Glaze vs. mix

1What light does inside paint

White light is a mix of every wavelength from violet (400 nm) to deep red (700 nm). Each wavelength gets its own treatment inside the paint. Light that enters can be:

• absorbed by a pigment particle, which turns it into a tiny bit of heat. That color is gone.
• scattered: bounced into a new direction, sometimes back up toward the surface.
• let through, untouched, to go deeper.

Watercolor paint is a thin, mostly transparent film. Most light goes down through it and hits the paper. Paper fibres scatter every wavelength strongly (that’s why paper is white), so the light bounces and heads back up through the paint. It crosses every layer at least twice, and many more times if it bounces between a layer and the paper. Whatever gets back out of the surface is the color you see.

Below, the same photons fall into both versions. Watch where they end up.

2Absorption: which colors each pigment removes

Each pigment has its own absorption curve. Hansa yellow absorbs violet and blue, below about 500 nm, and lets green through red pass. Orange absorbs further along: violet, blue and green, up to about 570 nm. Put both on the paper in any arrangement, and what survives is roughly what both let through: orange and red light.

Absorption alone can’t tell a glaze from a mix. Each pigment removes a share of the light passing through it, and shares multiply. If the orange lets 40% of some wavelength through and the yellow lets 50% through, 0.4 × 0.5 = 20% survives. That’s the same in either order, and the same with both mixed in one layer.

What changes is which pigment does the absorbing. Switch the photons above to one wavelength at 450 nm (blue). In the glaze, the orange on top catches almost all of it and the yellow underneath barely gets any. In the premix, the yellow and the orange split the work.

Make every pigment a perfect filter

3Scattering: why the order matters

Pigment particles also bounce light. In the glaze, all the orange particles sit on top. Some light bounces back out of the orange layer before it ever reaches the yellow. That light has only been through orange. It also never went deep, so it comes back lighter and a little chalky. In the premix, yellow and orange particles sit side by side at every depth, so light that bounces back early has still met both.

So scattering gives the top layer more say, and how much depends on the pigment:

• Transparent pigments (hansa yellow, transparent pyrrole orange, the quinacridones and phthalos) have tiny particles that scatter very little. They behave almost like filters, so a glaze comes out close to the premix.
• Opaque pigments (the cadmiums, cerulean, many earths) have larger particles that scatter a lot. On top, they cover what’s underneath. Underneath, they act like a second, colored sheet of paper. They send light back up through the top layer before it reaches the real paper.

Use two transparent paints Put an opaque yellow on top

This is why a glaze isn’t a sheet of colored glass. Glass only absorbs. Paint absorbs and scatters, the paper scatters, and light can bounce between the layers many times before it escapes or is absorbed. Kubelka–Munk theory, behind every number on this page, models exactly that [10][9]. It follows two streams of light, one heading down and one heading up. Absorption drains both streams, and scattering moves light from one stream to the other.

4Spectral reflectance: the curve that becomes a color

The share of light that comes back at each wavelength is the paint’s spectral reflectance, the physical fingerprint of its color. Your eye doesn’t see the curve itself. It has three kinds of cone cells, each adding up a broad slice of the spectrum, so the whole curve gets squeezed into three numbers [18]. That’s why two different curves can look almost alike. It’s also why a difference in the yellow-green, where the eye is most sensitive (around 555 nm), counts for more than one at the deep-red end.

5Dullness: the physics

A color is vivid when the light it returns is concentrated in one part of the spectrum, and dull when light from across the spectrum is mixed in. Colorimetry makes this precise with the excitation purity p_e [3][18]. On the CIE 1931 chromaticity diagram, draw the straight line from the white point W through the color C until it meets the spectral locus at D, the color of a single wavelength (the dominant wavelength). Then

White has p_e = 0; light of one wavelength has p_e = 1. Chroma C*, used elsewhere on this page, measures the same property on a scale spaced like human vision. A reflectance spectrum R(λ) can lose purity in two generic ways:

• Veil: add a spectrally flat component. R′ = v + (1 − v)R. The chromaticity slides along a straight line toward the white point (5.4 explains why).
• Mud: remove light from the band the color depends on. Multiply R by a transmittance that is low exactly where R is high. Luminance collapses; purity falls more slowly (5.5).

