Bond orientational order paramete

The distribution of the intermolecular vector is also of value in distinguishing between smectic A and smectic B phases with the latter having long range bond orientational order [23, 24]. At the local level we can define a bond orientational order parameter, PeCn) for molecule i at position q by [25]... 

Figure 3. (a) Two-dimensional, bond orientational order parameter average values in the molecular fluid layers of LI ecu confined in a multi-walled carbon nanotube of diameter D=9norder parameter values for the contact, second, third and fourth layers, respectively. The dotted line represents the bulk solid-fluid transition temperature, (b) Positional and orientational pair correlation functions in the unwraiqred contact layer of U CCU confined in a multi-walled carbon nanotube of diameter D=9.1< (5 nm) showing liquid phase at 7=262 K and crystal phase at 7=252 K. 

Free energy method The method relies on the calculation of the Landau free energy as a function of an effective bond orientational order parameter <1>, using GCMC simulations [5]. The Landau free energy is defined by,... 

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. 

Apart from the defined above bond-orientational order parameters (see eqns. (31) and (32)) it is useful to define the corresponding susceptibilities... 

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. 

Figure 14. The bond-orientational order parameters for the two-dimensional films of particles with o = 1.2 formed on the (100) fee plane with characterized by different corrugation of the gas - solid potential (Vb=0.2 (a) and 0.4 (b)).

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. 

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... 

Of particular interest is the specific bond-orientational order parameter given by... 

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...

Once the bond angle field is defined, a sixfold bond orientational order parameter field can be constructed,... 

In order to probe local solid-like fluctuations in the dense 2D WCA liquid, we measured the local sixfold bond-orientational order parameter... 

Pft(r) is the sixfold bond orientational order parameter density. 

The local sixfold bond orientational order parameter is defined in Eq. (3.3). g FpFj) is divided out of Eq. (3.15) in order to remove translational correlations from the bond orientational correlation function. In the homogeneous and isotropic liquid phase gl (r,F2) reduces to a function of Fj, only, which we will denote by g r), and a corresponding translation- and rotation-invariant quantity can be defined for the solid phase by performing suitable averages. 

As discussed above, the local sixfold bond orientational order parameter is a sensitive probe of the local geometry of 2D systems. This is illustrated by Fig. 34, which shows that distribution functions for the time-averaged 3584-particle WCA system at various densities. In the solid... 

Estimated Magnitudes of the Fourfold, Sixfold, and Twelvefold Bond Orientational Order Parameters for the Vertex Types Shown in Fig. 21... 

In Figs. 65-67 we have shown the fourfold, sixfold, and twelvefold bond orientational order parameter distribution functions for DRPs, calculated using only intact bonds. For comparison, we have also shown the corresponding random distributions, calculated using randomly chosen bond angles and the same (intact bond) coordination number statistics as the DRP system. As for the WCA system, the distributions for the DRP system have sharp peaks corresponding to particular vertex types (see Table III), features which do not appear in the random distributions. These observations support the tiling description we have developed. 

As for the WCA system, the fraction of sixfold-ordered particles in DRPs (0.547 0.004) is significantly less than the fraction of six-coordinated particles (/g = 0.7861 0.0018), confirming that the sixfold bond orientational order parameter is a more sensitive indicator of local geometrical disorder than is the coordination number. The average size of ordered clusters in DRPs is (s) = 30.4 0.8, and the normalized average number of ordered clusters is Ac/N = 0.0180 0.0005. These values are comparable to those measured in the dense WCA liquid near freezing (see Figs. 52 and 53). 

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.)...

As in the S phase, Sg lacks true long-range translational order because of the Peierls-Landau instability. Theoretically, the existence of molecular tilt implies that there must be a certain degree of bond-orientational order in the Sg phase. This has been verified experimentally by a high resolution synchrotron X-ray study of the transition from Sg to S, in a monodomain freely suspended film. It will be recalled that S, is a tilted hexatic phase. The bond-orientational order parameter Cg (as defined in (5.7.1), but appropriately modified to allow for the fact that the molecules are tilted) is plotted as a function of temperature in fig. 5.8.2. The weak bond-orientational order in evolves continuously into S, showing... 

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