  We develop an energetic model that captures the twisting behavior of
spindle-shaped polymer microparticles with nematic ordering, which display
remarkably different twisting behavior to ordinary nematics confined to
spindles. We have previously developed a geometric model of the twisting, based
on experimental observations, in which we showed that the twist pattern follows
loxodrome spirals [Ansell et. al., Phys. Rev. Lett., 123, 157801 (2019)]. In
this study, we first consider a spindle-shaped surface and show that the
loxodrome twisting behavior of our system can be captured by the Frank free
energy of the nematic with an additional term constraining the length of the
integral curves of the system. We then extend the ideas of this model to the
bulk and explore the parameter space for which the twisted loxodrome solution
is energetically favorable.
