Saddle-splay-induced periodic edge undulations in smectic-A disks immersed in a nematic medium

We report experimental studies on the phase behavior of binary mixtures of 1″,7″-bis(4-cyanobiphenyl-4'-yl)heptane (CB7CB) and 4,4-diheptyloxyazoxybenzene, which exhibit, apart from the nematic (N) and twist-bend nematic (N_{TB}) phases, the induced smectic-A (Sm-A) phase for weight fraction of CB7CB between 0.05 and 0.70. In planar nematic layers, the N_{TB} phase separates as droplets of tactoidlike planform; the chirality of droplets manifests in the optical dissimilarity between their opposite angular ends. Our main result is that, in the appropriate two phase region, Sm-A nuclei with positive dielectric anisotropy change over to disks immersed in the nematic above some electric field, their edges decorated by periodic bright spots, a result which was earlier reported in another binary system exhibiting the induced Sm-A phase [R. Pratibha and N. V. Madhusudana, Physica A 224, 9 (1996)10.1016/0378-4371(95)00311-8]. We develop a simple theory for the threshold of this distortion, which is a periodic undulation of the edge of the disk, demonstrating that it arises from saddle-splay elasticity of Sm-A, the low Sm-A-N interfacial tension unable to suppress the distortion. The observed increases in the number of bright spots with field, and with the radius of the disk at a given field, in both the experimental systems are also accounted for by the model. The distortion, which results in the most direct visualization of saddle splay in Sm-A, is also exhibited by disks nucleating on surfaces treated for homeotropic anchoring.