Understanding JAYPAK Plastic Paints Basic Recipe Prediction Model

Colour and Textile Products

Jay Instruments & Systems Pvt. Ltd.


Understanding the Plastic/paints basic recipe Understanding the Plastic/paints basic recipe prediction modelprediction model

Complex subtractive mixingRecipe Prediction

Adapted from Lecture notes of University of Manchester

Translucent Material

Transparent film on Opaque Support

Opaque Material


Complex Subtractive MixingComplex 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


Kubelka-Munk LawKubelka-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.


Kubelka-Munk TheoryKubelka-Munk Theory

Translucent Material

Transparent film on Opaque Support

Opaque Material


How does Kubelka-Munk Theory Work?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.


Kubelka-Munk – Key AssumptionsKubelka-Munk – Key Assumptions

• Light within the colorant layer is completely diffuse.• There cannot be a change in refractive index at the

sample’s 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.


Kubelka-Munk MeasurementsKubelka-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


Kubelka-Munk Theory – What sort of Samples?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)



• 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)λ 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.



• 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


K-M – Development and Validation (1)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.



• The colour ramp provides a knowledge of each sample’s 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.



• 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


K-M Saunderson Correction FactorK-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:

m

mi RKKK

KRR

,221

1,, 1

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


K-M DefinitionsK-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.


Kubelka-Munk – Saunderson Correction FactorKubelka-Munk – Saunderson Correction Factor

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

i

im RK

RKKKR

,2

,211, 1

)1)(1(

• 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.

• JAYPAK Software has facility to fine tune the number manually.


K-M – Predicting ReflectanceK-M – Predicting Reflectance

• 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 colorant’s unit absorptivity, kλ and unit scattering sλ, scaled by effective concentration, c, plus the absorption and scattering of the substrate (notated by subscript t):

22

, 21

S

K

S

K

S

KR i

i

i

R

R

S

K

,

2,

2

1

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


K-M Predicting ReflectanceK-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:

...

...

3,32,21,1,

3,32,21,1,

,

,

,

scscscs

kckckck

S

K

S

K

t

t

mix

mix

mix

...3,

32,

21,

1,,

s

kc

s

kc

s

kc

s

k

S

K

tmix


Kubelka-Munk Numerical Example (1)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.



• 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?



Wavelength

Sample 420 nm 560 nm

Y 0.216 0.872

M 0.384 0.146

B 0.167 0.163

W 0.768 0.882 Wavelength


Y 1.423 0.009

M 0.494 2.498

B 2.078 2.149

W 0.035 0.007

Calculate

R

R

S

K

2

1 2



• Determine the “unit K/S” – the contribution to K/S from unit concentration of each of the pigments, denoted by lowercase (k/s)λ.

• This is done by using the mixtures Y and M.• For Yellow

ww

yy

Y s

kc

s

kc

S

K

,,,

• A similar equation can be written for the M curve.

• To solve (k/s)λ,y at 420nm and 560nm

Wavelength Unit K/S Equation

420

560

035.0815.0185.0423.1,420

ys

k

007.0815.0185.0009.0,560

ys

k



• 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:

Wavelength


White 0.035 0.007

Y 7.538 0.018

M 3.410 18.323

• 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:



• If we rearrange the equation we obtain

m

my

yw

myB s

kc

s

kc

s

kcc

S

K

,,,,

1

wmm

wyy

wB s

k

s

kc

s

k

s

kc

s

k

S

K

,,,,,,

• This leads to the following mixing equations:

Wavelength Mixing Equation

420 2.078 – 0.035 = cy(7.538 - 0.035) + cm(3.410-0.035)

560 2.149 – 0.007 = cy(0.018 - 0.007) + cm(18.323-0.007)



• 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.


What is a Recipe?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

dyes pigments

Little or no scattersoluble in the mediumused in textiles, paper,wood etc.

Often scatter as well as absorbinsoluble in the mediumused in paints, inks, plastics etc


Recipe PredictionRecipe 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.

• JAYPAK 4808 CCMS software is a 8th generation software.



• 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


Light AbsorptionLight 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


Light ScatteringLight 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



• 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.



R

R

S

K

2

1 2

One-constanttheory

At each λ

K/S

Concentration

21

S

K

S

K

S

K

S

K

submix

(K/S)mix is computed at each λAnd then converted to R(λ)



Two-constanttheory

K

Concentration

S

Concentration

Kmix = K1+K2Smix = S1+S2

R(λ) is computed at each wavelength from Kmix, Smix and Rg

(the reflectance of the substrate)


Select a recipee.g. C1, C2, C3

Predict reflectance

Compute colour coordinates

Compare to the target

Within tolerance?Modify recipe print

Kubelka-Munk


CombinationsCombinations

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

))!(!(

!

rnr

nCrn

Example: n=20, r=3

1140rnC

n r=3 r=4

3 1 0

4 4 1

6 20 15

10 120 210

20 1140 4875

30 4060 27405

Where n = number of dyes in the permitted list and r= number of dyes allowed per recipe


Recipe CorrectionRecipe Correction

• Commercial CMP systems like JAYPAK Paint software 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).


Advantages of Computer Match Prediction :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


THANK YOU

Manish Kapadia.

Date post:	16-Oct-2014
Category:	Documents
Upload:	manish-kapadia
View:	106 times
Download:	1 times

Understanding JAYPAK Plastic Paints Basic Recipe Prediction Model

Documents