Flory-Huggins theory mean-field

The Flory-Huggins mean-field theory recovers the van t Hoff Law 4Fq. (1-72)] in the dilute limit (as 0—>0). At higher concentrations, the mean-field theory predicts that two-body excluded volume interactions make osmotic pressure proportional to the mean-field probability of monomer-monomer contact (0 ) ... 

The most provocative prediction of the PRISM-MSA theory was a non-classical relation between the critical temperature for pha% separation, and degree of polymerization, N, of the form T,. oc This unexpected result corresponds to a massive stabilization of the mixed phase (via a long range concentration fluctuation process) relative to Flory-Huggins mean field theory which predicts T oc N. Moreover, in the high temperature limit, XiNc/Xo oc thereby implying that the Flory-Huggins form is not re-... 

Of conrse, a nseful starting point of the discnssion still is the Flory-Huggins mean-field theory (13-16), where the excess free-energy density of mixing is written as... 

Flory-Huggins mean-field theory. A similar mean-field theory successfully describes thermodynamics of polymer blends and, with some modifications, diblock copolymers and their blends with homopolymers. 

In Eq. 2.35, the nonideal terms depend on only. Then polydispersity affects the ideal solution part, but not the nonideal part, in the Flory-Huggins mean-field theory. 

We shall in section 4.2 deal with regular solutions of small-molecule substances. The construction of phase diagrams from the derived equations is demonstrated. The Flory—Huggins mean-field theory derived for mixtures of polymers and small-molecule solvents is dealt with in section 4.3. It turns out that the simple Flory—Huggins theory is inadequate in many cases. The scaling laws for dilute and semi-dilute solutions are briefly presented. The inadequacy of the Flory-Huggins approach has led to the development of the equation-of-state theories which is the fourth topic (section 4.6) Polymer-polymer mixtures are particularly complex and they are dealt with in section 4.7. 

Several flaws in the Flory—Huggins mean-field theory have been discussed in the previous section. The theory does not explicitly show any volume dependence of the free energy. The so-called equation-of-state theories have in common that they are founded on the expression A = f T, V) where A is the Helmholtz free energy. An equation of state, p = f T, V), can be formulated because p = dA/dV)T. 

Serious flaws in the Flory-Huggins mean-field theory - e.g. that it does not consider volume effects - led to the development of a whole family of theories called equation-of-state theories. These models can predict both LCST and UCST, and negative mixing enthalpy and mixing volume. Some of the equation-of-state theories are specifically developed to account for specific interaction. 

Fig. 2 A graphical visualization of the Flory-Huggins mean field theory. The grating positions with white circles illustrate the solvent and the gray circles illustrate the PE. It s assumed that Each field has the same size, no overlap of fields or chains, all field positions are occupied, all polymer-polymer interactions are the same (all chain parts are the same)...

While the Flory-Huggins mean-field theory [13, 14, 15] of Sect. 2.1 describes the generic, qualitative behavior of incompressible polymer h-solvent mixtures, it invokes three important simplifications that restrict its application ... 

This phase separation has been observed experimentally and this prediction is one of the major successes of the Flory-Huggins mean field theory. In general the free energy of the solution depends on (/) as shown in Figure 12. 

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