Soft Matter at the Thermal Scale, Ch Frank elasticity, defects and smectic layers
Problem
A liquid-crystal sample is examined by two techniques.
((a)) Under crossed polarisers, one point in the texture shows 2 dark brushes meeting at it, and a second, separate point shows 8 dark brushes. Using the rule that a disclination of strength s shows 4∣s∣ brushes, state ∣s∣ for each point.
((b)) The same compound is cooled and its X-ray pattern is recorded. Above a certain temperature the pattern is a single diffuse ring; below it, a sharp peak appears at q0=1.9nm−1 with no higher-order peaks, the wide-angle pattern stays diffuse, and the sample still flows when sheared gently between the plates. Find the periodic spacing this peak corresponds to, and explain in one or two sentences why the low-temperature phase is a smectic A rather than a three-dimensional crystal.