Modelling Interactions of Light and Matter

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Modelling Interactions of Light and Matter

• Complex subtractive mixing
• Recipe Prediction
• Dr Huw Owens

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Complex Subtractive Mixing

• Most common and complex type of colour mixing is
  that when colorants scatter and absorb light.
• This is known as complex-subtractive mixing.
• For practical purposes, simplified equations,
  which are approximately correct, are used to
  describe complex-subtractive mixing.
• The most widely used of these equations were
  derived by Kubelka and Munk (1931, 1948, 1954)
• Colour Recipe Prediction Systems

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Kubelka-Munk Law

• Kubelka-Munk considered a translucent colorant
  layer on top of an opaque background.
• Within the colorant layer, both absorption and
  scattering occur.
• Kubelka and Munk made a simplifying assumption
  that the light either travels up or down
  perpendicular to the plane of the sample non
  preferentially.
• This has led to Kubelka-Munk theory being
  referred to as a two-flux theory.

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Kubelka-Munk Theory
Transparent film on Opaque Support
Translucent Material
Opaque Material
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How does Kubelka-Munk Theory Work?

• A pair of differential equations are solved.
• One for each direction of flux.
• This results in an equation that predicts
  internal reflectance from knowledge of the
  background reflectance, absorption and scattering
  properties of the colorant layer, and the
  thickness of the colorant layer.
• Samples are prepared on black and white
  backgrounds or on transparent materials (e.g.
  polyester film or glass) or at different
  thicknesses until the colorant layer becomes
  opaque.
• From these data various mathematical forms of the
  Kubelka-Munk theory can be derived.

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Kubelka-Munk Key Assumptions

• Light within the colorant layer is completely
  diffuse.
• There cannot be a change in refractive index at
  the samples boundaries.
• As in the Bouguer-Beer law, the measured
  transmittance is transformed to internal
  transmittance using the Fresnel equations.
• A similar transform is performed in Kubelka-Munk
  theory known as the Saunderson correction.
• Because the light is assumed to be diffuse K-M
  theory does NOT apply to metallic or pearlescent
  colorants or to colorant layers that change the
  degree of polarisation of the incident light
  significantly.
• K-M does NOT apply to fluorescent colorants
  though it can be used as a starting point for
  other approaches.

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Kubelka-Munk Measurements

• K-M theory applies to a single wavelength at a
  time.
• Practically, this means measuring samples with a
  spectrophotometer.
• Theoretically the geometry should be diffuse
  illumination and diffuse collection.
• This means that integrating sphere
  spectrophotometers have the closest geometry to
  this theoretical requirement

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Kubelka-Munk Theory What sort of Samples?

• K-M theory is used to develop mixing laws for
  three types of samples
• Translucent materials. (e.g. plastics, printing
  inks with appreciable scattering and paint
  samples not at complete hiding (i.e. not opaque))
• Transparent film on opaque diffusely scattering
  support. (e.g. photographic paper,
  continuous-tone prints using thermal transfer
  technologies)
• Opaque absorbing and scattering materials. (e.g.
  textiles, paint films and plastics at complete
  hiding and dyed paper)

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Kubelka-Munk Theory

• As an example, we will take opaque absorbing and
  scattering materials.
• Reflectance is transformed to the ratio of
  absorption (K) to scattering (S), (K/S)? known as
  K over S.
• This is a linear system so the scalability and
  additivity requirements apply to the individual
  absorption and scattering properties of
  individual colorants.
• This leads to the expression two constant
  Kubelka-Munk theory.
• For a colour ramp, the normalised absorption
  spectra would be nearly identical as would be the
  normalised scattering spectra.

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Kubelka-Munk Theory

• For materials where the colorants have negligible
  scattering properties in comparison to those of
  the supporting medium (textiles or paper), only
  the K/S ratio is used to characterise a colorant,
  leading to the expression single-constant
  Kubelka-Munk theory.
• Two-constant Kubelka-Munk Theory is always used
  for the coloration of paints and plastics whereas
  single-constant Kubelka-Munk theory is most often
  used for textiles or dyed paper.
• How do you know which one to use?
• Evaluate the scalability If the normalised K/S
  spectra are nearly identical then two constant is
  probably not required
• The opposite situation is true. As a consequence
  two-constant theory has been applied to textiles
  where single constant theory was inadequate

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K-M Development and Validation (1)

• To develop and validate a particular form of K-M
  theory for a given coloration system requires the
  same procedure described for simple subtractive
  mixing.
• Colour ramps are used to validate the scalability
  requirement and colour mixtures are used to
  validate the additivity requirement.
• For two-constant K-M theory separate coefficients
  for absorption and scattering are required which
  makes the techniques more complicated.
• Least-squares techniques are most commonly used,
  in which the two coefficients are estimated
  simultaneously using all of the samples forming a
  colour ramp.

