Industrial Color Physics (Springer Series in Optical Sciences)

Springer Series in

OPTICAL SCIENCES founded by H.K.V. Lotsch Editor-in-Chief: W.T. Rhodes, Atlanta Editorial Board: A. Adibi, Atlanta T. Asakura, Sapporo T. W. Hänsch, Garching T. Kamiya, Tokyo F. Krausz, Garching B. Monemar, Linköping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter, München

154

Springer Series in

OPTICAL SCIENCES The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes, Georgia Institute of Technology, USA, provides an expanding selection of research monographs in all major areas of optics: lasers and quantum optics, ultrafast phenomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics, applied laser technology, industrial applications, and other topics of contemporary interest. With this broad coverage of topics, the series is of use to all research scientists and engineers who need up-to-date reference books. The editors encourage prospective authors to correspond with them in advance of submitting a manuscript. Submission of manuscripts should be made to the Editor-in-Chief or one of the Editors. See also www.springer.com/series/624 Editor-in-Chief William T. Rhodes, Ph.D. Professor Emeritus, Georgia Institute of Technology Professor of Electrical Engineering and Associate Director, Imaging Technology Center Florida Atlantic University Boca Raton, FL 33431 USA E-mail: [email protected] Editorial Board Ali Adibi Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected] Toshimitsu Asakura

Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: ferenc [email protected] Bo Monemar

Hokkai-Gakuen University Faculty of Engineering 1-1, Minami-26, Nishi 11, Chuo-ku Sapporo, Hokkaido 064-0926, Japan E-mail: [email protected]

Department of Physics and Measurement Technology Materials Science Division Linköping University 58183 Linköping, Sweden E-mail: [email protected] se

Theodor W. Hänsch

Herbert Venghaus

Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: [email protected]

Fraunhofer Institut für Nachrichtentechnik Heinrich-Hertz-Institut Einsteinufer 37 10587 Berlin, Germany E-mail: [email protected]

Takeshi Kamiya Ministry of Education, Culture, Sports Science and Technology National Institution for Academic Degrees 3-29-1 Otsuka, Bunkyo-ku Tokyo 112-0012, Japan E-mail: [email protected]

HorstWeber Technische Universität Berlin Optisches Institut Straße des 17. Juni 135 10623 Berlin, Germany E-mail: [email protected]

Ferenc Krausz

HaraldWeinfurter

Ludwig-Maximilians-Universität München Lehrstuhl für Experimentelle Physik Am Coulombwall 1 85748 Garching, Germany and

Ludwig-Maximilians-Universität M Sektion Physik Schellingstraße 4/III 80799 München, Germany E-mail: [email protected]

Please view available titles in Springer Series in Optical Sciences on series homepage http://www.springer.com/series/624

Georg A. Klein

Industrial Color Physics

13

Georg A. Klein Herrenberg Germany

ISSN 0342-4111 e-ISSN 1556-1534 ISBN 978-1-4419-1196-4 e-ISBN 978-1-4419-1197-1 DOI 10.1007/978-1-4419-1197-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009940329 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Translated from German by Todd Meyrath and the author.

In memoriam of Dr. Hartmut K.A. Pauli

Preface to the English Edition

Colors arise only in the brain, normally originating from electromagnetic waves from the outside world.

This book is based on courses given by the author in the Department of Colors, Paints and Plastics at the University of Applied Sciences in Stuttgart and continued at the University of Applied Sciences in Esslingen, Germany. The development of color physics in industry began in the middle of the 19th century with the large-scale manufacturing of natural colors. Since that time, a great variety of new, especially synthetic, colorants have been produced in order to meet increasing demands for non-self-luminous colors with regard to color applications. The rapid progress in color physics and accompanying applications over the last three decades are the reasons for this work. Here, the fundamentals of color physics are outlined and the most important recent developments and applications in the color industry are discussed. In comparison to the first German edition,1 all chapters of the book have been revised and expanded with regard to effect pigments. After the introductory chapter, the optical fundamentals of absorbing and effect colorants are discussed. The exceptional spectral and colorimetric properties of effect pigments are detailed in combination with further characterizing parameters. Color spaces are presented as well as the efficiency of recent color difference formulas. In addition to the normal spectral measuring methods for absorbing colorants, modified procedures for effect colorations are outlined. The typical angle-dependent properties of pearlescent, interference, and diffraction pigments as well as mixtures with metallic pigments are clarified by measurements. In addition, criteria for estimation of measurement errors as well as the statistics of color difference values are detailed.

1 Klein, G.A.: “Farbenphysik für industrielle Anwendungen”, Springer, Berlin, Heidelberg (2004).

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The directional two-flux approximation with directional rays is introduced and allows for the creation of the optical triangle. Some characteristic triangle parameters enable the demonstration of the performance of the two-, three-, and multi-flux approximations up to a maximum of n = 256 fluxes. Based on these different concepts, the fundamentals of the classical color recipe prediction methods are described. These topics are completed with the most important modern methods for recipe prediction for all known sorts of colorations. This edition has only been realized through the kindness of many people, and I am happy to express my thanks to them. First, I owe a special debt of gratitude to Dr. Todd Meyrath (Ventura, CA, USA). Not only did he translate from German, he also offered numerous ideas – most of which have been realized in the text. Possible translation errors in the book should be attributed to the author. A lot of suggestions concerning radiative transfer came from friendly discussions with Dr. Hartmut Pauli (Basel, Switzerland). Many thanks go to Dr. Peter W. Gabel (Darmstadt, Germany), Dr. Alfried Kiehl (Velden, Germany), Dr. Changjun Li (Leeds, UK), Mike Nofi (Santa Rosa, CA, USA), Francis Powers (Shawbury, Shropshire, UK), Ian Wheeler (Leven, UK), Dipl.-Ing. Gerhard Wilker (Frankfurt am Main, Germany), and Dr. Klaus Witt (Berlin, Germany) for their encouragement and, in many cases, the kind permission to reproduce some light and scanning microscope images of colorants. Thanks to Klaus-Juergen Woyczehowski (Frankfurt am Main, Germany) for timely help with these; I very much appreciate his kindness. Last but not least, sincere thanks go to my family. Herrenberg, Germany May 2009

Georg A. Klein

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Light Sources, Types of Colorants, Observer . . . . . . . . . . . . . 2.1 Optical Radiation Sources and Interactions of Light . . . . . . . . 2.1.1 Visible Spectrum and Colors . . . . . . . . . . . . . . . . 2.1.2 Types of Light Sources . . . . . . . . . . . . . . . . . . 2.1.3 Technical Light Sources . . . . . . . . . . . . . . . . . . 2.1.4 Illuminants . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Geometric Optical Interactions . . . . . . . . . . . . . . 2.1.6 Interference of Light . . . . . . . . . . . . . . . . . . . . 2.1.7 Diffraction from Transmission and Reflection Gratings . . 2.2 Absorbing Colorants . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Types and Attributes of Absorbing Colorants . . . . . . . 2.2.2 Pigment Mixtures and Light Transmittance . . . . . . . . 2.2.3 Description of Color Attributes . . . . . . . . . . . . . . 2.2.4 Color-Order Systems . . . . . . . . . . . . . . . . . . . 2.2.5 Surface Phenomenon . . . . . . . . . . . . . . . . . . . 2.3 Effect Pigments . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Types of Metallic Pigments . . . . . . . . . . . . . . . . 2.3.2 Morphology of Metallic Particles . . . . . . . . . . . . . 2.3.3 Coloristic Properties of Metallic Pigments . . . . . . . . . 2.3.4 Sorts of Pearlescent and Interference Colorants . . . . . . 2.3.5 Interference Pigments Consisting of Multiple Layers . . . 2.3.6 Spectral Behavior of Pearlescent and Interference Colorants 2.3.7 Opaque Films Containing Absorbing and Effect Pigments . 2.3.8 Colors of Diffraction Pigments . . . . . . . . . . . . . . 2.4 Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Color Perception and Color Theories . . . . . . . . . . . 2.4.2 Color Perception Phenomenon . . . . . . . . . . . . . . . 2.4.3 Subtractive and Additive Mixing of Colors . . . . . . . . 2.4.4 Tristimulus Color-Matching Experiments . . . . . . . . .

11 11 12 14 18 23 25 34 38 43 43 48 51 57 59 63 65 69 76 82 85 91 101 104 109 110 114 116 120

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2.4.5 Determination of Tristimulus Values . . . . . . . . . . . . 2.4.6 CIE 1931 and CIE 1964 Standard Colorimetric Observers . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 125 129

3 Systems of Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments . . . . . . . . . . . . . . . . . . . 3.1 Systems of Standardized Tristimulus Values . . . . . . . . . 3.1.1 CIE 1931 Tristimulus Values . . . . . . . . . . . . 3.1.2 Chromaticity Coordinates and Chromaticity Diagram 3.1.3 CIE 1976 Color Spaces . . . . . . . . . . . . . . . 3.1.4 DIN99o Color Space . . . . . . . . . . . . . . . . . 3.2 Color Difference Metrics and Color Tolerances . . . . . . . 3.2.1 CMC(l:c) Color Difference Formula . . . . . . . . . 3.2.2 CIE94 Color Difference Expression . . . . . . . . . 3.2.3 CIEDE2000 Color Difference Equation . . . . . . . 3.2.4 Efficiency of Color Difference Formulas, CIE Color Appearance Models . . . . . . . . . . . . . . 3.2.5 Color Tolerances . . . . . . . . . . . . . . . . . . . 3.3 Color Constancy and Metamerism . . . . . . . . . . . . . . 3.3.1 Chromatic Adaptation and Color Constancy . . . . . 3.3.2 Index of Color Inconstancy . . . . . . . . . . . . . 3.3.3 Kinds of Metamerism . . . . . . . . . . . . . . . . 3.3.4 Special Metamerism Indices . . . . . . . . . . . . . 3.4 Specific Qualities of Colorants . . . . . . . . . . . . . . . . 3.4.1 Build-up and Coloring Potential . . . . . . . . . . . 3.4.2 Strength and Depth of Color . . . . . . . . . . . . . 3.4.3 Covering Capacity . . . . . . . . . . . . . . . . . . 3.4.4 Transparency and Coloring Power . . . . . . . . . . 3.4.5 Color Fastness and Turbidity . . . . . . . . . . . . . 3.4.6 Stability of Effect Pigments . . . . . . . . . . . . . 3.5 Chroma of Effect Pigments . . . . . . . . . . . . . . . . . 3.5.1 Coloristic Quantities of Effect Pigments . . . . . . . 3.5.2 Color Difference Equation for Metallics . . . . . . . 3.5.3 Chroma of Pearlesence and Interference Pigments . . 3.5.4 Mixtures of Effect Pigments . . . . . . . . . . . . . 3.5.5 Color Development of Diffraction Pigments . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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133 133 134 137 141 148 152 153 154 157

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159 161 164 164 166 169 172 175 176 179 185 189 192 197 200 201 206 209 214 221 228

4 Measuring Colors . . . . . . . . . . . . . . . . . . . . . 4.1 Measurement of Reflecting and Transmitting Materials 4.1.1 Measurement of Colors and Visual Judgment . 4.1.2 Measurement Geometries . . . . . . . . . . . 4.1.3 Sample Requirements . . . . . . . . . . . . . 4.1.4 Transparent, Translucent, Opaque Colors . . .

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233 234 235 239 248 250

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Contents

4.1.5 Color Matching . . . . . . . . . . . . . . . . . . 4.1.6 Acceptability and Tolerance Agreement . . . . . . 4.2 Measuring Methods . . . . . . . . . . . . . . . . . . . . 4.2.1 Tristimulus Colorimeter . . . . . . . . . . . . . . 4.2.2 Spectrophotometer . . . . . . . . . . . . . . . . . 4.2.3 Accuracy of Spectrophotometers . . . . . . . . . . 4.2.4 Reflectance and Transmittance of Layers . . . . . 4.2.5 Auxiliary Optical Methods for Effect Pigments . . 4.2.6 Fluorescent, Thermochromic, Photochromic Colors 4.3 Uncertainties of Spectral Color Measurement . . . . . . . 4.3.1 Qualitative Errors . . . . . . . . . . . . . . . . . 4.3.2 Quantitative Errors and Error Distribution . . . . . 4.3.3 Normal Distribution in Three and More Dimensions 4.3.4 Statistical Testing of Color Differences . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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252 255 257 257 259 263 266 271 274 278 279 280 284 287 292

5 Theories of Radiative Transfer . . . . . . . . . . . . . . . . . . . 5.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Basic Concepts and Definitions . . . . . . . . . . . . . 5.1.2 Absorption and Scattering . . . . . . . . . . . . . . . . 5.1.3 Single and Multiple Scattering . . . . . . . . . . . . . . 5.1.4 Radiative Transfer Equation . . . . . . . . . . . . . . . 5.1.5 Radiative Transfer in Plane Parallel Layers . . . . . . . 5.1.6 Phase Function for Anisotropic Scattering . . . . . . . . 5.2 Directional Two-Flux Approximation . . . . . . . . . . . . . . 5.2.1 Reflection and Transmission . . . . . . . . . . . . . . . 5.2.2 Optical Special Cases . . . . . . . . . . . . . . . . . . 5.2.3 Optical Triangle . . . . . . . . . . . . . . . . . . . . . 5.2.4 Determination of Optical Coefficients . . . . . . . . . . 5.3 Theory of Kubelka and Munk . . . . . . . . . . . . . . . . . . 5.3.1 Empirical Approach . . . . . . . . . . . . . . . . . . . 5.3.2 Exceptional Optical Cases . . . . . . . . . . . . . . . . 5.3.3 Determination of Optical Constants . . . . . . . . . . . 5.3.4 Boundary Layer Correction . . . . . . . . . . . . . . . 5.3.5 Limits of Kubelka–Munk Theory . . . . . . . . . . . . 5.4 Three-Flux Approximation . . . . . . . . . . . . . . . . . . . 5.4.1 Conception of Three-Flux Theory . . . . . . . . . . . . 5.4.2 Reflection and Transmission . . . . . . . . . . . . . . . 5.4.3 Optically Different Materials . . . . . . . . . . . . . . 5.4.4 Correction of Surface Effects . . . . . . . . . . . . . . 5.4.5 Special Cases of External Reflection and Transmission . 5.5 Approximation of Radiative Transfer by Multi-Flux Theory . . . 5.5.1 Exact Solutions for Internal Reflection and Transmission

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295 295 296 299 302 306 308 313 316 317 319 321 324 326 327 329 331 332 337 340 341 343 345 348 352 356 357

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5.5.2 5.5.3 5.5.4

Surface Boundary Corrections . . . . . . . . . . . . Total Reflection and Optical Extreme Cases . . . . . Boundary Conditions, Matrices of Reflection and Transmission . . . . . . . . . . . . . . . . . . Directional and Diffuse Reflection and Transmission Inclusion of Total Reflection . . . . . . . . . . . . . Corrected Optical Triangle . . . . . . . . . . . . . .

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370 372 373 377 379

6 Recipe Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Classical Method . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Calibration Series with Absorbing Colorants . . . . . . . . 6.1.2 Calibration and Reference Colorations for Effect Pigments 6.1.3 Determination of Absorption and Scattering Coefficients . 6.1.4 Optical Path and Albedo . . . . . . . . . . . . . . . . . . 6.1.5 Coefficients of the Phase Function . . . . . . . . . . . . . 6.1.6 Accuracy of Predicted Recipes . . . . . . . . . . . . . . . 6.2 Strategies for Recipe Prediction . . . . . . . . . . . . . . . . . . 6.2.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . 6.2.2 Spectrometric Strategy . . . . . . . . . . . . . . . . . . . 6.2.3 Colorimetric Method . . . . . . . . . . . . . . . . . . . 6.2.4 Balanced Color Differences . . . . . . . . . . . . . . . . 6.3 Realization of Recipes . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Selection of Suitable Recipes . . . . . . . . . . . . . . . 6.3.2 Sensitivity and Correctability of Color Recipes . . . . . . 6.3.3 Numerical Procedures for Recipe Correction . . . . . . . . 6.3.4 Databases . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Modified Expert Systems . . . . . . . . . . . . . . . . . 6.3.6 Neural Networks . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

381 382 384 390 393 397 399 400 403 404 409 414 417 420 421 424 427 431 435 441 445

Appendix . . . . . . . . . . . . . . . . . . . . . . A.1 Non-colored Applications of Effect Pigments A.1.1 Metallic Pigments . . . . . . . . . . A.1.2 Pearlescent Pigments . . . . . . . . A.2 Chromatic Adaption Transform CAT02 . . . A.2.1 Forward Mode . . . . . . . . . . . . A.2.2 Reverse Mode . . . . . . . . . . . . A.3 Two-Flux Approximations . . . . . . . . . . A.3.1 Directional Fluxes . . . . . . . . . . A.3.2 Diffuse Fluxes . . . . . . . . . . . .

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447 447 447 448 448 448 450 451 451 453

References in Alphabetic Order . . . . . . . . . . . . . . . . . . . . .

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Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.5.5 5.5.6 5.5.7 References

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Chapter 1

Introduction

Colored objects influence our lives each and everyday. Within physics the term color is used ambiguously and is in no way restricted to the field of optics. In solid-state physics, the origin of the colors of crystals is certainly of interest; in particle physics the so-called quarks carry electrical charges which are astonishingly called red, green, and blue, yet have no relation to any real colors! The well-known colors of light as well as the various phenomena of non-selfluminous colors belong to the field of optics. Non-self-luminous colors result if suitable colorants such as dyes or pigments are mixed into an organic or inorganic medium. This book confines itself to non-self-luminous colors, their physical and colorimetric qualities as well as their typical industrial applications. Modern colorants can be divided into two groups: absorption colorants and effect pigments. Non-self-luminous colors are only perceived when they are illuminated by light or other energy source. Being a perception, the color impression of the observer is thereby of central importance. Amazingly, our vision – from which we obtain about 80% of the information from our surroundings – is provided by nature with the capability of distinguishing more than 10 million colors. At this stage, we must stress that color is a perception; it is not a physical quantity but rather a purely psychophysical response, usually to visual light after entering the eye. It is, therefore, not measurable by normal engineering methods. It is possible, however, to describe colors in an objective manner by quantifying them with three distinct numbers. These numbers are called color values and are dimensionless quantities. The values can be interpreted, for example, as lightness, chroma, and hue of the relevant color. Certainly, the three values are only representatives of a specific color; they do not describe the color perception itself. From a geometrical point of view, these three color quantities can be taken as coordinates of a point in a three-dimensional color space. Hence, a color space can be considered as an isolated part of a three-dimensional Cartesian coordinate system. On the other hand, each space is obviously characterized G.A. Klein, Industrial Color Physics, Springer Series in Optical Sciences 154, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1197-1_1, 

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1 Introduction

by the coordinate system and the metrics chosen. Therefore, the so-called colorimetry describes the metrics between the color quantities in a predefined color space. Since color vision arises exclusively from a neural-induced perception and not directly from a physical effect, colorimetry is used to compare colors of similar or nearly identical hue, among other things. Our modern scientific and technical knowledge about colors follows a long historical development. The oldest surviving colored documents created by humans are the rock and cave paintings from the Paleolithic period found worldwide and dating from the time between about 40,000 and 8,000 years ago [1]. Already during this period, humans decorated their skin and cloths with natural colors. In the following epochs until the Egyptians, refinement was limited to merely the application of natural colors. Even from modern antiquity, no fundamental or new findings about color properties are known, simply some unsuccessful attempts for color theories can be found. The first systematic experiments with color phenomenon were made by Newton with sunlight in the year 1671: with a simple prism he proved that sunlight is comprised of colored parts. In 1704, he concluded: “For the Rays to speak properly are not coloured. In them there is nothing than a certain Power and Disposition to stir up a Sensation of this or that Colour”. His knowledge and experience in optics are concentrated in the work “Opticks”, in which he already began treating terms such as refraction, reflection, and interference [2]. In his color theory of 1810, Goethe criticized this apparent overemphasis of the physical aspects of color; in particular, he stressed the physiological components of color vision [3]. However, the physical statements of his theory were subsequently disproved. Fundamental scientific understanding of colors goes back to Young. In 1802, he created the spectrograph for measuring the intensity distribution of light sources. Young additionally introduced the term wavelength of light and established the first physically and physiologically founded trichromatic theory of color vision [4]. A physiologically determined systematic of color sensation was defined by Grassmann in 1853 with four empirical laws of additive color mixing. They form the basis of modern colorimetry [5]. Moreover, Grassmann was the first to characterize colors by quantities. These quantities are composed of numbers for hue, lightness, and saturation, which are similar to the above description. With these, he has introduced the principle of color values – an essential method in modern industrial color physics. The modern trichromatic theory was established already in 1867 by Helmholtz [6]; it is based on the earlier ideas of Young [4] and Maxwell [7]. With the so-called Young–Helmholtz theory, the psychophysically induced color stimulus can be described numerically by physical and physiological factors. The underlying trichromatic theorem – confirmed experimentally in 1964 – is that color stimulus is caused by the fact that the retina in the eye contains three receptors with photopigments of different sensitivities for the primary colors

1 Introduction

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red, green, and blue. Only these three receptors are responsible to initiate color sensation [8]. Independently in 1878, Hering formulated the so-called opponent color theory. According to this model, the rear of the retina contains three different channels for the opponent color pairs of red and green, yellow and blue as well as the opponent achromatic colors black and white, which are able to produce all color sensations [9]. These ideas were finally proven in 1956 [10, 11]. The retinex theory, introduced by Land in 1959, starts from the hypothesis that the retina and the visual cortex (located at the rear areas of both brain lobes) are connected by three separate channels. In these channels, the arriving neural signals are compared according to their trichromatic lightness contribution so that color sensation is stimulated [12, 13]. The three aforementioned color theories outlined here are not only of interest for research of color perception but also for the application of their established results to the evaluation of the visual and colorimetric properties of colors and colorants. Of great importance in this development is the so-called high colorimetry, formulated (in vector algebra) by Schroedinger in 1920, since then it is abbreviated as colorimetry. Our actual colorimetric methods are based on these ideas [14]. With this formalism, it is possible to make statements about the complete identity of color sensation. Moreover, we are able to obtain information about the similarity of visually different color impressions. These phenomena are relevant for color matching and comparison of nearly identical colors [15–17]. There has been great delay in the implementation of most of the above insights and accumulated knowledge into the industrial setting. About 80 years after the beginning of large-scale production of modified natural and increasingly synthetic absorption colorants, in 1931 the first conventions were set out by the Commission Internationale de L’Éclairage (CIE, International Commission on Illumination) to describe colors with uniform quantities [18]. With these decisions, an essential step toward better communication between manufacturers and users of absorption colorants was initiated. In the proposals of the CIE, ideas of the trichromatic theory have been adopted. This concept was thoroughly revised in 1974 and elements of the opponent color theory as well as colorimetry were taken into consideration. Two years later, parts of this program – named CIELAB system – were recommended for global application [19]. On the basis of the experiences achieved with these proposals, the recommendations have been built up and modified since this time [20–23]. Apart from the higher color physical requirements on today’s industrial colorants, their production, improvement, and processing are of primary interest. Among others, important properties are the total miscibility of the utilized colorants, components, and additives, their chemical and physical compatibility, as well as their durability against shear, pressure, or temperature, all of which might depend, for instance, on the chosen process technology. The principally desired characteristics for most technological applications include: high color

