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How to use
Pick a preset or move any slider (changing the radius, angle, contact-line mode, tracer count or seed restarts the drop; air, particle size and Marangoni stress act live). The large picture is a cross-section of the drop: red arrows are the evaporation flux (long at the rim), yellow arrows are the outflow at three depths, the dots are tracer particles and the brown layer on the glass is the deposit. The panel on the right (or below on a phone) is the top view of the stain forming. Fast-forward dries the drop at once. The graphs show the contact angle, radius and volume, the outflow profile, the deposit density against a uniform film, and the flux profile. Things to try: (1) the classic ring, then raise the humidity and see only the speed change; (2) switch the contact line to receding and the ring disappears; (3) raise the Marangoni stress above 6 to reverse the surface flow; (4) use stick-slip for concentric rings; (5) make the particles 2 um and look at the width of the ring. Space pauses, R restarts.
The science
A drop of coffee dries into a dark ring, not a uniform disc. The explanation (Deegan et al., Nature 389, 827, 1997) needs only two ingredients: the contact line is pinned to the surface, and evaporation is faster at the rim. This lab computes both from first principles and follows tracer particles through the resulting flow.
Diffusion-limited evaporation. Water leaves through the surrounding air by vapour diffusion. The total evaporation rate of a spherical-cap drop of contact radius R and contact angle θ (in radians, valid up to about 90 degrees) is (Hu and Larson 2002)
Q = π R D (csat − c∞) (0.27 θ2 + 1.30), c∞ = RH · csat, D = 2.12×10−5 (T/273.15)1.8 m2/s
with the saturation density from the Magnus fit psat = 610.94 exp(17.625 T/(T + 243.04)) Pa and csat = psatM/(RgT). The local flux diverges at the rim (the vapour has the shortest way out there):
J(r) = J0 (1 − r2/R2)−λ, λ = 1/2 − θ/π, J0 = Q (1 − λ) / (π R2)
J0 is fixed so that the flux integrated over the free surface equals Q. For small angles λ tends to 1/2 and the rim evaporates much faster than the middle (graph 4).
The drop. The shape is a spherical cap: apex height h0 = R tan(θ/2), volume V = π h0(3R2 + h02)/6 and profile h(r) = √(Rc2 − r2) − (Rc − h0), Rc = (R2 + h02)/(2h0). Volume is lost at rate Q/ρ. Three contact-line behaviours: pinned (R fixed, θ falls), receding (θ fixed, R shrinks) and stick-slip (pinned until θ drops to 42 % of its start, then R jumps inward to give θ = 72 % of the start, repeatedly).
The capillary flow. Take the liquid inside radius r. Its volume Vin(r) changes because of evaporation from that disc and because liquid flows out through its edge. The depth-averaged radial velocity is therefore
ū(r) = −[ dVin/dt + Φ(r)/ρ ] / (2 π r h(r)), Φ(r) = Q [1 − (1 − r2/R2)1−λ]
where Φ(r) is the evaporation inside radius r in closed form. When the edge cannot move, the liquid evaporated near the rim must be replaced from the centre, so the outflow is positive everywhere, and near the rim h is proportional to (R − r) so the outflow grows as (R − r)−λ: it diverges (graph 2 on log axes, with the guide line). The flow is much faster than diffusion (Péclet number in the readout), so every particle is carried out.
Depth structure and Marangoni flow. In the lubrication approximation the radial velocity across the depth (ζ = z/h) with a surface shear stress τ = −k μ ū/h is
u(ζ) = ū [ (k + 6) ζ/2 − 3 (k + 2) ζ2/4 ]
It has depth-average ū for every k (the net flow is fixed by mass conservation); k = 0 is the free-surface parabola (surface speed 1.5 ū), k = 6 makes the surface stagnant, and k above 6 reverses the surface flow, so fluid near the surface goes inward and the bulk carries a recirculation loop. Each tracer particle moves with this flow, with vertical motion that follows the streamline (continuity: ζ changes through film thinning and the receding surface) and adds Brownian motion with D = kT/(6 π η a). A particle is captured when the film is as thin as the particle (h below dp, or within dp/tanθ of the rim); at the end of the drying the particles still suspended are deposited where they are.
Numerical checks run in the page. (1) The dry time of a pinned drop from the time-stepping simulation against an independent quadrature over the contact angle, t = ∫ ρ (dV/dθ)/Q(θ) dθ; (2) the exponent of the outflow near the contact line against −λ (30 degrees); (3) the mass-balance residual of the computed flow at the rim (relative, must vanish); (4) the diffusivity measured from the mean-square displacement of 4000 simulated walkers against the Stokes-Einstein value; (5) the Magnus saturation density at 25 C against the tabulated 23.05 g/m3; (6) the depth-average of the Marangoni profile at k = 3 equals 1; (7) the free-surface speed vanishes at k = 6.
Simplifications. The drop is an exact spherical cap at every instant (valid for Bond number far below 1, i.e. R under about 2.7 mm for water, and for contact angles up to about 90 degrees; the 3 mm and 70 degree limits are at the edge of that range). The evaporation model ignores evaporative cooling and thermal Marangoni forces (they drive the real Marangoni loop, which here is a free parameter k), the slope factor in the flux, particle-particle interactions, the rise of the solute concentration (no gelation or DLVO effects) and the hindered motion of particles near the substrate. The outflow is computed exactly from mass conservation but the lubrication assumptions (small slopes) are not accurate close to the rim at the very end of drying. The sticking criterion is a simplification of capillary immersion forces. Material constants are for pure water in air at atmospheric pressure.