2.3 Beyond two fluxes
The Kubelka–Munk law keeps its bookkeeping simple by following only two fluxes of light, a flux here being just the light flowing one way: one runs down through the paint and the other comes back up. It also treats each flux as perfectly diffuse, its light spread evenly over every direction, and Saunderson’s correction assumes the same of the light that meets the surface from inside. In real paint some directions carry more light than others, and the surface lets light out when it meets it nearly head on but turns it back in when it meets it at a slant. The full description, called radiative transfer, follows the light direction by direction instead of lumping it into up and down [9].
In a tint the simple picture is close to the truth, because the white scatters the light so often that it really is spread close to evenly, and the model built on it reproduces measured tints closely (Section 2.6). In a glaze it is not, because nothing in the film scatters to even the light out. Light that crosses the film steeply passes through less pigment than light that crosses at a slant, and in the colors the pigment absorbs, the short, steep paths let through more light than the simple law allows for. Followed direction by direction, a glaze of the same lightness comes out a few units of chroma less vivid than the simple law makes it, and still far more vivid than a tint of the same lightness. For ease of computation the rest of the paper keeps the simple law, and the error is small: inside the model’s accuracy for tints, and a few units of chroma for glazes, so the glazes of Section 3.6 slightly overstate their chroma. The math box at the end of this lesson works through the calculation for four paints.
Radiative transfer and the glaze. The full description is the equation of radiative transfer [9]. In a flat layer of paint, the light at depth \(z\) traveling at an angle \(\theta\) to the downward vertical has an intensity \(I(z, \mu)\), where \(\mu = \cos\theta\) is positive for light going down and negative for light going up. Along its path a ray loses light to absorption and to scattering out of its direction, and gains light scattered into its direction from every other:
\[ \mu\, \frac{\partial I(z,\mu)}{\partial z} = -(\sigma_a + \sigma_s)\, I(z, \mu) + \frac{\sigma_s}{2} \int_{-1}^{1} p(\mu, \mu')\, I(z, \mu')\, d\mu' . \tag{10} \]
Here \(\sigma_a\) and \(\sigma_s\) are the absorption and scattering coefficients of the pigment particles per unit length of path, which follow from Mie theory for particles of a given size and refractive index [4], and \(p(\mu, \mu')\) is the phase function, the share of the light scattered from direction \(\mu'\) that goes into direction \(\mu\), averaged around the vertical and normalized so that \(\tfrac12 \int_{-1}^{1} p\, d\mu = 1\). The factor \(\mu\) on the left is the slant: a ray at angle \(\theta\) crosses a slice of thickness \(dz\) along a path of length \(dz/\mu\), so a slanted ray meets more pigment per unit of depth than a vertical one. The surface enters through the boundary conditions. At the top, the light going down in direction \(\mu\) is the light that came in from outside plus the fraction \(R_F(\mu)\) of the light coming up at the mirrored angle, where \(R_F\) is the boundary reflectance seen from inside. For a binder of index \(n\), every direction with \(\mu\) below \(\mu_c = \sqrt{1 - 1/n^2}\) lies beyond the critical angle and has \(R_F = 1\); for \(n = 1.5\), \(\mu_c = 0.745\), and only light within \(42^{\circ}\) of the vertical can leave. At the bottom, the ground reflects whatever reaches it.
