Huggins-Flory theory

For a solution of a polydisperse polymer with m polymer components in one solvent, the Flory-Huggins expression for the Gibbs energy of mixing per mole of lat- 

To avoid confusion with the polymer-solvent interaction parameter, the symbol gsp [Eq. (36)] is used instead of/, which is independent of concentration. Equations (37a)-(37c) are examples of expressions used for the temperature and composition dependence of gsp [27-29]. 

We can derive Eq. (38) for the solvent, and Eq. (39) for polymer component i, where r is number-average chain length of the polymer, which is defined by Eq. (40) and which is proportional to the number-average molecular weight of the polymer. 

The description of the influence of pressure on polymer-solvent phase behavior is also possible using a pressure-dependent gsp [27, 30-32). However, this approach is purely empirical. 

The most relevant theory for modeling the free energy of binary polymer mixtures is the Flory-Huggins theory, initially employed for solvent-solvent and polymer-solvent mixtures. This theory was independently derived by Flory [4, 5] and Huggins [6, 7]. The key equation (combined from discussions earlier in this chapter on entropy and enthalpy of mixing) is  

from the relationship, AGm = AHm — TASm, then AHm = 0102A l2-RTtyVr = 0102 12  

In some references, AGm is expressed as the term AGm/V in Eq. 2.24, in those cases AGm has units of cal/cc. Unless noted otherwise, AGm in this text has units of cal. Also, in some cases, p is assumed to equal 1.0 g/cc and is eliminated from the equation however, the imits remain and must be accounted for. 

As the molecular weights of the respective polymers increase, x uycr nd Buycr 0- Often, the density of both polymers is assumed = 1.0 g/cc, and p is eliminated from the equations (again units must be accoimted for). With p = 1.0 g/cc and equal molecular weight for the components, = 2/M mole/cc andfr = 2RT/M cal/cc. 

For the spinodal condition (Eq. 2.3) to be satisfied, the second derivative of Eq. 2.23 yields  

Nonideal thermodynamic behavior has been observed with polymer solutions in which A Hm is practically zero. Such deviations must be due to the occurrence ofa nonideal entropy, and the first attempts to calculate the entropy change when a long chain molecule is mixed with small molecules were due to Flory [8] and Huggins [9]. Modifications and improvements have been made to the original theory, but none of these variations has made enough impact on practical problems of polymer compatibility to occupy us here. 

The Flory-Huggins model uses a simple lattice representation for the polymer solution and calculates the total number of ways the lattice can be occupied by small molecules and by connected polymer segments. Each lattice site accounts for a solvent molecule or a polymer segment with the same volume as a solvent molecule. This analysis yields the following expression for A5m, the entropy of mixing A l moles of solvent with N2 moles of polymer. 

Equation (12-16) is similar to Eq. (12-1), except that volume fractions have replaced mole fractions. This difference reflects the fact that the entropy of mixing of polymers is small compared to that of micromolecules because there are fewer possible arrangements of solvent molecules and polymer segments than there would be if the segments were not connected to each other. 

Equation (12-16) applies also if two polymers are being mixed. In this case the number of segments r, in the /th component of the mixture is calculated from 

The entropy gain per unit volume of mixture is much less if two polymers are mixed than if one of the components is a low-molecular-weight solvent, because N is much smaller in the former case. 

In terms of the new symbols, the second virial coefficient is then given by the expression [cf. Eq. (3.78)]  

at a special temperature T = 9, A2 becomes equal to zero and the solution therefore becomes pseudoideal. Such solutions are also called theta solutions. The second virial coefficient is positive at temperatures higher than 9 and negative at lower temperatures. 

Essentially in the above modification, one interaction parameter x has been replaced with two new parameters j/ and 9, thus adding flexibility to the treatment. In doing so, a phenomenological approach of modeling has been adopted in place of the pseudolattice model we started with. 

Problem 3.7 Osmotic measurements on a high-molecular-weight polystyrene sample in cyclohexane at various temperatures yielded the following values for the second virial 

Determine for the polymer-solvent system, (a) the temperature at which theta conditions are attained, (b) the entropy of dilution parameter ijr, and (c) the heat of dilution parameter K at 27°C. [Specific volume of polymer = 0.96 cm /g molar volume of cyclohexane at 27°C = 108.7 cm /mol.] 

A schematic phase diagram of a symmetrical binary mixture is shown in Fig. 1 in a temperature versus composition representation. A symmetric polymer blend is characterized by two polymer components of the same molar volume and, therefore, with a 50% critical composition. Within mean field approximation the binodal and spinodal phase boundaries of a binary (A/B) incompressible polymer mixture are described by the Gibbs free energy of mixing AG according to 

The total volume of the solution is divided into no( = + nn2) cells, where and n2 represent the 

22 12 are the interaction energies between the nearest neighbor solvent molecules, 

Equation (103) gives the relation between the x parameter and the binary cluster integral of equation (19). When x= 1/2, the second virial coefficient A2 vanishes and the osmotic pressure is given by the ideal gas law  

The temperature at which = 1/2 is called the ideal (0) temperature and the Gaussian statistics prevails. 

For good solvents (x 0), the Flory-Huggins theory predicts that A2 is proportional to w and is independent of chain molecular weight. However the experimental data show that A2 Attempts to explain this molecular weight dependence of the second virial coefficient have been made by Flory and Krigbaum and others. A complete theory will not involve a lattice model and treat the averages more carefully. It is still an active subject. 

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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),... 

Figure 8.4 Experimental test of Flory-Huggins theory by Eq. (8.62) for the systems indicated. (From Ref. 3, used with permission.)...

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. 

Flory-Huggins Theory. The simplest quantitative model foi that iacludes the most essential elements needed foi polymer blends is... 

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. 

By using the liquid lattice approach to treat the random mixing of a disoriented polymer and a solvent, the so-called Flory-Huggins theory is often used to correlate the penetrant activity and the composition of the solution ... 

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. 

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