1.85 combined absorption and scattering (kubelka–munk analysis)
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.