5.1 Where flat light comes from: refractive-index contrast

Light is partly reflected wherever the refractive index n changes. At normal incidence, the Fresnel reflectance of a boundary between media of index n₁ and n₂ is

Away from absorption bands, n varies only slowly across the visible spectrum, so this reflected light is close to spectrally flat. It is white light returned before any pigment has had the chance to filter it. A paint film contains an enormous number of such boundaries: the top surface, every pigment particle and every paper fibre. Their reflections, summed over many particles, are what the scattering coefficient S describes.

Particle size matters as much as index contrast. For particles much smaller than the wavelength (the Rayleigh limit), a sphere of diameter d has scattering cross-section [2]

Three consequences follow.

• Index contrast enters through the last factor, which vanishes as m → 1. It is 0.125 for cadmium yellow in gum arabic and 1.2 × 10⁻⁴ for ultramarine in gum arabic: a factor of about a thousand.
• Size enters as d⁶. At a fixed volume of pigment, the number of particles scales as d⁻³, so the scattering coefficient S scales as d³: halving the particle size cuts S eightfold. Pigment makers exploit this. “Transparent” grades of some pigments, transparent iron oxides for example, are the same compound as the opaque grade, made far finer.
• Wavelength enters as λ⁻⁴, so fine particles scatter blue preferentially, the same physics that makes the sky blue.

For larger particles, Mie theory replaces the Rayleigh limit. Scattering per unit volume of pigment peaks for diameters of a few tenths of a micrometre, comparable to the wavelength [2], which is the size range opaque pigments are milled to.

5.2 How much flat light a layer returns: absorption racing scattering

Inside the paint, light is a diffuse radiation field. Kubelka and Munk reduced the radiative-transfer problem to two fluxes: I, travelling down (increasing depth z), and J, travelling up [10][9]:

K is the absorption coefficient, and S the coefficient for scattering from one flux into the other. (The photon boxes above simulate exactly this process, one photon at a time.) A layer thick enough to hide its background has a reflectance R_∞ that depends only on their ratio. This is the Kubelka–Munk remission function:

Inside a pigment’s absorption band, K ≫ S, so the floor of the reflectance spectrum is roughly S/2K: it rises in proportion to scattering. At 450 nm, transparent pyrrole orange has K/S ≈ 44, so R_∞ ≈ 1.1%. Semi-opaque pyrrole orange has K/S ≈ 8.5, so R_∞ ≈ 5.3%, almost five times higher.

5.3 The glaze: the top layer answers first

A top layer with reflectance R₁ and transmittance T₁ lies on a background with reflectance R_b (here, yellow on paper). Kubelka’s layering result gives the reflectance of the stack [9]:

The series is the sequence of inter-reflections between the two layers. The first term, R₁, is light scattered back within the top layer. In the top pigment’s absorption band, it is the floor from 5.2: never filtered by the yellow, and nearly flat. That’s the veil. Everything else crosses the top layer at least twice and is filtered by both pigments: the full trip.

In a premix, the same scatterers sit among yellow absorbers, so the effective K/S includes the yellow’s absorption everywhere. At 450 nm, the premix has K/S ≈ 13.6 against 8.5 for the orange layer alone, so its floor is 3.5% instead of 5.3%. The same scattering costs the premix less purity.

5.4 Why added light desaturates: Grassmann’s law

Lights add. By Grassmann’s laws of color mixture [7], the tristimulus values of a sum of lights are the sums of their tristimulus values: X = X₁ + X₂, and likewise for Y and Z. In chromaticity coordinates (x, y) = (X, Y) / (X + Y + Z), the mixture therefore lies on the straight segment between the two components, at their centre of gravity with weights Σ = X + Y + Z:

The full-trip light of the pyrrole-orange glaze has p_e ≈ 100%. Its spectrum is close to a step rising near 580 nm, and the red end of the spectral locus is nearly straight, so step-like light lands on the locus itself. The veil, filtered only partly and only by the orange, has p_e ≈ 50%. Their sum sits between the two, at about 79%. No light was removed to cause this loss of purity. Light was added, pointing toward white.

