Flory-Huggins theory, polymer volume fraction

Next we use the Flory-Huggins theory to evalute AG by Eq. (8.44). As noted above, the volume fraction occupied by polymer segments within the coi domain is small, so the logarithms in Eq. (8.44) can be approximated by the leading terms of a series expansion. Within the coil N2 = 1 and Nj = (1 - 0 VuNa/Vi, where is the volume of the coil domain. When all of these considertions are taken into account, Eq. (8.108) becomes... 

In practice, it is difficult to assign the number of repeating segments in solvent or polymer unambiguously. For this reason, it is usual in using Flory-Huggins theory to replace segment fraction (f) in equation (3) by volume fraction ( >). 

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 ... 

Figure 5. Examples of moment free energy (70) for Flory-Huggins theory of length-polydisperse polymers, with one moment density, p, retained. The parent is of the Schulz form (65), with pf = 0.03, Lu = 100 (hence p = p, /Lv = 3 x 10-4), and a = 2 (hence Lw = 150) the point pt = pj° is marked by the filled circles. In plot (a), the value of x = 0.55 is sufficiently small for the parent to be stable The moment free energy is convex. Plot (b) shows the cloud point, % 0.585, where the parent lies on one endpoint of a double tangent the other endpoint gives the polymer volume fraction p, in the shadow phase. Increasing x further, the parent eventually becomes spinodally unstable [x 0.62, plot (c)]. Note that for better visualization, linear terms have been added to all free energies to make the tangent at the parent coincide with the horizontal axis.

The deficiencies of the Flory-Huggins theory result from the limitations both of the model and of the assumptions employed in its derivation. Thus, the use of a single type of lattice for pure solvent, pure polymer and their mixtures is clearly unrealistic since it requires that there is no volume change upon mixing. The method used in the model to calculate the total number of possible conformations of a polymer molecule in the lattice is also unrealistic since it does not exclude self-intersections of the chain. Moreover, the use of a mean-field approximation to facilitate this calculation, whereby it is assumed that the segments of the previously added polymer molecules are distributed uniformly in the lattice, is satisfactory only when the volume fraction (f>2 of polymer is high, as in relatively concentrated polymer solutions. 

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. 

A general treatment of the dependence of M,. on the nature of solvent and the volume fraction of polymer was outlined by Ivin and Leonard184. Adopting the Flory-Huggins theory of polymer solutions to ternary systems, they derived the relation... 

The assumption of symmetry holds reasonably well for molecules of approximately the same size in polymer solutions, it is necessary to use volume fractions instead of mole fractions to maintain the symmetric form (Flory-Huggins theory). 

The terms between the brackets correspond to the osmotic contribution to the Gibbs free energy (AG), and they also constitute the standard expression for AG of the Flory-Huggins theory of polymer solutions [61], where < p is the volume fraction of polymer and the ratio of the equivalent number of molecular segments of solvent to polymer (usually expressed as the ratio of molar volumes of solvent and polymer). Xap is the Flory-Huggins interaction parameter of solvent and polymer and the last term of Equation 14.1 is the interfacial free energy contribution where y is the interfacial tension, the molar volume of solvent, and r the particle radius. T is temperature in Kelvin and R is the universal gas constant. 

The free volume dissimilarity provides one of the important conceptual features that is missing from the Flory-Huggins theory. It rationalizes (i) the observed phase separation on heating (ii) the strong entropic contribution to X that opposes mixing and (iii) the observed increase in x with volume fraction of polymer in certain systems. Qualitatively, we can write for the mixing of polymer and solvent at room temperature ... 

According to the precepts of the original Flory-Huggins theory, the interaction parameter Xi should be independent of the volume fraction of polymer. The measured values of Xi. however, usually display a strong dependence upon the volume fraction of polymer. To indicate that the interaction parameter is a variable, we will drop the subscript 1 and denote the variable by x-... 

The simple Flory-Huggins theory discussed above is based on a series of questionable assumptions lattice sites of equal size for solvent segments and polymer monomeric units, uniform distribution of the monomeric units in the lattice, random distribution of the molecules, and the use of volume fractions instead of surface-area fractions in deriving the enthalpy of mixing. Proposed improvements, however, have led to more complicated equations or to worse agreement between theory and experiment. Obviously, various simplifications in the Flory-Huggins theory are self-compensating in character. 

In deriving this equation, the chemical potentials in the melt of the two components are given by the Flory-Huggins theory [22]. In Eq. (11.6), Vu and Vi are the molar volumes of the chain repeating unit and diluent, respectively V2 is the volume fraction of polymer in the mixture and Xi is the Flory-Huggins interaction parameter [22]. The implicit assumption is made in deriving Eq. (11.6) that tree is independent of composition. The similarity of Eq. (11.6) to the... 

The Flory-Huggins theory uses the lattice model to arrange the polymer chains and solvents. We have looked at the lattice chain model in Section 1.4 for an excluded-volume chain. Figure 2.1 shows a two-dimensional version of the lattice model. The system consists of si,e sites. Each site can be occupied by either a monomer of the polymer or a solvent molecule (the monomer and the solvent molecule occupies the same volume). Double occupancy and vacancy are not allowed. A hnear polymer chain occupies N sites on a string of N-l bonds. There is no preference in the direction the next bond takes when a polymer chain is laid onto the lattice sites (flexible). Polymer chains consisting of N monomers are laid onto empty sites one by one until there are a total tip chains. Then, the unoccupied sites are filled with solvent molecules. The volume fraction of the polymer is related to rip by... 

Colligative properties reflect this difference. Figure 31.3 shows the vapor pressure of the solvent benzene over a solution containing rubber, which is a polymer (a) as a function of mole fraction, and (b) as a function of volume fraction. The vapor pressure of a small-molecule solvent over a polymer solution shows nonideal behavior when plotted versus mole fraction x. The Flory-Huggins theory described in the next section shows that a better measure of concentration in polymer solutions is the volume fraction cf>. 

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