Direct measurement of orientation correlations in a two-dimensional liquid-crystal system

The behavior of a two-dimensional orientation field has been studied directly in space and time. In a freely suspended thin film of tilted smectic liquid crystal the local projection onto the film plane of the molecular orientation averaged through the film forms a two-dimensional (2D) vector field C^(x,y) with local orientation Φ(x,y). This orientation field was imaged in real space by depolarized reflection microscopy. By probing the light at two distinct (x,y) locations in the image and employing cross-correlation intensity-fluctuation spectroscopy, we have measured the space-time behavior of ${\sigma }^2$(ρ,τ)≡〈‖Φ(0,0) -Φ(ρ,τ)${\Vert }^2$〉. At large τ, σ increases logarithmically with τ, consistent with the expected diffusional dynamics and thermally excited fluctuation spectrum of Φ. This logarithmic (Landau-Peierls) divergence confirms that this 2D orientational system is at its lower marginal dimensionality. In addition, by fitting the behavior of ${\sigma }^2$(ρ,τ), we extract the 2D orientational diffusion constants for splay and bend in both the smectic-C and smectic-I phases.