On the contact region of a diffusion-limited evaporating drop: A local analysis

Motivated by experiments showing that a sessile drop of volatile perfectly wetting liquid initially advances over the substrate, but then reverses, we formulate the problem describing the contact region at reversal. Assuming a separation of scales, so that the radial extent of this region is small compared with the instantaneous radius a of the apparent contact line, we show that the time scale characterizing the contact region is small compared with that on which the bulk drop is evolving. As a result, the contact region is governed by a boundary-value problem, rather than an initial-value problem: the contact region has no memory, and all its properties are determined by conditions at the instant of reversal. We conclude that the apparent contact angle θ is a function of the instantaneous drop radius a, as found in the experiments. We then non-dimensionalize the boundary-value problem, and find that its solution depends on one parameter 葦, a dimensionless surface tension. According to this formulation, the apparent contact angle is well-defined: at the outer edge of the contact region, the film slope approaches a limit that is independent of the curvature of bulk drop. In this, it differs from the dynamic contact angle observed during spreading of non-volatile drops. Next, we analyse the boundary-value problem assuming 葦 to be small. Though, for arbitrary 葦, determining θ requires solving the steady diffusion equation for the vapour, there is, for small 葦, a further separation of scales within the contact region. As a result, θ is now determined by solving an ordinary differential equation. We predict that θ varies as a-1/6, as found experimentally for small drops (a < 1 mm). For these drops, predicted and measured angles agree to within 10-30 %. Because the discrepancy increases with a, but 葦 is a decreasing function of a, we infer that some process occurring outside the contact region is required to explain the observed behaviour of larger drops having a > 1 mm