Models Flory-Huggins

The Flory-Huggins lattice model assumes that a polymer chain consists of a number of equivalent segments. The extension to polymer solvent interactions assumes that the polymer solution consists of a three-dimensional lattice and each lattice site is occupied either by a polymer segment or by a solvent molecule. Flory and Huggins calculated the entropy change of mixing as 

Here l and N2 are the number of solvent and polymer molecules, respectively, and the volume fractions 01 and p2 are defined by the expressions 

Several improvements to Equation (2D-4) have been suggested. Primarily these modifications involve a more exact treatment of the polymer chain in the lattice such as including the probability of overlapping chains. These improvements are not generally applied in view of the approximations inherent in the lattice model of the fluid and the marginal increase in accuracy resulting from these improvements. 

Flory (1942) noted that the combinatorial term is not sufficient to describe the thermodynamic properties of polymer-solvent systems. To correct for energetic effects, he suggested adding a residual term, ares, to account for interactions between lattice sites. 

Ideal solutions are defined as mixtures that have no volume or enthalpy changes upon mixing, but have an ideal entropy of mixing given by 

The solubility parameter, 6, is defined as the square root of the cohesive energy density. The cohesive energy density is the amount of energy per unit volume that keeps the fluid in the liquid state. An excellent approximation for the cohesive energy of a solvent, cn, is the heat of vaporization, which is the amount of energy that must be supplied to vaporize the fluid. The solubility parameter is calculated from 

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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... 

In practice, such a fractionation experiment could be carried out by either lowering the temperature or adding a poor solvent. In either case good temperature control during the experiment is important. Note that the addition of a poor solvent converts the system to one containing three components, so it is apparent that the two-component Flory-Huggins model is at best only qualitatively descriptive of the situation. A more accurate description would require a... 

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 ( ). 

Flory-Huggins model for polymer solutions, based on statistical thermodynamics, is often used for illustrating the behavior of polymer blends [6,7]. The expression for the free energy change... 

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. 

Rigby et al. (1985) showed using a Flory-Huggins model that for symmetric blends, the spinodal and critical temperatures decrease linearly with increasing content of a symmetric diblock for blends with equal volume fractions of homopolymers (with the same molecular weight). The condition for a linear decrease of the binodal was less restrictive, not requiring equal concentrations of homopolymer in the blend. 

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... 

Aldrich humic acid, equilibrium dialysis Aldrich and Fluka humic acid, Flory-Huggins model, Chin Weber 1989)... 

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... 

Fig. 9. Phase diagram ofthe thin film with surface parameters p O.2, g=-0.5 plotted in the plane of variables % 1, for polymers of chain length N=100 and for three choices of film thicknesses D=20 (diamonds), D=60 (crosses) and D=100 (squares). Broken curve shows the bulk phase diagram of the underlying Flory-Huggins model for comparison. Remember that lengths are measured in units of the size b of an effective monomer. From Flebbe et al. [58]...

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. 

Figure 2F-6. Correlation using a modified Flory-Huggins model (crosses) for the Acetone-Polystyrene (Mw = 19,800) system. Experimental data (squares) of Siow etal. (1972).

Regressed Interaction Parameters for Water-Dextran (Mn = 23,000)-Poly(ethylene glycol) (Mw = 6,750) at 273 K Using a Modified Flory-Huggins Model... 

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. 

Programs for all of the methods in Chapter 3 of the Handbook are provided in the software package. FORTRAN subroutines for the UNIFAC-FV, High-Danner, Chen et al., and Flory-Huggins models and for the calculation of the P-V-T behavior of pure polymer liquids are also provided. 

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