Lay a wash on dry paper and leave it alone. It dries with a crisp, darker line around its border, as if someone had outlined it. A spilled drop of coffee does the same on a countertop. Nobody painted that line. Where does it come from?
The short answer
The water in a drying puddle isn’t still. It flows outward, toward the edge, and it carries pigment with it.
The edge is stuck. On dry paper, the rim of the puddle catches on the paper’s fibres and stays put while the puddle dries. The puddle can only get flatter, not smaller.
The edge dries fastest. Water vapour escapes most easily from the thin rim, where open air is right beside it. Over the middle, the air is already damp from all the water around.
So water flows out to refill the edge, and floating pigment rides along. Most of it moves in the last moments of drying, when the puddle is thinnest. It piles up where the water ends: a dark line.
Fine, light pigments ride the flow all the way. Heavy grains that sink to the paper first, and fine ones the paper soaks up on the way, stay in the middle.
A puddle of paint drying
A puddle 2 cm across and 0.3 mm deep, from the side and from above.
Press “Watch it dry”. The water carries the grains outward, and they pile up at the edge into a dark line. Then try a heavy pigment such as cerulean, which sinks before the water can move it, or let the edge slide.
↑water evaporating, longer where faster→flow inside the puddlepigment grains (not to scale)pigment on the paper
Seen from above
What changes the edge
Each row dries the same paint with one thing about the puddle changed. Pigment, strength, paper and drying time are set on the right.
Puddle
Middle · edge
Outer 0.5 mm
ΔE
1What painters see
Curtis and colleagues, who built one of the first convincing watercolor simulations, describe it this way: on dry paper, the paper’s sizing and the water’s surface tension keep a brushstroke from spreading, and “the pigment migrates from the interior of the painted region towards its edges as the paint begins to dry, leaving a dark deposit at the edge” [1]. They call it a key effect that painters rely on, and one that paint makers take pains to keep in their formulas [1].
Physicists know it as the coffee-ring effect. Robert Deegan and colleagues at the University of Chicago worked out its cause in 1997 [2]. Their longer paper lists “enhanced edges in water color paintings” as an example, alongside coffee stains and the mineral rings on washed glasses [3]. They found rings form on glass, metal, plastic and ceramic, in drops from 1 mm to 15 cm, with solids from sugar molecules to 10 µm plastic spheres. Rings formed whenever three things held: the liquid wets the surface partly, its edge stays put, and it evaporates [2][3].
2The edge is stuck
A drop on a perfectly smooth surface would shrink as it dried, its edge sliding inward. Real surfaces have bumps and patches that snag the edge, and once it catches it stays. Physicists call it a pinned contact line. Watercolor paper, a mat of fibres coated with sizing, gives an edge plenty to catch on. Grains piling up at the edge then pin it harder still, so the drop holds its own edge in place [3].
Unpin the edge and the ring goes. Deegan’s group dried drops on smooth Teflon, where the edge slid inward as the drop shrank, and no ring formed [3].
3Why the edge dries fastest
Water leaves a puddle as vapour, and the vapour has to wander off into the room. Right at the water’s surface the air is saturated. A molecule that escapes from the middle of the puddle is surrounded by that damp air, and on its random walk it often lands back in the water. One leaving from the thin rim has dry air on one side, and is more likely to get away [3].
The math is the same as for electric charge on a metal object, which crowds onto sharp edges. That is why lightning rods are pointed. The rim of a puddle is a sharp edge, so evaporation there is strongest of all. Deegan’s group found it grows without limit toward the rim [2].
Two things painters may notice follow from this. The points of an irregular shape dry first and darkest: a clover-shaped coffee drop left its heaviest deposits at the tips. And two puddles side by side shade each other: where they are close, their damp air overlaps, and their facing edges dry lighter [3].
4Why the water flows outward
Surface tension keeps the puddle a smooth dome. With its edge stuck, the only way the dome can lose water is by getting flatter. A flattening dome drops a lot in the middle and hardly at all at the rim. But evaporation does the opposite: it takes the most at the rim. The rim is losing water much faster than its height is falling, so water has to come from somewhere. It flows in from the middle [2].
The flow is fastest near the edge, and it speeds up as the puddle thins: the same water has to cross the same distance through less and less depth. So most pigment arrives late. Marín and colleagues filmed particles reaching the edge and saw their speed shoot up in the last moments of drying. Particles that arrived early had time to settle into neat crystals, and the late ones jammed in a jumble [7]. Press “Watch it dry” and look at the second half.
Even evaporation still gives a ring. A flattening dome still drops more in the middle than at the rim, so some water still has to move out. Deegan’s group dried a drop set level in a water bath, which evens out evaporation, and it still left a ring, a wider one. Only when they made the drop evaporate mostly from the middle, through a lid with a hole over the centre, did the solids stay put [3]. In this model, “from the middle” matches the way a flattening dome comes down exactly, so the water doesn’t need to move at all.
Where the pigment ends up
Across the dried puddle, from the middle to the edge, for the settings above.
How much pigment dried at each distance from the middle. The spike at the right is the dark edge.
The scale is logarithmic: each step up is three times more. 1× is the paint spread evenly. The dashed line is where the pigment was when the puddle was laid down. There was more in the middle then, where the water was deeper.
5Grains that don’t make the trip
A grain only reaches the edge if it is still floating when the flow comes for it. Two things take grains out of the water early.
