymer solutions and introduced the interaction parameter This parameter increases as solvent power decreases hence, a thermodynamically good solvent is characterized by a low interaction parameter. In practice, most polymer-solvent combinations result in x-values ranging from 0.2 to 0.6. Moreover, the theory predicts that a polymer will dissolve in a solvent only if the interaction parameter is less than a critical value Xc. which, at a given temperature, depends on the degree of polymerization (x) of the dissolved polymer ° ... 

At temperatures above Tg, the magnitude of Vg° is a measure of the solubility of the probe in the stationary phase. From the Flory-Huggins treatment of solution thermodynamics, one can obtain the x parameter, which is a measure of the residual free energy of interaction between the probe and the polymer QZ,. 18). 

In polymer solutions or blends, one of the most important thermodynamic parameters that can be calculated from the (neutron) scattering data is the enthalpic interaction parameter x between the components. Based on the Flory-Huggins theory [41. 42], the scattering intensity from a polymer in a solution can be expressed as... 

There is an upper limit for the Flory-Huggins thermodynamic interaction parameter X if the polymer solution is homogeneous ... 

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. 

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 ). 

In this section the basic principles of membrane formation by phase inversion will be described in greater detail. All phase inversion processes are based on the same thermodynamic principles, since the starting point in all cases is a thermodynamically stable solution which is subjected to demixing. Special attention will be paid to the immersion precipitation process with the basic charaaeristic that at least three components are used a polymer, a solvent and a nonsolvent where the solvent and nonsolvent must be miscible with each other. In fact, most of the commercial phase inversion membranes are prepared from multi-component mixtures, but in order to understand the basic principles only three component systems will be considered. An introduction to the thermodynamics of. polymer solutions is first given, a qualitatively useful approach for describing polymer solubility or polymer-penetrant interaction is the solubility parameter theory. A more quantitative description is provided by the Flory-Huggins theory. Other more sophisticated theories have been developed but they will not be considered here. 

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