Fractionation of polydisperse particles in a receding floating film

Abstract

A thin volatile film carrying N particle species of different sizes evaporates on a deep immiscible liquid subphase. The spreading coefficient is positive, so nothing pins. The film ends at a receding front whose motion falls out of the film equations; no contact-line law is imposed. We solve the lubrication problem with a conservative depth-integrated treatment of species transport, and the front turns out to be a chromatograph. Every species piles into a concentration spike at the front, each held there in proportion to its P\'eclet number. The smaller, more diffusive species leaks continuously into the fluid that survives the front's passage; the larger species is laid down along the sweep path. The dried deposit is the time-integrated record of that sweep, and it is sorted by size: small-rich centre, large-rich mid-annulus. Pinned bidisperse droplets sort the other way. Depinned droplets on solids sort this way, but by a force balance; here, diffusivity contrast alone picks the direction. Size enters the transport problem only through the P\'eclet number, so diffusivity contrast is the only symmetry-breaking available to choose a direction: Marangoni stresses, colloidal interactions and wetting effects act on the magnitude and, by the symmetry of the transport operator, cannot set the direction. Adjacent-species band separations scale with the difference in inverse effective P\'eclet number, a law derived and measured here: the resolution of the chromatograph. That scaling points to a route to size fractionation of polydisperse nanoparticles in the few-nanometre regime, exactly where standard methods struggle.