We explore the structure of nuclei and topological defects during the phase transition in lyotropic chromonic liquid crystals (LCLCs). The LCLCs are formed by self-assembled molecular aggregates and show a broad biphasic region. The defects emerge as a result of two mechanisms. 1) Surface anisotropy mechanism that endows each N nucleus (tactoid) with topological defects thanks to tangential orientation of the director at the interface, and 2) Kibble mechanism with defects forming when differently oriented tactoids. Different scenarios of phase transition involve positive (N- in-I) and negative (I-in-N) tactoids with non-trivial topology of the director field and also multiply connected tactoids-in-tactoids configurations. The interface limiting a tactoid shows a certain number of cusps. The each cusp contains a point defect-boojum. The number of cusps shows how many times the director becomes perpendicular to the interface. We derive conservation laws that connect the number of cusps to the topological strength of defects in the N part of the simply- and multiply-connected tactoids. We demonstrate how the elastic anisotropy of the N phase results in non-circular shape of the disclination cores. A generalized Wulff construction is used to derive the shape of tactoids as the function of interfacial tension anisotropy in the frozen director. The shapes and structures of tactoids and topological defects demonstrate an important role of surface anisotropy in morphogenesis of phase transitions in LCs.

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