Flory-Huggins theory temperature

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. 

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... 

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. 

Equation-of-state theories employ characteristic volume, temperature, and pressure parameters that must be derived from volumetric data for the pure components. Owing to the availability of commercial instruments for such measurements, there is a growing data source for use in these theories (9,11,20). Like the simpler Flory-Huggins theory, these theories contain an interaction parameter that is the principal factor in determining phase behavior in blends of high molecular weight polymers. 

Figure 4 shows a plot of the static expansion factor (o ) as a function of the relative temperature 0/T, where a is defined as Rg(T)/Rg(0) and r is the number of residues that may be one monomer unit or a number of repeat units. When T < 0 (water is a good solvent for PNIPAM), the data points are reasonably fitted by the line with r = 105 calculated on the basis of Flory-Huggins theory [15]. Similar results have also been observed for linear polystyrene in cyclohexane [25,49]. The theory works well in the good-solvent region wherein the interaction parameter (x) is expected to be... 

Figure 11-1. Temperature dependence of the partitioning of benzene (5) between water (w) and //-alkanes (a) of different chain lengths octane ( ), decane ( ), dodecane (A), tetradecane (H), and hexadecane (O)-In (C), die left vertical scale pertains when Flory—Huggins theory is applied to both die a and w phases, whereas die right vertical scale pertains when Rory—Huggins theory is applied only to die a phase. The figure is taken from de Young and Dill [ 17] with permission...

In practice, the Flory-Huggins theory fails to predict many features of polymers solutions, either qualitatively or quantitatively, but remains widely used because of its simplicity. The Flory parameter x, assumed to be constant, often increases with interaction-energy scaled on kT, often exhibits a more complicated temperature dependence than 1/T (Flory, 1970). Such behavior stems from energetic effects, such as directional polar... 

The thermodynamic definition of the spinodal, binodal and critical point were given earlier by Eqs. (9), (7) and (8) respectively. The variation of AG with temperature and composition and the resulting phase diagram for a UCST behaviour were illustrated in Fig. 1. It is well known that the classical Flory-Huggins theory is incapable of predicting an LCST phase boundary. If has, however, been used by several authors to deal with ternary phase diagrams Other workers have extensively used a modified version of the classical model to explain binary UCST or ternary phase boundaries The more advanced equation-of-state theories, such as the theory... 

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... 

The lattice fluid equation-of-state theory for polymers, polymer solutions, and polymer mixtures is a useful tool which can provide information on equa-tion-of-state properties, and also allows prediction of surface tension of polymers, phase stability of polymer blends, etc. [17-20]. The theory uses empty lattice sites to account for free volume, and therefore one may treat volume changes upon mixing, which are not possible in the Flory-Huggins theory. As a result, lower critical solution temperature (LCST) behaviors can, in principle, be described in polymer systems which interact chiefly through dispersion forces [17]. The equation-of-state theory involves characteristic parameters, p, v, and T, which have to be determined from experimental data. The least-squares fitting of density data as a function of temperature and pressure yields a set of parameters which best represent the data over the temperature and pressure ranges considered [21]. The method,however,requires tedious experiments to deter-... 

Temperature dependence of y for mixtures of hydrogenated polybutadiene (88% vinyl) and deuterated polybutadiene (78% vinyl) and the calculated phase diagram from Flory-Huggins theory with Aa = Ab = 2000 and Vo = 100 A. The binodal is the solid curve and the spinodal is dashed. Adapted from N. P. Balsara, Physical Properties of Polymers Handbook (J. E. Mark, editor), AIP Press, 1996, Chapter 19. 

Temperature dependence of x for mixtures of polyisobutylene and deuterated head-to-head polypropylene and the calculated phase diagram from Flory-Huggins theory with... 

Since the mean-field Flory-Huggins theory puts everything that is not understood about thermodynamics into the x parameter, this parameter is experimentally found to vary with composition and temperature. For solutions of linear polystyrene in cyclohexane, the interaction parameter... 

The solution properties of copolymers are much more compHcated. This is due mainly to the fact that the two copolymer components A and B behave differently in different solvents, and only when the two components are soluble in the same solvent will they exhibit similar solution properties. This is the case, for example for a nonpolar copolymer in a nonpolar solvent. It should also be emphasised that the Flory-Huggins theory was developed for ideal Hnear polymers. Indeed, with branched polymers with a high monomer density (e.g. star-branched polymers), the 0-temperature will depend on the length of the arms, and is in general lower than that of a linear polymer with the same molecular weight. 

In the original formulation of the Flory-Huggins theory, xy was strictly an energetic parameter that was proportional to tne energy required to form an i-j bond from a i-i and j-j bond. It also had a simple 1/T temperature dependence and was independent of solution composition. Experimentally, Xii often has a large positive entropic component which arises, according to the Flory and lattice fluid (LF) theories, from differences in the equation of state properties of the pure components (1,10). 

As predicted by the Flory-Huggins theory, such a system shows a lower miscibility gap characterized by an upper critical point, at temperature Ta, which depends on both the oil and the amphiphile structure (Figure 3.11a). The critical composition is usually not far from the pure oil side. 

Equation 5.7 [10], where do, d[, 62 and d3 are fitting parameters, and d>B is the volume fraction of Component B, can produce all of the binary phase diagram types shown in Figure 5.1, when used either to lit experimental data in the context of the Flory-Huggins theory of thermodynamics or to express interaction energies calculated by atomistic simulations in a convenient manner as a function of the temperature and the component volume fractions. 

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