Order translational

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]). 

CS. The true two-dimensional crystal with chains oriented vertically exists at low T and high ir in the CS phase. This structure exhibits long-range translational order. 

Thennotropic liquid crystal phases are fonned by anisotropic molecules witli long-range orientational order and in many types of stmcture witli some degree of translational order. The main types of mesogen are Arose tlrat are rodlike or calamitic and Arose Arat are disclike or discotic. 

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.

As in crystals, defects in liquid crystals can be classified as point, line or wall defects. Dislocations are a feature of liquid crystal phases where tliere is translational order, since tliese are line defects in tliis lattice order. Unlike crystals, tliere is a type of line defect unique to liquid crystals tenned disclination [39]. A disclination is a discontinuity of orientation of tire director field. 

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

The nematic to smectic A phase transition has attracted a great deal of theoretical and experimental interest because it is tire simplest example of a phase transition characterized by tire development of translational order [88]. Experiments indicate tliat tire transition can be first order or, more usually, continuous, depending on tire range of stability of tire nematic phase. In addition, tire critical behaviour tliat results from a continuous transition is fascinating and allows a test of predictions of tire renonnalization group tlieory in an accessible experimental system. In fact, this transition is analogous to tire transition from a nonnal conductor to a superconductor [89], but is more readily studied in tire liquid crystal system. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

Crystals are sohds. Sohds, on the other hand can be crystalhne, quasi-crystal-hne, or amorphous. Sohds differ from liquids by a shear modulus different from zero so that solids can support shearing forces. Microscopically this means that there exists some long-range orientational order in the sohd. The orientation between a pair of atoms at some point in the solid and a second (arbitrary) pair of atoms at a distant point must on average remain fixed if a shear modulus should exist. Crystals have this orientational order and in addition a translational order their atoms are arranged in regular lattices. 

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

Smectic A and C phases are characterized by a translational order in one dimension and a liquid-like positional order in two others. In the smectic A phase the molecules are oriented on average in the direction perpendicular to the layers, whereas in the smectic C phase the director is tilted with respect to the layer normal. A simple model of the smectic A phase has been proposed by McMillan [8] and Kobayashi [9] by extending the Maier-Saupe approach for the case of one-dimensional density modulation. The corresponding mean field, single particle potential can be expanded in a Fourier series retaining only the leading term ... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

In crystals of any material, the atoms present are always arranged in exactly the same way, over the whole extent of the solid, and exhibit long-range translational order. A crystal is conventionally described by its crystal structure, which comprises the unit cell, the symmetry of the unit cell, and a list of the positions of the atoms that lie in the unit cell. 

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.)...

A second quantity used to describe the translational order is —s /kg, which is defined as... 

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... 

The conditions on the phase diagram for which this anomalous behavior occurs has been termed water s structurally anomalous region. Inspection of the order map (Figure 4) reveals a dome of structural anomalies within the temperature-density plane, bounded by loci of maximum tetrahedral order (at low densities) and minimum translational order (at high densities) as shown in Figure 5. Also marked on Figure 5 are regions of diffusive anomalies,... 

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