Critical points three-dimensional crystals

Two- and three-dimensional crystals.In two-dimensional (2D) systems the critical regions are of two kinds points at energy Eo at the edge of the B.Z. reminiscent of ID patterns in the electron distribution with a contribution in x and x as given by (7) and lines at energy Ej intrinsic to the 2D system. The effect... 

It was, however, observed that such systems under appropriate conditions of concentration, solvent, molecular weight, temperature, etc. form a liquid crystalline solution. Perhaps a little digression is in order here to say a few words about liquid crystals. A liquid crystal has a structure intermediate between a three-dimensionally ordered crystal and a disordered isotropic liquid. There are two main classes of liquid crystals lyotropic and thermotropic. Lyotropic liquid crystals are obtained from low viscosity polymer solutions in a critical concentration range while thermotropic liquid crystals are obtained from polymer melts where a low viscosity phase forms over a certain temperature range. Aromatic polyamides and aramid type fibers are lyotropic liquid crystal polymers. These polymers have a melting point that is high and close to their decomposition temperature. One must therefore spin these from a solution in an appropriate solvent such as sulfuric acid. Aromatic polyesters, on the other hand, are thermotropic liquid crystal polymers. These can be injection molded, extruded or melt spun. 

Micelles are loose aggregates of amphiphiles in water or organic solvents which form above a certain temperature (Krafft point) and concentration (critical micellar concentration, cmc). Below the Krafft temperature, clear micellar solutions become turbid and the amphiphile forms three-dimensional hydrated crystals. Below the cmc, micelles dissociate into monomers and small aggregates. Above the cmc, the micelles of an aggregation number n are formed n then remains stable over a wide concentration range . Table 1 gives some typical cmcs and three Krafft point values. 

In this paper we give an overview of the mean-field theory of phase transitions in coupled rotors with particular attention to the issues of reentiance, other quantum anomalies, and meta-stability. We comparatively analyze coupled planar rotors (two-dimensional model) and coupled linear rotors (three-dimensional). We show that the dipolar potential does not exhibit the reentrance anomaly, whereas the quadmpolar one does. The phase transition turns out to be second order in all cases except for the linear rotors in a quadmpolar potential where it is first order. We also investigate the effects of the crystal field in the case of the linear rotor model with quadmpolar potentials the crystal field causes the appearance of critical points which separate lines of the phase diagram where the transition is first order from regions where there is no... 

The X-ray crystal structure database led us to believe that peptide bonds adopt either the cis or trans conformation in native proteins [22,128]. However, NMR spectroscopy [143], and in a few cases, crystal structure analysis [144], provide encouraging experimental evidence of conformational peptide bond polymorphism of folded proteins. Furthermore, conformational changes in response to ligand binding, crystallization conditions and point mutations at remote sites are frequent. Consequently, the three-dimensional protein structure database contains homologous proteins that have different native conformations for a critical prolyl bond [12]. 

Kirkwood has pointed out that the density distribution function of a crystalline solid (the translational molecular distribution function) can be expanded in a three-dimensional Fourier series. The coefficients in this series are then identified as the order parameters of the crystalline phase. All these order parameters vanish discontin-uously at the first-order melting point. Empirically, there are no second-order melting transitions, nor do there seem to be any solid-liquid critical points. Though not a proven fact (as far as I am aware), it seems reasonable that crystal melting is always first order because all of the order parameters cannot vanish simultaneously and continuously before the free energy of the solid phase exceeds that of the liquid phase. 

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