The Flory-Huggins Model

The mole fractions of polymer and solvent are UpKnp + Hs) and UsKnp + . ), respectively. The volume fractions, (ps for the solvent and pp for the polymer, 

Equation (31.3) is incomplete because it does not account for excluded volume among distant segments, the possibility that one monomer may land on a lattice site that has already been occupied by a more distant monomer. Equation (31.3) overestimates the number of conformations iq. To account for excluded volume, PJ Flory (1910-1985), who won the 1974 Nobel prize in Chemistry for his contributions to polymer science, made the approximation that the volume excluded to each segment is proportional to the amount of space filled by the chain segments that have already been placed on the lattice, as if they were randomly dispersed in space. According to this approximation, (M -1) /Af is the fraction of sites available for the second monomer, (M 2)/M is the fraction available for the third monomer, and so on. Therefore a better estimate of the number of conformations available for one chain is 

Now we use the same logic for putting all of the n.p chains onto the lattice. Each of the rip chains has a first monomer unit. The number of arrangements Vfirst for placing the first monomer segments for all rip chains is 

To compute the total number of arrangements W(rip,ns) of the system of all rip polymer molecules and ris solvent molecules, multiply the factors together  

The factor Up in the denominator on the right-hand side of Equation (31.7) accounts for the indistinguishability of one polymer chain from another. The factor M-Nn.p) accounts for the indistinguishability of the solvent molecules. 

Here (/ a and 0b are the volume fractions of polymers A and B, and Va and Nb are the number of lattice sites occupied by an A and a B molecule, respectively. If Nb is set to unity, the above equation corresponds to polymer A dissolved in a small-molecule solvent B. //cbF is the energy cost per lattice site of moving an A polymer molecule from a reservoir of all A into a reservoir of all B. / is thus related to the contact energies, wn, by 

X is supposed to account for van der Waals interactions. For purely dispersive interactions between similarly sized molecules, using Eqs. (2-27) and (2-29), we expect that 

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 

Although the Flory-Huggins theory was derived from a lattice model in which units of polymer A and polymer B are-the same size (i.e., they each occupy a single lattice cell), the theory is readily generalized so that it can apply to realistic cases in which the volumes of monomers A and B are unequal  

For a polymer in a small-molecule solvent, it is convenient to define N = Na to be the polymer degree of polymerization and take Nb = = I for the solvent. It is 

Equation (2.14) can be considered a general expression valid for regular solutions, regardless of the size of the components (solvents or polymers). Note that the entropic term scales with the number of molecules, and the enthalpic term with the volume of the system hence, both terms scale differently with the size of the system, and implicitly with the size of the molecules comprising the system. To explore the effect of these size-related aspects on AG , Eq. (2.14) is rewritten as an intensive property, usually referenced to the volume corresponding to one mole of lattice sites, VV The number of lattice sites occupied by component i is given by its molar volume ratio, rj, defined as  

Equation (2.17) provides AG per mole of lattice sites, or per Vr volume units of mixture. It is valid for any type of r lar solution, regardless of the difference in size between the components of the mixture. In Eq. (2.17), the entropic term is always negative (ln0i 0, since 0j 1) and therefore favorable to the miscibility. The plot of this term versus composition is concave upwards for any system at any temperature. Miscibility depends therefore on the sign and magnitude of the enthalpic term. If B 0, the enthalpic term is also concave upwards and the plot 

Phase separation begins at the critical conditions, when the two minima of the AG versus j plot meet at the same point, and also meet the two inflection points limiting the concavity regions. Therefore, the following conditions apply [1-5]  

Applying Eqs (2.18) to (2.17), the following relationship is obtained for the critical conditions [2-5]  

There is, however, another important difference between the low- and high-molecular-weight mixtures seldom discussed in the literature according to the 

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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... 

To arrive at an expression for AS, we follow a series of steps which parallel-for a different model-the development of the Flory-Huggins model for AS,... 

The first qualitatively correct attempt to model the relevant chemical potentials in a polymer solution was made independently by Huggins (4, ) and Flory [6). Their models, which are similar except for nomenclature, are now usually called the Flory-Huggins model ( ). 

This equilibrium concentration c, or the corresponding mole fraction x, of EG, water and DEG in the interface can be calculated from the vapour pressure and the activity coefficient y, derived from the Flory-Huggins model [13-17], Laubriet et al. [Ill] used the following correlations (with T in K and P in mm Hg) for their modelling ... 

If we turn from phenomenological thermodynamics to statistical thermodynamics, then we can interpret the second virial coefficient in terms of molecular parameters via a model. We pursue this approach for two different models, namely, the excluded-volume model for solute molecules with rigid structures and the Flory-Huggins model for polymer chains, in Section 3.4. 

The most important factor controlling the morphologies generated is the location of the composition of the initial blend, < )mo with respect to the critical composition, < )M cnt (Figs 8.5 and 8.6). The latter may be calculated from the Flory-Huggins model as applied to a binary blend (step reactions) or a ternary blend (chain reactions), taking into account polydispersity (Kamide, 1990). The size of particles increases with the concentration of the component that forms the dispersed phase. Typically, for < )mo < < >M,crit, an increase in < )M0 will lead to an increase in both the volume fraction and the average size of dispersed phase modifier-rich particles. 

Figure 6 shows the respective data plotted according to (21) for a number of blends with different degrees of polymerization. The left plot shows the Soret coefficients as measured and the right one after normalization to the mean field static structure factor calculated from the Flory-Huggins model, cf. (7). Although the structure factors and the Soret coefficients of the different samples vary by more... 

