Brush cerulean or ultramarine across rough paper and it dries speckled, the color gathered into the paper’s tooth. Phthalo blue on the same paper dries as smooth as ink. Same water, same paper. What’s different?
The short answer
It comes down to how big and how heavy the pigment’s grains are.
Big, heavy grains sink. A grain falls through water at a speed that grows with the square of its size. Cerulean’s grains reach the paper in about half a second. Phthalo blue’s would take about eight hours, and the constant jostling of water molecules keeps them afloat anyway.
Sunk grains land in the hollows. The water’s surface is flat but the paper’s isn’t, so there is more water, and more grains, above each hollow. The grains that sink print the paper’s texture into the paint.
Clumps act like bigger grains. Some pigments, ultramarine most famously, stick together in clumps. A few big lumps can’t spread as evenly as millions of fine grains, so the wash looks speckled, even on smooth paper.
Fine pigments are still afloat when the water goes, and they spread out evenly with it.
A wash drying on textured paper
A wash 0.1 mm deep, seen from above and as a slice from the side.
The swatch is the dried wash, enlarged. Below it, watch a slice of the wash: heavy grains drop into the paper’s hollows, while fine ones just jiggle in place. Try cerulean, then phthalo blue.
40 µm per dot
grains, not to scale: bigger dots for bigger grainsWater level falls as the wash dries
Where the texture comes from
The same swatch with one cause at a time. ΔE is how far a spot’s color is from the average, on average.
Cause
Light · dark spot
ΔE
1What painters mean by granulation
Curtis and colleagues describe granulation as “a kind of grainy texture that emphasizes the peaks and valleys in the paper”. It varies from pigment to pigment, and it is strongest when the paper is very wet [1]. A close cousin is flocculation, “in which particles are drawn together into clumps usually by electrical effects”, and it looks much the same [1].
Bruce MacEvoy rates every paint he tests for granulation, from 0 (“liquid”, no texture) to 4 (“granular”). He brushes the paint out in a dilute wash and judges the texture by eye and by touch. His rating counts flocculation too, the “clumping of the pigment texture into tiny flakes or spots (for example, in ultramarine blue)” [4].
2Sinking: size matters twice
A grain sinking through water speeds up until the water’s drag balances its weight. George Stokes worked out that speed in 1851 [6]:
v = (ρgrain − ρwater) gd² / 18μ
Weight grows with a grain’s volume, d³, but drag grows only with its width, d. So the speed goes as d²: a grain 100 times wider sinks 10,000 times faster. Density counts too, but pigments’ densities only run from about 1.4 to 5.5 times water’s [3]. Grain sizes run from 0.05 µm to 50 µm or more, a thousandfold [3].
How long grains take to sink through a wash
Through 0.1 mm of still water, from Stokes’ law.
Each bar is how long that pigment’s grains take to sink to the paper. The dashed line is when the wash dries. Everything left of it lands on the paper; everything right of it is still floating when the water goes.
Grain sizes and densities are MacEvoy’s typical values [3]. Real paints have a spread of sizes around them. Water at 20 °C. Gum arabic thickens the water a little, which would slow every bar by the same factor. Faded bars: grains so small that Brownian motion keeps them mixed through the water (section 3).
3Too small to sink: Brownian motion
Water molecules batter every grain from all sides, and a small grain gets knocked about in a random walk. Einstein showed in 1905 how far such a grain wanders: the smaller it is, the farther [2]. Gravity pulls grains down, and the jostling spreads them back up. They balance when the grains are spread over a height kT / (weight in water). For phthalo blue’s 0.1 µm grains that height is 1.3 mm, thirteen times the depth of the wash. They stay mixed through the water, however long it takes to dry. For ultramarine’s 1 µm grains it is 0.6 µm: they go to the bottom.
That balance has a place in the history of physics, and a watercolor pigment is part of it. Jean Perrin tested Einstein’s theory by counting tiny grains of gamboge, a yellow watercolor pigment, at different heights in water. He found their numbers fell off with height just as predicted, and used them to measure Avogadro’s number [5].
4Why the grains end up in the hollows
Surface tension holds the top of a wash flat, but the paper under it is bumpy. Over a hollow the water is deeper, so it holds more grains. A grain that sinks lands roughly where it started, right below. So once the grains have had time to sink, each spot gets pigment in proportion to the water that was above it. The hollows come out darker and the bumps lighter: the paper’s texture, printed in paint.
