The Flory-Huggins theory

The classical Flory-Huggins theory assumes at the outset that there is neither a change in volume nor a change in enthalpy on mixing a polymer with a low-molecular-weight solvent [3-5] the influence of non-athermal 0) 

In order to calculate the entropy of the mixture, we first arrange all of the polymer molecules on the lattice. The identical solvent molecules are placed thereafter. If j 

The total number of ways of arranging all of the 2 polymer molecules, Q, is the product of the number of ways of arranging each of the 2 molecules in sequence. This fact and Eq. (9.3.2) yield 

Having arranged all of the polymer molecules, the number of ways of fitting all of the indistinguishable solvent molecules into the remaining lattice 

Because j does not appear in the last term on the right-hand side of Eq. (9.3.5), that term adds up to m — 1) 2 ln[(z — l)/uo]- Also, the first term can be replaced by Stirling s approximation  

The starting point of the Flory-Huggins theory (1942)10 11 is a model in which the polymer solution is represented by a set of non-intersecting chains drawn on 

By statistically evaluating the number of arrangements possible on the lattice, Flory and Huggins obtained an expression for the (extensive) configurational entropy changes (those due to geometry alone), AS, in forming a solution from / moles of solvent and 2 moles of solute  

Example 7.4 Estimate the configurational entropy changes that occur when 

Solution. Mj= 92, Ms = 104. Using these values and those given for the polymers, we get , = 500 g/M,. In the absence of other information, we must assume that the number of segments equals the number of repeat units for the polymers. Therefore, Xi = Mnihrii, where nii is the molecular weight of the repeat unit, mpg = 104, wppo =120. These quantities may now be inserted in Equations 6.10, 6.10fl, and 6.10i. The results are summarized below (/ =1.99cal/molK)  

The result for (c) illustrates why polymer-polymer solubility essentially requires a negative isH. 

An expression for the (extensive) enthalpy of mixing. A//, was obtained by considering the change in adjacent-neighbor (molecules or segments) interactions on the lattice, specifically the replacement of [1,1] and [2,2] interactions with [1,2] interactions upon mixing  

As mentioned earlier, a beautiful theory of polymer collapse in a solvent was provided long ago by Paul Flory in a theoiy currently known as the Flory-Huggins theory. The present understanding of this phenomenon (including that of protein folding) has been developed around this theory by modifying the same. We now discuss the theory. 

The free energy F for this model has two components an entropy term describing the number of arrangements of the chains that can exist on the lattice for a given 0, and an energy term describing the interactions between adjacent molecules. 

The entropy 5 of the solution can be obtained from the Boltzmann law 5=In where Q is the number of ways distinct solute-solvent configurations can be generated on the lattice. This is obtained by using combinatorial analysis. The final form of the entropy has a surprisingly simple structure  

The first term is related to the translational entropy of the chain is the chain concentration in dimensionless units, where N is the number of segments in the 

In fact, instead of considering the full entropy it is more convenient to focus 

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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... 

We shall devote a considerable portion of this chapter to discussing the thermodynamics of mixing according to the Flory-Huggins theory. Other important concepts we discuss in less detail include the cohesive energy density, the Flory-Krigbaum theory, and a brief look at charged polymers. 

We concluded the last section with the observation that a polymer solution is expected to be nonideal on the grounds of entropy considerations alone. A nonzero value for AH would exacerbate the situation even further. We therefore begin our discussion of this problem by assuming a polymer-solvent system which shows athermal mixing. In the next section we shall extend the theory to include systems for which AH 9 0. The theory we shall examine in the next few sections was developed independently by Flory and Huggins and is known as the Flory-Huggins theory. 

The lattice model that served as the basis for calculating ASj in the last section continues to characterize the Flory-Huggins theory in the development of an expression for AHj . Specifically, we are concerned with the change in enthalpy which occurs when one species is replaced by another in adjacent lattice sites. The situation can be represented in the notation of a chemical reaction ... 

In this section and the last, we have examined the lattice model of the Flory-Huggins theory for general expressions relating AHj and ASj to the composition of the mixture. The separate components can therefore be put together to give an expression for AGj as a function of temperature and composition ... 

