Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 4.4, pp. 453-455   | 1 | 2 |

Section Homogeneous smectic-A and smectic-C phases

P. S. Pershana*

aDivision of Engineering and Applied Science and The Physics Department, Harvard University, Cambridge, MA 02138, USA
Correspondence e-mail: Homogeneous smectic-A and smectic-C phases

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In the smectic-A and smectic-C phases, the molecules organize themselves into layers, and from a naive point of view one might describe them as forming a one-dimensional periodic lattice in which the individual layers are two-dimensional liquids. In the smectic-A phase, the average molecular axis [\langle {\bf n} \rangle] is normal to the smectic layers while for the smectic-C it makes a finite angle. It follows from this that the smectic-C phase has lower symmetry than the smectic-A, and the phase transition from the smectic-A to smectic-C can be considered as the ordering of a two-component order parameter, i.e. the two components of the projection of the molecular axis on the smectic layers (De Gennes, 1973[link]). Alternatively, Chen & Lubensky (1976[link]) have developed a mean-field theory in which the transition is described by a free-energy density of the Lifshitz form. This will be described in more detail below; however, it corresponds to replacing equation ([link] for the free energy [\Delta F (\psi)] by an expression for which the minimum is obtained when the wavevector q, of the order parameter [\psi \propto \exp [i{\bf q} \cdot {\bf r}]], tilts away from the molecular axis.

The X-ray cross section for the prototypical aligned monodomain smectic-A sample is shown in Fig.[link]. It consists of a single sharp spot along the molecular axis at [|{\bf q}|] somewhere between [2\pi / 2L] and [2\pi / L] that reflects the QLRO along the layer normal, and a diffuse ring in the perpendicular direction at [|{\bf q}| \approx 2 \pi / a] that reflects the SRO within the layer. The scattering cross section for an aligned smectic-C phase is similar to that of the smectic-A except that the molecular tilt alters the intensity distribution of the diffuse ring. This is illustrated in Fig.[link] for a monodomain sample. Fig.[link] illustrates the scattering pattern for a polydomain smectic-C sample in which the molecular axis remains fixed, but where the smectic layers are randomly distributed azimuthally around the molecular axis.

The naivety of describing these as periodic stacks of two-dimensional liquids derives from the fact that the sharp spot along the molecular axis has a distinct temperature-dependent shape indicative of QLRO that distinguishes it from the Bragg peaks due to true LRO in conventional three-dimensional crystals. Landau and Peierls discussed this effect for the case of two-dimensional crystals (Landau, 1965[link]; Peierls, 1934[link]) and Caillé (1972[link]) extended the argument to the mesomorphic systems.