The two-flux model comes out of equation (10) by adding up the light over each half of the directions, and the quantity to add up is the flux, the power crossing a horizontal plane per unit of its area. A ray at angle \(\theta\) meets a patch \(dA\) of the plane obliquely, and the patch presents to it only the area \(dA\cos\theta\), so light of intensity \(I\) arriving within a small solid angle \(d\Omega\) carries the power \(I\cos\theta\, d\Omega\, dA\) across the patch. A small solid angle is \(d\Omega = \sin\theta\, d\theta\, d\phi\), with \(\phi\) the bearing, the horizontal direction of travel, and since \(d\mu = -\sin\theta\, d\theta\) it is also \(d\mu\, d\phi\) in size. Adding over every downward direction, with the bearing contributing a factor \(2\pi\) because \(I\) does not depend on it, gives the downward flux, and the same sum over the upward directions gives the upward one:
\[ F_+(z) = 2\pi \int_0^1 I(z, \mu)\, \mu\, d\mu, \qquad F_-(z) = 2\pi \int_0^1 I(z, -\mu)\, \mu\, d\mu . \tag{11} \]
Multiply equation (10) by \(2\pi\) and integrate it over the downward directions, \(\mu\) from 0 to 1. The left side becomes \(dF_+/dz\), because the \(\mu\) the slant put into equation (10) is the same cosine that turns intensity into flux. The first term on the right becomes \(-(\sigma_a + \sigma_s) E_+\), with \(E_+ = 2\pi \int_0^1 I\, d\mu\). This has no factor \(\mu\) and is not the flux. It counts each ray by the light it carries whatever its slant, which is the right count for absorption: a ray crossing the slice \(dz\) at a slant is absorbed along the path \(dz/\mu\), and its flux \(I\mu\, d\Omega\) times \(\sigma_a\, dz/\mu\) is \(\sigma_a I\, d\Omega\, dz\), with the \(\mu\) gone. Write \(E_+ = u_+ F_+\), where
\[ u_+ = \frac{\int_0^1 I\, d\mu}{\int_0^1 I\, \mu\, d\mu} \tag{12} \]
is the average of \(1/\mu\) over the downward light, each direction weighted by the flux it carries: the length of path the downward light travels, on average, per unit of depth. In the last term of equation (10), change the order of the two integrals. For light arriving from a direction \(\mu'\), \(\tfrac12 \int_0^1 p(\mu, \mu')\, d\mu\) is the share of what the particles scatter out of it that goes into the downward half. Call \(\beta(\mu')\) the share that goes into the half the light did not come from, so that downward light keeps \(1 - \beta\) of its scattered light going down and upward light sends \(\beta\) of its scattered light down. Averaging \(\beta\) over each half, weighted by intensity, gives a back-scattered share \(b_+\) for the downward light and \(b_-\) for the upward, and the integrated equation is
\[ \frac{dF_+}{dz} = -(\sigma_a + \sigma_s)\, E_+ + \sigma_s (1 - b_+)\, E_+ + \sigma_s b_-\, E_- = -\sigma_a E_+ - \sigma_s b_+ E_+ + \sigma_s b_- E_- . \]
The downward flux loses what is absorbed and what is scattered back up. Light a particle scatters forward leaves one downward direction for another and is still in the downward flux, so it cancels out. The downward flux gains what the upward light scatters back down. With \(E = uF\), and the same steps over the upward half, where the left side comes out as \(-dF_-/dz\) because the upward flux grows toward the top,
\[ \begin{aligned} \frac{dF_+}{dz} &= -u_+ (\sigma_a + b_+ \sigma_s)\, F_+ + u_- b_- \sigma_s\, F_-, \\ -\frac{dF_-}{dz} &= -u_- (\sigma_a + b_- \sigma_s)\, F_- + u_+ b_+ \sigma_s\, F_+ . \end{aligned} \tag{14} \]
Nothing has been approximated yet, but the equations are not closed. The path factors \(u_\pm\) and the back-scattered shares \(b_\pm\) depend on how the light is spread over its directions, which is what equation (10) was needed to find, and they change with depth.
Kubelka and Munk closed the equations by assuming that the light is perfectly diffuse, with \(I\) the same in every direction of each half. Then equation (12) gives \(u_+ = u_- = 1/\tfrac12 = 2\), and \(b_+\) and \(b_-\) become one number \(b\), the back-scattered share for diffuse light, which belongs to the particles alone. Equations (14) become
\[ \begin{aligned} \frac{dF_+}{dz} &= -(K + S)\, F_+ + S F_-, \\ -\frac{dF_-}{dz} &= -(K + S)\, F_- + S F_+, \end{aligned} \qquad K = 2\sigma_a, \quad S = 2b\,\sigma_s , \tag{15} \]
which are the Kubelka–Munk equations, with \(F_+\) and \(F_-\) in place of the \(I\) and \(J\) used above. A particle that scatters equally in every direction sends half of what it scatters into the other half, \(b = \tfrac12\), and \(S = \sigma_s\); a particle that scatters mostly forward has a smaller \(b\). The 2 in \(K\) is the path factor of diffuse light: it crosses a slice along a path twice the slice’s thickness, on average, and so it is absorbed twice as fast per unit of depth as a vertical beam. Where the light is not diffuse, the true \(u\) and \(b\) differ from these values. Near the top, under the instrument’s nearly vertical beam, \(u\) is close to 1; in a glaze that does not scatter, the upward light loses its slanted rays as it rises and its \(u\) falls below 2. Other assumptions about the spread of directions give other factors, and Mudgett and Richards, solving equation (10) accurately for pigmented layers, derived empirical relations for \(K\) and \(S\) that make the two-flux model agree with the accurate solution as nearly as it can [52].
