Structural order parameters bond-orientational

Given that the supercooling of a liquid can lead to structurally distinct possibilities (the stable crystal or a glass), structural order parameters are especially valuable in understanding low-temperature metastabiUty. In particular, it has been demonstrated (van Duijneveldt and Frenkel, 1992) that the bond-orientational order parameters introduced by Steinhardt et al (1983) are well suited for detecting crystalline order in computer simulations of simple supercooled liquids. The bond-orientational order parameters are so named because they focus on the spatial orientation of imaginary bonds" that connect molecules to their nearest neighbors defined as above with... 

In contrast to the bond-orientational order parameters mentioned above, scalar measures for translational order [that is, of the tendency of particles (atoms, molecules) to adopt preferential pair distances in space] have not been well studied. However, a number of simple metrics have been introduced recently (Truskett et al., 2000 Torquato et al., 2000, Errington and Debenedetti, 2001) to capture the degree of spatial ordering in a many-body system. In particular, the structural order parameter t. 

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...

In simulation, we can also measure the structural order parameters that provides local structural arrangements. The orientational order among the bonds present in the system is characterized by local orientational order Qi which is given by... 

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L.133 Using two sets of backbone RDC data, collected in bacteriophage Pfl and bicelle media, they obtained order tensor parameters using a set of crystallographic coordinates for the structural model. This allowed the refinement of C -C bond orientations, which then provided the basis for their quantitative interpretation of C -H RDCs for 38 out of a possible 49 residues in the context of three different models. The three models were (A) a static xi rotameric state (B) gaussian fluctuations about a mean xi torsion and (C) the population of multiple rotameric states. They found that nearly 75% of xi torsions examined could be adequately accounted for by a static model. By contrast, the data for 11 residues were much better fit when jumps between rotamers were permitted (model C). The authors note that relatively small harmonic fluctuations (model B) about the mean rotameric state produces only small effects on measured RDCs. This is supported by their observation that, except for one case, the static model reproduced the data as well as the gaussian fluctuation model. 

The distributions of both segment and bond densities allow us to calculate any structural property of the mono-layer. The bond orientational probabilities (forward, lateral, and backward) as well as the bond order parameter of a chain as a function of bond number along the chain are given below. 

The order parameter used is the bond orientational order parameter, Qe, originally introduced by Steinhardt, Nelson and Ronchetti [89] and later used by van Duijneveldt and Frenkel [96] to simulate crystallization. It is sensitive to the overall degree of crystallinity in the system, irrespective of the crystal structure, i.e., it distinguishes the liquid from the crystal. The values of Qe for pure fee, body-centered-cubic (bcc), and for the liquid are 0.57452, 0.51069, and 0.0, respectively. Previous work [96] has shown that a defective fee crystal, which crystallizes from the liquid in simulations, has an order parameter below 0.5 and that finite-size effects lead to small positive values for the order parameter in the liquid phase. 

Preliminary results of Monte Carlo simulation [183] have also demonstrated the usefulness of the bond - orientational order parameters in determining the inner structure of films adsorbed on the (100) face of an fee crystal. 

Similar approach can be used to study phase transitions in films formed on other crystals, e.g., on the (110) faces of fee and bcc crystals of various metals. It is possible to define the appropriate bond-orientational order parameters suitable for determining the formation of registered and uniaxial structures. Such computer simulation studies can be very helpful in determining the role of the surface corrugation on the structure of adsorbed films and the nature of phase transitions between different adsorbed phases. 

Figure 26. Characterization of the inherent structures for the model calamitic system GB(3,5,2, 1) ( = 256). (a) Parallel radial distribution function g (/ ) for the inherent structures at all temperatures considered along the isochor at density p = 0.32. Note that the curves for the highest five temperatures are nearly superposed on each other. For others, the amplitude of the peaks gradually increases as the temperature drops, (b) Evolution of the 6-fold bond orientational order parameter 4>6 for the inherent stmctures with temperature at three densities. (Reproduced from Ref. 144.)...

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