Grafisk produktion och Tryckkvalitet Kubelka-Munk teori, Ink penetration och FWA

Sasan Gooran (VT 2004) Li Yang (VT 2003)

The K-M Theory of Reflectance • This theory was originally developed for paint films but works quite well in many circumstances for paper • It is not, however, terribly good for dyed papers (or very dark, unbleached papers) when light absorption reaches a high level • A limiting assumption is that the particles making up the layer must be much smaller than the total thickness

The K-M Theory of Reflectance • Both absorbing and scattering media must be uniformly distributed through the sheet • The theory works best for optically thick materials where > 50 % of light is reflected and < 20 % is transmitted

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K-M Theory

K-M Theory • K is the Absorption Coefficient ≡ the limiting fraction of absorption of light energy per unit thickness, as thickness becomes very small • S is the Scattering Coefficient ≡ the limiting fraction of light energy scattered backwards per unit thickness as thickness tends to zero

K-M Theory • The effect of the material in a thin element dx on iT and iR is to: decrease iT by iT(S + K) dx (absorption and scattering) decrease iR by iR(S + K) dx (absorption and scattering) increase iT by iR S dx (scattered light from iR reinforces iT) increase iR by iT S dx (scattered light from IT reinforces IR)

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K-M Theory • Therefore:

where x is measured from the bottom of the sheet, i.e. upwards in the figure, which affects the signs

K-M Theory

K-M Theory Define R = I/I0 as reflectance of sheet and r = iR/iT as reflectance of increment and

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K-M Theory

K-M Theory

K-M Theory Consider the limiting condition where X = ∞ , R = R∞ and R’ can take any value, since no light gets to it, so we can set R’ = 0. The left hand side of previous equation must equal ∞ , which means that the denominator must equal 0 and:

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K-M Theory By finding R∞ by measurement (using for example a black backing paper R’=0 and a very thick paper sheet) And measuring another reflectance K and S can be approximated. Desirable K small S “big” R0 The opacity = R can also be calculated. ∞

Ink Penetration

Ink Penetration

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Desirable properties • Paper should be of strong scattering (s) and little absorption (k)--it gives the paper good whiteness • Ink should be of strong absorption in certain wavelength band (say cyan in long wavelength)--it makes the ink have good color

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Ink Penetration

Ink Penetration

• It causes color shift ! • It leads to lower color saturation ! • It reduces the color gamut !

Ink Penetration

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Ink Penetration

Ink Penetration

Ink Penetration

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Hypothesis for the model • Fundmental physical parameters: Scattering coefficients (s) and absorption coefficients (k) for clean paper (sp,kp), pure inks (si,ki) and ink-paper mixtures (sip,kip) ; • two channel approximation: We consider the light goes into up hemisphere and down hemisphere; • Complete back scattering: light heading downwards is scattered into up hemisphere and vice versa.

Light propagation in the media

Mathematical descriptionLight propagation

• Differential equations for light propagation in the media (paper, ink, ink-paper mixture)

−

dI ↓ dz

= −(k + s ) I ↓ + sI ↑

dI ↑ dz

= −(k + s ) I ↑ + sI ↓

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Spatial distribution for s and k • Constant model:

s = s p + si

k = k p + ki

• Linear model:

s = s p + si

z − z0 z1 − z0

k = k p + ki

z − z0 z1 − z0

• Exponential model: α

s = s p + si e

z−z0 z1− z 0

α

k = k p + ki e

z−z0 z1− z 0

Solution for constant ink distribution • Differential equation: − s ⎤⎛ I ↓ ⎞ ⎛ I ⎞ ⎡( k + s ) ⎜ ⎟ d ⎜⎜ ↓ ⎟⎟ = ⎢ − (k + s )⎥⎦⎜⎝ I ↑ ⎟⎠ ⎝ I↑ ⎠ ⎣ s • Solve it:

−s ⎡(k + s) − A ⎤ =0 ⎢ − (k + s) − A⎥⎦ s ⎣ ⇔ A2 − ( k + s ) 2 + s 2 = 0 ⇒ A = ± k 2 + 2ks

Solution for constant ink distribution • General Solution:

I ↓ = a1e Az + a2 e − Az I ↑ = b1e Az + b2 e − Az • border conditions: a1,a2,b1 and b2 are determined by border conditions at z=z0,z1 and z2

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Compute Reflectance

R= =

I ↑ ( z2 ) I0 (1 − Rg X 2 )e − Az1 − (1 − Rg X 1 )e Az1 ( X 1 − Rg )e − Az1 − ( X 2 − Rg )e Az1

Simulations for ink penetration (linear ink distribution)

Simulations for ink penetration Linear vs. Constant

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Comparison between measurement and simulation

Simulations and Discussions • The print shows a stronger reflectance due to ink penetration: Rink >Rno; • Printed color becomes less saturated due to ink penetration 700

X ink =

∫R

ink

(λ )S (λ ) x(λ )dλ

400 700

≥

∫R

no

(λ )S (λ ) x(λ )dλ = X no

400

Simulations and Discussions • X0-Xink< X0-Xno; Y0-Yink< Y0-Yno; Z0-Zink< Z0-Zno; • The obtainable color gamut becomes smaller -- absolutely undesirable! • How to avoid it: paper coating (expensive), change inks (size of color particles, properties of solution); • Fundamental—Better understanding of the mechanism !

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photogloss Åpaper

multicopy paper Æ (ink penetration)

Chromaticity diagram photo gloss vs. multi copy

a*b*-diagram photo gloss vs. multi copy

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Lab diagram photo gloss vs. multi copy Copy paper

Glossy paper

Whiteness • Whiteness is a subjective perceived property. Most people consider that it increases when the material has a slightly blue tone. • Whiteness is in some way an aspect of color perception.

Whiteness W = Y + 800( xn − x) + 1700( yn − y ) W, Whiteness Y, the Y value x, y the chromaticity coordinates xn and yn the chromaticity values of the light source Y, x and y are calculated for observation angle 10 degrees and d65 illumination

The equation says that whiteness is composed of the luminance Y and a color term. The whiteness is increased when the material approaches blue and reduced when it is yellowish. See the diagram

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Whiteness

Whiteness line

(xn, yn)

The Fluorescent Whitening Agents FWA • Effective agents for increasing the whiteness impression of paper products • FWA’s are organic materials, which absorb ultraviolet radiation and transform it to visible light

Measurement of Fluorescent • Measurements are made with and without a cut-off-filter and the difference is calculated • The fluorescence appears at wavelengths of 440 to 500 nm. The difference in whiteness is taken as a measure of fluorescence.

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FWA

Measurement of Fluorescent Problems • The cut-off filter never completely removes the effect of a FWA. The reason: They are not only activated by UV, but also by blue visible light with wavelengths up to 420 nm. • Two different cut-off filters are used. One removes UV below 395 nm and the other removes all light below 420 nm.

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