The usual treatment of thermal vibrations in three-dimensional crystals estimates the Debye–Waller factor by integrating the thermal expectation value for the mean-square amplitude over reciprocal space (Kittel, 1963[link]): [W \simeq {k_{B}T\over c^{3}} \int_{0}^{k_{D}} {k^{(d-1)}\over k^{2}} \hbox{d}k, \eqno(] where c is the sound velocity, [\omega_{D} \equiv ck_{D}] is the Debye frequency and [d = 3] for three-dimensional crystals. In this case, the integral converges and the only effect is to reduce the integrated intensity of the Bragg peak by a factor proportional to [\exp (-2W)]. For two-dimensional crystals [d = 1], and the integral, of the form of [\hbox{d}k/k], obtains a logarithmic divergence at the lower limit (Fleming et al., 1980[link]). A more precise treatment of thermal vibrations, necessitated by this divergence, is to calculate the relative phase of X-rays scattered from two points in the sample a distance [|{\bf r}|] apart. The appropriate integral that replaces the Debye–Waller integral is [\langle [u({\bf r}) - u(0)]^{2}\rangle \simeq {k_{B}T\over c^{3}} \int \sin^{2} ({\bf k} \cdot {\bf r}) {\hbox{d}k\over k} \hbox{d} \{\cos ({\bf k} \cdot {\bf r})\} \eqno(] and the divergence due to the lower limit is cut off by the fact that [\sin^{2} ({\bf k} \cdot {\bf r})] vanishes as [k \rightarrow 0]. More complete analysis obtains [\langle [u({\bf r}) - u(0)]^{2}\rangle \simeq (k_{B}T/c^{2}) \ln (|{\bf r}|/a)], where [a\approx] atomic size. If this is exponentiated, as for the Debye–Waller factor, the density–density correlation function can be shown to have the form [\langle \rho ({\bf r}) \rho (0)\rangle\simeq |{\bf r}/a|^{-\eta}], where [\eta \simeq |{\bf q}|^{2}(k_{B}T/c^{2})] and [|{\bf q}| \simeq 2\pi / a]. In place of the usual periodic density–density correlation function of three-dimensional crystals, the periodic correlations of two-dimensional crystals decay as some power of the distance. This type of positional order, in which the correlations decay as some power of the distance, is the quasi-long-range order (QLRO) that appears in Tables[link] and[link]. It is distinguished from true long-range order (LRO) where the correlations continue indefinitely, and short-range order (SRO) where the positional correlations decay exponentially as in either a simple fluid or a nematic liquid crystal.

The usual prediction of Bragg scattering for three-dimensional crystals is obtained from the Fourier transform of the three-dimensional density–density correlation function. Since the correlation function is made up of periodic and random parts, it follows that the scattering cross section is made up of a [\delta] function at the Bragg condition superposed on a background of thermal diffuse scattering from the random part. In principle, these two types of scattering can be separated empirically by using a high-resolution spectrometer that integrates all of the [\delta]-function Bragg peak, but only a small part of the thermal diffuse scattering. Since the two-dimensional lattice is not strictly periodic, there is no formal way to separate the periodic and random parts, and the Fourier transform for the algebraic correlation function obtains a cross section that is described by an algebraic singularity of the form [|{\bf Q} - {\bf q}|^{\eta - 2}] (Gunther et al., 1980[link]). In 1972, Caillé (Caillé, 1972[link]) presented an argument that the X-ray scattering line shape for the one-dimensional periodicity of the smectic-A system in three dimensions has an algebraic singularity that is analogous to the line shapes from two-dimensional crystals.

In three-dimensional crystals, both the longitudinal and the shear sound waves satisfy linear dispersion relations of the form [\omega = ck]. In simple liquids, and also for nematic liquid crystals, only the longitudinal sound wave has such a linear dispersion relation. Shear sound waves are overdamped and the decay rate [1/\tau] is given by the imaginary part of a dispersion relation of the form [\omega = i(\eta / \rho)k^{2}], where [\eta] is a viscosity coefficient and [\rho] is the liquid density. The intermediate order of the smectic-A mesomorphic phase, between the three-dimensional crystal and the nematic, results in one of the modes for shear sound waves having the curious dispersion relation [\omega^{2} = c^{2}k_{\perp}^{2}k_{z}^{2}/(k_{\perp}^{2} + k_{z}^{2})], where [k_{\perp}] and [k_{z}] are the magnitudes of the components of the acoustic wavevector perpendicular and parallel to [\langle {\bf n} \rangle], respectively (De Gennes, 1969a[link]; Martin et al., 1972[link]). More detailed analysis, including terms of higher order in [k_{\perp}^{2}], obtains the equivalent of the Debye–Waller factor for the smectic-A as [W \simeq k_{B}T \int_{0}^{k_{D}} {k_{\perp} \hbox{d}k_{\perp} \hbox{d}k_{z}\over Bk_{z}^{2} + Kk_{\perp}^{4}}, \eqno(] where B and K are smectic elastic constants, [k_{\perp}^{2} = k_{x}^{2} + k_{y}^{2}], and [k_{D}] is the Debye wavevector. On substitution of [u^{2} = (K / B) k_{\perp}^{2} + k_{z}^{2}], the integral can be manipulated into the form [\int \hbox{d}u/u], which diverges logarithmically at the lower limit in exactly the same way as the integral for the Debye–Waller factor of the two-dimensional crystal. The result is that the smectic-A phase has a sharp peak, described by an algebraic cusp, at the place in reciprocal space where one would expect a true [\delta]-function Bragg cross section from a truly periodic one-dimensional lattice. In fact, the lattice is not truly periodic and the smectic-A system has only QLRO along the direction [\langle {\bf n} \rangle].

