The Smectic C Order Parameters

The smectic C order is characterized by the two parameters 6 and (j), and therefore the order parameter can be written as... 

Such molecules exhibit columnar mesophases and smectic C phases. In contrast to the corresponding tetracatenar compounds described above, the columnar phase is, in most of these swallow-tailed compounds, the low temperature phase with respect to the smectic C phase [58]. This inversion of thermotropic sequence with respect to the biforked compounds described just above has not found any explanation in relation with molecular parameters. The symmetry of the columnar two-dimensional lattice is found to be hexagonal, oblique and even rectangular [59] in some cases. For the latter case, the model proposed involves several molecules packed side by side to form the lattice unit, the different moieties being packed alternately in order to form a two-dimensional centred cell as shown in Fig. 11. 

Fig. 5.8.9. Variation of the supetfluid order parameter with distance from the centre of a quantized vortex filament as given by the Ginsburg-Pitaevskii equation (5.8.16). The same curve describes the variation of the tilt angle in the core region of the smectic C disclination. ...

Fig. 6.10 (a) TheOTetical (dash curve) and experimental (solid curve) deptaidence of the smectic A order paiametCT pi rai temperature Tna = 75 C for cholesteryl nmianoate [10]. (b) Free energy of a smectic A as a function of order parameter pi for different temperatures 10°C below the transition (cirrve 1), T = TjvA (cirrve 2) and 10°C above (curve 3)... 

In the smectic C phase a new feature appears, a uniform molecular tilt that is characterized by the two-component order parameter exp(i(p) [15], For simplicity, we can fix the azimuth angle cp and operate with a real order parameter 9 ... 

As we have seen, locally the smectic C layers are polar, belonging to pyroelectric class C2. Macroscopically SmC either forms a helical structore or does not. So, we can discuss a structure without helicity. In a sense, the formation of a helix is equivalent to formation of ferroelectric domains which would reduce overall macroscopic polarisation. Thus we can consider the (1) (very important) and (2) (additional) requirements fulfilled. As to the phase transition (3), we know that in the smectic A phase, even chiral, there is no polar axis, therefore that phase can be considered as a paraelectric phase. The two-component order parameter of the A -C transition is the same as the parameter of the A-C transition in an achiral substance, namely 9exp (i(p), where we recognise the tilt 9 and azimuth (p angles. The spontaneous... 

Due to low symmetry (C2) of the chiral smectic C phase, its theoretical description is very complicated. Even description of the achiral smectic C phase is not at all simple. In the chiral SmC phase two new aspects are very important, the spatially modulated (helical) structure and the presence of spontaneous polarisation. The strict theory of the SmA -SmC transition developed by Pikin [10] is based on consideration of the two-component order parameter, represented by the c-director whose projections ( 1, 2) = are combinations of the director compo-... 

We now consider a second-order phase transition. Simple examples include the smectic C-smectic A transition, which is characterized by a continuous decrease in an order parameter describing molecular tilt. The smectic A (lamellar)-isotropic transition can also be second order under certain conditions. In this case symmetry means that terms with odd powers of are zero. To see this consider the smectic C-smectic A transition. The appropriate order parameter is given by Eq. (5.23) and is a complex quantity. However, the free energy density must be real thus only even terms ijnjr and etc., remain. Then the free energy can be written as... 

Here, yr is the superconductor gap order parameter. It corresponds to the wave function of the superconducting pair in BCS theory and has the X Y symmetry of the smectic order parameter. The magnetic vector potential A comes analogous to the director n (m and e are the mass and charge of a single electron, fi Planck s constant, c the velocity of light and jU the magnetic permittivity). 

In the smectic C phase, the layer normal z and the director n represent the only two natural directions which we can define. That is, z and n are the only natural vectors in the medium. Using these we now want to construct an order parameter which is a vector, is chiral, has C2 symmetry, leads to a second order A -C transition and is proportional to 6 for small values of 6. We can do this by combining their scalar and vector products, multiplying one with the other. 

The principal parameter that distinguishes smectic-C from smectic-A is the tilt angle Oq. Because of the chiral character of the molecule, the tilt processes around the normal to the smectic layers, together with the transverse electric polarization P (cf Fig. 4.16). In the theories developed to describe the phase transition phenomena from the smectic-C phase to the smectic-A phase, the tilt angle is treated as a primary order parameter of the system, very much as the director axis n in the nematic or cholesteric phase, while P is regarded as a secondary one. 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

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