  Understanding and controlling the director field configuration, shape, and
orientation in nematic and cholesteric liquid crystals is of fundamental
importance in several branches of science. Liquid crystalline droplets, also
known as tactoids, which spontaneously form by nucleation and growth within the
biphasic region of the phase diagram where isotropic and nematic phases
coexist, challenge our current understanding of liquid crystals under
confinement, due to the influence of anisotropic surface boundaries at
vanishingly small interfacial tension and are mostly studied under quiescent,
quasi-equilibrium conditions. Here, we show that different classes of amyloid
fibril nematic and cholesteric tactoids undergo out-of-equilibrium order-order
transitions by flow-induced deformations of their shape. The tactoids align
under extensional flow and undergo extreme deformation into highly elongated
oblate shapes, allowing the cholesteric pitch to decrease as an inverse power
law of the tactoids aspect ratio. Energy functional theory and experimental
measurements are combined to rationalize the critical elongation ratio above
which the director-field configuration of tactoids transforms from bipolar and
uniaxial cholesteric to homogenous and to debate on the thermodynamic nature of
these transitions. Our findings suggest new opportunities in designing
self-assembled liquid crystalline materials where structural and dynamical
properties may be tuned by non-equilibrium phase transitions.
