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The combination of low elastic modulus, anisotropy, and responsiveness to external fields drives the rich variety of experimentally observed pattern formation in nematic liquid crystals under capillary confinement. External fields of interest in technology and fundamental physics are flow fields, electro-magnetic fields, and surface fields due to confinement. In this work we present theoretical and simulation studies of pattern formation of nematic liquid crystals disclination loops under capillary confinement including a branching processes from m=+1 disclination line to two m=+1/2 disclination curves that describes the post nucleation and growth regime of the textural transformation from radial to planar polar textures. The early post-nucleation and growth of emerging disclination loops in cylindrical capillaries is characterized using analytical and computational methods based on the nematic elastica that takes into account line tension and line bending stiffness. Using sub diffusive growth and constant loop anisotropy, we find that the solution to the nematic elastica is a cusped elliptical geometry characterized by exponential curvature variations. The scaling laws that govern the loop growth reflect the tension/bending elasticity balance and reveal that the loop dilation rate depends on the curvature and normal velocity of the disclination. The line energy growth is accommodated by decrease in total curvature. These finding contribute to the evolving understanding of textural transformations in nematic liquid crystals under confinement using the nematic elastica methodology.
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