Saunderson’s internal coefficient is the same assumption applied at the surface. Perfectly diffuse light meets the surface from below with the share \(2\mu\, d\mu\) in the directions between \(\mu\) and \(\mu + d\mu\), and the share the surface reflects is
\[ k_2 = \int_0^1 R_F(\mu)\, 2\mu\, d\mu , \tag{16} \]
which for \(n = 1.5\) is 0.596. Of that, \(1 - 1/n^2 = 5/9\) comes from the directions beyond the critical angle, all of which are reflected, and the remaining 0.04 from partial reflection of the rest. This is the 0.60 of Section 2.2. The external coefficient \(k_1\) is the boundary reflectance for the instrument’s beam, about 0.04 near normal incidence. Klein’s objection to the formula is that its two coefficients rest on two different assumptions, one directional and one diffuse, and that neither holds exactly for light that is partly both [39].
Equation (10) can be solved as accurately as one likes by replacing the integral over directions with a sum over a chosen set of directions, usually the points of a Gauss quadrature. It becomes a set of linear differential equations in the intensities along those directions, whose solution in a uniform layer is a sum of exponentials in depth, and the boundary reflectance is applied direction by direction at each boundary. This is the method of discrete ordinates set out by Chandrasekhar [9]; in the literature on paint and paper it is called the many-flux method. Mudgett and Richards applied it to pigmented layers and compared it with the two-flux model and with a four-flux model, which adds a collimated beam going down and one coming back to the two diffuse fluxes [52]. The two-flux model gave excellent results when the absorption was small compared with the scattering and the layer’s optical thickness was more than 5. The four-flux model did well at any thickness under collimated light, even with strong absorption. Vargas and Niklasson showed formally that the Kubelka–Munk model is a good approximation for optically thick films of particles that absorb weakly or not at all [74]. Klein’s preference for three or more fluxes in translucent layers [39] agrees with both.
The tints in these notes lie mostly where the two-flux model is good. A tint is a thick layer whose scattering comes mainly from titanium white, which at a normal loading scatters several hundred times per millimeter (Section 2.4); the instrument’s beam loses its direction within the first few scattering lengths, and below that the light is close to diffuse. In the bands a pigment passes, the absorption is small compared with the scattering, the case Mudgett and Richards found the two-flux model handles well. In the bands it absorbs strongly the ratio is not small, and that is where the model is least exact; the calibration, done on the same kind of layer, takes up part of the error there. The direct test is measurement. The modeled tints come within \(0.85\,\Delta E_{00}\) on average of the 26 ladders Okumura measured (Section 2.6), and whatever the two-flux approximation costs in a tint is inside that figure.
The glaze is the opposite case. A film of transparent pigment over white scatters almost nothing, so the light in it is never made diffuse by scattering. Light from the ground does start out diffuse, since the white scatters it in every direction, but it does not stay so. On the way up, a ray at angle \(\theta\) crosses a film of thickness \(d\) along a path \(d/\mu\), so the slanted rays are absorbed more than the steep ones, and only the steep ones, those within \(42^{\circ}\) of the vertical, can leave. The rest are turned back by total internal reflection and cross the film again at the same slant. The light that leaves on its first try has crossed the film along a path 1.14 times its thickness on average, against 2 for diffuse light; the light turned back crosses it at least twice more, each time at a slant. A film that does not scatter is simple enough to work out direction by direction. Let the film’s absorption coefficient along a ray be \(\alpha\), so that its optical depth is \(\tau = \alpha d\); let the ground be the model’s white, a diffuse reflector of reflectance \(R_g\) seen from inside the binder; and let the instrument’s beam come in at \(8^{\circ}\), which bends to \(\mu_0 = 0.996\) inside. A sphere instrument lights the sample diffusely and reads it at \(8^{\circ}\), and by reciprocity that gives the same reflectance as lighting it at \(8^{\circ}\) and collecting every direction. The beam reaches the ground with the fraction \((1 - k_1)\, e^{-\tau/\mu_0}\) of its light. Of the diffuse light the ground sends up, the fraction that leaves through the top on its first try, and the fraction that is reflected at the top and gets back down to the ground, are
\[ U(\tau) = \int_0^1 \bigl(1 - R_F(\mu)\bigr)\, e^{-\tau/\mu}\, 2\mu\, d\mu, \qquad D(\tau) = \int_0^1 R_F(\mu)\, e^{-2\tau/\mu}\, 2\mu\, d\mu , \tag{17} \]
and adding up the light over its round trips between the ground and the surface gives
\[ R = k_1 + \frac{(1 - k_1)\, e^{-\tau/\mu_0}\, R_g\, U(\tau)}{1 - R_g\, D(\tau)} . \tag{18} \]
For a film of no thickness, \(U = 1 - k_2\) and \(D = k_2\) by equation (16), and equation (18) is Saunderson’s formula over the bare ground. The two-flux glaze of Section 3.6 replaces every crossing of the film by the same exponential, \(e^{-uK}\) each way with \(K\) the coefficient for diffuse light; equation (18) keeps the spread of paths.