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1 Introduction

constancy against different climatic environments, electromagnetic, radioactive, and even cosmic radiation, as well as fastness – meaning resistance – against chemical changes. Beyond these, the innocuousness concerning toxicological and ecological criteria is increasingly important. For testing the aforementioned and additional coloristic properties, spectral and colorimetric methods are used. With the CIELAB system and later adjustments, colorimetric criteria are applied in color industry, for example: – to specify colors and colorants; – to characterize the typical color producing properties of colorants; – to quantify and assess the color differences of similar or nearly equal colors; – for suitable estimation and assessment of the altered color sensation caused by substitution of colorants, change in illumination or observer conditions; – for recipe prediction of colored samples of unknown colorant compositions. The treatment of these and similar applied problems in an economic fashion was realized only during recent decades. Three reasons are responsible for this: first, the development of complex microstructures in semiconductors (actually reduced now to nanodimensions); second, computers with faster computing speed and larger storage density since 1970; and third, the increased accuracy and simultaneously improved reproducibility of the developed components for instruments to measure colors. The resulting progress has led to extensive and fundamental knowledge about color physical problems of absorption and effect pigments. Until now, effect pigments have been intentionally excluded from our considerations. These flake-shaped particles with a diagonal dimension of about 5–100 μm produce exceptional color sensations, differing essentially from those of absorption colorants. Effect colors result optically either from metallic reflection, interference, or diffraction. Effect pigments are produced industrially to an increasing extent only since the middle of the last century. Two main reasons are responsible for this late distribution: first, effect pigments passed through an autonomous technical development in comparison to absorbing colorants; second, the completely unusual colors and optical interactions produced by this kind of colorant. Metallic reflection, interference, and diffraction are not comparable with selective absorption or scattering, which generate the used colors of absorbing colorants. Effect pigments can be divided into four groups: metallic, pearlescent, interference, and diffraction pigments. In Table 1.1 various sorts of modern colorants are listed together with their dominant color producing interactions. Typical for metallic pigments is the metallic gloss, which is absent in traditional absorption pigments. The beginning of metallic pigments goes back to the beating of gold; a skill practiced in Egypt as early as 5,000 years ago [24]. At that time, gold was beaten into foils with minimal thickness of 1 μm (1 μm

1 Introduction

5

Table 1.1 Modern sorts of colorants and main optical interactions Type of colorant

Colorant

Dominant light interaction

Absorption colorants

Dyes Absorption pigments

Absorption Absorption, scattering

Effect pigments

Metallic pigments Pearlescent, interference pigments Diffraction pigments

Reflection Interference Diffraction

corresponds to 10–6 m) for commodities and pieces of jewelry as well as parts of buildings with a precious and protective surface. Since about 200 BC in China, Japan, and India, Buddha statues were embellished with slices of gold. From there, the art of gold-beating reached Rome, where this skill was even mentioned briefly by Plinius Secundus the Elder in the first century AD; the topic is treated in more detail in the Lucca Manuscript. This document is a recipe collection gathered by Greek monks of the 9th century. The most important confirmed last historical discourse on the technology of gold-beating is represented by a manuscript from the 12th century of the German monk Theophilus . Two centuries after the founding of various centers in northern Germany, the town Nuernberg became the primary center of the gold-beating technology. During the period that followed, increasing demand for gold led to ever thinner metallic foils which unfortunately resulted in a greater failure rate. This is why they were embedded in protective binders ever since. From those flakes, the so-called gold flitter, over the last centuries led to today’s gold and metallic pigments. For reasons of economy and excessive demand, it was finally necessary to find a replacement for gold. This led to gold-colored brass, a material which shows an optical behavior similar to gold in the visible range. Brass is well known since the 3rd century BC in Babylon and Assyria, but vanished thereafter and was rediscovered in Roman times. Since the year 1820, brass has been produced on the large scale in Germany by alloying copper (56–90%) with zinc. A similar history is responsible for the replacement of bright silver–bronze, an alloy of silver with tin, by a less expensive metal: aluminum, which shows a metallic reflection comparable to that of silver–bronze. Pure aluminum has been obtained economically using the technology of melt-flow electrolysis since 1886 and can be produced in the shape of metallic pigments. Certainly aluminum pigments have shown an increased acceptance only since 1970, because since that time it has been possible to realize the desired formulations with this kind of effect pigment on the large scale. Aluminum pigments are preferentially applied for mono- and multi-layer coatings, plastics, or printing inks. Metal flakes for colorants – with mean diameters between 5 and 50 μm – have been called

6

1 Introduction

metallic pigments only for some decades. Now, metallic flakes are being produced in various geometrical shapes and even various colors with the addition of absorption pigments fixed on the surfaces of the particles. Pearlescent pigments, the second group of effect pigments, imitate the nacre luster of natural pearls. The colored effect seems to rise from the depths to the surface of the pearl and results from constructive interference at the various internal interfaces. In order to duplicate this color phenomenon, in Paris in 1656 Jacquin produced and processed the so-called essence d’orient (pearl essence) – a suspension consisting of tiny plates of guanine and hypoxanthine – acquired from fish scales [25]. The use of this cost-intensive production process ceased in 1920: during the following two decades some suitable inorganic compounds were crystallized with increasing success in layers of various thicknesses, which completely imitate the luster of pearls. Modern pearlescent pigments mostly consist of metal oxides fixed on natural or synthetic mica substrates. In the middle of the 20th century the economic upturn allowed for the development of pearlescent pigments in a similar range of application as metallic pigments. In addition to mono-layer pearlescent pigments, transparent or opaque twoand multi-layer pigments have also been realized. Their striking chromatic and brilliant colors result mainly from interference at the combined multi-layers: that is why they are called interference pigments. Sometimes the colors of interference pigments change greatly with the angle of observation. This is principally a consequence of the great difference in refractive index of the layers, which consist of different metals, metal oxides, or even liquid crystal structures [25]. The individual films have thicknesses between about 20 and 200 nm (1 nm corresponds to 10–9 m), therefore, on the high side, nearly the size of visible wavelengths, a requirement for optical interference among other things. The typical size of interference pigments extends from some microns to nearly one millimeter. This is about one to three orders of magnitude greater in size compared to absorption pigments. More recently, new types of effect colorants, called diffraction pigments, have been manufactured. Their production is based on knowledge and materials of nanotechnology. The color effect of these pigments is initiated by diffraction – again a consequence of the wave properties of light – and is extremely angle dependent. Compared with interference pigments, they even show extreme color changes ranging from violet up to red – over the entire visible spectrum. The diffraction effect is a consequence of the periodic grating structure engraved in the particles; the pigments have diameters of about 20–100 μm. Without a doubt all kinds of effect pigments increase enormously the available colors in industry, but the methods of characterizing and measuring them are less than satisfactory. This is no surprise given the fact that the methods for measuring and characterization of colors have thus far only be developed for absorbing colorants. These methods are in general unsuited to qualify and

1 Introduction

7

quantify the characteristic color properties and angle-dependent colors of effect pigments. The development of adequate measuring methods for these colorants is ongoing. It is known that new findings and applications in modern natural sciences are improved by interaction between theory and experiment. Since the introduction of digital computers after 1960, the methods of numerical simulation have greatly progressed. Right from the beginning, computers were applied in industrial color physics to simulate the light interactions of colorants. For an adequate optical theory, it is necessary to describe the relevant optical processes in colored materials in a suitable manner. Strictly speaking, the various single or combined optical interactions have to be taken into account. At least two optical principles are used for this: single and multiple scattering. Single scattering was first developed by Mie [26], who postulated that color is caused only by a single interaction of light with the electric charges of the colorant particles. In contrast, in multiple scattering light continues to interact until it is absorbed or it leaves the material. These ideas were finally formulated by Chandrasekhar [27]. Although both concepts of single and multiple scattering produce similar results, the formalism of Mie is to this day much too extensive and time consuming for industrial color applications. In industrial applications, the theory of radiative transfer of Chandrasekhar is mostly used, modified with a well-established method in physics: the continuous radiative field of the light inside and outside of a colored layer is divided into discrete light cones. Two cones represent a two-flux theory and so on, until we get the multi-flux theory with any positive integer number n > 2 of cones. The special case of two fluxes with diffuse radiation corresponds to the theory of Kubelka–Munk [28], which certainly has only limited validity. With finer adjustment of the subdivision, the cone model seems to correspond more closely with reality. At least the multi-flux formalism is a reliable apparatus in color physics and related fields. Although the relevant formalisms are merely approximations, with these and further approximations, the color industry has nonetheless gained economic advantage to a large extent. One of the most important applications of this procedure is the numerical recipe prediction for colorants. Since about 1970, this method has developed itself into an established and particularly efficient tool in the color industry. With this procedure, a color of unknown colorant composition can be matched with available colorants only by calculating their approximate concentrations. Recipe prediction is nowadays an easy and effective aid. This is in contrast to the past, when previous generations were forced to select some suitable samples only visually from extensive series of experiments in a long-term empirical manner. But now, classical recipe calculation delivers a multitude of equivalent or alternative color recipes, and with the added benefit of a considerable shorter development time. The experience with the multitude of varied recipes, generated by this method, has made it necessary to catalog them, for example, with

8

1 Introduction

regard to the constituent colorants. To manage the enormous number of ancient recipes, modern software databases or expert systems are used. Today, these systems are often combined with numerical recipe prediction, in particular, to match colors containing both absorption and effect colorants. Clearly, all existent theoretical concepts aside, the visual sensation of the observer will always be the final authority on all efforts in the color industry. The neural processes in the brain are known to behave nonlinearly; the neural signals coming from the retina are gathered and modulated in different areas of the brain lobes until the signals reach the visual cortex. Consequently they are extremely difficult and complex to detect. It is, without a doubt, desirable to bridge the gap to the understanding of the neurological and the neurophysiological processes in the brain responsible for color sensation. In this context, only interdisciplinary research is able to make a considerable step forward [29]. Otherwise nearly all methods for characterizing, qualifying, or comparing colors remain on the contemporary, empirical level. In any case, it is definite that a perceived color depends on three criteria: the source of light, the illuminated color sample, and the color sensation of the observer. These three factors together constitute the fundamentals of color physics, colorimetry, and relating industrial applications described in this book.

References 1. Guthrie, RD: “The Nature of Paleolithic Art”, University of Chicago Press, Chicago (2005) 2. Newton, SI: “Opticks: or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light; Also Two Treatises of the Species and Magnitude of Curvelinear Figures”, repr edition London 1704; Culture et Civilisation, Bruxelles (1966) 3. Goethe, JW von: “Zur Farbenlehre”, repr 1st vol ed Tuebingen in 1810, Harenberg, Dortmund (1979) 4. Young, T: “On the theory of light and colours”, Philos Trans Lond 92 (1802) 12 5. Grassmann, HG: “Zur Theorie der Farbenmischung”, Annalen der Physik 89 (1853) 69; original translation in English: Philosophical Magazine 7, Ser 4 (1854) 254; further: MacAdam, DL, Ed: “Selected Papers in Colorimetry – Fundamentals”, SPIE Milestone Series MS 77 (1993) 10 6. Helmholtz, H von: “Handbuch der physiologischen Optik”, 2nd rev ed, Voss, Hamburg (1896) 7. Maxwell, JC: “On the theory of compound colours and the relations of the colours of the spectrum”, Philos Trans R Soc Lond 150 (1860) 57 8. Brown, PK, Wald, G: “Visual pigments in single rods and cones of the human retina”, Science 144 (1964) 45 9. Hering, E: “Outline of a Theory of the Light Sense”, (1920), Transl.: Hurvich LM a. Jameson D; Harvard University Press, Cambridge, MA (1964) 10. Svaetichin, G: “Spectral response curves of single cone”, Acta Physiol Scand 39 Supply 134 (1956) 17 11. Hurvich, LM, Jameson, D: “Some quantitative aspects of an opponent colors theory”, I, II, III; J Opt Soc Am 45 (1955) 546, 602; ibid. 46 (1956) 405 12. Land, EH: “Experiments in color vision”, Sci Am 200 5 (1959) 84 13. Land, EH: “Recent advances in retinex theory”, Vision Res 26 (1986) 7

References

9

14. Schroedinger, E: “Grundlinien einer Theorie der Farbenmetrik im Tagessehen I., II., III.”, Annalen der Physik 63 (1920) 397, 427, 481 15. Richter, M: “Einfuehrung in die Farbmetrik”, 2nd ed, W de Gruyter, Berlin (1981) 16. Wyszecki, G, Stiles, WS: “Color Science”, 2nd ed, Wiley Classics Library, New York (2000) 17. Judd, DB, Wyszecki, G: “Color in Business, Science and Industry”, 3rd ed, Wiley, New York (1975) 18. CIE 1931: “Proceedings of the Eighth Session”, Cambridge, England 1931; Bureau Central de la CIE, Paris (1931) 19. CIE No 15.2: “Colorimetry”, 2nd ed, Commission Internationale de L’ Éclairage (CIE); Bureau Central de la CIE, Wien (1986) 20. CIE No 116: “Industrial Colour-Difference Evaluation”, Commission Internationale de L’Éclairage (CIE), Bureau Central de la CIE, Wien (1995) 21. CIE No 142: “Improvement to Industrial Colour-Difference Evaluation”, Commission Internationale de L’Éclairage (CIE), Bureau Central de la CIE, Wien (2001) 22. CIE No 15.3: “Colorimetry”, 3rd ed, CIE, Bureau Central de la CIE, Wien (2004) 23. Schanda, J, Ed: “Colorimetry: Understanding the CIE System“, Wiley, Hoboken, NJ (2007) 24. Humpl, I: “Blattgold”, C Winter, Heidelberg (1990) 25. Pfaff, G: “Special effect pigments”, in: Smith, HM, Ed: “High Performance Pigments”, Wiley-VCH, Weinheim (2002) 26. Mie, G: “Beitraege zur Optik trueber Medien, speziell kolloidaler Metalloesungen”, Annalen der Physik, 25, 4th ser (1908) 377 27. Chandrasekhar, S: “Radiative Transfer”, repr 1st ed 1950, Dover, New York (1960) 28. Kubelka, P, Munk, F: “Ein Beitrag zur Optik der Farbanstriche”, Zeitschr Techn Physik 12 (1931) 593 29. Backhaus, WGK, Kliegel, R, Werner, JS, Eds: “Color Vision: Perspectives from Different Disciplines”, W de Gruyter, Berlin (1998)

Chapter 2

Light Sources, Types of Colorants, Observer

In this chapter, the fundamental conditions for color production are discussed. In simplified terms, the visual color impression of non-self-luminous colors is ultimately due to three independent components: the light source, the colorants of the color pattern, and the observer. The color perception depends, therefore, on the specific properties of these factors. Factors of particular importance are as follows: – the spectral power distribution emitted of the light sources used; – the light interactions with the colorants of the color sample, especially the resultant absorption, scattering, reflection, transmission, as well as interference or diffraction; – the color perception capability of the observer. We will go into these three factors in more detail in the following sections. To begin with, we deal with the most important light sources used for color assessment and the most simple light interactions of colorants. The composition and the typical spectral properties of industrially applied colorant sorts are described in detail. The explanation of color sensation of the observer seems initially to be out of scope; however, this theme is necessary to consider for at least two reasons: first, some phenomena of the human color sense really stand out. These have to be taken into account during color assessment. Second, color perception is affected by the law of additive color mixing, upon which the entire colorimetry and the corresponding applications of industrial color physics are founded.

2.1 Optical Radiation Sources and Interactions of Light Without light there exists no color. The concept of color is bound to visible wavelengths. Therefore, we turn at first toward the typical properties of natural

G.A. Klein, Industrial Color Physics, Springer Series in Optical Sciences 154, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1197-1_2, 

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2 Light Sources, Types of Colorants, Observer

and man-made light sources because the spectral power distribution of an illuminant affects the color impression. Due to ever-present changes in natural daylight, such a source is unsuited for producing a consistent color sensation with an unchanging non-self-luminous color. On account of this uncertainty, we are forced to rely on man-made sources of constant and reproducible light emission – normally in the visible range. For a reliable assessment of colors in industry, the spectral power distributions of two commonly used man-made light sources have been standardized. The actual physical processes producing color appearances can be described by elements of simple geometrical optics as well as some effects of wave and quantum optics. In this section, these interactions are described in so far as they are significant for better understanding of the color physical properties of modern industrial colorants.

2.1.1 Visible Spectrum and Colors The electromagnetic spectrum covers an enormous range of wavelengths λ from, for example, values such as λ ≈ 1 fm (1 fm corresponds to 10–15 m) for cosmic radiation to λ ≈ 10 km for radio waves, therefore a range of around 19 orders of magnitude; see Fig. 2.1. On the other hand, the visible range of humans is only a small part of the spectrum of electromagnetic waves. Merely wavelengths in the very small interval from 380 to 780 nm are normally perceived by humans as visible light. Wavelengths at the left end of the range between

10–15

10–12

10–9

10–6

10–3

1

λ m

103

1 2 3 4 5

6 7 8 9 10

Violet Green Orange Blue Yellow Red 380 nm

f Hz

1021

1018

780 nm 1015

1012

109

106

Fig. 2.1 Spectrum of electromagnetic waves: 1 cosmic radiation, 2 gamma radiation, 3 Xray radiation, 4 ultraviolet radiation, 5 near-ultraviolet radiation; 6 infrared radiation, 7 radar waves, 8 VHF waves, 9 television waves, 10 radio waves

2.1

Optical Radiation Sources and Interactions of Light

13

380 and 440 nm are perceived as violet. With increasing wavelength, the color impression changes to blue, green, yellow, orange, and finally red. Red is perceived at wavelengths above 600 nm. The associated wavelengths are subject to individual variations in color perception. The so-called spectral colors are the purest producible colors. They are characterized by a wavelength width of less than 1 nm (i.e., with a laser). On the other hand, if the radiation contains nearly all wavelengths of the visible spectrum and of equal intensity, the resulting color impression is white light (e.g., white clouds). For the entire range of visible light between about 380 and 780 nm to be perceived, there must be sufficiently high intensity. Under normal illumination conditions, the wavelength interval that can be seen by humans is restricted to between about 400 and 700 nm. Many modern color measuring instruments work in this limited range [1]. Apart from its wave character, light also exhibits simultaneously particlelike properties. To this day, perhaps, this dualism is not understood in terms of everyday human experience. The corresponding particles of electromagnetic radiation, and therefore of visible light, are the so-called photons. Photons in the visible range carry a sufficient amount of energy for selective stimulation of the photosensitive pigments in the retina of the eye to initiate color impression. The eyes should, however, be protected from dangerous ultraviolet (UV) or infrared (IR) radiation. However, radiation of wavelengths near the visible range which cannot be directly perceived by humans can, in conjunction with suitable colorants, cause different physical effects. Luminescence colorants, for example, absorb energy at UV wavelengths and then emit most of this energy at longer wavelengths – usually within the visible or IR range. The needed transitions from energetic steady states occur either spontaneously by fluorescence or delayed by phosphorescence. The energy surplus is converted into molecular vibration energy and leads macroscopically to a temperature increase. The energy absorption of normal absorption colorants in the visible or IR range is also transformed into molecular vibration energy. If this energy conversion is accompanied by a color change, such kinds of colorants are denoted as thermochromic. Moreover, colorants are termed as phototropic, if the color change is only caused by energy absorption at visible wavelengths. On the other hand, excessively high UV or IR radiation energy is sometimes able to initiate irreversible molecular changes, which result in bleaching or total loss of color. In contrast to the above-mentioned colorants, overall the greatest percentage of industrially used colorants contain absorption and effect colorants. In absorption colorants, the incident energy is sufficiently high to initiate partial absorption or scattering. Physically, light absorption in colorant molecules occurs only for certain transitions between quantum energy levels – therefore, in special wavelength regions of visible light. The corresponding processes are called selective absorption or scattering. On the other hand, scattering of light depends on the electrical charge distribution and the geometry of the colorant

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2 Light Sources, Types of Colorants, Observer

particles. In contrast to absorption pigments, dyes do not normally show any scattering because the size of the isolated molecules in solution is too small for such an interaction. The light reflected in the direction of the eye initiates in the retina signals which are perceived usually as non-self-luminous colors. The color separation of pearlescent, interference, and diffraction pigments is a consequence of the wave nature of light. In order to produce a suited interference effect, the pigment particles are built up of layers with different refractive indices of which the light waves interfere constructively or destructively depending on optical path length. The layer thicknesses are smaller than the interfering wavelengths. In contrast, the particles of diffraction pigments show an embossed regular grating structure at which suited wavelengths are diffracted. The distance between two light-transferring slits is about 1 μm.

2.1.2 Types of Light Sources The perception and assessment of non-self-luminous colors requires illumination with a suitable light source. Depending on the mechanism of light generation, optical radiation sources have different spectral power distributions. On the basis of the emitted spectrum, illuminants can be divided into two categories: temperature and luminescence radiators. The most important luminous sources of both classes are given in Table 2.1. In the following, we discuss details of both categories because the illumination of color samples is our primary context. In the near future, semiconductor diodes and lasers are expected to replace, in part, the light sources used to date. Therefore, we also discuss on these sources even though they are, thus far, rarely applied in color industry despite a multitude of advantages. Table 2.1 Kinds of optical light sources Temperature radiator

Luminescence radiator

Natural

Artificial

Artificial

Sunlight, Scattered light of the Earth atmosphere, stars, galaxies

Blackbody radiator, incandescent lamp, arc lamp

Gas discharge tube, fluorescent lamp, light emitting diode (LED), source of coherent light (laser)

For a common characterization, optical radiation source output distributions are often compared with the spectral energy distribution or temperature of a so-called blackbody radiator – also denoted as blackbody or cavity radiator. At lower temperatures, metals emit heat energy in form of IR radiation; gradually, with increasing temperature, dark-red glow emanates. With a further increase

2.1

Optical Radiation Sources and Interactions of Light

15

of temperature, the color changes to orange and yellow, finally to bluish white. During this process, both the radiation energy and the brightness of the emitted light increase. The wavelength with the most energy shifts to smaller wavelengths, i.e., a blue shift. For an ideal blackbody, this radiation is generated inside the cavity of a blackbody radiator and the outside of the cavity walls absorb all external electromagnetic waves. The ideal situation, therefore, is the emission of only the cavity radiation according to its temperature. The radiation power S(λ,T)dλ of a blackbody radiator at wavelength interval dλ is given by the Planck law of radiation [2]: S(λ,T)dλ =

λ5

c1  dλ. · {exp [c2 (λT)] − 1}

(2.1.1)

The radiation constants c1 and c2 have values of c1 = 2πhc2 = 3.74185 × m2 and c2 = hc/k = 1.43884 × 10−2 m K. In Equation (2.1.1), the wavelength λ should be in units of meters and the temperature T in units of Kelvin. The radiation constants contain the velocity of light in vacuum c and the Boltzmann constant k. For derivation of Equation (2.1.1), Planck introduced h – now called the Planck constant. Figure 2.2 shows the spectral power distribution in wavelength given by Planck’s law. As can be seen, at a temperature of 500 K, the peak of spectral power is in the IR range. For an increase to much higher temperatures, i.e., to greater than 104 K, this peak shifts over the visible range into the UV range. For temperatures of about 7,600 and 3,700 K, the peak of the spectrum lies at the respective edges of the visible range. The Planckian formula (2.1.1) contains 10−16 W

109

Energy density / W.m–2.nm–1

Visual range g

a: 500 K b: 1,000 K c: 2,000 K d: 4,000 K e: 6,000 K f: 10,000 K g: 20,000 K

106 f e d

103

c b

100

a 10–3 101

102

103

104

Fig. 2.2 Spectral power distribution of blackbody radiator of different temperatures

λ

nm

16

2 Light Sources, Types of Colorants, Observer

two limiting cases: for short wavelengths Wien’s law of radiation and for long wavelengths Rayleigh–Jeans radiation formula. Before the quantum hypothesis was established, both laws lead to inconsistent infinite energies in the UV range (“UV catastrophe”). The primary assumption made by Planck is that a radiating system exchanges energy with the surrounding radiation field only with an integer multiple of the quantum energy E=

hc = hf . λ

(2.1.2)

