pigmented colorants: dependence on media and time - Cornell ...

The Kubelka-Munk equations were generalized to allow arbitrary mixtures of
pigments [Dun40]. If there are n multiple materials in the same layer with differ-
ent scattering and absorption coefficients (ie: multiple pigments within the same
layer of paint), they may be combined via linear weighting using their respective
concentrations ci:
Smixture(λ) =
Kmixture(λ) =
n
ciSi(λ)
i=1
205
n
ciKi(λ) (A.16)
Kubelka extended the 1931 work in two subsequent articles. He first solved the
differential equations of Equation A.4 for a finite thickness of paint (originally, the
infinite case was presented) [Kub48]. If the paint film has thickness x, then
Rx =
i=1
1
R∞ (R0
− R∞) − R∞ R0 − 1
R∞
(R0 − R∞) −
R0 − 1
R∞
1
e
Sx( R∞ −R∞)
1
e
Sx( R∞ −R∞)
(A.17)
Another form of this equation can be found using hyperbolic functions, allowing
for simpler computation:
Rx = 1 − R0(a − b coth bSx)
a − R0 + b coth bSx
(A.18)
where a =1+ K
S and b = √ a 2 − 1 as earlier. When the paint becomes thick enough
to hide the substrate, R0 → 0. Thus we have
Rx =
1
a + b coth bSx
(A.19)