Flory—Huggins theory assumptions

With natural assumptions of the Flory-Huggins theory, from Eq. (4.38) show that... 

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. 

When the extrapolation q - 0 is carried out in (6.47), noting that D(0) = 1, we obtain an expression that is identical to (6.23) derived in Section 6.1.1.3 on the basis of the Flory-Huggins free energy of mixing. This shows that the assumptions embodied in (6.47) are essentially equivalent to those in the Flory-Huggins theory. Using the approximation for the Debye function given by (5.33), Equation (6.47) can be written as... 

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). 

Free volume approach to the combinatorial entropy The combinatorial entropy of mixing can be more readily derived by a free volume approach which renders the assumptions inherent in the Flory-Huggins theory more transparently obvious. Anticipating what is to be presented in Section 3.3, vis-d-vis the equation-of-state theory, we present a brief account of this alternative derivation. 

S.4. The defects in the Flory-Huggins theory An explanation The derivation of the Flory-Hu ns theory was predicated on the assumption that the volume changes at occur on mixing are negligible and, indeed, the volume changes are often quite small ( 1%). It turns out, however, that what is important is not so much the overall volume but the free volume. 

One of the reasons for the failure of the ideal solution law is the assumption that a large polymeric solute molecule is interchangeable with the smaller solvent molecule. The law also neglects intermolec-ular forces since the heat of mixing (AH x) assumed to be zero. The Flory-Huggins theory attempted to remedy these shortcomings in the ideal solution law. > ... 

Precipitation data for several systems have proved the validity of Equation 8.47. Linear plots are obtained with a positive slope from which the entropy parameter /i can be calculated, as shown in Figure 8.4. Typical values are shown in Table 8.1, but /i values measured for systems such as polystyrene-cyclohexane have been found to be almost ten times larger than those derived from other methods of measurement. This appears to arise from the assumption in the Flory-Huggins theory that is concentration independent and improved values of /i are obtained when this is rectified. 

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. 

The p s are the probabilities that the available cell is already occupied after (x-1) segments have been added. The assumption of the Flory-Huggins theory is that this probability is just the average probability throughout the lattice and is just equal to the fraction of the cells that are still empty. 

This equation presumes additivity of the free energy of mixing and of elasticity. Eichinger [23] has questioned and tested this assumption. The mixing term can be evaluated using the Flory-Huggins theory (equation 3.83)... 

The Flory-Huggins theory is in fact nothing more than a two-component polymer version of the simple lattice gas model introduced in section 2. We divide the free energy into an entropic part, which is assumed to take the simplest perfect gas form, while the enthalpic part is estimated using a typical mean-field assumption. 

The assumption of unperturbed chain statistics. Implicit in Flory-Huggins theory is the assumption that the long-range chain statistics of polymer chains are ideal random walks. This is not to be expected a polymer chain in a solvent collapses as conditions are changed to bring about phase separation between the polymer and the solvent (Grosberg and Khokhlov 1994). One would expect a polymer chain in a mixture to do the same as the conditions for phase separation were approached (Sariban and Binder 1987). 

Extrapolating Flory-Huggins theory to the dilute limit (beyond the assumption of the theory) also provides... 

This equation is the well-known Flory-Huggins Equation. The power of a theory is reflected by its capability for further applications in various situations. In the next section, we will introduce how the assumptions of Flory-Huggins theory can be amended for its broad applications. 

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 above-mentioned deficiencies of the Flory-Huggins theory can be alleviated, in part, by using the local-composition concept based on Guggenheim s quasichemical theory for the random mixing assumption and replacing lattice theory with an equation-of-state model (Prausnitz et al., 1986). More sophisticated models are available, such as the perturbed hard sphere chain (PHSC) and the statistical associating fluid theory (SAFT) (Caneba and Shi, 2002), but they are too mathematically sophisticated that they are impractical for subsequent computational efforts. 

Even though the Flory-Huggins theory predicts an upper critical solution temperature and allows a qualitatively correct phase diagram to be calculated, it cannot predict the experimentally observed lower critical solution temperature observed for virtually all polymer solutions. The fundamentally incorrect assumption in the theory is that the volume of mixing of the solution is assumed to be zero. To remedy this problem and improve the predictive power of the theory of concentrated solutions, a full fteory for... 

It is convenient to partition polymer solutions into three different cases according to their concentration. Dilute solutions involve only a minimum of interaction (overlap) between different polymer molecules. The Flory-Huggins theory does not represent this situation at all well due to its mean-field assumption. The semi-dilute case involves overlapping polymer molecules but still with a considerable separation of the segments of different molecules. 

Solution activity data obtained by osmometry on dilute solutions showed that the second virial coefficient is dependent on molar mass, contradicting the Flory-Huggins theory. These problems arise from the mean-field assumption used to place the segments in the lattice. In dilute solutions, the polymer molecules are well separated and the concentration of segments is highly non-uniform. Several scaling laws were therefore developed for dilute (c < c is the polymer concentration in the solution, c is the threshold concentration for molecular overlap) and semi-dilute (c > c ) solutions. In a good solvent the threshold concentration is related to molar mass as follows ... 

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