In this formula, f is the frequency of the corresponding wavelength. Both the continuous spectra of temperature radiators and the discontinuous line spectra of luminescence radiators are based on the emission of light quanta that are identical with the already mentioned photons. Because the color of the spectrum emitted by a blackbody radiator changes with temperature, it is useful to introduce the term color temperature. With this quantity, the emitted light of a source is compared with that of a blackbody radiator and thus characterized. The color temperature of an illuminant corresponds to the temperature of the blackbody radiator which emits the maximum of light at the same color as the actual illuminant. Strictly speaking, only a temperature radiator can be assigned a color temperature. Other optical radiators, such as luminescence radiators, are characterized by a so-called similar or correlated color temperature. The most well-known temperature radiator is the Sun. Inside the Sun, at temperatures higher than 107 K, deuterium is converted to helium by nuclear fusion (Bethe–Weizsäcker cycle) [3]. For generation of 1 mol helium, the gigantic energy of 1.55 × 102 GJ is released. The total solar fusion energy results in a Sun surface with a mean temperature of about 5,800 K and an exceptionally high radiation power of about 63.3 MW/m2 . Merely a small fraction of this power, namely the so-called solar constant with value 1.37 kW/m2 , reaches the Earth’s atmosphere. This value reduces to about 1.12 kW/m2 if the Sun is at its zenith and the atmosphere is free of clouds. Already these considerations indicate the necessity of carrying out outdoor exposure tests of industrial colors (Section 3.4.5). Along the way to the Earth’s surface, the light interacts with particles of the atmosphere; its intensity is reduced by absorption and scattering. The light scattering is caused by the molecules in the air and this is responsible, for example, for the blue sky. According to the Rayleigh law J=

V π 2 (n − 1) 2 · · E · cos2 ϑ, N r 2 λ4

(2.1.3)

short wavelength blue light is scattered more strongly than the long wavelength red light because λ is contained in the denominator of this expression. The

2.1

Optical Radiation Sources and Interactions of Light

17

further quantities in Equation (2.1.3) are defined as follows: J the intensity of the scattered light, N the number of scattering particles per unit volume V, n the refractive index of the scattering medium, r the particle radius, E the amount of the electric field strength, and cos2 ϑ the phase function (see Section 5.1.5); ϑ denotes the scattering angle with regard to the incident intensity. The Rayleigh law is only valid for wavelengths λ which are longer than the particle radius r. In contrast to the blue color of the sky, sunrise and sunset are caused by scattering and absorption of light in the atmosphere, more precisely due to the air molecules as well as to aerosols (water drops, dust particles, etc.). On the long and nearly tangential optical path of the light through the layer of air, blue wavelengths are more strongly scattered and absorbed. The remaining blue light, therefore, reaches the observer on the Earth with considerably less intensity as compared with the much less scattered long wavelength red light. The Sun and the sky, therefore, appear reddish. In addition to dependence on daytime, received sunlight changes due to weather conditions, geographical latitude, season, and due to the approximately 11-year sunspot cycle. Accordingly, the color temperature of daylight is subject to substantial variations and takes values in the range of 5,500 K for direct sunlight to more than 14,000 K for blue zenith skylight. Simultaneously, the spectral power distribution changes. This is shown in Fig. 2.3 with curves normalized at the wavelength 555 nm. At this wavelength, the sensitivity of the human eye is at its highest (Section 2.4.6).

a

Relative spectral energy

b

400

c d e

500

600

λ

700

nm

Fig. 2.3 Relative spectral energy distribution curves of daylight, normalized at 555 nm: (a) cloud-free zenith skylight, (b) cloud-free north skylight, (c) overcast skylight, (d) medium daylight, and (e) direct sunlight [4]

18

2 Light Sources, Types of Colorants, Observer

The daylight variations mentioned above create extra complexities for the unambiguous visual and objective assessment of non-self-luminous colors and their typical properties. A single color sample can produce a completely different color impression simply due to a changing illumination condition. In the practice of color physics, it is necessary to use reproducible artificial light sources which have nearly constant spectral power distributions. These artificial light sources are often referred to as technical sources. This corresponds to a constant color temperature if aging effects are neglected. From an economic point of view, the sources should also have a reliable working life of more than 1,000 operating hours. These standards are fulfilled by most of the technical illuminators of importance in the color industry; in the following section, we turn toward such kinds of sources.

2.1.3 Technical Light Sources For solving coloristical problems of non-self-luminous colors, technical light sources are used exclusively. The main reason for this is the high reproducibility of the generated spectrum. In technical temperature radiators, a metal of high melting point is heated up by an electric energy supply to such an extent that a continuous spectrum is emitted in the visible range; this spectrum is similar to that of a blackbody radiator of the same temperature. A temperature radiator in widespread use is the tungsten filament lamp; its color temperature is essentially dependent on the filament thickness, the applied voltage, and the kind of gas filling of the bulb. The so-called tungsten–halogen bulb contains bromine or iodine which increases the light efficacy, the working life, as well as the color temperature from about 2,800 to 3,000 K. The tungsten filament lamp of color temperature 2,856 K is named standard illuminant A by the CIE. The gradual loss of filament thickness in normal tungsten lamps is slowed down by the included halogen: the vaporized tungsten combines with the halogen, cools down at the surrounded quartz bulb, and reaches – by convection – the hot filament surface; there it dissociates so that tungsten is removed. Tungsten filament lamps typically emit light of yellowish color. The accompanying spectral energy distribution is shown in Fig. 2.4. Furthermore, in a tungsten arc lamp with an argon atmosphere, both tungsten electrodes are heated by an arc discharge in such a way that a radiation distribution is produced similar to that of the tungsten filament lamp; the accompanying color temperature amounts to about 3,100 K. A carbon-arc lamp shows a color temperature of 6,000 K and a very high luminance of about 1.6 Gcd/m2 .1

1 The unit candela (cd) is defined as the luminous flux radiated from 1/60 cm2 of a blackbody with temperature 2.042 K.

2.1

Optical Radiation Sources and Interactions of Light

19

200

Spectral energy distribution S (λ)

A

150 D65 100

50

0 300

400

500

600

λ

nm

Fig. 2.4 Spectral energy distribution of a tungsten filament lamp (CIE standard illuminant A) and a UV-filtered xenon lamp (CIE standard illuminant D65)

The term luminescence radiators represents a group of radiators including so-called discharge lamps, as well as photoluminescence radiators [5, 6]. In contrast to temperature radiators, luminescence radiators emit either a line spectrum of discrete wavelengths or a band spectrum of broader wavelength intervals. Line spectra are exclusively generated by gas discharge lamps. The physical mechanisms for generating line spectra are first to accelerate charge carriers by an electric field; during collision with the gas atoms, excitation energy is transferred to the outer electron orbits of these atoms. On the basis of the quantized energy of the electron shells, the electron transition into the ground level results in emission of light with discrete wavelengths λ according to Equation (2.1.2). The actual temperature of luminescence lamps is clearly far lower than the surface temperature of temperature radiators of the same light color. Among the technical gas discharge lamps, only the filtered light of xenon or mercury vapor lamps is of major importance for visual assessment or spectral measurement of colors. The emission of light in gas discharge lamps is based on the same physical mechanisms as in luminescence radiators. At first an electric voltage pulse in a xenon atmosphere causes free charge carriers. In a highpressure xenon lamp under pulsed or constant voltage, the charged particles generate a nearly continuous spectrum in the visible range. This is accompanied by a small amount of UV radiation. The spectral energy distribution shows a flat peak at a wavelength of about 450 nm and the emitted spectrum shows only a

20

2 Light Sources, Types of Colorants, Observer

slight decrease in magnitude for longer wavelengths. Therefore, the perceived light appears slightly but insignificantly bluish; this can be seen in Fig. 2.4, curve D65. The radiation distribution is similar to that of diffuse daylight at midday on cloudless north sky, cf. Fig. 2.3, curve d. For these reasons, a UV-filtered xenon lamp of color temperature 6,500 K is used in the color industry to simulate midday light. The spectral energy distribution of this xenon discharge lamp is standardized; it is termed by the CIE as standard illuminant D65. The CIE recommends the use of xenon lamps with color temperatures of 5,000, 5,500, or 7,500 K, if the standard illuminant D65 is not available. The mercury vapor discharge lamp generates a line spectrum with emission wavelengths of 405, 436, 546, 577, and 579 nm, as well as in the UV range of 254, 314, and 365 nm. Because of the energetically high UV amount, this discharge lamp is utilized in a so-called light booth for visual assessment of fluorescence colorants, artificial color fastness tests, and in fluorescence microscopy. A light booth consists of a small one-sided open compartment with one or two small platforms to lay down the color samples to compare, as well as different non-glare light sources which can be individually switched on. The abovementioned emission wavelengths are also used for wavelength-scale calibrations of color measuring instruments. The UV fraction of the mercury spectrum is furthermore applied to stimulate the phosphorus in fluorescent lamps in order to initiate photoluminescence in the visible range. The spectral composition of the resulting band spectrum or the resulting light color depends on the chemical structure and the mixing ratio of the involved phosphorus. For color assessment of special importance, there is the cold white light of the fluorescent lamp CWF (identical with illuminant FL 2) and the light emission of the so-called three-band lamp TL 84 (identical with FL 11); the three-band lamp has radiation maxima at wavelengths of about 440, 550, and 610 nm; see Fig. 2.5. These wavelengths have spectral colors of blue, green, and red and cause trichromatic a neutral white light color. Fluorescent lamps are in widespread use only because of economic reasons: they have a luminous efficacy and a physical life which are about eight times higher than those of tungsten filament lamps, cf. Table 2.2. Light sources of principle importance in the near future are expected to be light emitting diodes (LEDs) and lasers, which greatly ripened technically in the 1960s. The central component part of an LED consists of a p–n semiconductor junction. A voltage between 1 and 15 V in conducting direction and a current of order 50 mA release photons in the p–n region. These photons are generated by an energy surplus from recombining electrons and defect electrons (holes). Available luminescence diodes doped with suited chemical compounds can emit quite monochromatic light with half-widths of 6–25 nm, for example, at wavelengths of 400 nm (gallium-nitride diode), 600 nm (gallium-arsenicnitride diode), and 660 nm (gallium-phosphide-zinc-oxide diode). The benefits of LEDs are the short switching time of about 5 ns, the small spectral

2.1

Optical Radiation Sources and Interactions of Light

21

Spectral energy distribution S(λ)

80

60

FL 11 40

FL 2 20

0

500

400

λ

600

nm

Fig. 2.5 Spectral energy distribution of fluorescent lamps: cool white fluorescent CWF (illuminant FL 2) and three-band lamp TL84 (illuminant FL 11) Table 2.2 Properties of five selected illuminants Color Correlated color rendering temperature/K index

Light efficacy/ lm/W

2,856

100

12

Tungsten filament lamp

CIE standard D65 illuminant D65, middle daylight

6,500

94

35

UV-filtered xenon lamp

Cold white daylight

FL 2

4,230

64

70

Fluorescent lamp CWF, cool white fluorescent

Bluish white daylight

FL 7

6,500

90

80

Broadband fluorescent lamp

4,000

83

90

Three-band lamp TL84

CIE CIE illuminant abbreviation CIE standard illuminant A, evening light

A

White daylight FL 11

CIE simulator

22

2 Light Sources, Types of Colorants, Observer

half-width of the emitted intensity, the high degree of optical efficiency, and the long working time. Disadvantages up to now have been the low illumination intensity in comparison to traditional light sources. Improvements in these respects, at the time of this writing, are the subject of ongoing research and development. The term laser is an abbreviation of “light amplification by stimulated emission of radiation.” A laser is, therefore, an optical amplifier which is based on the principle of stimulated emission of light. To initiate stimulated emission of light2 , an irradiating field, in some sense, forces the emission of light in atoms, molecules, or ions of gases, liquids, or solids. The incident field of frequency f has photon energy according to Equation (2.1.2). The primary requirement for stimulated light emission is that this photon energy is at least the natural energy difference of the medium. In such cases, the medium that has some population in an excited state has some return to the ground state. This emitted energy is incorporated into the incident field. In order to obtain an amplification of radiation by stimulated emission, the energetically higher levels or bands of a medium should maintain a state with a greater degree of filling. This greater electron number or population in the upper energy state is usually maintained using an energy supply such as flash lamps, other laser pump sources, currents in semiconductor laser, or atom/electron collisions. Typical classification of lasers is along the lines of the physical state of the gain medium: solid state (crystals), semiconductor, liquid (usually organic dyes), or gas lasers, for example. The ruby laser with emission wavelength of λe = 694.3 nm is a solid-state laser. Semiconductor lasers are, for example, indium-gallium-phosphide (In1-x Gax P) or aluminum-gallium-arsenide lasers (Alx Ga1-x As); the emitted monochromatic wavelength of each usually lies in the range 500 and 1,000 nm depending on the content x of the indicated element. Dye lasers are the dominant class of liquid lasers; typical laser mediums are dyes such as coumarin (460 nm ≤ λe ≤ 560 nm) or rhodamine (535 nm ≤ λe ≤ 630 nm). The most well-known gas lasers are the helium–neon laser (632 nm) and the argon-ion laser (wavelengths of highest intensity 488.0 nm and 514.5 nm). The advantages of lasers are clear, considering the outstanding features of the emitted light: constant frequency, highly monochromatic, spatial and temporal coherence, high beam directivity, and adjustable energy density. In the color industry, lasers have been successfully used for the determination of surface gloss, covering capacity and glittering of effect colorants, as well as size distribution of pigment particles in powders, among other things. In the following section, we concentrate on specific properties of light sources which are of special interest for colorimetric applications.

2 The opposite is

absorption, or, more precisely, stimulated absorption.

2.1

Optical Radiation Sources and Interactions of Light

23

2.1.4 Illuminants In the previous section, the physical basics of light emission and the applications in technical light sources have been introduced. Now, we direct our attention toward the special handling of light sources in colorimetry or color matching. Unquestionably, the vast variety of technical light sources complicates the unambiguous visual assessment of colors: colored objects are exposed to changing natural as well as artificial illuminations such as daylight, evening light, or fluorescence light. The change in illumination can alter the visual color impression (see below). The CIE has, therefore, recommended the most representative light sources to use for color assessment applications. For clearness and better communication, four terms should be identified. These terms describe the different kinds of light sources: 1. CIE illuminant: this corresponds to a theoretical source of a tabulated relative spectral power distribution S(λi ); 2. CIE standard illuminant: only two illuminants are standardized by the CIE, illuminant A and illuminant D65; 3. CIE source: corresponds to a technically realized CIE illuminant; 4. CIE simulator: is a technical source which approximately corresponds to the desired CIE illuminant. The first and second terms need some further explanation. The CIE specified several representative illuminants [7], among them are the following: a. three temperature radiators designated D50, D55, D75 with color temperatures of 5,000, 5,500, and 7,500 K, respectively; b. twelve fluorescent lamps designated from FL 1 to FL 12; the illuminants FL 1–6 emit line spectra, FL 7–9 broadband, and FL 10–12 narrowband spectra. Among these, FL 2, FL 7, or FL 11 are preferably used in colorimetry. In Fig. 2.5, only the spectral power distribution of illuminants FL 2 and FL 11 are shown; c. in the end, five high-pressure lamps designated as HP 1–5, of which two are sodium vapor lamps and three metal halide lamps; these are normally not significant in colorimetry but rather in lighting engineering. The two CIE standard illuminants are characterized by the following features: the spectral power distribution S(λi ) of standard illuminant A is given by the Planck law of radiation (2.1.1), whereas that of D65 is given by tabular values [8, 9]. These values correspond to the UV-filtered emission of a high-pressure xenon lamp shown in Fig. 2.4. Standard illuminant A is recommended for simulation of room light in the evening, D65 of midday light of color temperature 6,500 K.

24

2 Light Sources, Types of Colorants, Observer

The tabular values of a CIE illuminant are generally used for computation of color values. The corresponding simulation illuminant serves for visual assessment of color patterns. In other words, the real light source which is used to illuminate a color sample is, for the purpose of calculation, substituted by a theoretical simulation source, consisting of discrete wavelengths and power emission. This can clearly be a reason for deviation between the visual assessment and the colorimetric result. A further deviation can result from the so-called CIE standard colorimetric observers. This is not discussed until Section 2.4.6. The five most commonly used illuminants in colorimetry are D65, A, FL 2, FL 7, and FL 11; a selection of their properties is given in Table 2.2. While the standard illuminant A is assigned a true color temperature, the illuminant D65 and the fluorescent radiators have only correlated color temperatures. These are for the luminescence sources FL 2 –11: 4,230, 6,500, and 4,000 K, respectively. In most cases, the change of illumination also results in a change of perceived color. This can be caused either by the colorants themselves or by the light source used. If the spectral power distribution is responsible for color changes, this is attributed to the color rendering of the illuminating source. The yellowish light of the sodium vapor lamp HP 1, for example, bathes each chromatic color in a pale yellow. This is because sodium emits only two closely spaced wavelengths of 589.0 and 589.6 nm in the visible range. The CIE proposed the dimensionless color rendering index Ra to characterize the grade of color rendering of light sources [10–12]. This index takes values in the range 0 ≤ Ra ≤ 100. The sodium vapor lamp HP 1 has an Ra value of 20; this indicates that colors are quite distorted. In contrast, the CIE standard illuminant A takes the highest possible value of 100. An additionally used characteristic, which has a meaning in terms of energy, is the light efficacy of a radiator. This economic quantity is defined as the ratio of emitted luminous flux of a light source to the input power of unit lm/W.3 As can be seen from Table 2.2, the displayed fluorescence lamps show a higher light efficacy than temperature radiators of equal or lower color temperature. From a comparison of the spectral power distributions shown in Figs. 2.4 and 2.5, it is possible to understand how the change of a source alters the color impression. Consider Fig. 2.4: using source A, the color sample appears more yellow and red compared with illumination of a D65 simulator. This is because of the continuously increasing radiation energy characteristic of the source A from yellow to red wavelengths with quite small values at short wavelengths. Consider now Fig. 2.5: the same color sample is rendered bluish white with FL 2 source, or with FL 11 source, redder in comparison to D65 simulator. Colors 3 By definition, the unit lumen (lm) is the luminous flux which a point light source of emissiv-

ity 1 cd (candela) emanates evenly in all directions in a solid angle of 1 sr (steradian): 1 lm = 1 cd sr; 1 steradian corresponds to a solid angle Ω – of even circular cone with center point in a sphere of radius 1 m – which cuts an area of 1 m2 out of the sphere surface, cf. Fig. 5.1.

2.1

Optical Radiation Sources and Interactions of Light

25

illuminated with a D65 or FL 7 source are rendered and perceived in a similarly balanced way as under midday light. This is because of the nearly constant and high values of the corresponding emission spectra in the visible range.

2.1.5 Geometric Optical Interactions There are various possible interactions between incident light and atoms, molecules, particles, or crystals. Of these interactions, we are primarily interested in the color appearances that result. In a plane electromagnetic wave, the electric and magnetic vectors E and H are perpendicular to one another, and, in addition, mutually perpendicular to the propagation direction. The so-called wave vector k is oriented in the propagation direction. The electromagnetic wave carries the energy flux density in the direction of k, given by vector S (named as Poynting vector) and relation S = E × H.

(2.1.4)

Figure 2.6 shows the connection between the three vectors E, H, and S. In the figure, the electric and magnetic vectors are shifted a quarter wavelength in phase with respect to one another. E

S

H Fig. 2.6 Electric and magnetic field of a stationary wave

The amount of energy flux density S carried by such waves, also termed as flux density, or short flux, is the origin of interactions with the molecules or particles of colorants to produce colors. Color production of non-self-luminous colors can be caused by simple or multiple reflection, refraction, absorption, scattering, interference, and/or diffraction. When the wavelength of the light (order 0.4 to about 1 μm) is much smaller than the size of the objects that it interacts with (i.e., macroscopic objects), the light no longer behaves strongly as a wave, but rather propagates in straight lines according to geometrical optics. Reflection, refraction, absorption, or scattering can occur simultaneously if the light is incident on macroscopic boundary surfaces consisting of mediums with different optical densities. The polarization

26

2 Light Sources, Types of Colorants, Observer

of light can intensify normal color appearance. This can appear especially for some absorption pigments and liquid crystal pigments. Materials can be illuminated by directional, diffuse, or mixed light. Quite simple, but of great importance, is the directional reflection – also denoted as specular reflection. Directed reflection arises from directional light at smooth, polished, or glossy surfaces, for example at organic binders, synthetic polymers, glasses, metals, as well as colorations with absorption and effect pigments. According to the reflection law, the angles of the incident ϑi and reflected light ϑ r , measured with respect to the normal of the reflecting surface, are equal: ϑi = ϑ r . Additionally, both rays and the normal of the reflecting surface lie in the same plane, that of the paper in Fig. 2.7. It is important to note that for visual assessment of colorations of glossy surfaces one must strictly avoid observations in the direction of the specular angle. For visual inspection of absorption colorations with collimated light, the surface should be illuminated from the side and the observation performed perpendicular to the sample surface. In contrast, the visual assessment of effect colorations requires a sophisticated procedure, cf. Figs. 2.29 and 2.30.

ϑi

ϑr

Fig. 2.7 Depiction of the reflection law with incident and reflected beams, as well as angle of incidence, and specular angle, each from normal to the surface

If the medium behind the glossy surface is transparent and of different refractive index than the first medium, the beam is additionally refracted into this medium. The refracted ray deviates from the original direction, see Fig. 2.8a, b, due to the different indices of refraction in the two media. The angle of refraction ϑ 2 depends on the angle of incidence ϑ 1 and the ratio of refractive indices n2 /n1 of the adjoining mediums according to Snell’s law of refraction sin ϑ1 n2 = =n sin ϑ2 n1

(2.1.5)

(original W. Snel van Royen, 1621). In general, the refractive indices are also wavelength dependent, this normally results in violet light being refracted at a steeper angle than red light. This property is called dispersion [2] and results in prism effects.

2.1

Optical Radiation Sources and Interactions of Light

27

ϑ1

ϑ2

n1

n1

n2

n2

ϑ2

ϑ1

a)

b)

Fig. 2.8 Refraction of light at the boundary surface of (a) an optically thinner medium and (b) an optically denser medium

The reflected fraction r(μ, n) of the directional beam at the boundary surface follows from the Fresnel equation      μ − nw 2 μn − w 2 1 , (2.1.6) + r(μ,n) = 2 μn + w μ + nw where μ = cos ϑ ,

w2 = 1 − (1 − μ2 )n2

(2.1.7)

[13, 14]. Equation (2.1.6) can be derived from Maxwell equations of electrodynamics [14]. The reversal of light direction does not change the reflected fraction nor the law of refraction. The reversibility of the light path without change of effect is a general principle of geometrical optics [15]; this principle is used in color measuring methods among other things (Section 4.1.2). In the case of light incident perpendicular to the surface, the special reflected fraction is given by   n−1 2 . (2.1.8) r(1, n) = n+1 This follows from Equations (2.1.6) and (2.1.7) using μ = 1. For air with refractive index n1 ≈ 1.0, for example, and plastics or binders with a typical value n2 = 1.5, using n = n2 /n1 , the normal incidence reflected fraction is r = 0.04.4 In other words, under these conditions, 4% of the incident light is immediately 4 The cited refractive indices in this book represent values which – as usual – belong to the wavelength of the sodium line of 589.0 nm.

28

2 Light Sources, Types of Colorants, Observer

reflected from the surface of a colored sample; this amount is not available for further light interactions in the volume of a color sample. The reflectance of pure metals follows from Maxwell’s equations as well, provided that the complex refractive index nˆ is introduced: nˆ = n(1 + iκ).

