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Helicoidal dynamics of biaxial curved rods in twist-bend nematic phases unveiled by unsupervised machine learning techniques.

Uniaxial rods in a nematic phase diffuse preferentially in the direction parallel to the nematic director n[over ̂]. The nematic director field n[over ̂](r) of a chiral twist-bend nematic (N_{TB}) phase of achiral banana-shaped particles, recently discovered experimentally, displays a heliconical twist of given handedness and periodicity. Using simulations, we investigate the long-time macroscopic diffusion in N_{TB} phases, and find that the predilection of curved rods to diffuse in the direction of the twisting n[over ̂](r) yields a fascinating chiral dynamics along helices, even though achiral curved rods display Brownian motion with a nontrivial rototranslational coupling. We devise a machine learning protocol to characterize the helicoidal particle trajectories, finding that their pitch and radius are determined by the pitch and conical angle of the N_{TB} phase thereby connecting its structural and dynamical properties.