X-ray scattering experiments to test this idea were carried out on one thermotropic smectic-A system, but the results, while consistent with the theory, were not adequate to provide an unambiguous proof of the algebraic cusp (Als-Nielsen et al., 1980[link]). One of the principal difficulties was due to the fact that, when thermotropic samples are oriented in an external magnetic field in the higher-temperature nematic phase and then gradually cooled through the nematic to smectic-A phase transition, the smectic-A samples usually have mosaic spreads of the order of a fraction of a degree and this is not sufficient for detailed line-shape studies near to the peak. A second difficulty is that, in most of the thermotropic smectic-A phases that have been studied to date, only the lowest-order peak is observed. It is not clear whether this is due to a large Debye–Waller-type effect or whether the form factor for the smectic-A layer falls off this rapidly. Nevertheless, since the factor [\eta] in the exponent of the cusp [|{\bf Q} - {\bf q}|^{\eta-2}] depends quadratically on the magnitude of the reciprocal vector [|{\bf q}|], the shape of the cusp for the different orders would constitute a severe test of the theory.

Fortunately, it is common to observe multiple orders for lyotropic smectic-A systems and such an experiment, carried out on the lyotropic smectic-A system formed from a quaternary mixture of sodium dodecyl sulfate, pentanol, water and dodecane, confirmed the theoretical predictions for the Landau–Peierls effect in the smectic-A phase (Safinya, Roux et al., 1986[link]). The problem of sample mosaic was resolved by using a three-dimensional powder. Although the conditions on the analysis are delicate, Safinya et al. demonstrated that for a perfect powder, for which the microcrystals are sufficiently large, the powder line shape does allow unambiguous determination of all of the parameters of the anisotropic line shape.

The only other X-ray study of a critical property on the smectic-A side of the transition has been a measurement of the temperature dependence of the integrated intensity of the peak. For three-dimensional crystals, the integrated intensity of a Bragg peak can be measured for samples with poor mosaic distributions, and because the differences between QLRO and true LRO are only manifest at long distances in real space, or at small wavevectors in reciprocal space, the same is true for the `quasi-Bragg peak' of the smectic-A phase. Chan et al. measured the temperature dependence of the integrated intensity of the smectic-A peak across the nematic to smectic-A phase transition for a number of liquid crystals with varying exponents [\nu_{\|}] and [\nu_{\perp}] (Chan, Deutsch et al., 1985[link]). For the Landau–De Gennes free-energy density (equation[link]), the theoretical prediction is that the critical part of the integrated intensity should vary as [|t|^{x}], where [x = 1 - \alpha] when the critical part of the heat capacity diverges according to the power law [|t|^{-\alpha}]. Six samples were measured with values of [\alpha] varying from 0 to 0.5. Although for samples with [\alpha \approx 0.5] the critical intensity did vary as [x \approx 0.5], there were systematic deviations for smaller values of [\alpha], and for [\alpha \approx 0] the measured values of x were in the range 0.7 to 0.76. The origin of this discrepancy is not at present understood.