(2.1.9)

The quantity nˆ is divided into the real part n for the refraction and the imaginary part nκ describing the light absorption at the interface. The quantity κ is √ named attenuation coefficient. In Equation (2.1.9), i is the imaginary unit (i = −1). For directional light incident perpendicular to the metal surface, the reflected total amount is given by r(n, κ) =

(n − 1)2 + (nκ)2 . (n + 1)2 + (nκ)2

(2.1.10)

For κ = 0, this formula reduces to Equation (2.1.8). The product nκ is termed as absorption coefficient; some measured values of n, nκ , and r(n, κ) for metals used for metallic pigments are listed in Table 2.7. A further sort of reflection occurs if a light beam enters an optically thinner medium coming from an optically denser medium. Note that the refracted ray cannot exceed an angle ϑ 2 = 90◦ ; see Fig. 2.9. For a refractive index n = 1.5, the corresponding incidence angle is ϑ 1 = 41.8◦ . In general, from the law of refraction (2.1.5), it follows that rays, with angles of incidence with γcr ≥ arcsin (1/n), are totally reflected back into the optically denser medium. The quantity γcr is called the critical angle of total reflection or in short critical angle. If

ϑ2 n1 n2

ϑ1 γcr

γcr

Fig. 2.9 Critical angle γcr at a boundary surface of different refractive indices

2.1

Optical Radiation Sources and Interactions of Light

29

the total reflected light cannot immediately leave a colored layer, it participates further in the interactions with the color-producing particles until it is absorbed or leaves the layer. Total reflected rays of angles γ ≥γcr are called partly directed in this text. For unpolarized light, a further property follows from the law of refraction with regard to the refracted ray. In the case that the reflected and the refracted rays make a right angle, the light of the reflected ray is linearly polarized, in fact perpendicular to the plane of incidence; see Fig. 2.10. This special angle of incidence is denoted as Brewster angle ϑB and is given from the law of refraction (2.1.5) with ϑ 2 = 90◦ – ϑB : tan ϑB = n.

(2.1.11)

Fig. 2.10 A reflected beam of Brewster angle ϑB is linearly polarized perpendicular to the plane of incidence

ϑB

n1 n2

ϑ2

The Brewster angle depends only on the ratio of the refractive indices at both boundary surfaces. For a ratio of n2 /n1 = n = 1.5, the Brewster angle of ϑB = 56.31◦ results. The critical angle γcr and the Brewster angle ϑB are shown in dependence on n in Fig. 2.11. Polarized light produces always more intensive colors than unpolarized light; polarized light is generated in some absorption colorants and especially in liquid crystal pigments. In addition to that from Equation (2.1.6) and the above considerations, the reflection coefficient of directional light depends on the direction of polarization parallel to the plane of incidence. These properties also follow from Fresnel equations [14]. This is shown in Figs. 2.12 and 2.13 for a refractive index value of n = 1.5 in dependence on the angle of incidence ϑi . The outer reflectance coefficient at the boundary of the optically thinner medium begins to differ from one another for the two rectangular linear polarizations already for small angles

30

2 Light Sources, Types of Colorants, Observer 100

Angle / degree

80

60

γcr

40

ϑB 20

0 1.0

1.2

1.4 1.6 Refractive index n

1.8

2.0

Fig. 2.11 Critical angle γcr and Brewster angle ϑB in dependence of refractive index n

Outer reflection coefficient

1.0

0.8

Polarisation: parallel normal

0.6 n = 1.5 0.4

0.2

0

0

20

40

60

80

Angle of incidence ϑ1

Fig. 2.12 Outer Fresnel reflection factor as function of angle of incidence for polarization parallel and perpendicular to the plane of incidence

of incidence (Fig. 2.12). This difference increases strongly with the angle of incidence. The inner reflection coefficient at the inner boundary of the optically thicker medium shows the same behavior but is compressed into the angle range 0 < ϑ2 < 41.8◦ ; see Fig. 2.13. The high increase of this reflection coefficient is caused by the critical angle of total reflection. Consider now diffuse illumination instead of directional light. This alters the reflection character. The majority of natural and artificial light propagates diffusely. Because of this, it is, perhaps, most reasonable to measure and visually

2.1

Optical Radiation Sources and Interactions of Light

31

Inner reflection coefficient

1.0

Polarisation

0.8

parallel normal 0.6 n = 1.5

0.4

0.2

0

0

20

60 40 Angle of incidence ϑ2

80

Fig. 2.13 Inner Fresnel reflection factor as a function of angle of incidence for polarization parallel and perpendicular to the plane of incidence

judge color samples under diffuse illumination. Ideal diffuse light is in the forward direction inside an angle range of ±90◦ and of equal energy over the entire range of these angles. Because of that the radiation power, the optical interactions, and specially the reflection conditions at the boundary surfaces are changed. The reflection coefficients for diffuse radiation follow from energetic considerations leading to the relation 1 − rd∗ n22

=

1 − rd . n21

(2.1.12)

The quantity rd∗ denotes the reflection coefficient of diffuse light at the boundary of the optically thinner medium with n1 and rd stands for the reflection coefficient of the optically denser medium. In Fig. 2.14, the reflection coefficients for diffuse light are represented schematically by arrows for simplicity. For diffuse illumination from air of n1 ≈1.0 in a layer of refractive index n2 = 1.5, the appropriate reflection coefficients rd∗ = 0.09178 and rd = 0.59635 come from

Fig. 2.14 Reflection coefficients of diffuse radiation at a boundary surface of different refractive indices (schematically)

32

2 Light Sources, Types of Colorants, Observer 1.0 rd

Reflection coefficient

0.8

0.6

0.4

rd∗

0.2 r 0 1.0

1.2

1.4 1.6 Refractive index n

1.8

2.0

Fig. 2.15 Three sorts of reflection coefficients in dependence on refractive index n: r for directional illumination perpendicular to the surface, rd∗ for outer diffuse illumination, and rd for inner diffuse illumination of a material

the literature [13]. The reflection coefficients for diffuse light rd∗ and rd as well as for directional light at perpendicular illumination r are shown in Fig. 2.15 in dependence of the refractive index n. The boundary surface reflection certainly complicates the visual and measuring assessment of colored samples: this surface reflection is superimposed on the entire visual impression as well as the spectrometric measuring results. But the essential and interesting parts of color sensation are generated by the light interactions in the volume of a colored layer. For the following, we define the reflection of an optical medium as the amount of incident radiation energy which is backscattered from the volume and this is superimposed by the surface reflection energy. Correspondingly, the transmission is the amount of the incident light energy which overcomes the interactions in the volume and exits the second boundary surface of the optical medium. The accompanying energies are called reflection and transmission energy; these are abbreviated by WR and WT . The electromagnetic field of a light wave can drive vibrations of suitable charge carriers in atoms or molecules of absorption colorants. This additional absorption of energy leads sometimes to emission of secondary radiation; this process is denoted as scattering. The scattering is elastic if the wavelengths of the incoming and scattered light are equal. This is the case especially within non-self-luminous colors but also for Rayleigh and Mie scattering (Sections 2.1.2 and 5.1.3, respectively). Inelastic scattering, however, produces a change of scattered wavelengths such as in Raman and Brillouin scattering [16].

2.1

Optical Radiation Sources and Interactions of Light

33

In addition to scattering, a part of the incoming light energy is normally absorbed by the charge carriers of the colorant molecules and not scattered. This causes an increased molecular vibration energy and, therefore, a temperature increase of the coloration. The absorption of energy of amount WA is finally absent from the reflected, transmitted or scattered light. In the most colored layers, scattering and absorption occur simultaneously in different amounts, and additionally dependent on wavelength. The three mentioned energy components WR , WT , WA come from the incident energy Wi . Therefore, the energy conservation law, in our case, amounts to Wi = WR + WT + WA .

(2.1.13)

1 = R + T + A.

(2.1.14)

Division by Wi results in

The quantities R, T, A are denoted as follows:  R = WR Wi reflectance,

(2.1.14a)

 T = WT Wi transmittance,

(2.1.14b)

 A = WA Wi absorption.

(2.1.14c)

These quotients are, for simplicity, also denoted as reflection, transmission, and absorption but also reflection factor, transmittance factor, and absorption factor, respectively. The law of energy conservation is an axiom of physics and is of fundamental importance; in the above formulation, it plays a central role in the entire radiative transfer of optical systems. The energy conservation law is even valid for each single wavelength, therefore, valid independent of the irradiated spectral power distribution. Furthermore, this law is independent of any specifications concerning the interacting particles of the optical medium. From that follows a basic realization, which is of great importance for the further discussions in this text: The resulting values of reflection, transmission, or absorption are characteristic quantities of the optical medium; in our case, they are essentially caused only by the colorants of the chromatic color. Reflectance and transmittance are determined with suitable color measuring instruments (Sections 4.2.1 and 4.2.2); absorption and scattering are characterized by corresponding optical constants which follow from optical models (Sections 5.1.2 and 5.1.4). Absorption and scattering are generally the most

34

2 Light Sources, Types of Colorants, Observer

important quantities of absorption colorants. In effect pigments, however, the dominant processes are wavelength dependent such as interference or diffraction. The basic optical laws responsible for color production of pearlescent, interference, and diffraction pigments are discussed in the next two sections.

2.1.6 Interference of Light The colors produced by absorption colorants and metallic pigments are essentially based on processes such as reflection, absorption, and scattering and are occurring at the surface and in the volume of a colored sample. In contrast to that, impressive color effects are generated by interference of light waves in pigment particles composed of an appropriate sequence of layers. Interference is an effect caused by superposition of suitable waves (of, e.g., liquids, gases, electromagnetic fields, elementary particles). Interference of light is, for example, responsible for the colors of soap lamellas, oil films on water, or coated lenses [17]; the colors of opals, natural pearls, insect wings, or bird feathers are in addition based on interference. The colors of interference pigments result from light waves, which are reflected at the inner and outer layer boundaries and which superimpose with the incoming waves. The produced colors are controlled by the thickness and refractive index of the different layers among other things. The layer thicknesses vary from about 10 nm to 1 μm; see Fig. 2.40 [18]. Interference colors can be distinguished from normal absorption colors by the color change in dependence of the observation angle; this is in some way similar to diffraction colors. A necessary requirement for interference is the existence of coherence. Most of temperature and luminescence radiators emit incoherent waves, because the single atoms of the source oscillate independently from each other, only short wave trains are produced; between the single waves exists no constant phase relationship. Waves are coherent, if the time dependence of their amplitude is the same irrespective of a phase shift. In the case of harmonic waves, this means that the frequencies of the waves have to be the same; however, they can have a constant phase difference. Coherent sine-shaped waves must, therefore, be of equal frequency. Coherent waves can result, for example, from the reflection of a radiative field at a mirror. From this perspective, laser light is of nearly perfect coherence because it is amplified during multiple reflections. Interference pigments normally consist of at least two layers of different refractive indices to produce a suitable reflection at the boundary layer, cf. Equation (2.1.6). In Fig. 2.16, there is a simplified illustration of the reflection and interference of light at a plane parallel layer. The reflected waves of beams R1 and R2 interfere above the layer. The waves of R 1 initiated by R1 take a longer way to the surface and, therefore, have an optical path length difference G with regard to the waves of R 2 . In view of the different path lengths, the ratio of refractive

2.1

Optical Radiation Sources and Interactions of Light R1

R2

35 R'1

ϑ1

R'2

ϑ1

n1 n2 d

ϑ2 n3

Fig. 2.16 Interference at a plane parallel layer of different refractive indices

indices n = n2 /n1 , the refraction law (2.1.5), and the phase jump of λ/2 caused by reflection of the waves at the upper layer surface, the difference G is given by

G = 2d ·

 λ n2 − sin2 ϑ1 + , 2

(2.1.15)

where d stands for the layer thickness and ϑ1 for the angle of incidence. At some locations, the wave amplitude is increased by the so-called constructive interference of the incident and reflected waves. This occurs if G is an evennumbered multiple 2z of λ/2, fulfilling the condition: (2z − 1)λ = 4d ·

 n2 − sin2 ϑz ,

z = 1, 2, 3, ... .

(2.1.16)

Destructive interference occurs for  G = (2z + 1)λ 2,

z = 0, 1, 2, ... .

(2.1.17)

The positive integer z is called interference order. The amplifying light waves and, therefore, the corresponding colors can be modified according to Equation (2.1.16) by the material quantities d and n; this is used to a great extent for realization of different kinds of pearlescent and interference pigments. For color sensation, the additional change of color impression in dependence on the observation angle ϑz can be quite strange. The quantity ϑz belongs to the different interference orders z instead of the constant angle ϑ1 . Generally, the intensity of the amplified waves decreases strongly with the number of z.

36

2 Light Sources, Types of Colorants, Observer

Some special interference features result in the case that light is incident perpendicular to the surface of a layer. Colors of pearlescent and simple interference pigments are visually characterized by observation perpendicular to the colored surface; some important technical applications come out of this. From Equation (2.1.16) with ϑ1 = 0 for the dominant first interference order (z = 1), the simple result d = λ/4n follows. Furthermore, if the interference wavelength λ and the refractive index n are constant, the layer thickness for order z, given by  d = (2z − 1)λ 4n,

(2.1.18)

leads to constructive interference. A stepwise increase of the layer thickness by the quantity d leads to a distance of Δd = λ/2n between two adjacent intensity maxima. The same distance holds for neighboring minima. These considerations are especially used to reduce the reflection of optical systems by destructive interference. With the so-called optical coating, the optical surfaces are coated with an inorganic layer by physical vapor deposition; the refractive index nC of the coating has to satisfy the condition √ nC = n1 n2 , (2.1.19) where n1 and n2 are the refractive indices of air and of the optical material, respectively. Unfortunately, the effectiveness of reflection reduction is limited to a middle visible wavelength. To achieve a nearly regular reduced reflection of wavelengths in the entire visible range, a multi-layered coating on both sides of the optical element is necessary at the most 10 layers which are tuned with each other. This is a type of dielectric coating called an anti-reflection coating. The so-called dielectric mirror coating relies likewise on interference. For this, multi-layer coatings of alternately higher and lower refractive indices nh and n1 are produced; see Fig. 2.17. If each single layer has a constant optical thickness dh = λ/4nh or dl = λ/4nl , then the reflected waves of first order interfere constructively with the incoming light caused by the additional phase jump at the boundary of the optical denser medium. Therefore, a narrowband mirror results; the reflectivity depends on the difference of the refractive indices of the layers [17]. With change of the layer parameters, the width of the wavelength interval can be controlled. These considerations are also used to achieve suitable layer structures in pearlescent and interference pigments to produce a dominant color component. Interference colors can be further improved by multiple reflection of light between two semi-transparent mirrors at distance d, similar to the placement in a so-called Fabry–Pérot interferometer; see Fig. 2.18. One part of the reflected light is transmitted through the second layer. Multiple reflection leads to a great number of partial beams, which interfere behind the second layer. The layer distance d and the refractive index n of the medium between the layers are chosen in such a way that only a small wavelength band is transmitted; this construction

2.1

Optical Radiation Sources and Interactions of Light

37

Fig. 2.17 Layer sequences of a dielectric mirror with high and low refractive indices nh and n1

nh nl nh nl

dh =

λ 4nh

dl =

λ 4nl

Substrate

operates as an interference filter. For fixed distance d, this construction is also called an etalon. The deciding parameters for the interference efficacy are n and d of the intermediate layer. Because adjacent beams have the same geometrical optical path difference, constructive interference is given by zλ = 2nd cos αz ,

z = ±1, ± 2, ... .

(2.1.20)

The intensified wavelengths become, therefore, smaller with increasing angle of beam incidence α z . For a layer thickness d = 300 nm and refractive index n = 1.5, the wavelengths of violet and red light (400 and 700 nm) are separated for the first order by an angle difference of about 25◦ . Founded on this principle, d

L

LQ P

Q1 n Q

αz

Fig. 2.18 Fabry–Pérot interferometer: the partial waves of points Q, Q1 produce interferences of “equal inclination”

38

2 Light Sources, Types of Colorants, Observer

in particular, is the angle-dependent color effect of multi-layered interference pigments in liquid crystal structures; see Fig. 2.43.

2.1.7 Diffraction from Transmission and Reflection Gratings When light is incident on objects or boundaries with dimensions on the order of the wavelength of the light, the wave characteristics of light become very important. This is the situation, for example, at a diffraction grating. Upon passing through the grating, some portion of the light is deflected and/or is fanned out depending on the grating geometry and the wavelength. This phenomenon is an example of Fraunhofer diffraction if the light beam is incident nearly perpendicular to the grating [19]. In addition to macroscopic structured materials, diffraction pigments can be impressed with such grating structures. This leads to color effects which are also depending on the angle of observation. A simple macroscopic transmission grating is shown in Fig. 2.19, not to scale. The geometry is simply achieved by scratching or etching of equidistant grooves in a thin plane parallel glass sheet. With recent techniques, diffraction pigments can be impressed with nanostructures. Each constant groove gap d, also called grating period, contains a narrow light-transmitting slit. The reciprocal quantity of the grating period, g = 1/d, is denoted as the grating constant; its units are given in lines per millimeter (l/mm). At each of the slits – which are spaced with typically as much as 5,000 l/mm or more – the incident wave front

αz

αi

d

αi G1

αz G2

Fig. 2.19 Diffraction conditions of an optical transmission grating (schematically)

2.1

Optical Radiation Sources and Interactions of Light

39

is diffracted according to the Huygens principle, which means that each point of a wave front is assumed to be the origin of a new elementary wave [19]. Because diffraction occurs at all illuminated grating slits, a pattern of constructively and destructively interfered waves develops behind the grating. The path difference G1 + G2 of waves coming from two neighboring slits follows from Fig. 2.19 as G1 + G2 = d(n1 sin αi + n2 sin αz ),

(2.1.21)

where n1 and n2 stand for the refractive indices in front and after the grating and α i for the angle of incidence. The diffraction maxima occur for a path difference G1 + G2 equal to an integer multiple z of the wavelength λ. With Equation (2.1.21), the relation zλ = d(n1 sin αi + n2 sin αz ), z = 0, ± 1, ± 2, ...,

(2.1.22)

follows. The zeroth-order diffraction, equivalent to z = 0, contains all wavelengths of the irradiated light. Higher diffraction orders for z = 0 are generated on both sides of the zeroth order with symmetric intensity maxima and minima. The wavelengths contained in the incident light fan out by the grating in such a manner that the condition in Equation (2.1.22) is fulfilled. The grating deflects large wavelengths stronger than shorter wavelengths; this is the converse of a prism. Moreover, the diffraction spectrum of the first order is of higher intensity compared to that of a prism spectrum. In order to avoid an overlapping of the diffraction maxima of higher order, the incident angle α i and the grating period d can suitably be tuned with each other (see below). In addition to transmission gratings, reflection gratings are also commonly used – these are especially realized in diffraction pigments. These can be formed by vapor coating both sides of a nontransparent grating structure with an additional layer of a reflecting metal, cf. Fig. 2.20; this has the added bonus of a possible improvement in the mechanical stability of the thin plates. The waves reflected from the periodically arranged mirrors interfere with the incoming waves from the opposite direction. Consequently, the diffraction pattern already known from transmission gratings develops in front of the reflection grating. The cross section of the grooves can be produced as rectangular, triangular, or sine shaped. Reflection gratings are used in modern spectrophotometers and optimized diffraction pigments, among other things. Examples of diffraction particles are shown in Figs. 2.54 and 2.55. Concerning the reflection grating in Fig. 2.20, the path difference G1 – G2 of the reflected waves from adjoining grooves follows from the expression G1 − G2 = d( sin αi − sin αz ) .

(2.1.23)

40

2 Light Sources, Types of Colorants, Observer

Fig. 2.20 Diffraction conditions of an optical reflection grating (schematically)

αi G2

αi d

αz G1

αz

In analogy to the transmission grating, the diffraction maxima are given by zλ = d( sin αi − sin αz ) ,

z = 0, ± 1, ± 2, ... .

(2.1.24)

For z = 0, it follows α i = α 0 and the grating operates like a normal flat mirror, that is, the wavelengths are not separated. Generally, the use of gratings results in spectra of higher light intensity in comparison to prisms, but the incident intensity is distributed among all diffraction orders. Among them, is the maximum for z = 0, which is of highest intensity, and also generally not of interest (see Fig. 2.21). This disadvantage can be bypassed using a so-called echelette grating5 ; see Fig. 2.22. On the basis of the special geometry of such a grating, nearly the entire intensity of the diffracted light can be concentrated into a particular diffraction order, as required. To achieve this aim, the cross section of the grooves has to meet the following requirements: 1. The desired diffraction order zB is not zero; the reflecting grating elements should be tilted by an angle α B above the grating base; this angle is called the blaze angle, after the corresponding production method known as the blaze technique.

5 echelette:

French, small ladder.

2.1

Optical Radiation Sources and Interactions of Light

41

Δα

Δβ

Fig. 2.21 Angle differences of diffraction maxima

2. The angle half-width α of the diffraction fringe, caused by this kind of grating element, has to be matched with the angle distance β between two successive diffraction maxima; see Fig. 2.21. In view of the first condition, it is useful to consider the angles from the grating base normal. The blaze angle is, therefore, given by the simple relation  αB = (αi − αz ) 2,

(2.1.25)

where α i and αz are the angle of incidence and the diffraction angle of order z, respectively. In addition, these two angles are combined with Equation (2.1.24), to give zB λB = d · (sin αi − sin αz ) , Grating normal

zB = 0, ± 1, ± 2, ... .

Face normal

αB αi αz b

αB d

Fig. 2.22 Echelette grating with blaze angle α B

(2.1.26)

42

2 Light Sources, Types of Colorants, Observer

The quantity λB is called blaze wavelength and zB denotes the blaze order. According to Equation (2.1.25), the blaze angle depends on the incident and the diffraction angles. For this reason, echelette gratings can be realized with different blaze angles. From the last two expressions, the condition zB λB = d · [ sin αi + sin (2αB − αi )] ,

zB = 0, ± 1, ± 2, ...

(2.1.27)

follows. It is independent of a diffraction angle. For improved understanding, we differentiate now between two types of light incidence – from which consequently follow different blaze angles. In the first case, we assume an illumination in the direction of the grating normal; see Fig. 2.22. In this case, the angle of incidence is α i = 0 and it follows from Equation (2.1.27):   1 zB λB (2.1.28) αB = arcsin , zB = 0, ± 1, ± 2, ... . 2 d In the second case, the light is incident in the direction of the face normal, the so-called Littrow or autocollimation configuration. In this case, then the incident angle is equal to the blaze angle α i =α B given by   zB λB (2.1.29) αB = arcsin , zB = 0, ± 1, ± 2, ... . 2d Note that because an echelette grating of sine-shaped cross section can be approximated by a series of successive isosceles triangles, the outlined considerations are also applicable to that geometry. From the second condition above, some further geometrical conclusions result. With the designations given in Figs. 2.20 and 2.22, the relation cos αz b = d cos (αz − αB )

(2.1.30)

can be derived. In other words, the triangle geometry depends on the blaze angle as well as the angle αz of diffraction order z. Additionally, according to Equation (2.1.30), the width b of a step is specified with the groove distance d. If the illumination is carried out parallel to the grating normal, from Fig. 2.22, we have the intermediate result α z = 2α B . The condition b=d·

cos 2αB cos αB

(2.1.31)

therefore follows. On the other hand, for illumination parallel to the face normal, the condition α z = α B is given and the simple formula b = d · cos αB

(2.1.32)

2.2

Absorbing Colorants

43

follows from Equation (2.1.30). In this case, the triangle of the echelette grating is right angled. The utilizable wavelength range of such kinds of gratings is roughly limited to the interval 0.7λB ≤ λ ≤ 2λB . Absorption and scattering dissipation reduce the efficiency of a blaze grating to about 70% [20]. Accordingly, the blaze technique enables the focusing of about 70% of the influx into a desired z = 0 diffraction order. Each diffraction pigment is normally optimized with regard to the first diffraction order z = ±1. This diffraction spectrum can be observed symmetrically with respect to the direction of illumination, and is followed by further orders of lower intensity, cf. Section 3.5.5. These principles apply not only to macroscopic gratings but also to diffractive pigment particles; see Figs. 2.55 and 2.56.