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K-M Development and Validation (2)

• The colour ramp provides a knowledge of each
  samples concentration, and a knowledge of the
  scattering and absorption of the substrate plus
  white colorant to make the sample opaque (usually
  titanium oxide).
• In cases where it is difficult to separate the
  absorption and scattering properties of the
  colorant from the white colorant (such as yellow
  colorants) mixtures with black are produced.
• If carefully prepared samples are produced with
  known recipes, these can be used to calculate the
  absorption and scattering coefficients for all of
  the colorants simultaneously.

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K-M Development and Validation (3)

• BUT These least-squares techniques require that
  there be a linear relationship between
  concentration and the scalar.
• The Application of K-M theory rarely includes the
  final step of relating theoretical and effective
  concentrations for each colorant.
• This may result in recipe prediction errors when
  a colorant is used over a wide range of
  concentrations.
• This may be remedied by using least squares to
  estimate a scalar for each sample forming the
  colour ramp using the estimated absorption and
  scattering coefficients.
• Any curvature between theoretical and effective
  concentrations is fit appropriately

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K-M Saunderson Correction Factor

• As K-M theory assumes there is no refractive
  index change, measured reflectance is converted
  to internal reflectance using Saunderson
  correction

• Where K1 is the Fresnel equation coefficient for
  collimated light and K2 is the reflection
  coefficient for diffuse light striking the
  surface from inside.

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K-M Definitions

• Theoretical concentration Concentration
  measured by a user such as the concentration of a
  dye in a dye-bath. This is equivalent to the
  user controls of a generic colour model.
• Effective concentration Concentration
  determined from colorant measurements of the
  coloured material. This is equivalent to the
  scalars of a generic model.

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Kubelka-Munk Saunderson Correction Factor

• Once internal reflectance has been calculated
  using the K-M mixing law, measured reflectance is
  finally calculated

• For specular excluded or bidirectional
  geometries, the separate K1 term is removed from
  the equation. K1 is usually around 0.04 because
  most coatings and plastics have refracted indices
  of 1.5. K2 usually varies between 0.4 and 0.6
  and can be optimised to improve scalability or
  linearity between theoretical and effective
  concentrations.

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K-M Predicting Reflectance

• For opaque materials, K-M found that internal
  reflectance, R?,i, depended on absorption, K?,
  and scattering, S?. Reversing this equation
  gives the well-known relationship between (K/S)?
  and R?,i.

• Reflectance should be between 0-1. K and S only
  appear as a ratio. The (K/S) ratio of a mixture
  is an additive combination of each colorants
  unit absorptivity, k? and unit scattering s?,
  scaled by effective concentration, c, plus the
  absorption and scattering of the substrate
  (notated by subscript t)

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K-M Predicting Reflectance

• For each component in the mixture, both the
  absorption and scattering properties need to be
  known.
• For materials such as textiles where the
  colorants do not scatter in comparison to the
  substrate, the mixing equation is simplified so
  that we only need too know the ratio of
  absorbance to scattering

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Kubelka-Munk Numerical Example (1)

• Sample W contains white pigment only
• Sample Y contains 18.5 yellow in white
• Sample M contains 13.6 magenta in white
• Sample B, which is brown contains unknown
  percentages of the yellow, magenta and white
  pigments
• Find the colorant recipe of the brown sample.

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Kubelka-Munk Numerical Example (2)

• Usually, we would use two-constant Kubelka-Munk
  equations for paint systems.
• We will make two assumptions
• Chromatic pigments have relatively small amounts
  of scattering in comparison with the white
  pigment
• Saunderson correction is omitted
• Thus we will use single-constant K-M
• First we need to select two suitable wavelengths
• 420nm and 560nm?

