Flory-Huggins liquid crystals

For the isotropic phase, one has y = x, which implies that y may take values greater than x. This is forbidden in the lattice model. Fortunately, we apply here the Flory-Huggins model for the isotropic phase and we do not need to use the quantity y. In the liquid crystal phase y < x and the lattice model holds. 

Are all quantitative predictions of the thermodynamics of liquid crystals correct. If not stop here. The reason for this step is that die theory (Flory-Huggins lattice model) also predicts the occurrence of the isotropic to nematic phase transition in liquid crystals. If the theory had predicted correctly the properties of glasses but had failed for liquid crystals we would have had to abandon it, especially since in both cases the cause of the transition is ascribed to the vanishing of the configurational entropy. Alternatively the correctness of the prediction for liquid crystals argues for the correctness of the prediction for glasses. Since we have not been stopped by steps 3 and 4 we proceed to step 5. 

A theoretical phase diagram for mixtures of polymer and liquid crystals has been calculated by combining Flory-Huggins free energy for isotropic mixing (10) and Maier-Saupe free energy for nematic ordering (11.12) as follows ... 

For the more frequent case where one or more pure crystals can reach stability on cooling with the liquid solution, a entectic phase diagram results. The free enthalpy of the pure crystal must then reach the chemical potential of its component in the solution. For equilibrinm, the Flory-Huggins expression of equation 38 was used, with the free enthalpy of fnsion represented by AGf = A/ff ATm/Tm°- Assuming that the molecules 2 make up the larger, macromolecular solute, one can write... 

Because of the liquid-crystal-like order, the viscosity of the block copolymer is usually high and is non-Newtonian with reference to dependance on shear rate. As the repulsive interaction energy or the Flory-Huggins interaction parameter increases, the temperature dependence of viscosity decreases. For styrene-diene polymers, the activation energy of flow in the melt state is similar to that of polystyrene. As the interaction gets smaller the distinction between melt state and disordered state disappears. 

Recently, phase behaviour of mixtures consisting of a polydisperse polymer (polystyrene) and nematic liquid crystals (p-ethoxy-benzylidene-p-n-butylani-line) was calculated and determined experimentally. The former used a semi-empirical model based on the extended Flory-Huggins model in the framework of continuous thermodynamics and predicted the nematic-isotropic transition. The model was improved with a modified double-lattice model including Maier-Saupe theory for anisotropic ordering and able to describe isotropic mixing. ... 

The first class of blends to be analyzed is that of a homogeneous, disordered liquid phase in equilibrium with a pure crystalline phase, or phases. If both species crystallize they do so independently of one another, i.e. co-crystallization does not occur. With these stipulations the analysis is relatively straightforward. The chemical potentials of the components in the melt are obtained from one of the standard thermodynamic expressions for polymer mixtures. Either the Flory-Huggins mixing expression (7) or one of the equation of state formulations that are available can be used.(8-16) The melting temperature-composition relations are obtained by invoking the equilibrium requirement between the melt and the pure crystalline phases. When nonequilibrium systems are analyzed, additional corrections will have to be made for the contributions of structural and morphological factors. 

For a numerical calculation, we introduce the temperature parameter t defined by t = llxi=k TfUo). We then have four parameters characterizing our systems itp, the number of segments on a flexible polymer n, the number of segments on a liquid crystal Xa. the attractive interaction (Maier-Saupe) parameter between liquid crystals x = lA, the polymer-liquid crystal interaction (Flory-Huggins) parameter whose the origin is the dispersion forces. We here define the nematic interaction parameter a = Xa/X-From Eq. (3), we can obtain the values of the order parameter S(t, < ) related to a certain temperature r and concentration (p. The nematic phase appears at fo(l- ) = 4.55 in Eq.(3) [15]. We then obtain the nematic-isotropic transition (NIT) temperature (tni) as a function of the polymer concentration 0 ... 

In this edition, previous material has been generally updated. In view of commercial developments over the decade, the discussion of extended-chain crystals has been increased and a section on liquid-crystal polymers has been added. The discussion of phase behavior in polymer-solvent systems has been expanded and the Flory-Huggins theory is introduced. All kinetic expressions are now written in terms of conversion (rather than monomer concentration) for greater generality and ease of application. Also, in... 

Mean field theories can also be extended to the phase behavior of smectic A liquid crystal and flexible polymer blends [63, 70-74] by combining the Flory-Huggins theory for isotropic mixing and Kobayashi-McMillan theory [32, 75] for smectic A ordering of liquid crystals. 

The first term in braces represents the Landau-type free energy of crystal solidification of each component in which the individual free energy of the constituents is weighted by the respective volume fractions to ensure that these potentials vanish at the extreme limits of zero crystallinity or if a component is non-crystallizable. The second term represents the entropic part of the free energy of mixing of the amorphous constituents. In the third term, Xaa corresponds to the amorphous-amorphous interaction parameter of Flory-Huggins that characterizes the stability of the liquid phase. 

Key words thermodynamics, miscible, Flory-Huggins, glass transition, liquid crystal polymer. 

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