2.2 Absorbing Colorants After considering different light sources and basic light interactions in the previous section, the second fundamental component for color producing of non-self-luminous colors is the color sample containing the colorants. Colorants can be divided into two groups, depending on the dominant mechanism of color production: classical absorption colorants and modern effect pigments (cf. Table 1.1). Industrially applied absorption colorants consist mainly of synthesized colors of inorganic and organic compounds, and in some rare cases of modified natural colors. In this section, we characterize dyes and absorption pigments, describe the most important color attributes and the accompanying coloristic properties as well. Some of these properties are correlated with specific spectral features of the corresponding coloration. Additionally, color-order systems will be outlined. In some application cases, such systems offer an overview of the diverse-generated absorption colors. Each of these color order systems is grouped according to preset characteristic criteria. On one such system is based a special CIE color space. It is also shown that the color impression results from light interactions in the volume as well as at the surface of a color pattern. Properties of effect pigments are described in the next section, which show a typical color development in dependence of the angle of observation, a dependence that is absent in absorption colorants.

2.2.1 Types and Attributes of Absorbing Colorants According to a rough classification, colors can be produced by 15 different physical mechanisms [21]. In the case of non-self-luminous colors, these processes are entirely attributable to energetic interactions of electromagnetic waves with the bounded electrons of atoms, molecules, particles, or crystallites of the color-producing material. With regard to absorption colorants, we first outline

44

2 Light Sources, Types of Colorants, Observer

some typical phenomenological properties of absorption pigments and afterward those of dyes. The color origin of absorption pigments can be reduced to absorption and scattering processes. Such kinds of light interactions are called selective if they are effective in a narrow range of wavelengths in the visible spectrum. Light is scattered by pigment particles if at least two conditions are met: first, the particle dimensions and the wavelengths of the irradiated light are on the same order of magnitude; second, the colorant molecules show spatially separated electric charge distributions – the so-called multipoles. Details of these are given in Sections 5.1.2 and 5.1.3. Chromatic absorption pigments are also called colored pigments or sometimes merely pigments. They consist of inorganic and organic compounds which, in either case, form crystallites of sometimes different types. The crystallite dimensions are typically of the same order of magnitude lesser as the wavelengths of light. Because of the charge separations in multipoles, the morphological non-uniform crystallites sometimes conglomerate to form differently shaped particles with sizes from 10 nm to as much as 1 μm in size. All sorts of pigments – absorption as well as effect pigments – are insoluble in a binder or any polymer matrix. The pigments are usually uniformly dispersible in such materials although some need a suited additive for that. These colorants are used not only, for example, for mass coloration of plastic materials, fibers, paper but also for coloring of lacquers, pastes, and coatings of solids of quite different surfaces. In synthetic high polymers, the pigments tend to congregate in the amorphous regions and, therefore, additionally change the mechanical, thermal, or even electrical properties of the compound. Of great importance – with regard to color physical applications – are the covering capacity (also called hiding power), color strength, and tinting power. Requirements for optimized incorporation of pigments are sufficient wettability, dispersibility, as well as compatibility of the binding materials used, polymers, or additives, among other things. Inorganic pigments consist of oxides, sulfides, sulfates, silicates, chromates, carbonates, and metal complexes, for example [22]. Compared to organic pigments, they are preferentially used on account of their distinct optical scattering power and the typical high hiding power. In comparison with organic pigments, due to their simple and stable molecular structure, inorganic pigments tend to have better rheological behavior as well as increased weather resistance. Characteristically, inorganic pigments are mostly dull colors with lower color strength but with greater hiding power compared to organic colorants. A multitude of absorbing colorations with desired color properties are, therefore, achieved only with mixtures of inorganic and organic pigments. With regard to the chemical structure, organic pigments are divided into azopigments, polycyclic pigments, and anthraquinone pigments. The outstanding feature of organic pigments is their distinctive colorfulness or chroma in comparison to inorganic pigments. This generally comes with a high color strength.

2.2

Absorbing Colorants

45

Compounds of polycyclic structures, compared to molecules with azo-groups, tend to disperse better in polymer binders, have a lower migration tendency, as well as higher weather durability [23]. Table 2.3 shows the most commonly used modern inorganic and organic absorption pigments, differentiated with regard to the kind of the dominant color production mechanism. White and black pigments assume a special role in industrial color physics. White pigments stand out due to their nearly ideal scattering of light in the visible range. They are preferentially used for pure white, increase in the lightness of any coloration, or covering of a background. On the other hand, the characteristic feature of black pigments is the dominant absorption. These pigments are applied mainly in tiny amounts for black colors and for darkening of absorption colors. With calibration series of white and black pigment mixtures, the calibration of the lightness scale is, therefore, carried out between minimal scattering (pure black pigment) and maximal scattering (pure white pigment). An analog consideration holds for the case of absorption (Section 6.1.1). It is further the case that a really tiny amount of a black pigment improves the interference color of transparent pearlescent or interference pigments; this is because the black amount absorbs the complementary color produced of interference pigments. Table 2.3 lists also fluorescence and phosphorescence pigments. These are summarized by the term luminescence pigments. Both sorts absorb light normally at lower wavelengths in the range of X-rays, UV, or the visible spectrum; they often also absorb energy from electron beams. After some short time, a part of the absorbed quantum energy is reemitted but with longer wavelengths (lower energy) according to Equation (2.1.2). Fluorescence radiation is emitted spontaneously and phosphorous radiation, in particular, has a delay in the reemission. In contrast to phosphorescence (Section 4.2.6), fluorescence emission stops immediately after illumination. Unlike absorption pigments, dyes are completely soluble in a solvent or in a polymeric medium. The coloristic properties are essentially based on absorption. On account of the absence of scattering, dyes are normally transparent – notable exceptions are dark dyes in high concentrations. Dyes consist nearly exclusively of organic chemical structures. Typical groups of atoms with covalent bonds such as >C=C<, >C=O, or –N=N– (ethylene, carboxyl, or azo group, respectively), called chromophores, are typically responsible for the coloration of the organic compounds. The π-electrons6 of these covalent bonds absorb a fraction of the incoming light waves. The incorporation of further molecular groups, the so-called auxochromes, causes a color shift either to

chemical compounds π -electrons form a pair of electrons which belong together to two different atoms.

6 In

Color production

Selective scattering/ selective absorption

Pigment

Absorption pigment, colored pigment Oxides Iron-II-oxide, iron-III-oxide, chromium-III-oxide, chromium-IV-oxide, chromium-VI-oxide pigments Mixtures with oxide pigments: cobalt blue, cobalt green, zinc iron brown Chromate pigments Chromium yellow, chromium orange, chromium green; chromium titanate, molybdate red, molybdate orange; copper chromate Iron blue pigments Iron cyanide blue Cadmium, bismuth pigments, ultramarine pigments

Inorganic

Typical examples

Table 2.3 Typical modern inorganic and organic absorption pigments

Azo pigments Mono-pigment, disazo-pigment, β-naphthole pigment, naphthole AS pigment, benzimidazolon pigment, disazo condensation pigment, metal complex pigment Polycyclic pigments Phthalocyanine, carbazole, quinacridone, perylene, perinone, pyrrolo/pyrrole, thioindigo pigments Anthraquinone pigments Anthrapyrimidine, flavanthrone, pyranthrone, anthanthrone, dioxazine, triarylcarbonium, quinophthalone pigments

Organic

46 2 Light Sources, Types of Colorants, Observer

Color production

Scattering

Absorption

Selective absorption, emission within 10 ns

Selective absorption, emission after 1 ms

Pigment

White pigment

Black pigment

Fluorescent pigment

Phosphorescent pigment

Pure phosphors Alkaline tungsten salts Other phosphors Alkali halogenoids, alkaline earth oxides, alkaline earth sulfides, Barium oxide: doped with Na, Mn, Ce, Sn, Cu, Ag; zinc phosphide, cadmium phosphide, gallium sulfide, zinc sulfide, cadmium selenide, gallium phosphide

Fluorite, uranylic salts, salts with metals of rare earths

Carbon black, iron oxide black, copper chromate

Titanium dioxide (anastas, rutil), zinc sulfide, zinc oxide pigments

Inorganic

Typical examples

Table 2.3 (continued)

Pure phosphors Carbazoles Other phosphors Fluorescent organic substances, embedded in crystalline matter

Oxinaphthaldazine, disazomethine

Aniline black

White plastic powders

Organic

2.2 Absorbing Colorants 47

48

2 Light Sources, Types of Colorants, Observer

higher wavelengths as with –NH2 and –OH groups (amino and hydroxyl groups, respectively) or to lower wavelengths with –NH–CO–CH3 (ethylene-acid-amide group), for example. Dyes are subdivided into natural and synthetic compounds; see Table 2.4. Dyes found in nature are generally known by common names like indigo, crimson, saffron, or alizarin, although they are prepared partial artificially today. For reasons of product constancy, generally only synthetic dyes are applied. Among synthetic dyes are azo-dye, anthraquinone dye, indigo dye, cationic dye, phthalocyanine dye, polymethine dye, triphenylmethane dye, xanthene dye, and fluorescence dye [24, 25]. Owing to their molecular double bond structures, dyes show a poorer fastness to light and weather resistance in comparison to pigments. The basic molecular interactions of fluorescence dyes and pigments are described in the literature [26]. Table 2.4 Typical examples of natural and synthetic dyes Dye

Typical examples

Natural

Indigo, crimson, saffron, alizarin

Synthetic

Azo-dye, anthraquinone dye, indigo dye, cationic dye, phthalocyanine dye, polymethine dye, triphenylmethane dye, xanthene dye, fluorescence dye

2.2.2 Pigment Mixtures and Light Transmittance The wavelengths of the visible range which are not absorbed by colorants make up a so-called complementary color. This color is the perceived color of the absorbing dye or pigment; see Table 2.5. It can be seen in the table Table 2.5 Spectral range of absorption, light color, and perceived complementary color of absorption colorants Rough spectral range of absorption/nm

Light color of the spectral range

Perceived complementary color of the colorant

380–440 440–480 480–490 490–500 500–560

Violet Blue Green blue Blue-green Green

Yellow-green Yellow Orange Red Crimson

560–580 580–595 595–605 605–750 750–780

Yellow-green Yellow Orange Red Crimson

Violet Blue Green blue Blue-green Green

2.2

Absorbing Colorants

49

that the light and complementary colors reverse their roles at a wavelength of about 560 nm. Over and above that, in Table 2.19 are given some pairs and triples of complementary colors; some of them are partly demonstrated in Color plate 1.7 These relationships concerning complementary colors are valid for uniform spectral power distributions of the irradiated light similar to a xenon discharge lamp. Complementary colors are arranged opposite (according to Hering color opponent theory) in a color plane of the CIELAB system; see Color plate 4. As mentioned in the previous section, the realization of a desired color is normally achieved with mixtures of different sorts of colorants. Apart from coloristic aspects, concerns over the compatibility and color fastness of the components are in the forefront. Colorations of lacquers, plastic materials, or textile fibers consist usually of mixtures with three up to six or more different colorants. Mixtures of different fluorescence dyes can be distorted by fluorescence extinction if the quantum emission of one dye in the visible range causes a fluorescence stimulation of a second dye in this range. For the realization of a great number of different color shades, a selection of at least 25 up to about 130 chromatic, white, and black absorbing pigments has to be made and as much as of 110 effect pigments. Among the colored absorption pigments, normally yellow and red should preponderate in comparison to violet, blue, or green shades; additionally, two to four brown colorants are normally taken into consideration. Similar considerations should be taken into account for dyes. The predicted color should match the reference color as exact as possible. In the majority of cases, there are additional properties, such as high color constancy or low metamerism, which need to be fulfilled. It is, therefore, necessary to have a broad experience with the composition of the colorant components and binding materials for mixing in order to achieve the coloristic, processing, or basic performance conditions required with regard to the reference color. Color samples are termed as transparent, translucent, or opaque in dependence on their light transmittance properties. Requirements for transparent colors are not only pure absorption and complete solubility of the used colorants but also the refractive indices of the applied colorant matrix (e.g., solvents, binding agent, or plastic materials) have to agree with those of the used colorants. Transparent optical media are characterized by the wavelength-dependent absorption coefficient. This coefficient follows from the spectrometric measurements of the transmittance and a suitable approximation of radiative transfer in optical media (Section 5.1). A transparent chromatic layer is perceived by way of not only transmitted light but also reflected light. The accompanying reflectance comes merely

7 Color plates

are inserted between Chapters 3 and 4.

50

2 Light Sources, Types of Colorants, Observer Table 2.6 Light transmittance and spectral quantities of color samples Measuring quantities

Determination quantities

Examples

Transparent

Directed, diffuse reflectance and transmittance

Absorption coefficient K

Liquids, inorganic, organic glasses

Translucent, translucent nonopaque, translucent glimmering

Directed, diffuse reflectance and transmittance

Absorption coefficient K, scattering coefficient S, phase function p

Binders, foils of partially crystalline high polymers, paper, opal glasses

Opaque

Directed, diffuse reflectance

Absorption coefficient K, scattering coefficient S, phase function p

Emulsion paints, lacquers, plastics, textiles, ceramics, leather

Light transmittance

from reflection at the illuminated outer and inner boundary surfaces which is caused by differing refractive indices. The magnitude of the reflected portion follows from Equations (2.1.6), (2.1.7), (2.1.8), (2.1.9), (2.1.10) or (2.1.12). In Table 2.6, three different sorts of color samples with regard to the degree of light transmittance are listed along with some spectral measuring and determination quantities. Translucent color patterns such as turbid films, paper, or opal glass are characterized by selective absorption and simultaneous scattering. Translucent media are only light transmitting on a small scale; nonetheless, a colored background is not completely covered. The light transmittance decreases with increasing layer thickness, this is similar to transparent layers. Some organic pigments of light yellow, orange, or red shade are translucent as are some pearlescent and liquid crystal pigments. The scattering of light occurs at the irregular surfaces or edges of the pigment particles and caused by the electromagnetic interactions with the multipoles of the pigment particles. In general, scattering of translucent colors depends on the pigment sorts used, the chemical structure of the surrounds, the refractive indices, and the structure of the boundary surfaces. Translucent colorations are particularly of interest for such coloristic applications, in which the same hue is to be realized simultaneously in a nearly transparent and almost opaque finish. In addition to reflectance and transmittance, the characteristic quantities of translucent color samples are turbidity and covering capacity (Sections 3.4.3, 3.4.4, and 3.4.5). The theoretical modeling of radiative transfer in translucent systems can actually be quite difficult. These complications are caused by the combined processes of absorption and scattering, where the latter is anisotropic in some

2.2

Absorbing Colorants

51

cases. On the basis of the different optical effects, translucent materials can be subdivided into the following groups: ideal translucent: a directional ray is partly absorbed and diffuse scattered; it exits the medium with lower intensity; translucent diaphanous: the primary ray in the optical medium is surrounded by a halo8 ; the primary ray is more attenuated than in the ideal translucent case; translucent gleaming: the energy of the primary ray is spread out in an intense halo; there is virtually no primary ray remaining at the second boundary surface. These different translucent states are best modeled with a multi-flux approximation. For this, in addition to absorption and scattering coefficients of the optical medium, some additional optical quantities such as the phase function p, for example, need to be taken into consideration (Section 5.1.5). The majority of natural and artificial non-self-luminous colors are opaque; they, therefore, appear impenetrable from the rear – a background is covered completely by this kind of colors. In all opaque systems light scattering is normally the dominant process, and it is accompanied by more or less distinct absorption. Scattering occurs isotropically without a preferred direction or, the other extreme, anisotropic with a specific preferred direction. Increasing scattering lowers the light transmittance and vice versa. The covering capacity of a coloration is, above all, caused by the scattering power of the pigment particles. The single spectrometric measuring quantity of opaque materials is the reflectance. The reflectance, in general, depends on the measuring angle for many kinds of effect pigments. Extreme optical states result in mixtures of absorption and effect pigments with different light transmittance. These are most often implemented in lacquers or plastic materials for consumer goods.

2.2.3 Description of Color Attributes Based on the discussion in the previous sections, it should be clear that the color of absorbing colorants is caused by light interactions such as scattering and absorption. On the other hand, the produced visual color sensation is by no means completely characterized by those. The variety of different combinations of wavelengths as well as the accompanying power distributions reaching the eye results in many colors, which can be described by different color attributes. For better understanding of the context, we use an analogy from acoustics. 8 This is a circular light spot encircling the primary ray; the halo of the Moon, for example, results of light refraction at the ice crystals in the atmosphere.

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2 Light Sources, Types of Colorants, Observer

A pure acoustic tone is unambiguous, physically described by its frequency and amplitude. The same tone, produced from different music instruments, is however surrounded by a typical sound spectrum. In a similar way, each nonself-luminous color is produced not simply by a single light frequency but also from various spectral values of the visible spectrum.9 The different wavelengths simultaneously entering the eye are perceived as a single color impression and not as separate single-colored wavelengths. This entire color impression can be verbally described by characteristic terms. The description or comparison of colors is used in color industry not only from physical or colorimetrical points of view but also according to coloristical attributes [27]. The following enumeration, which is not to be considered complete, gives some conventional terms for characterizing the color impression of non-selfluminous colors. The accompanying definitions and assemblies are, however, not uniformly used in the literature: hue, shade: the color property which is mainly described by adjectives such as red, green, yellow, blue; hue and shade are terms equivalent to color tone, or tint, see below; relative color strength, relative tinting strength: the color economy of an available colorant material relative to an arbitrarily chosen colorant of equal or similar color; color depth, color intensity: the color distinctiveness, which increases with enhancement of the same colorant amount; dullness: a characteristic color feature which can be described by an existing amount of gray or black; it is the converse to brightness. The quantitative determination of relative color strength is standardized based on colorimetric criteria (Section 3.4.2). These instructions underlie the formalism of the CIELAB or DIN99o color spaces. The following terms are also used to denote a color point in color space: lightness: a measure for the reflectance of a non-self-luminous colorant; the reflection ratio of absorption colorants is standardized by the so-called gray scale in which black is assigned to the value of 0 and white to the value 1; fluorescent and effect colorants often have reflectance values greater than 1, caused by the arbitrary fixing of the gray scale; selfluminous sources such as light sources or the sky are characterized by the term brightness rather than lightness; saturation: a quantity describing the colorfulness of colorants which does not change by further increase of the colorant concentration; shade, color tone: this corresponds to hue; equivalent terms are tint or tinge; 9 This metaphor is not to be confused with synesthesia, in which a physiological stimulus induces a further stimulus, for example, colors are associated with music and vice versa.

2.2

Absorbing Colorants

53

chroma: an absorption colorant loses colorfulness with lower saturation; chroma depends on both saturation as well as lightness because a tiny amount of black reduces the colorfulness of a color.10 Common for characterization of color impressions are also adjectives and their comparative forms; their meaning is somewhat equivalent to the terms already given above. Here are listed properties in opposing pairs: colored, chromatic: any color, such as yellow, orange, red, or green; it is analogous to the term hue; uncolored, achromatic: colors like white, gray, or black; light, bright: a color with a high lightness amount; light colors have a high reflectivity; dark: colors of small or even zero reflectance; brilliant, clear, pure: colors of both high hue and high lightness but missing white amount; pale, dirty, dull: colors of both minor hue and minor lightness. A great number of non-self-luminous colors containing absorption colorants can be directly manufactured with properties in the ranges of the last 12 listed extreme color states. Some of the cited characteristics correlate also with typical special features of the corresponding spectral reflection or transmission curve traces. The measured spectrum can sometimes be interpreted as a kind of “finger print” of the accompanying color. Some of these properties can be visualized by characteristic spectral reflection curves of absorption pigments, for example, in Figs. 2.23–2.27. Some of these features can also be found by the angle-dependent spectral reflection of effect pigments. The spectral reflectance of a chromatic absorption color is generally characterized by either a significant maximum (e.g., violet, blue, green) or a steep rise to higher wavelengths (e.g., yellow, orange, red) in the wavelength range of the complementary color or rather natural color; see Fig. 2.23. The highest absorption exists in a wavelength range of a distinct reflection minimum; backscattering, however, dominates in the range of maximal reflection or the plateau region of the relevant pigment. In contrast, an achromatic color shows a nearly constant spectral reflectance; see Fig. 2.24. Ideally scattering white is represented in the visible range by a constant reflection of 1.0 and ideal black of 0.0; gray takes intermediate values depending on the content of white or black. The decrease of spectral reflection for white or gray with shorter wavelengths is due to the tail of the UV absorption of the underlying TiO2 pigment. The impression of white results exclusively 10 These four color terms can be visualized by geometrical quantities in both mentioned color

spaces and further ones; see Sections 3.1.3 and 3.1.4.

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2 Light Sources, Types of Colorants, Observer

R(λ) c 0.8

d e

0.6 b a 0.4

0.2

0

400

500

600

λ

700

nm

Fig. 2.23 Spectral reflectance curves of chromatic pigments: (a) blue, (b) green, (c) yellow, (d) orange, and (e) red

R(λ)

a 0.8

0.6 b 0.4

0.2 c 0

400

500

600

700

λ

nm

Fig. 2.24 Spectral reflectance curves of achromatic colors: (a) white, (b) gray, and (c) black

2.2

Absorbing Colorants

55

R(λ) 2bt 0.8

2dk

1bt

0.6

0.4 1dk 0.2

0

400

500

600

700

λ

nm

Fig. 2.25 Spectral reflectance of bright and dark pigments: 1bt bright blue, 1dk dark blue, 2bt bright yellow, and 2dk dark yellow

R(λ)

0.8 a 0.6 b 0.4 c 0.2

0

400

500

600

700

λ

nm

Fig. 2.26 Spectral reflectance of different gray colors: (a) light gray, (b) middle gray, and (c) dark gray

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2 Light Sources, Types of Colorants, Observer

R(λ) 2br 0.8 2du 1br 0.6

0.4 1du 0.2

0

400

500

600

700

λ

nm

Fig. 2.27 Spectral reflectance of brilliant and dull pigments: 1br brilliant blue, 1du dull blue, 2br brilliant yellow, and 2du dull yellow

from regular light scattering over the entire visible range. This is the case for examples such as water vapor, clouds, or snow. Ideal black, on the other hand, is caused by absorption of entire visible wavelengths. Light (or dark) colors stand out either due to a high (low) reflection maximum or a widened (narrowed) plateau region; a light (dark) gray shows a shift of the reflection level to white (black); see Figs. 2.24 and 2.26. Most of the color-order systems and the colored calibration samples for recipe prediction are based on colored pigment mixtures with white and black colorants. It is remarkable that the human eye provided by nature is more sensitive to color differences between dark colors (violet, blue, green) in comparison to light colors (yellow, orange, red). This is correlated with a low maximum respective high reflectance plateau range. The higher and steeper the rise of the curve to the plateau region or the smaller the half-width of the reflection maximum, the more the color is perceived as brilliant, clear, or pure. Conversely, pale, dirty, or dull colors possess a spectral reflection characterized by a lower and broader maximum or overlapping maxima, compare in Fig. 2.27 curve 1br with 1du for a blue pigment. The spectral reflectance of a dull yellow, for example, shows a more flat peak profile and a lower plateau range compared to a brilliant yellow, cf. curves 2br, 2du in Fig. 2.27. The terms outlined up to now are merely used to describe some typical coloristical properties of classical absorption colors or colorants. This vocabulary

2.2

Absorbing Colorants

57

cannot be directly carried over to effect pigments. Rather than absorption and scattering, effect pigments produce new sorts of colors on account of different optical mechanisms; identical terms have, therefore, different meanings for effect pigments than for absorption colorants. Metallic pigments, for example, produce a characteristic metallic lightness, brilliance, or hue extinction. This ambiguity is discussed in Sections 2.3.3 and 3.5.1.

