Flory—Huggins theory interaction parameter

Equation (12-23) suffers from the same limitations as the simple solubilty parameter model, because the expression for Wm is derived by assuming that in-termolecular forces are only nondirectional van der Waals interactions. Specific interactions like ionic or hydrogen bonds arc implicitly eliminated from the model. The solubility parameter treatment described to this point cannot take such inler-actions into account because each species is assigned a solubility parameter that is independent of the nature of the other ingredients in the mixture. The x parameter, on the other hand, refers to a pair of components and can include specific interactions even if they are not explicitly mentioned in the basic Flory-Huggins theory. Solubility parameters are more convenient to use because they can be assigned a priori to the components of a mixture, x values are more realistic, but have less predictive use because they must be determined by experiments with the actual mixture. 

Mathematically, the sorption data for the amorphous acrylic pol)nners which will be considered here cannot be correlated by a single parameter sorption Isotherm. In the Flory-Huggins theory, the parameter is the Interaction parameter, which for the simplest possible case characterizes the enthalpy of mixing which results from the Intermolecular bonding mismatch between polymer and water. A two parameter sorption Isotherm provides an excellent vehicle for data treatment. The two parameters can be identified as the interaction parameter and a clustering parameter. 

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. ... 

In polymer solutions or blends, one of the most important thennodynamic 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 [4T, 42], the scattering intensity from a polymer in a solution can be expressed as... 

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. 

What is the physical significance of the x parameter in the Flory-Huggins theory How is it related to solute/solute and solute-solvent interactions ... 

The Flory-Huggins theory of polymer solutions has been documented elsewhere [26, 27]. The basic parameters necessary to predict polymer miscibility are the solubility parameter 6, the interaction parameter %, and the critical interaction parameter ( ) . 

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 ... 

Equation-of-state theories employ characteristic volume, temperature, and pressure parameters that must be derived from volumetric data for the pure components. Owing to the availability of commercial instruments for such measurements, there is a growing data source for use in these theories (9,11,20). Like the simpler Flory-Huggins theory, these theories contain an interaction parameter that is the principal factor in determining phase behavior in blends of high molecular weight polymers. 

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... 

Figure 4 shows a plot of the static expansion factor (o ) as a function of the relative temperature 0/T, where a is defined as Rg(T)/Rg(0) and r is the number of residues that may be one monomer unit or a number of repeat units. When T < 0 (water is a good solvent for PNIPAM), the data points are reasonably fitted by the line with r = 105 calculated on the basis of Flory-Huggins theory [15]. Similar results have also been observed for linear polystyrene in cyclohexane [25,49]. The theory works well in the good-solvent region wherein the interaction parameter (x) is expected to be... 

In practice, the Flory-Huggins theory fails to predict many features of polymers solutions, either qualitatively or quantitatively, but remains widely used because of its simplicity. The Flory parameter x, assumed to be constant, often increases with interaction-energy scaled on kT, often exhibits a more complicated temperature dependence than 1/T (Flory, 1970). Such behavior stems from energetic effects, such as directional polar... 

The Flory-Huggins theory [20, 22-26, 42, 43] considers the change in potential energy in going from the pure states of polymer and liquid to a mixture thereof. In the original form of this theory, the interaction parameter, %, between liquid and polymer was defined as... 

For polymers, x is usually defined on a per monomer basis or on the basis of a reference volume of order one monomer in size. However, x is usually not computed from formulas for van der Waals interactions, but is adjusted to obtain the best agreement between the Flory-Huggins theory and experimental data on the scattering or phase behavior of mixtures (Balsara 1996). In this fitting process, inaccuracies and ambiguities in the lattice model, as well as in the mean-field approximations used to obtain Eq. (2-28), are papered over, and contributions to the free energy from sources other than simple van der Waals interactions get lumped into the x parameter. The temperature dependences of x for polymeric mixtures are often fit to... 

The lattice fluid equation-of-state theory for polymers, polymer solutions, and polymer mixtures is a useful tool which can provide information on equa-tion-of-state properties, and also allows prediction of surface tension of polymers, phase stability of polymer blends, etc. [17-20]. The theory uses empty lattice sites to account for free volume, and therefore one may treat volume changes upon mixing, which are not possible in the Flory-Huggins theory. As a result, lower critical solution temperature (LCST) behaviors can, in principle, be described in polymer systems which interact chiefly through dispersion forces [17]. The equation-of-state theory involves characteristic parameters, p, v, and T, which have to be determined from experimental data. The least-squares fitting of density data as a function of temperature and pressure yields a set of parameters which best represent the data over the temperature and pressure ranges considered [21]. The method,however,requires tedious experiments to deter-... 

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... 

Additionally, the parameters A and B are often found to depend weakly on chain lengths and composition. Shortcomings of the Flory-Huggins theory are usually lumped into the interaction parameter x- The Flory-Huggins equation (with all the corrections combined in x) contains all of the thermodynamic information needed to decide the equilibrium... 

Since the mean-field Flory-Huggins theory puts everything that is not understood about thermodynamics into the x parameter, this parameter is experimentally found to vary with composition and temperature. For solutions of linear polystyrene in cyclohexane, the interaction parameter... 

Although the Flory-Huggins theory is sound in principle, several experimental results cannot be accounted for. For example, it was found that the x parameter depends on the polymer concentration in solution. Most serious is the fact that many polymer solutions (e.g., PEO) show phase separation on heating, when theory predicts that this should occur only on coohng. Another complication arises from specific interactions with the solvent, for example hydrogen bonding between the polymer and solvent molecules (e.g. with PEO and PVA in water). Aggregation in solution (a lack of complete dissolution) may also present another problem. 

Equation 5.7 [10], where do, d[, 62 and d3 are fitting parameters, and d>B is the volume fraction of Component B, can produce all of the binary phase diagram types shown in Figure 5.1, when used either to lit experimental data in the context of the Flory-Huggins theory of thermodynamics or to express interaction energies calculated by atomistic simulations in a convenient manner as a function of the temperature and the component volume fractions. 

FIGURE 6.17 Solubility of a homopolymer according to the Flory-Huggins theory. Variables are the excluded volume parameter ft (or the polymer-solvent interaction parameter y), the net volume fraction of polymer q>, and the polymer-to-solvent molecular volume ratio q. Solid lines denote binodal, the broken line spinodal decomposition. Critical points for decomposition (phase separation) are denoted by . See text. 

Vapor sorption isotherms in a polymer as computed from Flory-Huggins theory by using interaction parameters shown. 

In a poor solvent, one parameter is no longer sufficient and now the three-body terms must be taken into account, not only implicitly but also explicitly. The second basic parameter required to study chains in the vicinity of TF is the repulsive three-body interaction c. The comparison made in Chapter 14 between the Flory-Huggins theory and the continuous model suggests calculating c by the formula... 

In dilute polymer solutions, the polymer molecules are isolated from each other by regions of pure solvent, i.e., the polymer segments are not uniformly distributed in the lattice. In view of this, the Flory-Huggins theory is least satisfactory for dilute polymer solutions and only applies to concentrated solutions or mixtures. Furthermore, the interaction parameter introduced to account for the effects of polymer-solvent contact interactions is not a simple parameter and should contain both enthalpy and entropy contributions. Additionally, as noted earlier, it has also been shown to be dependent on the solution concentration. 

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