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

Since the Flory-Huggins theory provides us with an analytical expression for AG , in Eq. (8.44), it is not difficult to carry out the differentiations indicated above to consider the critical point for miscibility in terms of the Flory-Huggins model. While not difficult, the mathematical manipulations do take up too much space to include them in detail. Accordingly, we indicate only some intermediate points in the derivation. We begin by recalling that (bAGj Ibn ) j -A/ii, so by differentiating Eq. (8.44) with respect to either Ni or N2, we obtain... 

A far more satisfactory test of the Flory-Huggins theory is based on the chemical potential. According to Eqs. (8.13) and (8.20),... 

The above argument shows that complete overlap of coil domains is improbable for large n and hence gives plausibility to the excluded volume concept as applied to random coils. More importantly, however, it introduces the notion that coil interpenetration must be discussed in terms of probability. For hard spheres the probability of interpenetration is zero, but for random coils the boundaries of the domain are softer and the probability for interpenetration must be analyzed in more detail. One method for doing this will be discussed in the next section. Before turning to this, however, we note that the Flory-Huggins theory can also be used to yield a value for the second virial coefficient. 

To use the Flory-Huggins theory as a source for understanding the second virial coefficient, we return to Eq. (8.53), which gives an expression for jui -jui°. Combining this result with Eq. (8.79) gives... 

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... 

The quantity in parentheses on the right-hand side is reminiscent of the expression AH - T AS, with the quantity 1/2R a contribution from the configurational entropy of the Flory-Huggins theory. Since our objective is to incorporate a solvation entropy into the discussion, we add the latter -in units of R for convenience-to 1/2R ... 

The wolume fraction emerges from the Einstein derivation at the natural concentration unit to describe viscosity. This parallels the way volume fraction arises as a natural thermodynamic concentration unit in the Flory-Huggins theory as seen in Sec. 8.3. 

Experimental values of X have been tabulated for a number of polymer-solvent systems (4,12). Unfortunately, they often turn out to be concentration and molecular weight dependent, reducing their practical utility. The Flory-Huggins theory quahtatively predicts several phenomena observed in solutions of polymers, including molecular weight effects, but it rarely provides a good quantitative fit of data. Considerable work has been done subsequentiy to modify and improve the theory (15,16). 

The well-known Flory treatment [50-52] of the en-thropic contribution to the Gibbs energy of mixing of polymers with solvents is still the simplest and most reliable theory developed. It is quite apparent, however, that the Flory-Huggins theory was established on the basis of the experimental behavior of only a few mixtures investigated over a very narrow range of temperature. Strict applications of the Flory-Huggins approach... 

For more than two decades researchers have attempted to overcome the inadequacies of Flory s treatment in order to establish a model that will provide accurate predictions. Most of these research efforts can be grouped into two categories, i.e., attempts at corrections to the enthalpic or noncombinatorial part, and modifications to the entropic or combinatorial part of the Flory-Huggins theory. The more complex relationships derived by Huggins, Guggenheim, Stavermans, and others [53] required so many additional and poorly determined parameters that these approaches lack practical applications. A review of the more serious deficiencies... 

The formation mechanism of structure of the crosslinked copolymer in the presence of solvents described on the basis of the Flory-Huggins theory of polymer solutions has been considered by Dusek [1,2]. In accordance with the proposed thermodynamic model [3], the main factors affecting phase separation in the course of heterophase crosslinking polymerization are the thermodynamic quality of the solvent determined by Huggins constant x for the polymer-solvent system and the quantity of the crosslinking agent introduced (polyvinyl comonomers). The theory makes it possible to determine the critical degree of copolymerization at which phase separation takes place. The study of this phenomenon is complex also because the comonomers act as diluents. 

The polymer solubility can be estimated using solubility parameters (11) and the value of the critical oligomer molecular weight can be estimated from the Flory-Huggins theory of polymer solutions (12), but the optimum diluent is still usually chosen empirically. 

According to the Flory-Huggins theory of polymer solutions, if the mixing process were driven only by an entropic gradient (nonpolar solvent) the solubility coefficient... 

Chiou, C.T. and Manes, M. Application of the Flory-Huggins theory to the solubility of solids in glyceryl trioleate, / Chem. Soa, Faraday Trans. 1, 82(l) 243-246, 1986. 

Aspler and Gray (65.69) used gas chromatography and static methods at 25 C to measure the activity of water vapor over concentrated solutions of HPC. Their results indicated that the entropy of mixing in dilute solutions is mven by the Flory-Huggins theory and by Flory s lattice theory for roddike molecules at very nigh concentrations. 

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