2.2.4 Color-Order Systems It is quite remarkable that the color sense of humans is capable of distinguishing about 10 millions of colors. It is further important to note that, in addition to the astonishing number of colors, a color can have different attributes; therefore, it seems essential to have systematic classifications for such a tremendous diversity of colors. Such classification systems serve not only as an overview but also as an improved communication about non-self-luminous colors, particularly for their use in difficult applications. From each of the established color-order systems, series of color samples exist for visualization. The antiquated color classifications are certainly not ordered after uniform criteria [28, 29]. The chronologically first systems were arranged based on features, which included the actual knowledge about colors. Among the contemporarily used color-order systems are to differentiate roughly two groups: systems on coloristic and on colorimetric basis. In the following, we sketch the common four representative examples of both groups. The most important coloristic-order system is that of Munsell, established in 1905. It is simultaneously comprised of various companion features such as, for example, precise nomenclature or systematic extension capability for additional colors [30]. The corresponding color patterns are ordered according to the three properties hue H, value V, and chroma C in three dimensions in a cylindrical coordinate system. The true significance of this collection are the colored patterns which display a nearly constant color difference between neighboring colorations. This was developed without any colorimetric background – an amazing for the beginning of the 20th century. In spite of the visually nearly equal color steps, this color-order system was not accepted until 1976 as a basis of the CIELAB system. Experience with the Munsell system has brought out critical disadvantages. Particularly unfavorable are the non-uniform perceived color differences of bright and dark colors. In remedy of these shortcomings, the Munsell system was supplemented by an additional color pattern in 1990. The natural color system (NCS) was worked out in Sweden and is standardized there; it is based on the opponent color theory of Hering and is arranged in pairs of the four chromatic colors red, green yellow, and blue as well as the achromatic colors white and black [31]. The patterns are organized in color

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2 Light Sources, Types of Colorants, Observer

spectra and color triangles after the three criteria blackness s, colorfulness c, and shade Φ. They are grouped in the form of an atlas of 1,741 color patterns. In spite of this great number, the NCS system is restricted in its applicability. It incompletely covers the color space, and additionally, colors with glossy surface are inapplicably marked. Moreover, the arrangement pattern is visually non-uniform. Without a measuring device, this color system is, therefore, only suited for rough orientation in color space. The so-called color index (C.I.) is suited for identification of colorants but represents no real color-order system as in the discussion above. It simply consists of a list with unsystematically assembled dyes and absorption pigments. These are partially denoted according to the origin (generic names) and according to the chemical constitution with natural numbers (constitution numbers) [32]. This listing offers a certain aid only in a few cases, if, for example, some of the colorants listed in the C.I. are to exchange with one another. Strictly speaking, the widely distributed color registry RAL 840 HR in Germany is also not a color-order system in a broader sense. This registry was created to ensure that industrial colors are classified uniformly with defined names and numbers for better communication between public agencies and producers of consumer goods. The color patterns were originally vaguely arranged in view of coloristic criteria; in the meantime, a modification was undertaken using the so-called DIN color chart, see below. It should be added that contemporary color systems also consist of patterns which change uniformly by a suitable technical parameter such as the concentration of the used colorants, for example. The corresponding inconstant changes of hue, lightness, or chroma certainly make color perception and color comparison more difficult. A visual ascertainment of color differences is not ensured by such systems. In contrast to the order arrangements above, the DIN color chart is a colorimetric-based color system [33]. It comes from the CIE chromaticity diagram; see Section 3.1.2 and Color plate 2. The color patterns are arranged with regard to three properties, similar to the Munsell system. These color properties are denoted as darkness D, hue T, and saturation S. They correspond to the ∗ , and chroma C∗ of the CIELAB system (Section 3.1.3). lightness L∗ , hue Hab ab The OSA-UCS system (Optical Society of America Uniform Color Scales) is principally used in the USA. It consists of a total of 588 color patterns, from which 12 colors at a time are positioned in the corners and one in the center of a cubic octahedron [34]. With this system, a multitude of visual equidistant color scales can be established; visually equidistant means that adjacent color patterns always show the same visually perceived color difference independent of color point in color space. This system cannot be directly transformed into the CIELAB system; the same holds for the Munsell system, NCS system, and the DIN color chart. A reliable determination of color differences is impossible on the basis of the OSA-UCS system.

2.2

Absorbing Colorants

59

Accordingly, it is advantageous to fall back on a universal applicable colororder system, which, for chromatic as well as achromatic colors, has the implicit metric of a color space. The RAL design system is organized on such considerations. It is based on the structure of the CIELAB system. The RAL design system enables, with good approximation, the visual determination of color differences. The corresponding color collection is established as RAL design atlas [35]. There exists also a collection of 70 RAL effect colors, which cannot be classified as a definite color-order system.

2.2.5 Surface Phenomenon The color attributes assembled in the next to last section are more or less an expression of a subjective color perception. As already mentioned in Section 2.1.5, the entire color impression is certainly composed of at least two components: the light interactions in the volume and at the boundary surface of a colored sample. In other words, the reflection or transmission from the volume is superimposed on the boundary surface reflection. This reflection is caused by different refractive indices at the boundary surface. Such kinds of boundary surfaces are present in paints, coatings, plastic materials, emulsion paints, or ceramics. This distinction cannot be made for undefined surfaces such as of textiles, uncoated papers, plasters, or suede leather. The kind of surface reflection is also influenced by the structure of the surface. A given color can exhibit a surface structure between two extremes: matt: appears a rough (or structured) surface; this appearance is caused by completely (or partly) diffuse surface reflection; high-glossy: appears a very smooth reflecting surface; such a surface induces only directed reflection, which is called specular reflection, specular gloss, or simply gloss. This behavior is described by the reflection law. Both the above surface properties should generally be interpreted as color attributes. For the illumination of a (color) pattern with directional light, the transition between matt, half-matt, glossy, or high-glossy surfaces can be characterized by the resultant diffuse or directed reflection with the aid of the reflection indicatrix. This is a plane polar diagram which represents the angle-dependent intensity distribution of the light reflected from the surface. In reality, an indicatrix is in three dimensions. Four different indicatrices are shown schematically in Fig. 2.28, each caused by a different sort of surface reflection. The diffuse and directional reflection amounts are given by the shape of the envelope curve and enveloping surface of the intensity distribution.

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2 Light Sources, Types of Colorants, Observer

Fig. 2.28 Reflection indicatrices of different surfaces: (a) matt: purely diffuse reflection, (b) half-matt: dominant diffuse, minor directed reflection, (c) glossy: predominant directed and minor diffuse reflection, and (d) high glossy: exclusive specular reflection

A diffuse reflecting surface of high roughness is achieved by various treatments, for example, with the help of suited polymer matrices or ceramics or pigment crystallites standing out non-uniformly from the boundary surface. In addition, the use of embossing dies is possible with regular geometrical surface forms such as micro-spheres and micro-prisms or irregular linen and other fabric structures. Further methods are sanding, etching, or physical vapor deposition (PVD). Certainly, the realization of ideal matt surfaces is exceedingly difficult. A matt surface appears normally somewhat lighter than a high-glossy surface of the same material. This is caused by the higher reflection coefficient for diffuse light in comparison to that of directional and perpendicular incident light, cf. Equations (2.1.8) and (2.1.12). The higher diffuse reflection is nearly independent of the absorption colorant sort. In the case of directional illumination, however, the reflection is only influenced by the degree of surface roughness. The matt surface of brilliant but dark-colored samples sometimes causes a minimal color shift. This phenomenon comes from the superimposed scattering of pigments near the surface. The entire diffuse reflection at the surface of a colored layer can, therefore, be composed of three components: – immediate reflection at the boundary surface; – scattering from particles near the surface; – anisotropic or isotropic scattering from the volume. The most of color physical problems are associated with the color component originated from the volume because this shows the interesting light interactions with the contained colorants. The result of a color measurement is, of course, composed of the volume and surface components. This result must, therefore, be corrected with regard to the surface boundary effect. This is achieved through the implementation of the various theoretical concepts detailed in Sections 5.3, 5.4, and 5.5, among other things. If the boundary surface contains no colorant particles, then the reflection is given by the refractive index of the binder. The pigments can, however, be

2.2

Absorbing Colorants

61

located in or near the surface, for example, by high pigment volume concentration (PVC), blooming, or flooding. In such cases, the arithmetic mean of the refractive indices of the involved materials is to be used. Such a surface boundary can have further color features, due to the fact that the refractive index of most colorants is dispersive. This effect is named bronzing; it can be present in printing inks, lacquers, plastic materials, or textiles, and can be avoided by appropriate measures. For unambiguous coloristic assessment of matt to high-glossy samples, it is absolutely necessary to avoid the glare caused by specular reflection. Therefore, the sample containing absorption colorants is always illuminated laterally at an angle of 45◦ and viewed perpendicular to the surface; see Fig. 2.29a. The reversed arrangement is also possible, but rarely used. In commercial light booths with lamps of different illuminants (normally D65, A, FL 2, FL 11 simulators), the light sources are arranged laterally and glare-free as shown in the mentioned figure. The color sample should be surrounded directly by a middle gray and the assessment should be performed by keeping the so-called CIE reference conditions, cf. Section 3.2.2. In contrast, the visual evaluation of the surface gloss is achieved with a fixed light source and constant line of vision by tilting of the sample over a specular Light source

Observer

Specular reflection

a)

Color pattern

> 60°

b)

Fig. 2.29 Two different configurations of the three factors for color impression of absorption colors: (a) for coloristical assessment and (b) for evaluation of surface gloss

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2 Light Sources, Types of Colorants, Observer

reflection angle range; see Fig. 2.29b. A light booth should, therefore, possibly have a tiltable support with angle scaling. The quantitative determination of surface reflection is carried out separately using a gloss meter; the reflected intensity is registered at several variable angles between normal to the surface and greater than 60◦ . In the case of curved substrate surfaces, for instance, car body components, cans, bottles, or tubes, it is possible to pursue simultaneously gloss, color blending, and change of gloss in dependence of the observation angle. To achieve comparable results, the radius of curvature of the substrate and the thickness of the coating have to be the same for all color samples of a collection. In contrast, coated curved substrate surfaces are absolutely unsuited for instrumental color measurements. The evaluation of the surface reflection of colorations containing effect pigments is particularly critical. The optical properties of a metallic paint coating, for example, are essentially caused by specular reflection at the flake-shaped metal particles. The flakes are primarily arranged parallel to the substrate surface. Therefore, specular reflection dominates, but it is superimposed by diffuse reflection coming partly from the rough particle edges. For metallic pigments, the angular distribution of the reflection depends also on the illumination angle as well as the angle of observation. As already explained in Sections 2.1.6 and 2.1.7, the color physical properties of effect pigments are also extremely angle dependent. Compared to absorption colorations, the visual assessment of such colorations needs a much more sophisticated procedure, cf. Fig. 2.30. The observation for effect pigments is also normally performed using a fixed light source, but the classical method needs two identical colorations, one kept horizontal and the other rotated by an angle < 45◦ from this position; see Fig. 2.30a. This procedure allows for the observation of the angle-dependent change of lightness intensity. For flake-shaped metallic pigments, this angledependent change in reflected intensity is typical; it is the so-called lightness flop. The color changes of interference and diffraction pigments between the two extreme observation angles is, however, termed as color flop. Figure 2.30b shows the modern arrangement for visual evaluation of effect colorations. In this configuration, the light source and the observer are fixed. Vertical movement of a color sample simultaneously changes both the angle of illumination and angle of observation. Therefore, the observation of the color flop, for example, needs only one color pattern and is carried out at the same surface spot. We have arrived thematically at the transition from absorption to effect pigments. These colorants have been used industrially since about 1970 and to an increasing extent. In the following sections, we discuss the structure, morphology, and the spectral properties of the most important sorts of these modern colorants.

2.3

Effect Pigments

63

Light source Observer

a)

Effect Color pattern

b)

Fig. 2.30 Two different configurations for assessment of colorations containing effect pigments with fixed observer: (a) classical method (tilting): lightness flop and (b) modern method (vertical movement): color flop

2.3 Effect Pigments Effect pigments have broken new ground in color physics, especially with regard to industrial-scale research, development, and application. The color production of effect colorants is predominantly caused by anisotropic processes like single or multiple reflection, interference, or diffraction. These processes are unrealized in absorption colorants. The generic term “effect pigment” is inadequate because colors generated by absorbing colorants are also based on an optical “effect”. However, flake-shaped pigment or shorter flake pigment is an accurate expression for this sort of colorants. The generated color impression of effect pigments is extremely angle dependent. This is a function of both the illumination and the observation direction. Effect pigments result in quite strange color

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2 Light Sources, Types of Colorants, Observer

sensations for human color sense because our sense has evolved to perceive only colors from absorption colorations. In a practical sense, effect colorants need more extravagant manufacture methods in comparison to absorption pigments and also a more extensive characterization, measuring techniques, application, and processing. All types of effect pigments consist of flake-shaped particles with a large range of typical lateral dimensions between 1 μm and 1 mm. This is more than 10 – 1000 times larger than that of absorption pigments. The flake thickness has values between 10 nm and 1 μm. On account of the flake form, the resultant color effect is increasingly distinct with a more uniform morphology and the more the particles are oriented in the binder parallel to the substrate or surface. Like absorption pigments, effect colorants can be of inorganic as well as organic nature. With regard to the processes of color production, they are divided into four groups (cf. Table 1.1). For historical reasons, they are named: – metallic pigment; – pearlescent pigment; – interference pigment; – diffraction pigment. In the literature, pearl luster pigments are often subsumed to the interference pigment classification [36]. The laws used to describe the optical properties of metallic pigments are essentially geometrical optics; all other sorts of effect pigments are generally described by wave optics. Metallic pigments consist normally of a metal or an alloy of metals. The typical metallic gloss is increasingly brilliant the more uniformly the flakes are oriented parallel to the boundary surfaces of a coating. The so-called metallic effect is mainly a consequence of the directional and diffuse reflection at the surface and the edges of the flakes. In contrast, pearlescent pigments consist of two or more layers with a high index of refraction difference; the values normally range from 1.5 to 2.9. The mostly used substrate is mica, but also metals or metal oxides are often applied. The specific pearl luster depends on the permutation of the layers. This luster originates from single or multiple reflections at the layer boundaries followed by interference of the light waves. Differing optical layers are behind the general function of interference pigments; this does not require the use of a mica substrate. An interference pigment subgroup is the so-called optically variable interference pigments. The high ratio of refractive indices, in conjunction with different layer thicknesses, fans out the first or more interference orders in such a way that a variety of interference colors are to observe angle dependence.

2.3

Effect Pigments

65

The grating structure of diffraction pigments deflects the incoming light. The resulting color effect can be attributed also to the wave nature of light. The substrate consists of a highly reflecting or even ferromagnetic substance. The substrate is vapor-coated symmetrically on both sides with several materials known from nanotechnology. The ferromagnetic particles can be oriented with an external magnetic field before the crosslinking of a binder. The produced unusual but often impressive colors require that effect pigments undergo a more subtle and closer examination and handling compared with absorption colorants.

2.3.1 Types of Metallic Pigments Metallic pigments, generally consisting of metal flakes, are employed mainly on account of the metallic reflection from the flake-shaped particles. This kind of reflection consists of superimposed specular and diffuse components which produce unusual color effects compared with absorbing pigments. Conventional metal flakes have mean lateral dimensions ranging from about 5 μm to nearly 50 μm, whereas the thickness varies between roughly 100 nm and 1 μm. In some extreme cases, the particles have dimensions which are up to 10 times higher. The ratio of thickness to diameter of the particle is called the form factor and it extends from 1:50 to about 1:500. Metal flakes are used in paints, lacquers, plastic materials, and inks; they are also employed in chemical products and for sinter metals, building materials, explosives, or pyrotechnics, for instance, as functional or chemically reactive particles. In this text, we are mainly interested in the metallic reflection. This property is caused, in simplified physical terms, by the fact that individual metal atoms can easily release the bonding electrons. In lattice arrangements, the metal atoms completely loose the valency electrons. These electrons form the electron gas which is distributed among the remaining ions so that each ion is fixed on a corresponding lattice position. On account of its interaction with the electrons, an external light wave from a normal source cannot penetrate the very dense electron gas. The majority of the light is rather reflected and the remaining part is absorbed within a very small penetration depth. Reflection and absorption produce the typical metallic brilliance and characteristic natural color of metals [14]. The change in electron gas density at the metal surface results in light dispersion in the visible range, among other things. This causes a light-, gray-, or low-colored metallic effect. The theoretical reflectivity of metals is given by Equation (2.1.10). In Table 2.7, the indices of refraction n, absorption nκ, reflection r(n,κ), and the melting temperature Tm are shown; these are for metals that are most commonly used for metal pigments. In this listing, the metals are arranged according to decreasing reflectivity. The given optical

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2 Light Sources, Types of Colorants, Observer

Table 2.7 Reflectivity, refractive index, absorption index for perpendicular incident light at wavelength of λ = 589.3 nm and T = 293 K as well as melting point of metals used for metallic pigments [37]

Metal

Reflectivity r (n, κ)

Refractive index (n)

Absorption index (nκ)

Melting point Tm /K

Ag Al Au Cu Zn Ni Fe Mo Ti W

0.99 0.912 0.888 0.804 0.768 0.664 0.586 0.575 0.565 0.524

0.052 1.181 0.280 0.493 2.74 1.71 2.91 3.40 2.09 2.83

3.91 6.99 2.91 2.80 5.77 3.61 3.58 3.56 3.11 3.02

1,235 933 1,336 1,356 693 1,726 1,809 2,890 1,933 3,683

quantities are for the middle of the visible Na wavelength of λ = 589.3 nm; they will generally have different values for other wavelengths. The melting temperatures of the given metals are higher than those of the highest flow temperature of normal polymer melts. The refractive index is also subject to dispersion; in cases of Ag, Au, and Cu it is valued n < 1. The phase velocity cp is, on account of cp = c/n, inside these three metals higher than the velocity of light c in vacuum; this is not in contradiction to the special theory of relativity because only the group velocity – with which energy or signals propagate – cannot exceed the velocity of light c. Clearly, the actual reflectivity is lower than the theoretical one. The real reflectivity depends on the details of morphology of the particles, especially the – surface grade and edge roughness; – particle size and particle size distribution; – flake thickness; – pigment orientation in the material of application. In the context of metallic pigments, it is useful to emphasize the fact that historical laxness in naming remains an issue. Terms such as aluminum bronze or silver bronze instead of aluminum pigments, gold bronze or even the more general metal bronzes, are still in widespread use today by pigment manufacturers, colorists, and even in modern literature [38–40]. Actually, aluminum bronze consists of copper alloyed at most 12% aluminum; silver bronze is an alloy of silver and tin; gold bronze is a solid blend of suitable amounts of copper and zinc; genuine bronze, however, consists of copper alloyed with tin.

2.3

Effect Pigments

67

Apart from dispersion and particle size, the character of the metallic color impression is influenced by further details. Among them are the chosen metal or alloy, the manufacturing and processing method of the flakes, and the wettability in binders. In the following, we give a survey of the influence of these parameters with regard to the visual perceived metallic effect. First of all, the chosen metal is responsible for the brilliance of the natural metallic pigment. The metallic colors change from light white (Ag, Ni), white (Al – as a substitute for Ag), bluish white (Zn), orange-yellow (Au), and reddish (Cu) to gray (Fe, Mo, Ti, W). These color attributes relate only to the metal gloss and, therefore, have a different meaning from that which was outlined for absorption pigments in Section 2.2.2. Additional natural metallic colors can be realized using mixtures of metallic pigments: Ni flakes lighten and Fe particles gray the metallic effect. Also alloys such as brass (alloy of Cu and Zn) are suited for gold-yellowish to reddish color, for example. Metallic pigments are manufactured with mixtures or alloys only if the required metallic effect is not achievable using only conventional metal flakes. The most commonly used metallic pigments from modern point of view are given in Table 2.8. Uncoated metal flakes based on natural metals have the most diversity. Particles with especially even surfaces can be manufactured using the PVD method. For this, a polymer foil is vapor coated in vacuum with a metal and afterward crushed at temperatures far below the glass transition temperature of the polymeric material. These flakes are termed as crushed PVD films. Due to modern developments, even colored metallic flakes can be manufactured. In these cases, a colored component is superimposed on the Table 2.8 Classification of metallic pigments Metallic pigment

Typical examples

Uncoated metal flakes

Aluminum (“aluminum bronze”, “silver bronze”), copper, zinc, copper/zinc alloys (70/30, 85/15, 90/10: “gold bronzes”), copper/aluminum alloys (4–12% Al: aluminum bronze), iron (austenitic steel, max. 11% Cr), nickel, tin, silver, gold, titanium

Crushed PVD films

Foils of polyethylene terephthalate, polystyrene, or polypropylene: vapor coated with aluminum, chrome, magnesium, copper, silver, gold (PVD procedure)

Flakes coated with absorption pigments

Inorganic or organic pigments in silicon dioxide or acrylate coating fixed on aluminum or “gold bronze” flakes as substrate (CVD procedure)

Partially oxidized and oxide-coated metal flakes

Partially oxidized from the surface aluminum flakes, copper flakes, zinc/copper flakes (“fire colors”); coated aluminum flakes: iron-III-oxide, tin oxide, zirconium oxide, iron titanate, cobalt titanate

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2 Light Sources, Types of Colorants, Observer

metallic effect. There are two different methods for manufacturing. In the first, absorption pigments are suitably fixed at the surface of the flakes by chemical vapor deposition (CVD) [38, 39]. The second method is based on metal flakes which are partly oxidized from the surface, or coated with metal oxides. Such metal flakes produce an intense color effect, sometimes called “fire colors.” These intense colors are caused by multiple reflections at the transparent outer layers with varying refractive indices; the multiple reflected waves then interfere. Technically, such colorants are not metallic pigments in the strict sense, rather just a multi-layered interference pigment on top of a metal substrate. In most cases, the original metallic brilliance of the metal substrate is lost. It must be noted that the chosen manufacturing method is of great influence to the resultant coloristical properties of metallic pigments. For shaping of natural particles, the melt is pressed through a narrow nozzle of suitable geometry with high pressure. Due to high flow velocities in the nozzle channel, often up to 400 m/s, the velocity gradient leads to an atomizing of the melt into round and contracting tiny droplets outside of the nozzle tip. After cooling, they are transformed into flake-shaped particles; these are the resulting metal flakes. The grinding of the solid particles is performed by two different processes. The wet milling method of Hall uses white mineral thinner for liquid phase. This simultaneously has the benefit of preventing dust explosions. In order to prevent clumping or welding of the particles, long-chained fatty acids are often added, typically 3–6% oleic or stearic acid. The dry milling method of Hametag, however, works in a N2 atmosphere containing at most 5% O2 . For this method, 4–6% palmitinic acid or stearic acid is often used as separators. The mentioned long-chained fatty acids, on the other hand, coat the entire surface of the resulting flakes; this certainly changes their wettability especially against binding materials. Because the end groups of the fatty acids are dependent of surface tension and behave either hydrophilic or hydrophobic, the metal flakes flood or disperse uniformly in the binding agent. Figure 2.31 gives an

Metal flakes

Coating

Substrate a) Leafing

b) Non–leafing

Fig. 2.31 Two different distributions of metallic particles in a painted layer: (a) leafing near the surface and (b) non-leafing or nearly uniform distribution

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illustration of two flake arrangements in a binder, the first is called leafing and the second non-leafing. The tendency to form the leafing arrangement is stronger with the lowering of the wetting with the surrounding polymer. Leafing flakes result from coating with stearic or palmitinic acid; non-leafing particles are usually obtained with oleic acid, for example. It is worth mentioning that the sort of wetting is generally correctable afterward with suitable additives. If the flakes are dispersed in the melt state of plastic materials, the leafing is generally avoided due to the high structural viscosity of the melt.11 A measurement method for determining the leafing behavior is detailed in the literature [41]. Leafing pigments incorporated in transparent binding agents show high brilliance due to a uniform and constant reflection. The abrasion resistance of the surface film is often reduced and, therefore, an additional top coat is usually applied. In contrast, the parallel and compact flake arrangement often behaves like an optical or mechanical barrier: in addition to visible wavelengths, it reflects UV and IR radiation and can also inhibit the diffusion of gases or vapors. Leafing flakes are, therefore, often used for reflection of UV and IR radiation as well as corrosion prevention pigments. Further non-colored applications of metallic pigments are given in Appendix 7.1.1.

