Description
We study microfluidic separation of helical particles with opposite chirality in pressure-driven flow through a cylindrical capillary. Chirality couples the local shear to an axial drift relative to the carrier flow, so left- and right-handed helices acquire equal-magnitude drift in opposite directions, enabling axial separation downstream. We show that adding a small oscillatory modulation to an otherwise steady pressure drive provides an additional, frequency-tunable handle: the AC contribution to the drift has a resonance-like dependence on rotational diffusivity, so the drift can be selectively amplified for particles whose rotational relaxation rate matches the forcing rate. This offers a practical way to enhance separation performance while retaining a simple channel geometry and mild shear conditions relevant for biological helices.
On the modelling side, we start from Jeffery’s equation for an axisymmetric body in a linear Stokes flow and include isotropic rotational Brownian motion. At the ensemble level this yields a Fokker–Planck equation for the orientation probability density on the unit sphere. In the small-tilt regime near the channel axis we project this equation onto the first spherical harmonic to obtain a closed Debye-type evolution equation for the transverse first moment of the orientation, which quantifies the shear-induced radial bias of the ensemble. For a combined steady and oscillatory shear profile, this reduced equation can be solved analytically; after averaging over many oscillation periods we obtain an explicit drift law that separates into a steady contribution and an AC-enhanced contribution.
We discuss how the effective chiral coupling coefficient entering the drift law can be estimated from hydrodynamic calculations or calibrated in steady shear. Combining the predicted drift with axial and radial translational diffusion yields compact design rules for capillary length, mean flow rate, and modulation amplitude that ensure both good axial resolution and negligible wall contact. Finally, we outline how Langevin simulations of Jeffery-plus-noise dynamics in a Poiseuille profile can be used to validate the reduced theory and guide microfluidic designs for separating synthetic helices, bacterial flagella, or other chiral filaments.