Flory—Huggins binary interaction parameters

An example of calculations is shown in Fig. 18.12 as the Huggins-Flory binary interaction parameter, x versus temperature (SANS data from Krishnamoorti et al. 1996). The extrapolated Arrhenius dependence suggests that the system is miscible up to about 118 °C. 

Symbols with a tilde are scaled [189]. Note that d depends on the reduced independent variables, P and T. Minimization of the entropic contribution to the Huggins-Flory binary interaction parameter leads to Hildebrand s dependence ... 

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. 

Parameter r measures the number of segments of a molecule for the term v in the Flory-Huggins equation. Parameters q, and are surface areas that are interchangeable for all except strongly hydrogen-bonded water and alcohols. Parameters r, q, and are pure-component molecular structure parameters. The combinatorial is dependent only on pure-component parameters. The residual depends additionally on binary interaction parameters and x ,... 

The lattice theories are the oldest and most frequently used to interpret and to predict the thermodynamic properties of multicomponent systems containing polymers. The Huggins-Flory lattice theory is the best known. To use the theory one must know the temperature, pressure and concentration dependence of the enthalpic and entropic contributions to the binary interaction parameter, P, ) + P, , ) / T. 

As the reported values of the thermodynamic parameters indicate, the largest pool of data is based on the Huggins-Flory relation. This is only to be expected since the theory and related to it concept of the binary interaction parameters, either B or was introduced to polymer science more than a half century ago, in 1941 to be precise. Even their well-recognized complexity of functional dependence, viz. B = B(T, P, ( ), MW, MWD, molecular stmcmre, stresses,...), does not discourage efforts to continue using this approach against all odds. 

The values of the 5 -components can be calculated for any chemical substance from the tabulated group and bond contributions. Once 8j is known for both polymers in the blend, the Huggins-Flory binary thermodynamic interaction parameter, can be calculated from ... 

It may be interesting to note that assuming equal degrees of polymerization, N., of both blend components, Eq. 12.25 yield a simple relationship between the binary interaction parameter and the molecular weight (as expressed by R) B N/RT = 2. Thus, within the framework of the Huggins-Flory theory, system will be miscible when B N/RT < 2, and immiscible when B N/RT > 2. This is schematically shown in Figure 12.24. 

According to Flory-Huggins theory, both the location and size of the different regions depend on the Flory-Huggins binary interaction parameters, which are parameters that characterize the interaction between pairs of compounds. Thus, we have the solvent-polymer parameter (X23), the nonsolvent-Polymer (X13), and solvent-nonsolvent (Xi2)- The subscripts, as shown also in the figure, refer to the nonsolvent, solvent, and polymer. ... 

The first theoretical modeling of polymer blends was carried out by Helfand and his colleagues [25-27]. Their lattice theory approximated the polymer 1-polymer 2 interactions via the Huggins-Flory binary parameter, Xrt-The derivation provided expressions for the concentration gradient across the interface, as well as for the interfacial tension coefficient, Vi2, and the interphase thickness, Al ... 

Blends of AMS-AN copolymers with PVC are used to increase the heat resistance of the product. Blends of PVC and AMS-AN copolymers were formed by precipitation with THF solution into methanol. Miscible blends were found for copolymers of AMS-AN with AN composition containing 11.9-30.0 wt%. Phase separation temperatures were noted by annealing and DSC methods. Phase separation temperatures below 130°C were found. Using the characteristic EOS parameters listed in Table 3.4 and the phase separation temperatures that were measured, the AP values were calculated for the blend system. These values were found to be AP nvc = 4.30 cal/cc APamsvc = 0-26 cal/cc and APan,ams = 8.6 cal/cc. The Flory-Huggins interaction parameters at 130°C were calculated and B Nyc = 4.22 cal/cc B msvc = 0-37 cal/cc Bams,an = 8.04 cal/cc. The binary interaction parameter of the copolymer-homopolymer blend for AMS-AN copolymer and PVC homopolymer for various AN compositions in the copolymer at various volume fractions in the blend is shown in Figure 3.7. 

Similar expressions hold for the volatile monomers, which at high dilutions can be treated independently. At low conversions, the infinite dilution is not applicable and the multicomponent Flory-Huggins equation [Eq. (26)], for example, should be used instead. Here the y, are the volume (or segment) fractions of the various components, i = 1 up to Ny being the volatile components and the polymer being component Ny - -1 the yy, are their mole fractions, Vy the molar volumes, and the Xij the binary interaction parameters. 

Empirical Description of the Binary Interaction Parameter. As noted earlier, the most important contribution to for blends of high polymers is due to enthalpic effects. Contrary to the assumptions of the original Flory-Huggins treatment, there now is experimental evidence that indicates that x 2 dependent on concentration and has a temperature dependence other than the inverse relation of Eq. (15) [33]. 

Biros, et al. [47] compared the predictions of both the equation of state and modified solubility parameter approaches for the dependence of the binary interaction parameter on temperature, pressure, solvent chain length, and polymer flexibility. Both treatments gave qualitatively similar predictions with values of x based on solubility parameters always lower than the equatiiS of state predictions. Quantitative agreement between the two approaches could be achieved if an entropic correction term, represented by a in Eq. (31), were employed. Thus, this modified Flory-Huggins approach should provide a good representation of the equilibrium thermodynamic state of the blends. 

The Huggins-Flory type segmental, binary interaction parameters, B l], are tabulated. A similar approach, based on the Hildebrand solubility... 

Huggins-Flory type binary interaction parameters... 

Schwahn, D. Willner, L. Phase Behavior and Flory-Huggins Interaction Parameter of Binary Polybutadiene Copolymer Mixtures with Different Vinyl Content and Molar Volume. Macromolecules 2002,35, 239-247. 

Thermodynamic descriptions of polymer systems are usually based on a rigid-lattice model published in 1941 independently by Staverman and Van Santen, Huggins and Flory where the symbol x(T) is used to express the binary interaction function [16]. Once the interaction parameter is known we can calculate the liquid liquid phase behaviour. 

The insertion of Eqs. (7) and (9) into Eq. (4) provides an expression which can be integrated analytically (Appendix B). The Flory-Huggins interaction parameter X can be used either as an adjustable parameter, or can be obtained from phase equilibrium data for the binary mixture polymer + water. 

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). 

Usually, the Flory—Huggins interaction parameters %ij are positive quantities smaller than unity. " " " There are, however, cases in which %ij has negative values. Because the interaction of a protein with water or urea is exothermic, the above parameters are expected to be negative. Indeed, according to the van Laar expression for the heat of mixing in a two component system, the Flory—Huggins interaction parameter 7 is proportional to the heat of mixing in the binary system i—j. Furthermore, as demonstrated in a previous section, for the water + lysozyme + urea mixture J2i > 0, and eq 22 in which 2.5 leads to... 

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... 

It has been shown (3) using Scott s ternary solution treatment (25) of the Flory-Huggins theory, that the overall interaction parameter between the volatile probe (1) and the binary stationary phase (2,3) is given by... 

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