2.3.2 Morphology of Metallic Particles As already mentioned, the metallic character is especially influenced by the morphology of the flakes. The wet milling procedure of Hall predominantly produces particles of irregular and uneven surfaces along with high edge errors – they are, therefore, quite accurately called cornflakes. Typical aluminum cornflakes are shown in Fig. 2.32. The picture was taken with a scanning electron microscope (SEM). The irregular particle edges result from fracture from other flakes during the milling process within the ball mill. The uneven surfaces are caused by abrasion of the coarse edges of colliding and pressed particles. At the particle surfaces shown in Fig. 2.32, it is possible to discern some remains of broken edge zones and indentation traces of striking balls. Both the uneven surface structure and the rugged particle edges are responsible for the typically increased diffuse light scattering of cornflakes. The light reflected from such flakes is composed of a directional and a superimposed scattering component. The accompanying indicatrix corresponds to that of Fig. 2.28c. The amount of diffuse reflected light is increased, therefore, with more uneven particle surfaces and rugged flake edges. With increasing particle scattering, there is a reduction of brilliance, that is, “it turns gray.”

11 Structural viscosity means that the viscosity has a nonlinear dependence on shear velocity;

this is in contrast to constant Newtonian viscosity.

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Fig. 2.32 SEM picture of typical cornflake particles of aluminum (source: Eckart Werke GmbH, Velden, Germany)

Sophisticated grinding techniques with metal brushings or polishing pastes have been used since about 1980 to reduce this brilliance loss. These techniques allow for the manufacture of thicker metal flakes with relatively plane surfaces as well as even or rounded edges. Particles of this kind are often called silver dollars, of which an example is shown in Fig. 2.33. These sorts of flakes reflect light with a dominant directional component accompanied by a considerably lower scattering. A paint film with silver dollars and clear top coat is, therefore, lighter and more brilliant than the one with cornflakes of the same metal and identical particle size distribution. Silver dollar pigments are preferentially employed in high-quality systems such as automotive coatings, the so-called metallics. They are also used in high polymer materials and printing inks. Shear stable flakes of upward 10 times the particle thickness of cornflakes can be realized in order to withstand processing techniques of high pressure or high shearing stress (e.g., ring main pumps, injection molding, blow molding, or extrusion). Particles with nearly plane surfaces and high brilliance can also be manufactured by PVD coating of thin polymer foils. They do, however, exhibit an undefined breaking edge. A representative example is shown in Fig. 2.34. In addition, it is possible to produce metallic pigments of constant thickness and of geometric regular shapes. These sorts of particles are called glitter flakes. To describe the glitter effect, the terms sparkle and glittering are also used. Both expressions are often utilized simultaneously, although they have somewhat different meanings.

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Fig. 2.33 SEM picture of silver dollar pigments of aluminum (source: Eckart Werke GmbH, Velden, Germany)

Fig. 2.34 Metallic pigments of PVD metal coated and broken polymer films (source: Eckart Werke GmbH, Velden, Germany)

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Sparkle is the property that single flake particles are directly recognized visually due to their expanded planar reflection surfaces. This is typically the case for mean particle sizes of about 30 μm and larger. The metal flakes are perceived separately on account of the surface reflection contrasting to the darker surroundings. They behave as isolated microscopic mirrors. In contrast to glittering, see below, neither the morphology nor the lateral dimensions of sparkle particles are necessarily uniform. The phenomenon of sparkle (also referred to as sparkling, optical roughness, micro-brightness, glint, or diamonds) is present in quite an assortment of effect pigments and is particularly observable with directional light; see also Section 3.5.1. In contrast, glittering is exclusively due to particles of uniform and regular geometry. These particles are manufactured in the form of squares, rectangles, rhombs, or circles by cutting or punching-out from metal foils or metallized polymer foils. In Fig. 2.35 an example of nearly quadratic aluminum glitter flakes cut from ribbons with a band knife is shown. Glitter flakes are of nearly uniform lateral dimension compared with conventional manufactured metallic pigments in ball mills. They are produced with sizes of 50 μm up to as much as 2 mm. They are generally at least 10 times thicker than usual cornflake or silver dollar particles. With regard to quadratic and rectangular glitter flakes, often specifications such as “2 × 2” or “2 × 4” are used. These can be interpreted as follows:

Fig. 2.35 Almost quadratic glitter flakes cut with a band knife from foil strips of aluminum [38] (source: Rapra Technology Ltd, Shawbury, UK)

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multiplication of each of these numbers with one thousandth inch (2.54 × 10–3 cm) gives the middle lateral extension of the corresponding particles. This means, for the given examples, that the glitter flakes have dimensions of 50 × 50 μm and 50 × 100 μm, respectively. It is also worth noting that with extreme process control in the Hall technique, it is also possible to produce spherical pigment particles with diameters as large as 700 μm. Such particles have, however, not the appearance of glitter flakes [38]. Colored metallic pigments offer an exceptional optical enlargement of usual metal pigments. These consist either of metal flakes covered by absorption pigments or metal flakes partially oxidized from the surface or oxide-coated metal particles. In the first case, the metal particles are coated with silicon dioxide or a polyacrylate containing inorganic or organic absorption pigments. In Fig. 2.36, there is an SEM photograph of aluminum flakes with a silicon dioxide coating of embedded inorganic pigment particles. This picture shows not only the coarse surface structure of the flakes but also the size proportion of the absorption pigments having only a few nanometers to the around two orders of magnitude larger sized metal flakes. From this difference in size, the typically higher scattering power of absorption pigments compared to effect pigments is also understandable. Colored metal effect pigments consist also of partially oxidized or oxidecoated flakes, which are manufactured using the CVD procedure. Partially

Fig. 2.36 Aluminum flakes coated with silicon dioxide and embedded inorganic pigment particles (source: Eckart Werke GmbH, Velden, Germany)

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20

100

10

50

Cumulative frequency curve

Relative frequency %

oxidized flakes of aluminum, copper, or zinc/copper are generally preferred. Iron-III-oxide-coated aluminum flakes are also common. All of these pigments produce particularly impressive reddish colors – and are, therefore, also called fire colors. These unusually striking colors are due to the combination of three effects: first, absorption and scattering of copper, zinc, or iron-III-oxide; second, interference caused by the layer construction; third, metallic reflection of the particles. Already the concept of the glitter flakes suggests that the character of the metallic effect is particularly dependent on the particle size. All particles, however, are not of the same size; therefore, the metallic effect is in general influenced by the particle size distribution. On the other hand, the size distribution can be controlled partly by the milling and sieving process; from these manufacturing steps, an asymmetrical particle size distribution with respect to the mean results. This distribution has a higher amount of smaller particles, cf. Figs. 2.37 and 2.38. The mean, in this context, is usually denoted by the quantity d50 or D[v; 0.5]. This quantity, for example, d50 = 25 μm, indicates that 50% volume of all particles are higher or lower in size than this near center value. The subtle properties of the metallic character even depend on the width of the distribution. In addition to the use of scanning electron microscopy, the laser granulometry method has proven quite useful for the determination of the mean particle size of effect pigments and the accompanying size distribution. The laser method is founded on scattering and diffraction of the particles suspended in a suitable liquid. This so-called laser granulometrical operation is suited only as a relative method, this is, not as an absolute method.

d50 0 10–1

0 100

101

102

d /μm

Fig. 2.37 Curves of particle size distribution and of cumulated frequency of an aluminum cornflake pigment with same value of d50 ≈ 20 μm as in Fig. 2.38

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20

100

10

50

Cumulative frequency curve

Relative frequency %

The size distribution of effect pigments is characterized by the so-called width W, which is given by the quotient W = (d90 − d10 )/d50 > 1. The quantities d10 and d90 mean that 10 and 90% volume, respectively, of the particles have lateral dimensions of Φ ≤ d10 and Φ ≤ d90 . A narrow distribution with low W produces a more brilliant metallic effect in comparison to a broader distribution of equal d50 value; see Figs. 2.37 and 2.38. The concentration of small particles is lower in a narrow distribution than in a broad distribution. However, the greater the number of small-sized particles present, the more gray and paler the metallic effect appears. Of course, with smaller flakes edge scattering preponderates, and in addition, the parallel configuration of the flakes is more distorted than with larger sized particles. The metallic reflection of typically narrowly distributed silver dollar pigments is consequently more distinct than that of traditional cornflakes. Cornflakes show a significantly broader particle size distribution. Among modern metallic pigments, silver dollar flakes of aluminum are the most light and most brilliant particles. According to Figs. 2.37 and 2.38, there is always an elevated number of particles smaller than 5 μm.12 This increased content of fine particles is unavoidably caused by the grinding process and is quite typical for the manufactured metal powders. The fine particles give rise to a scattering component which reduces the metallic brilliance. Finally, the production of the metallic effect is also influenced by the orientation of the particles in the polymeric surrounding medium. The particle

d50 0 10–1

100

101

102

0 d /μm

Fig. 2.38 Curves of particle size distribution and of cumulated frequency of an aluminum silver dollar pigment with same value of d50 as in Figs. 2.37

12 Particles

smaller than 5 μm contribute to particulate (dusty) matter.

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alignment is primarily determined by the processing techniques used. The maximum possible metallic reflection is achieved by complete flake parallelism to the bounding faces of a plane parallel layer; this is equivalent to a mirror. Local perturbations of particle alignment parallel to the paint coating can be caused by turbulent motion of the solvent during evaporation; such kinds of heterogeneity lead to mottling or flocculation. Both kinds of visible non-uniformities are also known from coatings with absorption pigments. Lacquers with low solid-state content display a more distinctive metallic effect than systems with higher solidstate content (so-called high solids). This is because the lower viscosity makes the orientation of the flakes easier. During processing of metallic or other flake-shaped pigments in printing inks, the absorbability of the stock can influence the parallel orientation of the flakes. But the slower the diffusion rate of the solvent, the more time available for parallel alignment of the particles. In high polymer materials, the flake pigments orient themselves according to the prevailing flow conditions. The alignment of the particles direct after surface formation is nearly maintained during cooling time and concurrent volume contraction. The collision of two melt fronts causes a flow line, at which the particles orient – visibly – parallel to the flow fronts. Flow lines can be avoided generally with suitable manipulation of the fusion in the mold [38].

2.3.3 Coloristic Properties of Metallic Pigments The geometry, morphology, and reflection of metallic pigments produce a variety of unusual color effects which need to be described in detail. The most important coloristic characteristics of metallic pigments are given in Table 2.9. From this apparent arbitrary division it is possible to discern two groups: in the first group, the indicated effect properties increase with expanding lateral dimension of the particles and for the second group, they decrease. This inverse behavior and the complex color attributes of metallic pigments demand a closer examination in comparison to the more simple properties of absorption pigments. First, the five special features of the first group will be discussed. The metallic character or metallic gloss depends on the ratio of the directional to the diffuse reflection. Both components are directly proportional to the quotient of the surface to the edge length of the metallic flakes. The higher this ratio, the more distinctly developed the metallic character. Small particles normally give rise to a higher edge scattering and, therefore, produce a poor and gray metallic effect. Conversely, flakes with a lateral dimension greater than 30 μm show a striking metallic character. In general, the narrower the particle size distribution, the more distinct the metallic luster.

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Table 2.9 Correlation between particle size and coloristical properties of metallic pigments (other used terms in brackets) Assessment of characteristic particle diameter d50 , non-leafing pigments Characteristic of metallic pigment

≤10 μm: fine pigments Cornflakes

≥30 μm: coarse pigments Silver dollars, glitter flakes

Metallic character (metallic gloss) Brightness (brilliance) Reflection brightness (whiteness) Sparkle, glittering (optical roughness) Graininess (coarseness, texture)

Insignificant, matt, gray

Quite distinct, brightened

Low Minor

High Enhanced

Invisible

Visually noticeable

Low

Visible

Light

Dark

Enhanced Enhanced

Low Low

High

Minor

Lightness flop (flop, flip, travel, two-tone) Hue extinction Covering capacity (hiding power, opacity) Distinctiveness of image (DOI)

The brilliance of metallic pigments is, in essence, caused by the directional reflection component. Metallic flakes, therefore, appear all the more brilliant the higher the directional reflected light amount and the lower the diffuse reflection component. The brilliance, and consequently the directional reflected component, is reduced with smaller and more uneven surfaces of the pigments, with broader particle size distributions and with greater irregularities in particle orientation in the surrounding polymer material. The reflection brightness or whiteness characterizes the lightness of the metallic effect. This property is a measure of the total reflected light and is composed of the directional and diffuse components. Consequently, reflection brightness differs from lightness of absorption pigments, which is due to scattering and absorption. In connection with effect pigments, the term reflection brightness has, therefore, a quite different meaning from the lightness of absorption colorants (although both quantities are determined with the same measuring and evaluation system). In fact, the spectral reflection of metallic pigments behaves similar to that of white/black mixtures of absorption pigments. Metallic pigments certainly show a key difference: the spectral reflection depends on the angle of illumination and direction of observation. This angle dependence is shown clearly, for example, by the reflectance of an aluminum cornflake pigment in Fig. 2.39. In this illustration, the reflectance

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R (%) 150

100 15 25 50

45 75 110

0 400

500

600

λ /nm

μas /degree

Fig. 2.39 Spectral reflectance of a cornflake aluminum pigment in dependence of the wavelength λ measured at aspecular angles of μas = 15º, 25◦ , 45◦ , 75◦ , 110◦ ; illumination angle β = 45◦

R is plotted in dependence of wavelength λ for five standardized aspecular measuring angles μas and an illumination angle of β = 45◦ .13 With increasing measuring angle, the reflectance behaves similar to that of light white and middle or dark gray, cf. Figs. 2.24 and 2.26. For angles steeper than μas ≈ 75◦ , the reflectance remains at a low level, the metallic effect appears dark gray [42, 43]. On the other hand, the reflectance increases exponentially for the aspecular angle of μas = 0◦ . This drastic change in brightness in dependence on the observation angle is characteristic for metallic flakes; it is termed as lightness flop or short flop (see below). The reflectance increase near the specular angle can be further raised, within limits, with a narrower particle size distribution or altogether larger sized flakes. Measured reflectance values higher than 100% are absolutely usual and produce no inconsistency with conservation of energy. Because of the lack of suited and corrosion-resistant metallic standards, the reflectance scale is based on the reflectance of a white (scattering) standard which is interpreted as a reflectance of 100% (Sections 2.2.3 and 4.1.2). Real effect pigments can have reflectance values up to about R = 800%.

13 For

nomenclature and counting of angles, see Section 4.1.2, Figs. 4.5 and 4.6.

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Metallic pigments of irregular lateral dimension greater than about 30 μm produce a phenomenon termed as sparkle in the last section. This effect is also called sparkling or optical roughness. For particle dimensions of this size or larger, the naked eye can distinguish single flakes of nearly equal orientation at the surface of a coating or plastic material. This is caused by the increased intensity of directionally reflected light at the larger particle surfaces (similar to parallelized crystals of an illuminated snow surface). As a consequence of the distorted orientation of the flakes, sparkling changes in dependence on both the illumination and observation angle. Leafing flakes develop no sparkling in dependence of the particle size because nearly all flakes have an alignment parallel to the surface of the coating or the plastic material. Sparkle is not restricted to metallic pigments and can be present in other sorts of flake-shaped pigments. This striking feature depends, e.g., on the size, the sort, the surface curvature, and the content of the effect pigment in a coloration. If the illumination is switched from directional to diffuse, the sparkling disappears completely and turns into a kind of graininess of the particles. Under this lighting condition, only a kind of fixed snowing picture is observable near the surface of the coating. This phenomenon is also called coarseness, texture, or vivid “salt and pepper” and is independent of observation angle. The impression of graininess depends on the size and type of the flakes, orientation irregularities, or clustering of the pigments during processing. Now, consider the second group of properties listed in Table 2.9. The abovementioned lightness flop (also light to dark flop, flop, flip, travel, two-tone) can be regarded as the most important and visually most striking property of metallic pigments. This term characterizes the decrease in lightness that a metallic coloration shows under diffuse illumination between two extreme angles of observation, especially at angles for perpendicular observation and an angle greater than 60◦ with regard to the normal of the surface (cf. Fig. 2.30a).14 The lightness flop is influenced by three primary factors: first, the flakeshaped pigment morphology, second, the surface and edge formation, and third, how well the flakes are ordered parallel to the surface boundaries of a layer. During observation perpendicular to the surface, the light enters directly the eye after interactions with the flakes. The observer registers the reference lightness. Now, for angles nearly parallel to the surface, the light path through the layer is longer. For this reason more interactions can take place with the particles of the layer, that is, at the surface or edges of the flakes. Due to this, light is increasingly scattered or reflected out of the line of observation and, therefore, only a small amount of the interacting light reaches the eye. This results in a registering of a reduction of lightness in comparison to the perpendicular observation.

observation means, in practice, that observation angles of β ν = ±20◦ , referred to the vertical, are permitted.

14 Perpendicular

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This flop is termed as light if only a small lightness difference is observed between both extreme observation angles. It is, however, called dark if a large lightness difference is registered. For finer and irregularly shaped flakes (e.g., cornflakes), for broader accompanying particle size distribution, and for greater particle disorder in the layer, the flop is lighter. Conversely, the flop is the more dark, for greater evenness of the particle surfaces, for more rounded flake edges (silver dollar flakes), for narrower size distributions, and for more consistent parallel orientation of the particles to the surface boundary of the layer. The lightness flop can be altered afterward to some degree by adding inorganic or organic absorption pigments (see Table 2.10). Table 2.10 Lightness and color flop of effect colorations

Flop kind

Flop character

Effect pigment/ absorption pigment

Lightness flop

Light

Metallic pigment

Metallic pigment and absorption pigment Dark

Metallic pigment

Metallic pigment and absorption pigment Color flop

Colored

Absorption pigment

Absorption and pearlescent pigment Interference pigment

Diffraction pigment

Typical examples Fine non-leafing cornflakes, eventually with orientation distortion medium Scattering, inorganic, and organic; for example, titanium dioxide, chromium titanate Coarse non-leafing silver dollar flakes, normal silver dollar flakes, PVD flakes Transparent, inorganic, and organic Blue with green flop: phthalocyanine pigments of neutral color flop; blue with red flop: α/-phthalocyanine pigments Combination of hiding absorption and transparent pearlescent pigments, nano-titanium dioxide Interference pigments with extreme color flop: LCP, flakes coated with silicon dioxide, aluminum oxide, iron-III-oxide, magnesium fluoride Aluminum or nickel substrate PVD coated with chromium/magnesium fluoride or magnesium fluoride/silicon dioxide

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The phenomenon of lightness flop can be examined quantitatively by a gloss meter. The usual measuring angles μν = 20◦ , 60◦ , 85◦ with regard to the normal of the surface are certainly not sufficient. This is already clear from the angle-dependent reflectance measurements in Fig. 2.39. A reliable characteristic value describing the lightness flop of a single-layered metallic formulation is the so-called flop index; however, for clear-over-base paints, the so-called metallic value is used (Section 3.5.1). As mentioned, some metallic systems are mixed with absorption pigments for coloring or covering of the background. These systems have a more or less colored flop; see Table 2.10. Mixtures with absorption pigments always result in a lighter flop compared to the natural metallic pigment. This effect is generally called color flop. Mixtures of pearlescent and metallic pigments normally produce a lighter flop than the corresponding single natural pigments. However, the brilliance is usually raised at perpendicular observation. This behavior can be reversed with suitable pigment combinations. The color flop of interference and diffraction pigments is generally accompanied by a distinct color change. This change is mainly due to the interference or diffraction maximum of first order (Sections 3.5.3 and 3.5.5). A particular kind of color flop is given by a mixture of a metallic flakes with titanium dioxide, provided that nanoparticles of the rutil modification are used. The observable minor color shift caused by the nanoparticles is generated by the selective scattering of waves in the region of blue wavelengths; this special phenomenon is called frost effect [44]. Such a layer has a yellowish color at top view but for a flat angle of observation, a bluish color impression results. Accordingly, the shorter blue wavelengths are more scattered than the other longer wavelengths, in agreement with Rayleigh’s law (2.1.3). This comparatively low effect is also observed with other absorption or pearlescent pigments of lateral dimensions in the nanometer range. Altogether, the color flop of a formulation can be controlled within limits by adding small amounts of nanoparticles. Certainly, the frost effect lowers the entire brightness of the relevant coloration. The next property of interest in Table 2.9 is the hue extinction. This feature characterizes the capability of a metallic pigment to change or to cover completely the natural color of an absorption pigment incorporated into a formulation. In the literature [39], this ability is called “hue saturation,” which is again misleading, because this term implies a saturation of the relevant absorption colorant. The extent of the hue extinction is related to the covering capacity of the metallic particles. In this case, for metallic particles this term has the same meaning as it does for absorption pigments (Section 3.4.3). Both the hue extinction and the hiding power are improved with smaller cross sections of the particles, broader particle size distributions, larger flake thicknesses, and denser packing of the flakes.

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Finally, there is the so-called distinctiveness of image (DOI). In addition to lightness flop and hue extinction, this property lists among the central characteristics of single as well as clear-over-base paints of metallic coatings. The DOI value represents the uniformity of the observable metallic reflection. It depends mainly on the particle size and orientation of the flakes in a layer. The distinctiveness of image is again improved by smaller particles and more highly parallelized grade of the flakes. The DOI value of large-sized flakes nearly correlates with the glittering effect. The determination of the DOI value by measurements is outlined in Section 3.5.1.

