Draw a stripe of black ink, let it dry, and brush your paints across it. Some vanish over the stripe. Others leave a pale film on top of it. What decides which?
The short answer
You see the black line through paint when light gets down to the ink and back out. A paint hides the line by sending light back up before it gets there, and that takes particles that bounce light. How much a particle bounces depends on three things:
Refractive index contrast. Light bounces off a particle only if the particle bends light differently from what surrounds it. Cadmium yellow (refractive index 2.3) in gum arabic (about 1.48) bounces a lot. Ultramarine (1.5) nearly matches gum arabic, so in the binder it barely bounces at all.
Particle size. For a given amount of pigment, particles about a quarter of a micrometre across bounce the most. Much smaller ones let light slip past; much larger ones are too few.
How much paint. Any paint laid thin enough is see-through, and almost any paint laid thick enough hides.
Darkness can hide a line too. A heavy wash of a dark, transparent paint can get almost as dark as the ink. That hides the line by absorbing light, not by bouncing it, and the paint is still transparent.
The black-line test
Three bands of one paint across a stripe of black ink: half, once and twice the amount.
Three strokes of one paint across a black line. If you can still see the line through the paint, the paint is transparent; if the paint covers it, it’s opaque. Try different paints and amounts.
Band
Paper · ink
Luminance
Contrast ratio
ΔE
Reads as
1What the black line measures
Painters test transparency exactly this way. MacEvoy’s ratings on handprint come from painting a diluted stroke over two black marker lines, then drawing a third line across the dry paint to compare. His scale runs from 4, “transparent: no visible masking”, to 0, “very opaque: extensive or complete masking” [9].
The test works because the ink takes away almost all the light that reaches it. Over the ink, the only light you see is light the paint itself sent back before it got that deep. Over white paper, you also see light that went down through the paint, bounced off the paper and came back up. Kubelka–Munk theory gives both [6][7]:
over black: R = R0 over paper: R = R0 + T² Rpaper / (1 − R0Rpaper)
R0 is the light the paint layer sends back by itself, and T is the share it lets through. The paint industry turns this into a number, the contrast ratio: the luminance over black divided by the luminance over white [3]. A perfectly transparent paint scores 0. By convention, a paint “hides” at 0.98, which is used to define hiding power in standard tests [1][5].
On this page, hansa yellow at the middle band’s starting amount scores 0.10 and cadmium yellow 0.52. To reach 0.98, cadmium yellow needs about 5 units of pigment, nine times that starting amount. Hansa yellow would need about 30, far more than a brush can lay down: in practice it never hides the line.
Two curves that meet
Luminance of the paint over white paper and over black ink, as the paint gets thicker.
As the paint gets thicker, its color over the black line and over white paper get closer together. Where the two lines meet, the black line is hidden.
Uses the paint and amount picked above; the colored bars mark the test card’s three bands. Over paper, more paint absorbs more light, so the solid curve falls. Over ink, more paint sends more light back before it reaches the ink, so the dashed curve rises. Where they meet, the line is hidden. A dark transparent paint can bring the two together by dragging the top curve down to the ink instead.
2Index contrast: why particles bounce light
Light is partly reflected wherever it crosses into a material that bends it differently, measured by the refractive index n. At one boundary, light arriving straight on is reflected in the share
R = ((n1 − n2) / (n1 + n2))²
A pigment particle is a boundary all the way around. Here are typical indices, from MacEvoy’s table of watercolor pigments [8], with the share each boundary reflects in dry gum arabic (taken as 1.48) and in air:
Material
n
In gum arabic
In air
Phthalocyanines, arylide (hansa) yellows
1.4
0.08%
2.8%
Ultramarine
1.5
0.005%
4.0%
Chinese (zinc) white, quinacridones
2.0
2.2%
11%
Cadmium yellow
2.3
4.7%
16%
Titanium white (rutile)
2.7
8.5%
21%
In gum arabic, an ultramarine boundary reflects a thousand times less than a cadmium one. That’s why ultramarine glazes clean and cadmiums cover. Air is another story: every pigment contrasts strongly with it. MacEvoy points out that dried watercolor is porous, with pigment touching air, and that by index ratio alone “nearly all watercolor pigments are opaque in that situation” [8]. They still look transparent because there is so little of them. A wash is a thin scatter of particles on paper, not a solid film. (What the air does to color as a wash dries is the subject of Why paint dries duller.)
3Size: the scattering sweet spot
The second factor is size, and here the full answer is Gustav Mie’s 1908 solution for light scattered by a sphere [10]. This page computes it with Bohren and Huffman’s standard algorithm [2]. Two limits frame the result:
Particles much smaller than the wavelength scatter weakly. Rayleigh showed that a tiny sphere’s scattering grows as its diameter to the sixth power [11]. For a fixed amount of pigment, the number of particles goes as one over diameter cubed, so the scattering of the whole lot still grows as diameter cubed. Halving the size cuts it eightfold.