A modified Cahn-Hilliard (CH) model [114] is used for the theoretical analysis of the impact of thermal diffusion on phase separation by taking into account an inhomogeneous temperature distribution, which couples to a concentration variation via the Soret effect. The Flory-Huggins model is used for the free energy of binary polymer-mixtures. The composition is naturally measured in terms of volume fraction 0 of a component A, which can be related to the weight fraction c by... 

So far two models have been employed to rationalize the solvation process the classical solution model, either the mole-fraction scale or any other concentration scale, and the Flory-Huggins model. The question is where to use which theoretical model to interpret the results of partitioning experiments, in which solute molecules distribute between two phases, a and ft. If the two phases are at equilibrium at the same temperature and the same pressure, /z = /xf. After rearrangement and applying Eq. (11-8), we can write... 

The first mean-field theories, the lattice models, are typified by the Flory-Huggins model. Numerous reviews (see, e.g., de Gennes, 1979 Billmeyer, 1982 Forsman, 1986) describe the assumptions and predictions of the theory extensions to polydisperse and multicomponent systems are summarized in Kurata s monograph (1982). The key results are reiterated here. 

The application of the Flory-Huggins model to liquid-liquid equilibria is discussed in Section 2F. 

For the Flory-Huggins model of the activity, Equations (2C-4)-(2C-6), the data are reported in terms of a concentration dependent chi parameter. Data must be taken or extrapolated to the 6 = 0 limit in order to remove solution structure effects. 

The WFAC, activities, and partial pressures can be calculated from the Flory-Huggins model if a value of the interaction parameter is available. 

In recent years, there have been substantial efforts to measure the composition dependence of the interaction parameter X12 (Bates et a/., 1988 Han et al, 1988) - see Fig. 4.4, p. 79 - and thus to refine our understanding of this theory. The original and customary derivations of the Flory-Huggins model introduced an interaction parameter that was considered to be independent oi 4>i but it has long been recognized that this is not the typical case (Flory, 1970). We depart here from the customary derivations to lay a groundwork for a basic reconsideration of that composition dependence. For example, we don t insist here that X12 is the traditional Rory-Huggins i2 parameter. 

Here we develop a derivation of the Flory-Huggins model that is broader than the customary derivations. The derivation below doesn t, at an initial stage, express various quantities related to polymeric materials on a per monomer basis, as is customary. The chief reason for this is that such a definition at an initial stage is typically based upon lattice modeling of the problem, and we wish here to avoid premature idealizations. 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

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. 

The Flory-Huggins model differs from the regular solution model in the inclusion of a nonideal entropy term and replacement of the enthalpy term in solubility parameters by one in an interaction parameter x- This parameter characterizes a pair of components whereas each S can be deduced from the properties of a single component. 

Fig. 3.14.4. Variation of the molar Gibbs free energy of mixing with composition of a binary mixture, as determined according to the Flory-Huggins model, Eq. (3.14.24), with = 0.1 and for various values of A jRT. Tick marks indicate location of minima.

A comparison of freezing temperature lowering via the mechanisms for the disruption of surface and for the Flory-Huggins model is shown in Fig. 34. Our result, Eq. (27), is similar to that obtained by Solms and Rijke (1971). In their case a polymer system where x < i is used to test the results of the theory. 

Figure 3.34 Left Adsorption data for benzyl alcohol (BA, symbols) at different temperatures, 20 o, 30 , 40 o, 50 A, and 60 °C. The solid lines are the correlations given by the Flory-Huggins model. Right Clausius-Clapeyron plots for BA, with data at different loadings, 10 o, 20 , 30 o, and 40 g/1 A. Reproduced with permission from I. Quidones, J. C. Ford and G. Guiochon, Chem. Eng. ScL, 55 (2000) 909 (Figs. 7 and 8).

Figure 4.6 Competitive isotherms of a ternary mixture of benzyl alcohol (BA), phenyl-2-ethanol (PE) and methyl-benzyl alcohol (MBA) on Cig-sUica with MeOH/H20 as the mobile phase, at different relative concentrations. Adsorbed amormts of (a) benzyl alcohol, (b) 2-phenylethanol, (c) 2-methyl benzyl alcohol, (d) activity coefficient in the mobile phase versus concentration. In Figures (a), (b), (c), the open circles are for equal parts of BA, PE and MBA, the diamonds for three parts of BA, 1 part PE and 1 part MBA), the triangles for one part of BA, one part of PE and three parts of MBA and the stars for the singlecomponent isotherms. The solid line is the Flory-Huggins model and the dashed lines are the IAS model isotherms. In Figures (d) and (e), the circles are for BA, the squares for PE, the triangles for MBA and the diamonds for the solvent. Reproduced from I. Quinones, J. Ford, G. Guiochon, Chem. Eng. Set, 55 (2000) 909 (Figs. 12,13 and 14).

The free energy of mixing (per mole of lattice sites) of a polymer solution (according to the Flory-Huggins model) is... 

Selecting Solvents with the Flory-Huggins Model... 

Section 16.3 reviews the basics of polymer thermodynamics, discusses the differences compared to thermodynamics of systems having only low-molecular-weight compounds, and finally gives an overview of the Flory-Huggins model, which has been considered one of the cornerstones of polymer thermodynamics. 

The solubility parameter is a very important property in science and has found widespread use in many fields and not just in the smdy of polymer-solvent thermodynamics. It is connected to the Flory-Huggins model as well, as explained in Section 16.3.3.2, but can also be used independent of it, as discussed in Sections 16.3.3.1 and 16.3.3.3. Several handbooks and reference books provide extensive lists of solubility parameters of numerous chemicals.The solubility parameter is defined as... 

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