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Kubelka-Munk Numerical Example (3)
Calculate
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Kubelka-Munk Numerical Example (4)

• Determine the unit K/S the contribution to
  K/S from unit concentration of each of the
  pigments, denoted by lowercase (k/s)?.
• This is done by using the mixtures Y and M.
• For Yellow

• A similar equation can be written for the M
  curve.
• To solve (k/s)?,y at 420nm and 560nm

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Kubelka-Munk Numerical Example (5)

• The assumption made is that the total amount of
  paint is one arbitrary unit.
• The table of unit value K/S values is as follows

• The brown sample has unknown amounts of the three
  pigments
• If we set the white concentration to cw
  (1-cy-cm) then we only have two unknowns to find

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Kubelka-Munk Numerical Example (6)

• If we rearrange the equation we obtain

• This leads to the following mixing equations

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Kubelka-Munk Numerical Example (7)

• Solving the equations we obtain cy 0.2197 and
  cm 0.1168.
• As cw 1 cy -cm the final percentages can be
  calculated by dividing each value by the sum of
  the concentrations.
• Thus the recipe for the brown sample is 21.9
  yellow, 11.68 magenta and 66.35 white.
• Mixtures of three coloured pigments in white can
  be treated similarly, but the calculations are
  more complicated.

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What is a Recipe?

• In the coloration industry, the term recipe is
  used to refer to a set of colorants (including
  their concentrations) that when applied correctly
  produce a certain colour.

colorants
pigments
dyes
Often scatter as well as absorbinsoluble in the
mediumused in paints, inks, plastics etc
Little or no scattersoluble in the mediumused
in textiles, paper,wood etc.
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Recipe Prediction

• Recipe prediction, or match prediction, is the
  process of generating a recipe to match a desired
  or target shade.
• Recipe prediction can be performed by a trained
  colourist but the process can be time consuming
  and inaccurate.
• When computer software is used to predict the
  recipe then the term computer recipe prediction
  is used.
• The first commercial computer recipe prediction
  systems were produced in the 1960s. Products are
  now widespread and sophisticated.

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Kubelka-Munk Theory

• Computer recipe prediction systems require a
  mathematical model that can relate the
  concentrations of
• The Kubelka-Munk theory characterises each
  colorant by absorption and scattering
  coefficients and is the basis for most commercial
  computer recipe prediction systems.

Absorption coefficient KScattering coefficient
- S
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Light Absorption

• Light absorption is greatest for small particle
  sizes
• For large agglomerates the pigment at the centre
  never sees any light
• Light fastness improves with increasing particle
  size

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Light Scattering

• Increases with increasing refractive index ratio
• Is optimum for particles with a particle diameter
  of approximately 220nm
• For very small particles blue light is scattered
  predominately

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Kubelka-Munk Theory

• Characterises each colorant at each wavelength by
  absorption (K) and scattering (S) coefficients.
• Provides a models for how the colorants behave
  optically when mixed together.
• Requires a database to compute K and S.
• Predicts reflectance from recipe information.

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Kubelka-Munk Theory
At each ?
K/S
One-constanttheory
Concentration
(K/S)mix is computed at each ? And then converted
to R(?)
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Kubelka-Munk Theory
S
Concentration
Smix S1S2
Kmix K1K2
R(?) is computed at each wavelength from Kmix,
Smix and Rg (the reflectance of the substrate)
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Select a recipee.g. C1, C2, C3
Kubelka-Munk
Predict reflectance
Compute colour coordinates
Compare to the target
Within tolerance?
Modify recipe
print
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Combinations

• The number of possible recipes rises rapidly as
  the number of possible colorants is increased.

Example n20, r3
Where n number of dyes in the permitted list
and r number of dyes allowed per recipe
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Recipe Correction

• Commercial CMP systems include a recipe
  correction system.
• Recipe correction is the process of correcting an
  existing recipe. For a batch process this may
  mean adding colorant (it may not be possible to
  take colorant out).

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Advantages of Computer Match Prediction

• The number of samples that need to be made to
  arrive at a satisfactory match can be reduced.
• The full range of combinations can be explored
• The final recipe may be less expensive
• The final recipe may be less metameric
• The final recipe may be more light/wash fast