Similar integrated intensity measurements in the vicinity of the first-order nematic to smectic-C transition cannot easily be made in smectic-C samples since the magnetic field aligns the molecular axis [\langle {\bf n}\rangle], and when the layers form at some angle [\varphi] to [\langle {\bf n}\rangle] the layer normals are distributed along the full 2π of azimuthal directions around [\langle {\bf n}\rangle], as shown in Fig.[link]. The X-ray scattering pattern for such a sample is a partial powder with a peak-intensity distribution that forms a ring of radius [|{\bf q}| \sin (\varphi)]. The opening of the single spot along the average molecular axis [\langle {\bf n}\rangle] into a ring can be used to study either the nematic to smectic-C or the smectic-A to smectic-C transition (Martinez-Miranda et al., 1986[link]).

The statistical physics in the region of the phase diagram surrounding the triple point, where the nematic, smectic-A and smectic-C phases meet, has been the subject of considerable theoretical speculation (Chen & Lubensky, 1976[link]; Chu & McMillan, 1977[link]; Benguigui, 1979[link]; Huang & Lien, 1981[link]; Grinstein & Toner, 1983[link]). The best representation of the observed X-ray scattering structure near the nematic to smectic-A, the nematic to smectic-C and the nematic/smectic-A/smectic-C (NAC) multicritical point is obtained from the mean-field theory of Chen and Lubensky, the essence of which is expressed in terms of an energy density of the form [\eqalignno{ \Delta F (\psi) &= {A\over 2} |\psi |^{2} + {D\over 4} |\psi |^{4} + {\textstyle{1\over 2}} [E_{\|}(Q_{\|}^{2} - Q_{0}^{2})^{2} + E_{\perp} Q_{\perp}^{2} \cr &\quad + E_{\perp \perp} Q_{\perp}^{4} + E_{\perp \|} Q_{\perp}^{2} (Q_{\|}^{2} - Q_{0}^{2})]| \psi ({\bf Q})|^{2}, &(}] where [\psi = \psi ({\bf Q})] is the Fourier component of the electron density: [\psi ({\bf Q}) \equiv {1\over (2 \pi)^{3}} \int \hbox{d}^{3}{\bf r} \exp [i({\bf Q \cdot r)}]\rho ({\bf r}). \eqno(] The quantities [E_{\|}], [E_{\perp\perp}], and [E_{\|\perp}] are all positive definite; however, the sign of A and [E_{\perp}] depends on temperature. For [A\gt 0] and [E_{\perp}\gt 0], the free energy, including the higher-order terms, is minimized by [\psi ({\bf Q}) = 0] and the nematic is the stable phase. For [A\lt 0] and [E_{\perp}\gt 0], the minimum in the free energy occurs for a nonvanishing value for [\psi ({\bf Q})] in the vicinity of [Q_{\|} \approx Q_{0}], corresponding to the uniaxial smectic-A phase; however, for [E_{\perp}\lt 0], the free-energy minimum occurs for a nonvanishing [\psi ({\bf Q})] with a finite value of [Q_{\perp}], corresponding to smectic-C order. The special point in the phase diagram where two terms in the free energy vanish simultaneously is known as a `Lifshitz point' (Hornreich et al., 1975[link]). In the present problem, this occurs at the triple point where the nematic, smectic-A and smectic-C phases coexist. Although there have been other theoretical models for this transition, the best agreement between the observed and theoretical line shapes for the X-ray scattering cross sections is based on the Chen–Lubensky model. Most of the results from light-scattering experiments in the vicinity of the NAC triple point also agree with the main features predicted by the Chen–Lubensky model; however, there are some discrepancies that are not explained (Solomon & Litster, 1986[link]).

The nematic to smectic-C transition in the vicinity of this point is particularly interesting in that, on approaching the nematic to smectic-C transition temperature from the nematic phase, the X-ray scattering line shapes first appear to be identical to the shapes usually observed on approaching the nematic to smectic-A phase transition; however, within approximately 0.1 K of the transition, they change to shapes that clearly indicate smectic-C-type fluctuations. Details of this crossover are among the strongest evidence supporting the Lifshitz idea behind the Chen–Lubensky model.


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