2.1 Reflection, absorption and scattering

Light comes in tiny packets called photons, and a photon that reaches a layer of paint can do one of four things. It can bounce off the top surface without entering; this is the gloss and does not change the wavelength of the photon. Otherwise it goes in, where it meets pigment particles. A particle can absorb the photon, scatter the photon, or let the photon pass undisturbed. To scatter a photon is to send it off in a new direction. A photon that is left undisturbed by the pigment eventually reaches the substrate and is either reflected or absorbed by it. Absorption is what gives a pigment its hue: a pigment absorbs some wavelengths far more than others, and the ones it spares are the color it sends back. Scattering is what makes a paint opaque and what sends the colored light back to the eye. A pigment that scatters strongly is opaque; a pigment that scatters weakly is transparent and in a thick layer is dark, because the light that enters it is mostly absorbed before anything sends it back.

Figure 8(a) sketches the four ways light interacts with paint. The simplest case is a glaze, a thin layer of transparent paint over a white ground, or a watercolor wash on white paper. Nothing in the layer scatters much, so the light goes straight down to the ground and straight back up, and the paint acts as a colored filter. Each slice of the layer takes away the same share of whatever light reaches it, the way each pair of sunglasses in a stack dims what comes through by the same proportion. So the reflectance, the share of the light that comes back at a given wavelength, falls steeply with the first bit of pigment and more and more gently after that, as panel (b) shows. This is the Beer–Lambert law.

At the other extreme is a layer thick enough that the ground no longer matters, the masstone of Section 1.1. Inside it the light zigzags from particle to particle, losing some to absorption at every step, and the share that finds its way back out depends on a single number: how strongly the paint absorbs divided by how strongly it scatters. Color scientists write this ratio \(K/S\), with \(K\) for absorption and \(S\) for scattering, after the theory Kubelka and Munk published in 1931 [40]. A paint that absorbs twice as much and scatters twice as much reflects exactly the same, because its light is turned around twice as often, so each leg of its zigzag is half as long, and on each leg it loses the same share as before.

The curve in panel (c) turns \(K/S\) into reflectance, and it has a shape every painter knows from the palette. It leaves white nearly vertically: near white, the first trace of an absorbing pigment removes a large share of the light, and each later addition removes much less. It approaches black slowly, so once a mixture is dark, more black changes it little. The dots on the curve are tints of carbon black in titanium white. The first tenth of a percent of black dims the light inside the paint by about a tenth, while going from half black to pure black takes it only from 9 percent to 3. This steep start is the painter’s rule that a little black goes a long way in a tint. Here the reflectance inside the paint means what the paint sends back before the surface reflection is added, the subject of the next lesson.

Figure 8. (a) The four things light can do in a layer of paint: reflect off the surface, be absorbed by a particle, be scattered back out, or pass through to the ground and back. (b) A glaze over a white ground and over a gray one. The horizontal axis is the amount of pigment the light crosses, its absorption times the thickness of the layer; the light comes back weaker the more pigment it crosses, falling fast at first and then slowly. (c) A thick layer: the light that comes back depends only on the paint’s absorption divided by its scattering, \(K/S\). The horizontal scale is logarithmic: each labeled mark is ten times the one before. The dots are tints of carbon black in titanium white, from 0.1 percent black to pure black, at \(550\,\mathrm{nm}\), computed with the model.

A particle scatters light because light bends, and partly reflects, wherever it crosses from one material into another, here from the binder into the pigment and out again. How much a material slows light is called its refractive index, and the bigger the mismatch between pigment and binder, the more sharply the light is turned. Scattering is mostly a function of the refractive index of the pigment relative to the binder and the particle size of the pigment. As for size, a given amount of pigment scatters most when its particles are a little under half the wavelength of the light inside the binder, and less when they are much smaller or much larger [50, 4]. Titanium dioxide is made at that size and has a refractive index of 2.7 against the binder’s 1.5, and no other white scatters as much [8]. A phthalocyanine has particles a third or less of the best size, and it absorbs so strongly that what little scattering its particles do is swamped [31]; it colors the binder much as a dye would.

A pigment brings two properties to a paint at every wavelength, how much it absorbs and how much it scatters, and both grow in proportion to the amount of pigment. Its hue comes from the pattern of its absorption across the spectrum: a yellow pigment absorbs the blue and violet and little else, a blue pigment absorbs the red and yellow. Its opacity comes from its scattering. A masstone shows only the ratio of the two, so a masstone alone cannot say how a paint will mix. The math box at the end of this lesson gives both laws and where they come from.

