Structural order parameters ordering phase diagram

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 construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

Having obtained a snapshot of the phase diagram, it is then usual to make up a number of well-defined samples at given concentrations in either water or D2O. Once these samples have equilibrated (which can take days), they can be investigated by X-ray diffraction methods and, if made up in D2O, NMR. The former technique, in conjunction with the microscopy, can identify the phase and give structural parameters, while the, latter technique can given some idea of the phase type and the order within it from the dipolar coupling constant obtained. 

Fig. 71. Order parameter square of the c(2x2) structure (lower part) plotted vs. temperature at constant coverage, as obtained from interpolation of data taken at constant chemical potential. Rill dots denote temperatures T j% where the order parameter square has dropped to 50 % of its low temperature value. These data are for R — -1. R —

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