Sinking. A heavy grain sinks through still water at a speed set by Stokes’ law, which grows with the square of its size [8]. Cerulean’s 10 µm grains sink through 0.1 mm of water in half a second, and phthalo blue’s would take about eight hours. Once on the paper, a grain stays. The Granulation page is about where these grains land.
Soaking in. Watercolor paper soaks up part of the water, and fine grains go into the fibres with it, wherever they happen to be. MacEvoy explains staining the same way: small particles creep into the spaces between the paper’s fibres and lodge there [5][6].
So the model predicts that heavy, granulating pigments make weaker edges than fine ones, and that quick drying helps them make one. I haven’t found a measurement of this for paint, so treat it as a prediction. Deegan’s group did suspect sinking: their 1 µm spheres reached the edge more slowly than their theory said, and they named sedimentation as one possible reason [3].
How fast the ring grows
Share of the pigment that has reached the edge, against time.
Both scales are logarithmic. For a thin drop whose edge stays put, Deegan’s group derived the ring’s share as [1 − (1 − t/tf)(1+λ)/2]2/(1+λ), which starts as a power law in time [3]. The dots are this page’s model with nothing sinking or soaking in. They land on the line, which checks the code. For water drops on glass, the group measured an early exponent of 1.3 ± 0.1 against a predicted 1.37 [2].
6What painters do about it
For a soft edge, don’t give the water an edge to stick. Wet the paper past where the wash will end. The paint then fades into damp paper instead of stopping at a dry rim, which is the soft-edged look of wet-in-wet [1].
Take away the water that will end up at the edge. The flow ends where the last water sits, often in a bead at the bottom of a tilted wash. Touching that bead with a damp brush lifts the pigment-laden water before it dries there.
Use it. A dark edge separates shapes cleanly, and repeated wet-on-dry glazes stack their edges. Curtis and colleagues suspected that this is part of why glazing looks luminous [1].
Backruns are the same flow at a new edge. Water that spreads back into a drying wash pushes pigment ahead of it and leaves branching shapes with severely darkened edges [1]. MacEvoy rates how readily each paint does this as its “blossom” [6].
7Flows that run the other way
Evaporation cools a drop, and cooler liquid has higher surface tension. If the top of the drop ends up cooler than its rim, the surface is pulled toward the top, and a circulation sets in that carries particles back toward the middle. This is a Marangoni flow. Hu and Larson showed it can reverse the ring entirely: drops of octane left their deposit in the middle [4]. In water, traces of surfactant usually suppress it, which is why ordinary water drops make rings [4]. Deegan’s group saw a weak version in their own water drops [3]. Watercolor paints contain wetting agents and gum arabic, and how those change the flow isn’t something this page models.
8What this model leaves out
A round puddle. Real strokes are long and irregular, and corners and points dry faster than straight edges [3].
A thin, simple drop. The water keeps a smooth dome shape, and each grain moves at the flow’s average over the depth. In a real puddle, grains near the paper move more slowly than grains near the surface.
Grains that land stay put. A grain that reaches the paper is taken as stuck, though the fast late flow could drag some along.
Soaking in, simplified. The paper takes a fixed share of the water, evenly across the puddle, and takes floating grains with it in proportion. Sizing, paper weight and how wet the paper already was all change that share, and “half” is a guess.
No Marangoni flow, no gum arabic. Gum arabic makes the water thicker, slowing everything down.
Illustrative pigments. Grain sizes and densities are MacEvoy’s typical values [5], and each pigment gets a spread of sizes around them. The colors come from the same illustrative absorption curves as the other pages.
The math
The puddle is a thin spherical cap, radius R = 10 mm and centre depth h0 = 0.3 mm, so its contact angle is θ ≈ 2h0/R = 3.4°. With the edge pinned, its height is h(r, t) = h0(1 − τ)(1 − ρ²), with ρ = r/R and τ = t/tf: the dome flattens at a steady rate [3].
Evaporation from the open air follows J ∝ (1 − ρ²)−λ with λ = (π − 2θ)/(2π − 2θ) = 0.49 [3]. “Evenly” is λ = 0. “From the middle” is λ = −1, which has the same shape as the dome’s own fall. Water soaking into the paper is a share s of the loss, spread evenly.
The flow comes from conservation of water in each ring of the puddle: rhv̄ = −∫0r(J + ∂h/∂t)r′dr′ [3]. With the evaporation normalised to the puddle’s loss, the depth-averaged speed is
For s = 0 this integrates in closed form: u1+λ = 1 − (1 − u01+λ)(1 − τ)−(1+λ)/2. A grain starting at ρ0 reaches the edge at τ = 1 − (1 − u01+λ)2/(1+λ), and summing over the starting depth gives Deegan’s ring growth law in the chart above. With soaking, the page integrates the same equation numerically and checks it against the closed form.
Leaving the flow. Each grain starts at a random point in the water. It sinks at its Stokes speed, v = Δρgd²/18μ [8], with sizes spread log-normally (σ = 0.5 in ln d) about MacEvoy’s typical values. The falling surface pushes it down too. It lands when it reaches the paper. Soaking carries a floating grain into the paper at a rate q/h, where q is the water soaking in per unit area and h the depth above the grain.
The ring fills from the edge inward. Each band holds a pile of grains at 60% solids as high as the water was there when the grains arrived. Grains that arrive early, when the water is still deep, pile high and narrow. Late ones spread inward [3]. Strength 1 is paint that is 1/30 pigment by volume.
Colors come from one Kubelka–Munk layer of the pigment on paper, at the amount found at each spot. ΔE is CIEDE2000, between the middle’s average and the darkest 0.1 mm band near the edge.
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