Flory’s model

The elastic contribution to Eq. (5) is a restraining force which opposes tendencies to swell. This constraint is entropic in nature the number of configurations which can accommodate a given extension are reduced as the extension is increased the minimum entropy state would be a fully extended chain, which has only a single configuration. While this picture of rubber elasticity is well established, the best model for use with swollen gels is not. Perhaps the most familiar model is still Flory s model for a network of freely jointed, random-walk chains, cross-linked in the bulk state by connecting four chains at a point [47] ... 

Figure 5.10 Gelation theories (Schematic), (a) Flory s model (molecular functionality / = 3). (b) Percolation on a square lattice. In each case, a spanning gel molecule is embedded in the sol. (From Ref. 23.)...

Flory-Huggins model[166] is the start point of the models on gels. Most of the models are derivatives or extension of the model for describing critical phenomena[164, 60]. Doi et al. studied deformation process of ionic gels in electric fields[127]. They carefully extended the Flory s model in variety of conditions. Osada et al. studied cooperative binding of surfactant molecules into the ionic gels[128, 129], They are also based on Flory s model. 

The spinodal curve and the critical points (including multiple critical points) only depend on few moments of the molar-mass distribution of the polydisperse system while the cloud-point curve the shadow curve and the coexistence curves are strongly influenced by the whole curvature of the distribution function. The methods used that include the real molar-mass distribution or an assumed analytical distribution replaced by several hundred discrete components have been reviewed by Kamide. In the 1980s continuous thermodynamics was applied, for example, by Ratzsch and Kehlen to calculate the phase equilibrium of a solution of polyethene in supercritical ethene as a function of pressures at T= 403.15 K. The Flory s model was used with an equation of state to describe the poly-dispersity of polyethene with a a Wesslau distribution. Ratzsch and Wohlfarth applied continuous thermodynamics to the high-pressure phase equilibrium of ethene [ethylene]-I-poly(but-3-enoic acid ethene) [poly(ethylene-co-vinylace-tate)] and to the corresponding quasiternary system including ethenyl ethanoate [vinylacetate]. In addition to Flory s equation of state Ratzsch and Wohlfarth also tested the Schotte model as a suitable means to describe the phase equilibrium neglecting the polydispersity with respect to chemical composition of the... 

At this stage, it may be appropriate to stress a point made earlier, namely the questionable use of Flory s model for evaluating the free energy of mixing for molecules which are not chainlike and not very different in size. This will apply to the compounds Z-j,... 

Flory s model applies to an athermic solution, meaning that the excess molar enthalpy is null, so the excess molar Gibbs energy is simply written as follows, in light of relation [3.125] ... 

Staverman uses Flory s model as a starting point, but introduces a form factor in the number of complexions, taking account of the fact that the molecules are only in contact at their surfaces. This enables him to construct a new expression for number of possibilities of introducing the molecule of polymer and the molecules of solvent. Thus, the excess molar entropy term of conformation is altered, and becomes ... 

Flory s model has the limitation of assuming that all polymeric chains have the same length (same degree of polymerization) however, longer chains have higher hydrodynamic diameters and show increased difficulty for disentanglement and orientation during the formation of the mesophase. Another limitation of this model... 

Since its introduction, Flory s model has found success at describing the qualitative behavior of copolymer crystallization however, it typically does not compare well quantitatively to experimental data [11-13]. Richardson et al. [11], then later Alamo et al. [12] examined the melting temperatures of ethylene copolymers with varying counit contents. As illustrated in... 

Figure 11.1, the experimentally determined copolymer melting temperature decreases with increasing counit content, in agreement with Flory s model however, the numerical values of the measured melting temperatures are significantly below those calculated from Equation 11.2, and the dependence of on counit content is much stronger. 

Although the copolymers shown in Figure 11.1 were crystallized very slowly to allow most favorable conditions for equilibrium crystallization at each temperature, and the reported melting temperatures, measured by dilatometry, were determined by locating the temperatures where the melting curves merged with the liquidus line [11], these conditions were still far removed from the thermodynamic equilibrium upon which the Flory model was built. Furthermore, the formation of thick crystallites that correspond to the equilibrium melting temperature in Flory s model is an extremely rare occurrence. Even if they were present, their minute quantity would make detection difficult. Thus, the... 

The results of Flory s model are in qualitative agreement with experimental results. As the molecular weight of a polymer increases (i.e., as x increases), the value of the critical volume fraction V2 j moves closer to 0 and the value of Xc approaches 0.5. Since the critical temperature increases as the molecular weight increases, the higher molecular weight samples are less soluble in a given solvent under similar conditions. Note that for the modest value of x = 100, V2 j is approximately 0.1 and Xc is 0.6. 

This analysis shows that Flory s model implies much more detailed assumptions about the liquid forces than only the assumed deformation of fluctuation domains. The effect of that assumption alone is estimated by Eq. (50) for real networks in equilibrium. Flory s additional assumption (ii) that C, the centre of the fluctuation domain undergoes displacement affine with the strain together with assumption (iv) about the distribution of fluctuations around 0, leads to non-affine displacement of 0, the equilibrium position in the phantom network. An alternative assumption could have been that the displacement of 0 is affine with the strain while that of C is not. This is perhaps even more probable because with increasing strain the chain forces will increase in the direction of strain while the liquid forces will not. Also the additional assumption (iv) about the distribution of fluctuations around 0 implies a detailed model of the / p, -configuration and the / reai-configuration. 

---