The spread is what changes the color. A change in the mean path alone would only change how thick a film is needed for a given lightness, and a glaze is compared with a tint at equal lightness. A spread of paths makes the reflectance an average of exponentials, \(\langle e^{-\alpha L} \rangle\) over the paths \(L\), and in a band the pigment absorbs strongly that average is ruled by the shortest paths, which let through far more light than a single path of the mean length would. At equal lightness the glaze worked out direction by direction reflects more in its absorbed band and is less pure. Table 2 gives the numbers for the four paints of Figure 18, computed both ways from the same absorption spectra, with the thickness set to reach each lightness and the same surface reflection at the top, \(k_1 = 0.035\), so that only the treatment of the light inside the film differs.
Table 2. Glazes over white at \({L^{*}} = 50\) and \(65\) computed two ways, by the two-flux model of Section 3.6 and direction by direction with equation (18). The tint is the same paint mixed with white to the same lightness. The last column is the share of the glaze’s lead over the tint that the direction-by-direction calculation removes.
| aze \({C^{*}_{ab}}\) | |||||
|---|---|---|---|---|---|
| 4-5 paint | \({L^{*}}\) | tint \({C^{*}_{ab}}\) | two-flux | by direction | lead removed |
| Quinacridone magenta | 50 | 55.2 | 86.7 | 81.3 | 17% |
| 65 | 46.3 | 67.4 | 62.4 | 24% | |
| Phthalo green | 50 | 45.8 | 76.0 | 72.0 | 13% |
| 65 | 46.8 | 82.1 | 75.5 | 19% | |
| Dioxazine purple | 50 | 44.4 | 71.8 | 65.6 | 23% |
| 65 | 34.6 | 49.7 | 46.4 | 22% | |
| Burnt sienna | 50 | 24.9 | 49.5 | 43.3 | 25% |
| 65 | 21.6 | 41.8 | 35.4 | 32% |
The glaze loses 3 to 7 units of chroma, and its lead over the tint shrinks by 13 to 32 percent, about a fifth on average. At a lightness of 50 the quinacridone magenta glaze has a chroma of 81 against the tint’s 55, where the two-flux model gave 87; phthalo green has 72 against 46, where it gave 76. The lead is 12 to 29 units direction by direction, against 15 to 35 by the two-flux model. The two-flux glaze moves by about as much with the surface coefficient alone: with \(k_2\) moved from 0.4 to 0.7, the magenta glaze at \({L^{*}} = 50\) goes from 91 to 84 and the phthalo green from 79 to 74. Both calculations leave out the little scattering a real glaze does, at its particles and where it meets the ground, which would take a little more from the glaze, and neither has been checked against a measured glaze.
The rest of the paper keeps the two-flux model with Saunderson’s correction for every layer, glazes included. It needs two numbers per wavelength for each paint and mixes paints by adding them, and it is the model the coefficients were extracted with, so that the extraction and the prediction share their errors. A many-flux model would need the phase function of each pigment, which reflectance measurements do not give, and a calibration of its own. For the tints and the mixtures with white, which make up most of Parts 3 and 4, the error this brings in lies inside the model’s measured accuracy of about one unit of color difference. For the glazes it is a few units of chroma, about a fifth of the gap between glaze and tint, and always in the same direction, so the glaze numbers of Section 3.6 are best read as upper bounds; the glaze stays far purer than the tint at both lightnesses compared. The masstones of the most transparent paints are a separate matter. Their trouble is a film that does not hide its card (Section 2.6), and no model of the light inside the film fixes that without a measurement of the same film over black.
Sources
[4] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983.
[9] S. Chandrasekhar, Radiative Transfer, Clarendon Press, Oxford, 1950; reprinted by Dover, New York, 1960.
[39] G. A. Klein, Industrial Color Physics, Springer Series in Optical Sciences 154, Springer, New York, 2010.
[52] P. S. Mudgett and L. W. Richards, Multiple scattering calculations for technology, Applied Optics 10 (1971), 1485–1502.
[74] W. E. Vargas and G. A. Niklasson, Applicability conditions of the Kubelka–Munk theory, Applied Optics 36 (1997), 5580–5586.