Phase transitions overlap functions

We have already stated that the a-[3 transition of quartz may be described as a A transition overlapping a first-order transition. The heat capacity function for the two polymorphs is thus different in the two stability fields, and discontinuities are observed in the H and S values of the phase at transition temperature T rans cf section 2.8). For instance, to calculate the thermodynamic properties of ]8-quartz at T = 1000 K and P = bar, we... 

KJ/mol and a gauche-trms energy difference of 2.5 kJ/mol). The constants D, and y [see Eq. (2)] represent the usual Morse oscillator parameters. The nonbonded terms e and a [Eq. (3a)] represent the Lennard-Jones parameters, b and C [Eq. 3b)] are related to the overlap and dispersion of the atoms i and j, and A is a parameter related to the position and well-depth of the interaction. The bending force constant is, and 6o [Eq. (4)] indicates the equilibrium value of the angle formed by the three atoms of interest. The above potential energy functions [Eqs. (2-5)] have been demonstrated to yield good spectroscopic, thermodynamic, and kinetic data, as well as to provide the atomistic details of temperature-dependent phase transitions for crystalline polymers. 

FIG. 14 Phases observed in aqueous solutions of surfactants m-2-m, 2Br as a function of m and m. The concentrations of the investigated solutions were between 0.1 and 10 wt %. Transition temperatures are indicated whenever a phase transition was observed. The different types of worm-like micelles differ by their overlap concentrations lower than 0.5 wt % for the longest micelles, between 0.5 and 2 wt %, and between 2 and 10 wt % (from Ref. 55). 

Prigodin and Efetov [70] studied the insulator-metal transition of these quasi-one-dimensional conducting polymers using a random metallic network (RMN) model to represent weakly connected, fibrous bundles of metallic chains. In this zero temperature model, the phase transition from insulating to metallic behavior is a function of the cross section of electronic overlap between fibers (a) and p = pLu c- the product of the localization radius (Tioc) and the concentration of cross-links... 

In polymer solutions, liquid-liquid (L-L) demixing is another common phase transition besides crystallization. The thermodynamic boundary conditions for both of them behave as the functions of polymer concentrations and temperatures, demonstrated as phase diagrams. The schematic L-L binodal and liquid-solid (L-S) coexistence curves in polymer solutions and their interception are shown in Figure 13.2. The illustrated L-L binodal contains an upper critical solution temperature. Some other solutions also contain binodals with a lower critical solution temperature. When the L-S curve intersects with the L-L curve in the overlapping temperature windows, both curves are terminated at the intersection point, which is referred to as the monotectic triple point. 

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