Particles much larger than the wavelength each scatter about as much light as their shadow, but there are few of them for their weight, and they throw most of that light forward, on into the paint, rather than back.
In between there is a peak. The figure plots how much light is turned back per unit volume of pigment, for the indices in the table above.
Scattering by particle size
Mie theory at 550 nm, averaged over a modest spread of sizes.
How strongly a pigment’s grains bounce light, by grain size. Grains about a quarter of a micrometre across bounce the most. Grains that bend light very differently from the gum around them, like titanium white and cadmium, bounce far more than grains that blend in, like ultramarine.
Particles surrounded by
The y axis is the transport scattering cross-section per unit volume, 3Qsca(1 − g)/2d: the light scattered, weighted by how far it is turned from straight ahead. That is the part that can send light back toward the viewer. Both axes are logarithmic. Spheres stand in for real particle shapes.
In gum arabic, the high-index pigments peak between about 0.2 and 0.3 µm: titanium white at 0.23 µm, cadmium yellow at 0.27 µm. Per unit of pigment, 0.25 µm titanium white turns back about 17 times more light than 0.05 µm or 2 µm titanium white. Ultramarine’s curve hardly leaves the floor, because its index matches the gum. Switch the surroundings to air and every curve jumps, ultramarine’s by more than a thousand times.
Pigment makers use both levers. The same compound can be sold as a transparent or an opaque grade by grinding it to different sizes. MacEvoy notes that paint transparency depends mostly on the average particle size and the thickness of the paint layer [8].
4Hiding by absorbing
A line can also disappear because the paint over it is nearly as dark as the ink. Try ultramarine at a strong amount: over paper it returns only about 4% of the light, so the contrast ratio rises toward 1 even though the particles hardly bounce anything. The industry’s contrast ratio counts that as hiding. A painter wouldn’t call such a paint opaque, because thinned down it lets the paper glow through again.
Absorption is also why strong transparent pigments seem to “cover” in masstone. Phthalo blue’s tinting strength is about 40 times ultramarine’s [8], so it goes dark at a small amount of pigment. Hiding by absorption fades with dilution. Hiding by scattering lasts much further: thinned cadmium still leaves a pale film over the line.
5Why watercolor cares
In watercolor, the paper is the light source. Light goes down through the paint, bounces off the paper and comes back colored. A transparent paint passes that light twice and keeps the white of the paper behind the color. An opaque paint sends some light back before it reaches the paper, so it reads lighter, chalkier and flatter. That’s often useful, for a sky or a highlight painted over a darker wash. It also means layer order matters. An opaque paint on top covers what’s underneath, as Glaze vs. mix shows.
6What this model leaves out
Illustrative pigments. The paint curves have the right kind of absorption and roughly the right opacity, but they aren’t measurements of real tubes, so the contrast ratios are a demonstration, not a rating of your paints.
Spheres, one at a time. Mie theory treats each particle as an isolated sphere. Real particles are faceted and clumped. When they crowd together, as in a dense film, they shade each other and scatter less than the sum of their parts. Coatings scientists track this with the pigment volume concentration [12].
Absorbing particles. The Mie figure uses real refractive indices. A strongly absorbing pigment also scatters differently, and the paint curves handle its absorption separately, through K.
Granulation. Heavy pigments settle into the hollows of the paper, so a real band over a black line is patchy rather than even [8].
The math
A layer with absorption K and scattering S (already multiplied by the amount of pigment) reflects R0 and transmits T[6][7]:
a = 1 + K/S, b = √(a² − 1) R0 = sinh(bS) / (a sinh(bS) + b cosh(bS)), T = b / (a sinh(bS) + b cosh(bS))
On a background reflecting Rg, the layer reflects R0 + T²Rg/(1 − R0Rg). The test card uses Rg = the paper, and Rg = 4% at every wavelength for the ink. The contrast ratio is Yink/Ypaper, with Y the CIE luminance under daylight.
Mie scattering. For a sphere of diameter d in a medium of index nm, the size parameter is x = πd nm/λ and the relative index m = np/nm. The series of Mie coefficients gives the scattering efficiency Qsca and the asymmetry g, the mean cosine of the scattering angle [2]. The code reproduces Bohren and Huffman’s test case (m = 1.55, x = 5.213: Qsca = 3.10543, Qback = 2.92534) and the Rayleigh limit Qsca = (8/3)x⁴((m² − 1)/(m² + 2))².
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