5.5 Mud: absorption adds

Mixing pigments in one layer adds their coefficients: K = Σ c_iK_i, and likewise for S [6]. Orange’s complement, phthalo blue, has its large K exactly in 580–700 nm, where orange’s reflectance lives. The full-trip light in that band is multiplied by roughly e^−2cK_blue, so luminance falls fast while chromaticity moves comparatively little. In the mud row above, 0.07 units of blue drop the luminance factor from 29% to 9.8%, while purity only falls from 96% to 86%.

Scaling R by a constant would darken the color and leave its chromaticity, and so its purity, unchanged. In colorimetric terms, brown is orange at low luminance. It reads as brown rather than as dim orange only next to brighter surroundings; it is a related color. Each additional pigment in a mix adds another absorption band, so a mix of three or four absorbs a little of everything: the same sinking of the curve’s top.

5.6 Wet versus dry, gloss versus matte

Watercolor is a lean paint: there is little gum arabic between particles, so the dried film is porous. Coatings scientists describe such films as lying above the critical pigment volume concentration. There, air-filled voids appear, and the extra air–pigment interfaces increase diffuse scattering and lighten the color [8]. While the paint is wet, water (n = 1.333) fills those voids and the paper’s pores, and index contrast drops. A cellulose boundary reflects 0.6% in water against 4.7% in air (figure 5.1). Light is scattered less, travels deeper, and is more likely to be absorbed before it escapes. The wet film looks darker and more saturated.

Two models of this wetting effect are in the literature. Twomey, Bohren and Mergenthaler attribute it to stronger forward scattering when index contrast drops [17]. Lekner and Dorf attribute it to total internal reflection inside the liquid film, which gives the surface more chances to absorb [11]. Measurements on particulate layers found that neither is sufficient alone [19].

The top surface adds its own term. A gum arabic–air boundary reflects about 3.7% at normal incidence. On a glossy surface, that light leaves in the mirror direction and is invisible from most viewing angles. On a matte (rough) surface, it is scattered into every direction and lays a white veil over everything. Light leaving the paint is also partly reflected back down at the surface, with an internal diffuse reflectance of about 0.6 for n ≈ 1.5, which sends it through the pigment again. Saunderson’s correction combines both effects [14]:

Here k₁ is the external and k₂ the internal surface reflectance. A glossy surface viewed away from the mirror direction loses the leading k₁.

In short, the purity of a glaze is set by three things. The first is the floor S/2K of its top layer in that pigment’s absorption bands. The second is how many flat-reflecting boundaries the light meets before it is filtered: particle index contrast, particle size, and the air in a dry film. The third is the surface. To keep a glaze clear, put low-scattering pigments on top (fine particles, index close to the binder), put any opaque pigments underneath, keep each layer to few absorbers, and glaze over dry paint in one pass.

6What this model leaves out

• Re-wetting. Glaze over yellow that isn’t completely dry, or brush over it twice, and some yellow lifts into the orange. That part becomes a premix.
• Wet mixes can separate. In a premix, heavy particles settle faster than light ones, and some pigments clump together (flocculate). The mix may not stay even as it dries.
• Surface reflection. The main swatches use bare Kubelka–Munk. Section 5.6 shows what the paint–air surface adds [14].
• Transparent paint on paper. For a weakly scattering layer on a strongly scattering support, models that also track internal reflections at the paint–air surface, such as Clapper–Yule, often fit measurements better than plain Kubelka–Munk [4][13]. Kubelka–Munk has also been fitted to watercolor directly, across different dilutions [16].
• Illustrative pigments. The curves have their absorption edges and opacity in the right places, but they aren’t measurements of real tubes. Treat the numbers as a demonstration, not a prediction for your paints.