Order parameter translational

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. 

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... 

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 ... 

Figure 1. Mass-density profiles for ice/water interfaces (solid lines) basal interface (a) and (2110) interface (b). Translational order parameter f(z) changing from 1 in crystal bulk region to 0 in liquid phase is shown by short-dashed lines.

The translational order parameter permits an estimate of the width of the interfaces. The 10-90 width is defined to be the length over which a specific interfacial order parameter changes from 10% to 90% of the bulk solid value. We have estimated the 10-90 widths of the interfaces using a fit by a simple hyperbolic tangent function, used frequently in earlier studies [17]. In the case of the mass-density profile, the translational order parameter may be extracted from a fitting procedure,... 

Other order parameters can be used to characterize ice/water interfaces via change in average density [17, 19] and diffusivity [17] across the interface. Local order parameters can also be defined which depend only on the position of a reference molecule. It was shown in [40] that the local tetrahedral order parameter, which reflects the change in tetrahedral environment, leads to a 10-90 width of 11 A for the basal ice-water interface which is in good agreement with the estimated value from the translational order parameter. 

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...

A water molecule is spatially correlated with its nearest neighbors due to hydrogenbonding. That is, we expect to find a number of water molecules at a nearly fixed distance and also at a relatively fixed orientation, as discussed in Chapter 1. In order to quantify this spatial order one often defines a translational order parameter as. 

The value of pi is usually taken as translational order parameter, see Section 6.3. 

In the first case, the uniaxial symmetry is conserved and the field stabilizes thermal fluctuations and increases the orientational order parameter both in the nematic and smectic A phases. A qualitative picture is shown in Fig. 1.21 taken from [80). With an increasing field the orientational order parameter S grows and the Tna temperatiure increases. The translational order parameter, being coupled with 5, also increases. 

Figure 6. Thermodynamic and structural quantities for the YK fluid with a = 3.3. Left column thermal expansion coefficient ap (units of k /e), isothermal compressibility Kp (units of o /e) and constant-pressure specific heat Cp (units of b) as a function of T along the isobar P = 2.5. For conventional liquids, oip, Kp, and Cp monotonically increase with T and ap > 0. Right column translational order parameter —sz (units of ks), bond-order parameter ge [89). and self-diffusion coefficient D ((units of cr (e/m / )), where m is the particle mass) as a function of P along the isotherm T = 0.06. For conventional liquids, —sz and ge increase with P while D decreases monotonically. Data are from Ref. [88].

Fig. 4. (a) Translational order parameter for rigid dumbbells with interatomic separation A/rr = 0.50 as a function of density for temperatures T = 0.30, 0.40, 0.42, 0.44, 0.46, 0.48, 0.50 and 0.60 (from top to bottom) (b) radial distribution functions (g (r)) for a fixed temperature T = 0.30 and several densities. The colors correspond to the densities where the translational order parameter increases (red), decreases (black) or increases (blue) under increasing density, as in (a). 

Here 5 is the usual nematic order parameter, T is the purely translational order parameter and the parameter cr characterizes a coupling between orientational and translational ordering. 

The contribution of translational order parameters to the anisotropy of physical properties of liquid crystals has not been studied in detail. Evidence suggests that there is a very small influence of translational ordering on the optical properties, but effects of translational order can be detected in the measurement of dielectric properties. There are strong effects in both elastic properties and viscosity, but the statistical theories of these properties have not been extended to include explicitly the effects of translational order. 

In addition to the purely orientational and translational order parameters, the (Pl(cos )> and mixed-order parameters, (PL(cos )cos(27m2/d)). These describe the... 

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