Order parameter, equilibrium phase diagrams

The coupling of the order parameter to the temperature gradient also leads to unexpected excursions along the concentration axis in the case of off-critical mixtures. As a consequence, equilibrium phase diagrams lose their usual meaning in thermal nonequilibrium situations, and even an off-critical blend with a temperature above the binodal can be quenched into phase separation by local heating with a laser beam. 

The order of a transition can be illustrated for a fixed-stoichiometry system with the familiar P-T diagram for solid, liquid, and vapor phases in Fig. 17.2. The curves in Fig. 17.2 are sets of P and T at which the molar volume, V, has two distinct equilibrium values—the discontinuous change in molar volume as the system s equilibrium environment crosses a curve indicates that the phase transition is first order. Critical points where the change in the order parameter goes to zero (e.g., at the end of the vapor-liquid coexistence curve) are second-order transitions. 

Connections to other types of phase diagrams can be obtained if order parameters are exchanged for intensive variables. Figure 17.2 is replotted with the order parameter V as the ordinate in Fig. 17.36. The diagram predicts the phases that would exist for a molar volume fixed by a rigid container at different temperatures. The tie-lines connect equilibrium molar volumes at the same temperature... 

With these results in view of derived formulae (26)-(29) for free energies we can study the temperature dependence of hydrogen solubility in the PtHx. HX phases, define the equilibrium value of order parameter, investigate the phase transitions in considered system with increasing temperature, establish the conditions of their realization, evaluate the energetic constants of all components of chemical reaction (1), construct phase diagram of the system. Below we shall examine these problems. 

Fig. 28. Averaged order parameter profiles cf>av(Z,x) plotted vs the scaled distance Z=z/2 b from the left wall at z=0 for four different scaled times T after the quench as indicated, for a scaled distance D =D/2 b=60. Choosing a rescaled distance L /2 b=600, and a discretization AX=1.5, Ax=0.05, the resulting equations are solved by the cell-dynamics method. The results shown are for parameters h1=y=4, g =-4, and averaged over 2000 independent initial conditions, corresponding to random fluctuations in a state with J( )av(Z,0)dZ=0. The parameters Iq and g were chosen such that both walls prefer A but one is still in the non-wet region of the equilibrium surface phase diagram of the corresponding semi-infinite system. From Puri and Binder [145]...

Fig. 9. Two-parameter ordering phase diagram for a system of 500 identical hard spheres (Truskett et ai, 2000 Torquato et ai, 2000). Shown are the coordinates in structural order parameter space (r, ) for the equilibrium fluid (dot-dashed), the equilibrium FCC crystal (dashed), and a set of glasses (circles) produced with varying compression rates. Here, r is the translational order parameter from (26) and is the bond-orientational order parameter Q( from (25) normalized by its value in the perfect FCC crystal ( = Each circle...

The preceding two conditions ensure that the system is in both thermodynamic and mechanical equilibrium. Figure 1.6 shows the results of quantitative calculation of the coexistence of all the phases. Here, we show four phases in all gas, liquid, MS, and SS phases. From the phase diagram, it is quite evident that, at a fixed temperature, the density (or order parameter) difference between F and SS is larger than that between F and MS. As we know from the extensive study of the gas-liquid system showing that surface tension between two phases depends strongly on the order parameter difference, one can qualitatively conclude that the surface tension between the F and SS phases is larger than that between the F and MS phases. More accurate quantitative results for surface tension are discussed in the next section. 

The original GB parametrisation with p = 2 and v = 1 is probably the most thoroughly studied one [5,11-16] and both Monte Carlo and molecular (tynamics methods have been employed [11-20,64] to obtain the equilibrium phases generated under a variety of thermodynamic conditions and to construct, at least in part, its phase diagram [14-16]. Isotropic, nematic and smectic B phases have been found and their order and molecular organisation have be determined. Curiously, direct simulation of the 4 site L uiard-Jones potratial which was originally fitted to yield this choice of GB parameters does not... 

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