Bond orientational ordering

There has been much activity in the study of monolayer phases via the new optical, microscopic, and diffraction techniques described in the previous section. These experimental methods have elucidated the unit cell structure, bond orientational order and tilt in monolayer phases. Many of the condensed phases have been classified as mesophases having long-range correlational order and short-range translational order. A useful analogy between monolayer mesophases and die smectic mesophases in bulk liquid crystals aids in their characterization (see [182]). 

Figure C2.2.4. Types of smectic phase. Here tire layer stacking (left) and in-plane ordering (right) are shown for each phase. Bond orientational order is indicated for tire hexB, SmI and SmF phases, i.e. long-range order of lattice vectors. However, tliere is no long-range translational order in tliese phases.

The transition from smectic A to smectic B phase is characterized by tire development of a sixfold modulation of density witliin tire smectic layers ( hexatic ordering), which can be seen from x-ray diffraction experiments where a sixfold symmetry of diffuse scattering appears. This sixfold symmetry reflects tire bond orientational order. An appropriate order parameter to describe tlie SmA-SmB phase transition is tlien [18,19 and 20]... 

Birgeneau R J and Ulster J D 1978 Bond-orientational order model for smeotio B liquid orystals J. Physique Lett. 39 399-402... 

K. J. Strandburg, ed. Bond Orientational Order in Condensed Matter Systems. New York Springer, 1992. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

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]... 

Strandberg KJ (1992) In Strandberg KJ (ed) Bond orientational order in condensed matter systems, chap 2. Springer, Berlin Heidelberg New York... 

Fig. 36 Three-dimensional map of the local bond orientational order. The bond order is expressed by the density-scale given in the legend to the right... 

Bondline readout, 7 122 Bond number (Bo), 15 687t Bond orientational order, of liquid crystalline materials, 15 85 Bonds, fullerene, 12 233-234 Bond strength, in thermal bonding, 17 510 Bone, 7 273t. See also Bones citric acid in, 6 632t Bone fractures, 3 725 treatment, 3 725... 

Liquid crystal display technology, 15 113 Liquid crystalline cellulose, 5 384-386 cellulose esters, 5 418 Liquid crystalline conducting polymers (LCCPs), 7 523-524 Liquid crystalline compounds, 15 118 central linkages found in, 15 103 Liquid crystalline materials, 15 81-120 applications of, 15 113-117 availability and safety of, 15 118 in biological systems, 15 111-113 blue phases of, 15 96 bond orientational order of, 15 85 columnar phase of, 15 96 lyotropic liquid crystals, 15 98-101 orientational distribution function and order parameter of, 15 82-85 polymer liquid crystals, 15 107-111 polymorphism in, 15 101-102 positional distribution function and order parameter of, 15 85 structure-property relations in,... 

Figure 1 Two basic types of ordering typically found in liquids, (a) Bond-orientational order describes the tendency of molecules to form well-defined angles between the fictitious bonds that can be drawn between the molecule of interest and two of its nearest neighbors, (b) Translational order describes the tendency of molecules to adopt preferential interparticle separations. (Adapted from Ref. 30.)...

Figure 3 shows an ordering map for this Lennard-Jones system, with the translational order t (of Eq. [2]) plotted against the bond-orientational order Qs (of Eq. [5]). It can be observed that the data, collected over a wide range of temperatures and densities, collapse onto two distinct equilibrium branches... 

Hexatic smectic mesophase in which the director is perpendicular to the layers with the long-range hexagonal bond-orientational order. 

Note 4 The structure of a smectic B mesophase is characterised by a D6h point group symmetry, in the Schoenflies notation, by virtue of the bond orientational order. 

Bond Orientational Order. In some cases, although the lattice of points of high density of molecular centers parallel to the planes are not correlated from layer to layer, the two principal directions of the lattice are the same for all layers. In these inult-rials. the interactions between the planes do not prevent the planes from translating relative to each other, but do prevent I hem from rotating relative to each other. 

Overbeek, G.A. Honig, D. Wolthaus, L. Gnabe, M. Mobius, D. Observation of bond orientational order in floating and transferred monolayers with Brewster angle microscopy. Thin Solid Films 1994, 242, 26. 

Podgornik R, Strey HH, Gawrisch K, Rau DC, Rupprecht A, Parsegian VA. Bond orientational order, molecular motion, and free energy of high-density DNA mesophases. Proc. Natl. Acad. Sci. U.S.A. 1996 93 4261 266. 

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. 

Within a smectic layer, a state of order called hexatic can exist that is intermediate between the liquid order of the smectic A, and the crystalline order of crystalhne smectic B. This state of order characterizes the hexatic smectic-B phase. Flexatic order consists of long-range bond orientational order, but no long-range positional order. This is illustrated... 

In the crystalline version of smectic B, there is positional as well as bond-orientational order, and this ordering is called hexagonal, as opposed to hexatic. In the tilted version of the hexatic smectic B, there are two different directions the tilt has been found to take with respect to the hexagonal bond orientation, namely toward nearest and toward next nearest neighbors, producing the two different smectics I and F (see Fig. 10-27). Similarly, there are two different tilted crystalline hexagonal phases, smectics J and G. Even in smectic crystals, the molecules retain freedom to rotate cooperatively about theii axes, hence smectic ciystals are not as solid as most simple crystals. Nature is wonderfully creative in contriving... 

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. 

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