International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 4.4, pp. 455-456   | 1 | 2 |

Section 4.4.3.3. Surface effects

P. S. Pershana*

aDivision of Engineering and Applied Science and The Physics Department, Harvard University, Cambridge, MA 02138, USA
Correspondence e-mail: pershan@deas.harvard.edu

4.4.3.3. Surface effects

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The effects of surfaces in inducing macroscopic alignment of mesomorphic phases have been important both for technological applications and for basic research (Sprokel, 1980[link]; Gray & Goodby, 1984[link]). Although there are a variety of experimental techniques that are sensitive to mesomorphic surface order (Beaglehole, 1982[link]; Faetti & Palleschi, 1984[link]; Faetti et al., 1985[link]; Gannon & Faber, 1978[link]; Miyano, 1979[link]; Mada & Kobayashi, 1981[link]; Guyot-Sionnest et al., 1986[link]), it is only recently that X-ray scattering techniques have been applied to this problem. In one form or another, all of the techniques for obtaining surface specificity in an X-ray measurement make use of the fact that the average interaction between X-rays and materials can be treated by the introduction of a dielectric constant [\varepsilon \approx 1 - (4\pi \rho e^{2} / m\omega^{2}) = 1 - \rho r_{e} \lambda^{2}/\pi], where [\rho] is the electron density, [r_{e}] is the classical radius of the electron, and [\omega] and [\lambda] are the angular frequency and the wavelength of the X-ray. Since [\varepsilon\lt 1], X-rays that are incident at a small angle to the surface [\theta_{0}] will be refracted in the material toward a smaller angle [\theta_{T} \approx ({\theta_{0}^{2} - \theta_{c}^{2})}^{1/2}], where the `critical angle' [\theta_{c} \approx {(\rho r_{e}\lambda^{2} / \pi)}^{1/2} \approx 0.003\hbox{ rad }(\approx 0.2^{\circ})] for most liquid crystals (Warren, 1968[link]). Although this is a small angle, it is at least two orders of magnitude larger than the practical angular resolution available in modern X-ray spectrometers (Als-Nielsen et al., 1982[link]; Pershan & Als-Nielsen, 1984[link]; Pershan et al., 1987[link]). One can demonstrate that for many conditions the specular reflection [R(\theta_{0})] is given by [R (\theta_{0}) \approx R_{F} (\theta_{0})|\rho^{-1} \textstyle\int \hbox{dz} \exp (- iQz) \langle \partial \rho / \partial z \rangle |^{2},] where [Q \equiv (4\pi / \lambda) \sin (\theta_{0})], [\langle \partial \rho / \partial z \rangle] is the normal derivative of the electron density averaged over a region in the surface that is defined by the coherence area of the incident X-ray, and [R_{F} (\theta_{0}) \approx \left({\theta_{0} - \sqrt{\theta_{0}^{2} - \theta_{c}^{2}}\over \theta_{0} + \sqrt{\theta_{0}^{2} - \theta_{c}^{2}}}\right)^{2}] is the Fresnel reflection law that is calculated from classical optics for a flat interface between the vacuum and a material of dielectric constraint [\varepsilon]. Since the condition for specular reflection, that the incident and scattered angles are equal and in the same plane, requires that the scattering vector [{\bf Q} = \hat{z} (4\pi / \lambda) \sin (\theta_{0})] be parallel to the surface normal, it is quite practical to obtain, for flat surfaces, an unambiguous separation of the specular reflection signal from all other scattering events.

Fig. 4.4.3.2(a)[link] illustrates the specular reflectivity from the free nematic–air interface for the liquid crystal 4′-octyloxybiphenyl-4-carbonitrile (8OCB) 0.050 K above the nematic to smectic-A phase-transition temperature (Pershan & Als-Nielsen, 1984[link]). The dashed line is the Fresnel reflection [R_{F} (\theta_{0})] in units of [\sin (\theta_{0}) / \sin (\theta_{c})], where the peak at [\theta_{c} = 1.39^{\circ}] corresponds to surface-induced smectic order in the nematic phase: i.e. the selection rule for specular reflection has been used to separate the specular reflection from the critical scattering from the bulk. Since the full width at half maximum is exactly equal to the reciprocal of the correlation length for critical fluctuations in the bulk, [2/\xi_{\|}] at all temperatures from [T - T_{\rm NA} \approx 0.006\hbox{ K}] up to values near to the nematic to isotropic transition, [T - T_{NA} \approx 3.0\hbox{ K}], it is clear this is an example where the gravitationally induced long-range order in the surface position has induced mesomorphic order that has long-range correlations parallel to the surface. Along the surface normal, the correlations have only the same finite range as the bulk critical fluctuations. Studies on a number of other nematic (Gransbergen et al., 1986[link]; Ocko et al., 1987[link]) and isotropic surfaces (Ocko, Braslau et al., 1986[link]) indicate features that are specific to local structure of the surface.

[Figure 4.4.3.2]

Figure 4.4.3.2 | top | pdf |

Specular reflectivity of ∼8 keV X-rays from the air–liquid interface of the nematic liquid crystal 8OCB 0.05 K above the nematic to smectic-A transition temperature. The dashed line is the Fresnel reflection law as described in the text.

References

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