That gives three practical rules, all of which the swatch above follows:
Rough paper granulates most. Deeper hollows mean a bigger difference in water, and in pigment.
A wash that stays wet longer lets more grains sink. In the model this matters for the in-between pigments, burnt sienna and single grains of ultramarine, which take a minute or two. The heaviest sink in seconds anyway.
Strong paint hides it. Twice the pigment in a hollow is a big difference in a pale wash, but in a dark one both spots are already near the deepest color the paint can make. MacEvoy rates granulation in a dilute wash [4].
5Clumps: why ultramarine looks speckled
Pigment grains tend to stick together. MacEvoy notes that most pigments clump into aggregates 5 to 50 times larger than their single grains, and that paint makers mill them with dispersants to break the clumps up [3]. Ultramarine is the classic case: MacEvoy’s granulation rating counts clumping in, and names ultramarine as the example [4].
Clumping does two things. A clump sinks far faster than its grains would one by one, so even fine ultramarine reaches the paper. And it cuts the number of separate pieces. A spot that should hold N pieces actually gets a random count that varies by about √N. At strength 0.4, with 4,000 fine grains in a 40 µm spot, that is a 1.6% wobble, invisible. With 65 clumps, it is 12%. With cerulean’s 4 big grains, it is 50%: a speckle that shows up even on smooth paper.
6Separation: two paints that part ways
Mix a granulating pigment with a fine one and they part ways on the paper. The heavy one sinks into the hollows, the fine one spreads evenly, and the dried wash shows two colors where the brush held one. Curtis and colleagues call this separation: denser pigments settle earlier than lighter ones [1]. Ultramarine with burnt sienna is a classic granulating gray. Here both granulate, at different rates.
7What this model leaves out
Grains moving after they land. Here a grain stays where it touches down. Real grains can roll, and the last water draining off the bumps can sweep them into the hollows. That may be why painters find granulation strongest in very wet washes [1]. In this model a wetter wash helps only by giving grains longer to sink.
The last of the water. Grains still floating at the end are spread evenly. Really they go wherever the last water goes, and that water has its own flows, the same ones that make dark edges.
The paper. Its bumps are made-up noise with a typical height of 4, 12 or 25 µm for the three surfaces. I couldn’t find published measurements of watercolor paper’s surface to use instead.
One size per pigment. Each pigment has one grain size, MacEvoy’s typical value [3], and a clump is always five grains across and half water. Real paints have a spread of sizes, and how much they clump depends on the formula.
Still water. No brush strokes, no tilting, no gum arabic.
Illustrative colors. The cobalt and cerulean absorption curves are added for this page and, like the others, are shaped like the real pigments but not measured.
The math
Sinking. A grain of diameter d and density ρp sinks at v = (ρp − ρw)gd²/18μ, with μ = 1.0 mPa·s for water at 20 °C [6]. A clump is a sphere five grains across, half pigment and half water, so its density above water’s is half a grain’s.
Brownian motion. The Stokes–Einstein diffusion coefficient is D = kT/3πμd[2]. Gravity and diffusion balance in an exponential profile with scale height ℓ = kT/(Δρgπd³/6), Perrin’s sedimentation equilibrium [5]. In the side view, grains with ℓ greater than the wash’s depth bounce off the paper instead of landing.
Landing. The water’s surface is flat, 100 µm above the paper’s mean height h̄, so the depth at each spot is w = 100 µm − (h − h̄). In the time t the wash stays wet, a grain sinks L = vt. Each spot collects the grains that started within L of it, min(L, w). The rest stay afloat and are spread evenly at the end. A falling water surface pushes floating grains down but doesn’t change which ones reach the paper in time, so it drops out.
Counting. Paint at strength 1 is 1/30 pigment by volume. A 40 µm spot under 100 µm of water at strength a holds N = a × (1/30) × 100 × 40² µm³ of pigment, divided by the pigment in one grain or clump. Its actual count is drawn from a Poisson distribution with that mean.
Paper. White noise, blurred with a Gaussian at the bump size and at a third of it, mixed 1 : 0.35 and scaled to the relief.
Colors. Each spot is one Kubelka–Munk layer on paper with its own amount of each pigment. Texture is the mean CIEDE2000 difference between each spot and the swatch’s average color, at this magnification. At reading distance the eye averages away the finest speckle.
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