TipThe math behind this lesson

Absorption, scattering and the two laws. Mie theory gives the scattering of a sphere of any size and refractive index [50, 4]. Per unit volume of pigment the scattering is greatest when the particle’s diameter is a little under half the wavelength inside the binder, which for visible light and a binder of index 1.5 is 0.2 to \(0.3\,\mu\)m. It falls off for particles much smaller, which scatter per unit volume of pigment as the cube of their size, and for particles much larger, whose scattering grows only as their surface. Titanium dioxide is made at the best size and has a refractive index of 2.7 [8]; a phthalocyanine’s particles are about \(0.1\,\mu\)m [31].

The Beer–Lambert law describes a film without scattering. If \(K\) is the absorption coefficient at some wavelength, the fraction of light absorbed in crossing a thin slice of thickness \(dz\) is \(K\,dz\), so the transmittance of a film of thickness \(d\) is \(T = e^{-Kd}\), and over a ground of reflectance \(R_g\) the film reflects

\[ R = R_g\, e^{-2Kd}, \tag{7} \]

the factor 2 because the light crosses the film down and back. \(K\) is proportional to the concentration of pigment, so \(Kd\) is the amount of pigment the light crosses, and for two pigments in one film the exponents add. The law is exact for light crossing a clear film straight down and straight back; light returning from a diffuse ground crosses at all angles and travels a little further, which in effect thickens the film slightly. It governs glazes, watercolor washes and inks on paper [3].

Kubelka and Munk’s law describes a layer that scatters [40]. It follows two fluxes of diffuse light, \(I\) going down and \(J\) coming up, through a layer with an absorption coefficient \(K\) and a scattering coefficient \(S\) per unit thickness, both depending on wavelength. In a slice \(dz\) the downward flux loses \((K + S) I\,dz\), part absorbed and part turned around, and gains \(S J\,dz\) from the upward flux turned around; the upward flux does the same in reverse. For a layer thick enough that the ground does not matter the solution is

\[ R_\infty = 1 + \frac{K}{S} - \sqrt{\left(\frac{K}{S}\right)^{2} + 2\,\frac{K}{S}}, \qquad \frac{K}{S} = \frac{(1 - R_\infty)^{2}}{2 R_\infty}, \tag{8} \]

the second form being the first solved for the ratio. Only the ratio appears. For small \(K/S\) the reflectance falls as \(1 - \sqrt{2K/S}\), the vertical start of the curve in Figure 8(c), and for large \(K/S\) it falls as \(S/2K\), the slow tail. The square root belongs to multiple scattering in general and does not depend on the two-flux simplification: in any thick layer that scatters strongly and absorbs weakly, light wanders a long way before it escapes, so the fraction lost grows like the square root of the absorption [39]. Kubelka’s 1948 solution for a layer of finite thickness over a ground joins the two laws; as the scattering goes to zero it becomes equation (7), and as the thickness grows it becomes equation (8) [3]. For the carbon black tints of Figure 8(c), at \(550\,\mathrm{nm}\), the first tenth of a percent of black takes the internal reflectance from 98 to 89 percent, the next nine tenths take it to 70, and going from 50 percent black to 100 percent takes it only from 9 to 3. A pigment contributes \(K(\lambda)\) and \(S(\lambda)\), each proportional to its amount, and a masstone gives only their ratio, through the second form of equation (8).

Sources

[3] R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 4th ed., Wiley, 2019.

[4] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983.

[8] G. Buxbaum and G. Pfaff (eds.), Industrial Inorganic Pigments, 3rd ed., Wiley-VCH, Weinheim, 2005.

[31] W. Herbst and K. Hunger, Industrial Organic Pigments: Production, Properties, Applications, 3rd ed., Wiley-VCH, Weinheim, 2004.

[39] G. A. Klein, Industrial Color Physics, Springer Series in Optical Sciences 154, Springer, New York, 2010.

[40] P. Kubelka and F. Munk, Ein Beitrag zur Optik der Farbanstriche, Zeitschrift für technische Physik 12 (1931), 593–601.

[50] G. Mie, Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen, Annalen der Physik (4) 25 (1908), 377–445.