1. Combined absorption and scattering
(Kubelka–Munk analysis)
Most opaque coloured objects illuminated
by
white light
produce diffusely reflected
coloured radiation
by the combined processes of light
absorption and light scattering.
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2. Kubelka–Munk analysis
 Consider the simple
case of a light beam
passing vertically
through a very thin
 pigmented layer of
thickness dx in a paint
film (Figure 1.28). We
consider separately
 the downward (incident)
and upward (reflected)
components of the
incident light
 beam, assuming that the
absorption coefficient is
represented by K and
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scattering
2
 coefficient by S.

3.  At the same time, the
 The downward flux (intensity I) upward flux (intensity J) is:
is:
 – decreased by absorption =
 – decreased by absorption = – – kJ dx
KI dx
 – decreased by scattering =
 – decreased by scattering = – – SJ dx
SI dx
 – increased by backscatter =
 – increased by backscatter = + + SI dx (from the radiation
SJ dx (from the radiation proceeding downward),
proceeding upwards, of which J summarised by Eqn 1.21:
is the intensity),
 which is summarised by Eqn
1.20:
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4. Kubelka–Munk analysis
 Solution of these differential equations depends
on the
 boundary conditions applied,
 but in the absence of scattering (S = 0) leads to
the Lambert–Bouguer law for the
 downward flux.
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5. Kubelka–Munk analysis
 For an iso-tropically absorbing and
scattering layer of infinite thickness
 (or at least so thick that the background
layer reflection is negligible), it leads to
the
 widely used Kubelka–Munk expression (Eqn 1.22):
where R∞ = Jo/Io is the reflection factor
at the surface
for a sample of infinite thickness.
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6. Kubelka–Munk analysis
 The K, S and K/S values provide
 the colour technologist
 with functions which, in principle, are additive
 And linearly related to concentration of dyes and pigments
 in solid substrates.
 For example,
 for a dyed substrate where the scattering is
 attributed entirely to the textile substrate
 and therefore does not vary with dye concentration [D],
 we have a particularly simple
 form of concentration dependence (Eqn 1.23):
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7. Kubelka–Munk analysis
 where Kf and Kd are the light absorption
coefficients
 for the fibre and dye respectively,
 at the wavelength of measurement,
 and Sf is the scattering coefficient of the fibre at
the same wavelength.
 dye concentration [D],
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8. Kubelka–Munk analysis
Limitation
 Although this
relationship has certain
limitations (for example,
 when dealing with
 highly exhausting acid
dyes on wool
 and when taking
measurements near the
wavelength
 of maximum absorption),
 good linearity is
observed
 (Figure 1.29; the raw
data
 from which this plot is
derived is shown in
8 Figure 1.31).
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9. Kubelka–Munk analysis MAJOR
Limitation
 Some of the major limitations to the Kubelka–Munk type
of analysis are that
 it deals with diffuse monochromatic radiation
 and handles only two fluxes
 (diffuse light travelling upwards or downwards)
 through a homogeneous absorbing and scattering medium.
 The light loss through edges is thus neglected,
 as are the surface
 and the totally internally reflected components
 of the incident light beam.
 Other assumptions
 such as the uniform distribution of the dyes or pigments,
 and the lack of interactions between them, are also not realised.
 Such factors lead to a nonlinearity of the
 Kubelka–Munk function when measured over wide concentration
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ranges.