2.3.4 Sorts of Pearlescent and Interference Colorants The color production of pearlescent and interference pigments is essentially based on constructive interference. Pearlescent pigments imitate the nacre luster of natural pearls. The brilliant colors and unique luster are due to superimposed light interactions such as absorption and multiple reflection at different boundary surfaces of the particles. Interference pigments generate substantially more brilliant colors than pearl luster pigments. The terms pearlescent and interference pigments are often wrongly used as synonyms for both pigment sorts, although only pearlescent pigments have the typical luster of natural pearls coming from the depths to the surface. The particles of pearl luster and interference pigments consist of layers with different refractive indices. The layer thicknesses are on the order of magnitude of visible wavelengths. The entire flake thickness varies from about 30 nm to 1 μm and the mean lateral dimension of the transparent to opaque pigments extends – similar to metallic flakes – from about 5 to 300 μm. Pearlescent, interference, and diffraction pigments can also exhibit sparkling. This sizedependent phenomenon is, in this case, generally darker than the sparkle of metallic pigments because the reflection is caused only by the difference of refractive indices instead of metallic reflection. The color-producing properties of pearlescent, interference, or diffraction pigments are dependent on the particle geometry, especially on the interfering light interactions, and the color physical conditions in the particle surrounding. According to Equations (2.1.15), (2.1.16), (2.1.17), (2.1.18), (2.1.19), and (2.1.20), the interference laws are functions of the wavelength λ, the refractive index n at the interface, the thickness d of each single layer, and the observation or interference angle α z . These parameters can be tuned in such a way that only the first interference order (z = 1) occurs in the range of normal observation angles. In this case, the higher orders can be neglected or are simply not to observe. The incoming light waves are partly reflected at each boundary surface of a particle, cf. Fig. 2.16. The constructive interference of the waves

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produces the matt to high brilliant color appearance. The interference color is observed at the specular angle on the illumination surface and the complementary color is transmitted by transparent flakes. Each single pigment, therefore, behaves as an interference filter, which the incoming light waves split up into a reflected interference component and a complementary transmitted or absorbed component. As a result of the different refractive indices at the interfaces as well as the morphology of the particles, the following light interactions together contribute to the color appearance of pearl luster and interference pigments: – constructive interference produces the sometimes intense colors which change in dependence of the angle of observation; the different colors and the angle dependence are given by the refractive indices, the thicknesses, and the number of layers; – single reflection at interfaces causes part of the glossiness; – multiple reflection at the different interfaces of the transparent or translucent layers causes the typical luster “from the depths” of pearlescent pigments; – scattering at the edges or rough surfaces of the particles leads to matt interference colors; – absorption reduces the brilliance; it depends on the layer material. Table 2.11 shows the customary substances from which pearlescent and interference pigments are composed. The materials are ordered according to increasing values of refractive index nD . In comparison, air at a pressure of 1.0 bar at room temperature T = 298 K has a refractive index of only n = 1.000272. The pigment particles consist of at least three different Table 2.11 Refractive index n of substances used for pearlescent and interference pigments; λ = 589.3 nm, T = 298 K [40, 45, 46] Substance

Special terms

Refractive index n

Synthetic high polymers MgF2 Proteins SiO2 Alumina silicate CaCO3 Al2 O3 Guanine, hypoxanthine Pb(OH)2 2PbCO3 BiOCl Fe3 O4 TiO2 α-Fe2 O3

Organic materials Magnesium fluoride Proteins Inorganic glasses Mica, muscovite Aragonite Aluminum oxide Natural pearl essence Basic lead carbonate Bismuth-oxide chloride Magnetite Anastas/rutil Hematite

1.35–1.70 1.384 1.40 1.458 1.50 1.68 1.768 1.85 2.00 2.15 2.42 2.5/2.7 2.88

·

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materials. Because the relevant substances are anisotropic crystalline, the indicated n values represent a mean. The refractive indices of the substances given in Table 2.11 are subject to dispersion. Table 2.12 is arranged by pigment classes: pearl luster, interference, and diffraction pigments. In other literature, these are often gathered arbitrarily under the collective name: special effect pigments [18, 40]. Our further discussion relates to the pigment classification and terms given in Table 2.12. This fixes a unified nomenclature for all sorts of pearlescent, interference, and diffraction pigments. Pearlescent pigments of the simplest structure are substrate free and form flake-shaped single crystals. The flakes consist of natural fish scales (75–95% Table 2.12 Effect pigments producing colors founded on wave properties of the light: pearlescent, interference, and diffraction pigments Sort of effect pigment

Unsystematic names

Pearlescent pigment

Platelet-like single crystals

Mica-based pearlescent pigments Interference pigment

Special interference pigments

Liquid crystal pigments (LCP) Optical variable interference pigments (OVIP)

Extended interference films Diffraction pigment

Grating pigments

Typical examples Natural pearl essence, basic lead carbonate, bismuth-oxide chloride, α-iron-III-oxide, titanium dioxide, mixed-phase pigments of aluminum oxide, manganese-iron-III-oxide Substrates: natural or synthetic muscovite layers: titanium dioxide (rutil or anastas), iron-III-oxide, chromium-III-oxide, silicon dioxide (multi-layer principle) Substrates: aluminum oxide, silicon dioxide, iron-III-oxide chromium, silicon–aluminum–boron silicate; Layers of iron-II-oxide-hydroxide, iron-III-oxide, chromium-III-oxide, titanium dioxide, chromium phosphate; chromium, iron-II-/iron-III-oxide, iron titanate, silver, gold, molybdenum Polysiloxanes in cholesteric phase, cross-linked in layers Substrates: aluminum, aluminum oxide, iron-III-oxide, silicon dioxide, glass flakes; Layers: aluminum, chromium, iron-III-oxide, magnesium fluoride, silicon dioxide, titanium dioxide Multi-layer film consisting of polyacrylates, polypropylene with polyethylene terephthalate, polystyrene, or polycarbonate Al substrate with symmetrical PVD layers of MgF2 or Cr/MgF2 Ferromagnetic: Ni substrate with symmetrical PVD layers of MgF2 /Al or Cr/MgF2 /Al

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guanine, 5–25% hypoxanthine), basic lead carbonate, bismuth-oxide chloride, or α-iron-III-oxide, as well as mixed phases of aluminum oxide or manganeseiron-III-oxide. On account of the lack of a mechanically stabilizing substrate, these pearl pigments will not easily survive shear and pressure flow of technical processing. They are similar to ground shells and are, therefore, limited to a small range of applications [40]. However, for the most part, the flake-shaped pearl and interference pigments consist of a symmetrically coated substrate of at least one additional layer. The most used substrates are natural or synthetic mica, aluminum, aluminum oxide, chromium, iron-III-oxide, or synthetic silicon dioxide. The lamellate center should have a great difference in refractive index compared to that of the coated layers. Pearlescent pigments are mostly based on mica substrate. This is a native depositing layer silicate, named muscovite, with total molecular formula KAl2 [(OH, F)2 AlSi3 O10 ]. Instead of natural muscovite, also synthetic mica is increasingly being used. The synthesized material shows a more uniformlayered structure and produces, therefore, more brilliant interference colors. Apart from metal oxides, further coating substances are fluorides or silicon dioxide, also cobalt and iron titanate, chromium phosphate, silver, gold, molybdenum, or chromium [18, 46–48]. The symmetric permutation of layers of the flakes is achieved by normal chemical procedures or vacuum evaporation coating by the PVD or CVD method [49, 50]. Examples of pearl luster pigments based on natural muscovite are shown in Color plates 7 and 8.

2.3.5 Interference Pigments Consisting of Multiple Layers Flake-shaped pigments based on muscovite or other substrates can be modified to produce a variety of further impressive colors. Intensified interference colors and simultaneously selective absorption, analogous to colored pigments, can be achieved by coating mica with two or more metal oxides of different refractive indices. Such kinds of flakes are also called combination pigments. Compared to single-coated mica pigments, they show more brilliant and brighter interference colors, as well as a more distinct color flop. The color flop is called distinct if a variety of interference colors are to observe. Conversely, the color flop is less distinct if only few interference colors are angle dependent to perceive. The color flop is again more distinct than with a mixture of the natural pearlescent and absorption pigments. Because of the underlying two different color production mechanisms, combination pigments are also termed as two-color pigments or color-flop pigments.15 In dependence on the observation angle, 15 Not

only combination pigments produce a color flop but rather interference pigments with other layer combinations as well as diffraction pigments, cf. Table 2.10., Section 2.3.3.

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2 Light Sources, Types of Colorants, Observer

Fig. 2.40 Cross section of a symmetrical multi-layer pigment; permutation of layers from the centered mica substrate: titanium dioxide, silicon dioxide, titanium dioxide; overall layer thickness about 500 nm (source: Merck KGaA, Darmstadt, Germany)

either the gloss is composed of the absorption and interference color or the non-self-luminous color of the absorption pigment is dominant. The most important combination pigments are TiO2 -coated mica flakes vapor-coated with further oxides such as α-Fe2 O3 (hematite modification), Fe3 O4 , or Cr2 O3 . As shown in Fig. 2.40, the muscovite substrate can be covered with two different oxide layers of TiO2 and SiO2 . On account of the six boundary layers with different refractive indices, 15 interference combinations are possible. These produce extreme interference colors which are superimposed by the absorption colors of both oxides. Mica pigments vapor-coated with TiO2 and top-coated with other metal oxides such as TiO2–x , TiOx Ny , FeTiO3 , or nanocarbon particles embedded in TiO2 generate silver gray or black pigments of improved compatibility with colored pigments. Transparent mica pigments can be produced in form of nano-sized particles by a suitable precipitation method of oxides or oxide hydrates; these colorants are called transparent colors. Moreover, modified process engineering allows for the manufacture of pigments of minor gloss [41]. The top-coat of an interference pigment can also consist of a pure metal. A substrate of a metal (Al or Cr) or metal oxide (Al2 O3 , Fe2 O3 , SiO2 ) is coated with a glassy layer of another refractive index (TiO2 , MgF2 ), as well as an evaporated semitransparent metal top-coat (Cr, Ni, or Al). The inner and outer reflecting layers form together a Fabry–Pérot etalon, cf. Fig. 2.18. This structure causes increased interference intensities and a sharp color flop by multiple reflection. An interference pigment showing a distinct color flop is generally named as optical variable interference pigment (OVIP). An example of such an OVIP of layer composition MgF2 /Al/MgF2 is shown in Fig. 2.41. In this SEM picture, it is particularly interesting to see the quite

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Fig. 2.41 SEM photograph of an optical variable interference pigment with symmetrical layer composition of MgF2 /Al/MgF2 ; cf. Color plate 9 (source: Flex Products Inc, Santa Rosa, CA, USA)

even surfaces and the sharp edges of the pigment particles. The even surfaces indicate also internally even surface boundaries and, therefore, brilliant interference colors followed by a distinct color flop. The sharp edges indicate that the flakes are broken at low temperatures and sifted out. The particles shown in Fig. 2.41 have a mean lateral dimension of d50 ∼ = 20 μm and a thickness of about 100 nm. An impression of the color flop produced by this pigment is given by comparing the two pictures of Color plate 9. Both photos show the same image field of a light microscope in bright- and dark-field illumination; bright field means illumination from the top of the surface and dark field means interference from the side of the color sample. In this case, the colored flop changes from green to violet (see also Section 3.5.3). Interference pigments of special shish-kebab layer structures and consisting of organic polymers are termed as liquid crystal pigments (LCP). A liquid crystal state is arranged without exception of rod- or elliptical-shaped polymer molecules consisting of suitable dipole moments or polarizing groups. The corresponding textures are optically anisotropic and are, for example, used in seven segment displays for more than four decades. The accompanying texture is called liquid crystalline or mesomeric phase because the molecular conformation corresponds neither to a random liquid nor to an ordered crystalline state.

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There exist altogether three different liquid crystalline textures. The corresponding molecular configurations are termed as nematic, smectic, and cholesteric: – nematic: the molecules are aligned parallel to their longitudinal axes, are arbitrary slidable in the axial direction, and are rotatable around the longitudinal axes independent of one another; – smectic: the molecules are oriented with their long axis perpendicular to each monomolecular sheet; the movement of the molecules is restricted to rotations around the longitudinal axes; – cholesteric: the molecules are arranged parallel to one another and grouped in layers; the layers are systematically turned toward each other by a specific angle; the mobility of the molecules is the same as in the nematic phase. These molecular configurations are schematically sketched in two dimensions in Fig. 2.42, but are to interpret as three-dimensional textures.

Nematic

Smectic

Cholesteric

Fig. 2.42 Schematic representation of rod-shaped polymer molecules in nematic, smectic, and cholesteric phase

The manufacture of liquid crystalline pigments of suitably layered structures normally starts with polymer molecules in the nematic phase. This state is converted into layers of cholesteric texture – for example, with silicones – at higher temperatures in the presence of a chiral additive. The single parallel layers are then cross-linked and fixed using UV radiation; see Fig. 2.43. Each layer is twisted by a constant angle relative to the adjacent layer. The nearly identical molecular conformation with regard to the initial layer is attained after passing the so-called pitch height p of about 100 nm. On account of the systematically twisted molecular conformation inside the pitch, the transmitted light is circularly polarized by this structure. The pigment particles have thicknesses

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Fig. 2.43 Parallel and cross-linked layers of polysiloxane molecules in cholesteric texture forming a half pitch of a liquid crystal pigment (schematically)

of about 5 μm, the lateral dimension normally ranges from about 7 to 90 μm. Liquid crystalline sparkling pigments can be realized up to about 500 μm. An entire pitch works optically like a Fabry–Pérot etalon [51]. The partial reflection at the pitch surfaces and the rotated molecular arrangement causes several optical effects which greatly affect the interference behavior of these pigments: – a kind of gloss coming from the depths, which is attenuated in pearl luster pigments because the light must pass through various pitches and layers; – on account of reflections between adjacent pitch surfaces, the interference wavelength λ changes according to Equation (2.1.20) for z = 1 and p = 2d; the wavelength decreases with increasing angle of observation from the vertical; an example of transparent liquid crystal pigments is shown in Color plate 10; – caused by a low pitch value, only one single interference order is observable within ±90◦ ; it is accompanied by a distinct color flop; – the reflected polarized light increases the brilliance of the interference color; – because the particles are transparent, the total color impression can be influenced by the background color or mixed absorption pigments. Furthermore, modern developments use even pure metals, metal alloys, metal oxides, or borosilicate glass for substrates of interference pigments; see Table 2.13. An example of an uncoated glass substrate and coated with TiO2 is shown in Fig. 2.44. The differences in refractive indices at the interfaces cause not only unusual brilliant and pure interference colors but also sparkle effects which are even colored. Pigments of other compositions, thicknesses, and permutation of layers cause impressive color flops. These occur nearly throughout the entire visible spectrum (see Section 3.5.3). Finally, there are the so-called extended interference films (cf. Table 2.12, previous section). The manufacturing is carried out by co-extrusion of transparent polymer foils with different refractive indices (1.48 ≤ n ≤ 1.60). The films are partly colored with absorption pigments. The original low interface reflectivity is improved by semitransparent silvering of some internal foils. The full foil sequence has a thickness of up to 400 μm and contains a maximum

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2 Light Sources, Types of Colorants, Observer Table 2.13 Some optical variable interference pigments

Substrate

Coating

Peculiarities

Al2 O3

Depending on coating thickness d TiO2 Fe2 O3 Lateral dimension 5 μm ≤ φ ≤ 30 μm

Colors: “Silver,” yellow over blue-green until blue; “Bronze,”, “Copper,” red interference colors; sparkle pigments for φ ≥ 30 μm

Ca–Al borosilicate

TiO2 , Fe2 O3 , SiO2 : SiO2 /TiO2 in multi-layers: d ≤ 1 μm, 20 μm ≤ φ ≤ 200 μm

Substrate of high transparency, “pure” interference colors without scattering, multicolored sparkle effect

SiO2

TiO2

Colored flop: “gold-silvery” to greenish, green-blue to dark-blue; more distinct with rutil compared to anastas

Fe2 O3

Layer configuration: Fe2 O3 /SiO2 /Fe2 O3 / SiO2 /Fe2 O3

Colored flop: violet to orange-“gold”; top coating of Fe2 O3 , corrosion resistant

Metals and metal alloys

Mono-layers: Al, Zn, Cu–Zn alloys; SiO2 /Al/SiO2 , α-Fe2 O3 /Al/α-Fe2 O3 Multi-layers: From the surface partially oxidized and oxides of metal flakes

Fabry–Pérot etalons; cf. Table 2.8

Fig. 2.44 (a) Uncoated borosilicate substrate, (b) coated with titanium dioxide layers; total flake thickness of about 300 nm (source: Merck KGaA, Darmstadt, Germany)

of 70 films. The multi-layer films are crushed at temperatures below the lowest glass transition temperature of the polymers used and are sieved in different fractions.

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2.3.6 Spectral Behavior of Pearlescent and Interference Colorants Pearl luster and interference pigments are either nearly transparent or opaque. The most important representatives of both groups are shown in Table 2.14. In this section, we limit ourselves to describing only transparent particles because these colorants have more interesting color shades. First, we discuss the dependence of the color impression on the background color; thereafter, we explore methods for covering coatings. As already mentioned, the interference color appears in the direction of the specular angle; this corresponds to the interference angle of first order. The complementary color is transmitted by the transparent particles and absorbed by opaque flakes. Table 2.14 Examples of nearly transparent and opaque pearlescent and interference pigments Light transmittance

Examples

Nearly transparent

TiO2 pigment, Fe2 O3 combination pigments on mica substrate; Fe2 O3 coating of Al2 O3 substrate; coated SiO2 flakes; liquid crystalline polysiloxanes

Opaque

Reduced TiO2 on white mica; multi-layer pigments: Al flakes coated with SiO2 , Fe2 O3 ; Al or Cr platelets vapor coated with MgF2 , Cr, or Ni

The total color impression of transparent interference pigments is strongly influenced by the chosen background; this is clearly shown in connection with Fig. 2.45. Because a black background absorbs the complementary color, only the interference color is observed above the coating. The interference color corresponds to the mass tone of the pigment. On account of the additional light scattering, which is generated to a greater or lesser degree by the corners or edges of the pigment particles, the surface of such coatings appears in fact dark, but rarely absolute black. On the contrary, a white background scatters, to a large extent, the transmitted complementary color and is only slightly absorbed. The interference color is then superimposed upon by the scattered amount of the complementary color in direction of the specular angle. In all other directions, only the scattered complementary color is observed. Consequently, over a black background, the natural colors of an interference pigment are observed and over a white background, the covering capacity. These uncolored background surfaces allow also for the determination of the reflectance and transmittance of a transparent or translucent layer from reflection measurements (Sections 3.4.3 and 4.2.4). In the case of a colored background, the interference color is clearly superimposed upon by that color. Because the interference and absorption colors can

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2 Light Sources, Types of Colorants, Observer Interference color

Complementary color

Black

White

Colored

Fig. 2.45 The color impression of colorations with transparent interference pigments is affected by the chosen absorbing or scattering background

be either identical or different, a variety of color flops are possible. At curved surfaces (which are always to be avoided for color measurements), the interference and the absorption color can be observed simultaneously. Transparent interference pigments mixed with colorants of complementary mass tone result in a white color according to the laws of additive color mixing. Mixtures of transparent interference pigments with other pigments of similar color produce more dull colors like the single colorants. As an alternative to a white background, a scattering silver-colored metallic or pearlescent pigment of d50 value less than 10 μm can be used. Such parameters for the pearlescent pigment normally ensure a high DOI value. Preferred bright silvery pigments for this purpose are mica flakes coated with TiO2 or aluminum cornflakes. For light gray layers FeTiO3 -mica pigments are suitable. On the basis of the scattering contribution or broad particle size distribution, the gloss is reduced. Aluminum pigments have also the tendency to reduce the chroma of pearlescent and interference pigments. This is in analogy to white in mixtures of colored absorption pigments. In the following, we consider in more detail the color systematic and the corresponding spectral reflection which result from thickness changes and composition of the layers. For this, we restrict ourselves to the pigments with layer compositions which were already discussed in the previous section, among other things. They are as follows: – titanium dioxide evaporated mica particles; – mica flakes coated with iron-III-oxide;

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93

– two mica-based combination pigments of titanium dioxide in rutil modification: top coated with iron-III-oxide or with chromium-III-oxide; – liquid crystal pigments consisting of polysiloxanes.

These pigments are typical representatives of other pearlescent and interference pigments. This goes especially for the accompanying colorimetric behavior of these pigments which show astonishing colorimetric parallels (Section 3.5.3).

2.3.6.1 Titanium Dioxide Evaporated on Mica Substrate An example of a pearlescent pigment on a mica substrate coated with rutil is shown in the upper half of Color plate 7. This picture was taken under bright field illumination. From the top view, the particles produce a yellow color impression over a black background. The non-uniform colors of the particles are caused by different layer thicknesses and by flakes tilted with regard to the image plane. This is further elucidated with the example of the red pearl luster pigment in Color plate 8. In this case, the particles in the same field are recorded in using bright and dark field illumination. The comparatively high-valued refractive index of titanium dioxide (n = 2.5 or 2.7) together with the low value of muscovite (n = 1.5) offers suitable conditions for developing neatly ordered interference colors. However, if the mica substrate is already flake shaped, TiO2 crystallizes in a thin film which is only suited for interference. Among the three possible crystal modifications of titanium oxide – rutil, anastas, and brookit – the rutil structure is preferred in this case. The dependency of the interference color on the titanium-dioxide-layer thickness is given in Table 2.15 (cf. Table 2.5). Each color impression over black background changes with increasing layer thickness from metallic “silver” over copper-red to green. These are no spectral colors because the accompanying colors are, in each case, composed of several adjoining interference wavelengths. The different wavelengths are selected by varied layer thicknesses of the metal oxide and by particles tilted differently toward the image plane (see Color plates 7 and 8). Under diffuse illumination, these pearl luster pigments produce the spectral reflectance over black background shown in Fig. 2.46. With increasing layer thickness, the peak shifts to longer wavelengths and simultaneously broadens. In the figure, this begins in the violet region. The reflectance minimum also moves to longer wavelengths, in the figure starting from the yellow range. The spectral reflectance of the “silver” pigment corresponds to that of a light gray hue or to that of aluminum cornflake pigments. This suggests the substitution of the metallic aluminum pigment by the cheaper pearlescent pigment in suitable cases.

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2 Light Sources, Types of Colorants, Observer

Table 2.15 Interference colors depending on layer thickness of titanium dioxide on mica substrate (cf. Table 2.5)

Layer thickness titanium dioxide/nm

Platelet thickness/nm

Interference color – inherent color over black background

Transmitted complementary color

40–60 60–80 80–100 90–110 120–130 100–140 120–160

120–140 140–160 230–250 250–270 280–300 310–320 370–390

“Silver” Yellow Red Copper-red Violet Blue Green

“Silver” Blue Green Blue-green Yellow-green Yellow Red

Now, assume that the peaks in Fig. 2.46 can be interpreted as interference wavelengths of first order. Clearly, this is not exact, but sufficient for the following estimations. If we use the values of the relevant refractive indices of Table 2.11 and assume perpendicular observation, then Equation (2.1.18) delivers an approximate value for the thickness of the titanium oxide layer. For the R (%) Layerthickness in nm 80

a: 40 – 60 b: 60 – 80 c: 80 – 100 d: 120 – 130 e: 100 – 140 f : 120 – 160

60

40

b a c d

20

e f

0

400

500

600

λ nm

Fig. 2.46 Spectral diffuse reflectance of mica pigments of different TiO2 coating thicknesses; measurements over black background, de:8 geometry16

16 The corresponding de:8 measuring geometry (see Section 4.1.2) simulates diffuse illumination conditions in closed rooms.

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95

last five flakes indicated in Table 2.15, this approximate calculation results in values which are in the given thicknesses intervals. The systematic changes in spectral reflectance are characteristic features of the basis pigment series: the spectral reflectance curves measured under diffuse illumination are “fingerprints” of the relevant pearl pigments. This is an analog to absorption colorants. These fingerprints are sometimes utilized for analyzing mixtures with further pigments. Using directional illumination and angle-dependent reflectance measurements, particularly important information about the properties of the relevant interference pigments can be obtained. For the moment, we restrict the discussion to only one illumination angle although the colors of pearlescent, interference, and diffraction pigments depend on the illumination as well as the observation angle. The five spectral reflectance curves in Fig. 2.47 were measured with the green pearlescent pigment, which has a peak at 507 nm in Fig. 2.46, curve (f). The film of pearl luster pigments over a black background is illuminated at an angle β = 45◦ and the spectral reflectance is measured at aspecular angles μas = 15◦ , 25◦ , 45◦ , 75◦ , and 110◦ . As can be seen, the peak reflectance reduces dramatically with increasing measurement angle and it shifts to slightly shorter wavelengths. For observations near the specular angle, the reflectance exceeds values higher than 1.0 or 100%. The chosen five measuring angles are by no means sufficient to gather the entire color dynamics of pearlescent and interference pigments [52, 53].

2.3.6.2 Iron-III-Oxide on Mica Substrate Crystalline α-Fe2 O3 has the highest refractive index of n = 2.88 among the substances listed in Table 2.11. As mentioned in the previous section, the total color impression of the layer composition with mica results from superposition of the layer-dependent interference color and the absorption color of the hematite. Iron-III-oxide produces a red to reddish-brown absorption color which is further modified by the thickness of the outer layer, see lower picture of Color plate 7. The color impression changes with increasing layer thickness from bronze colored to copper to purple or even reddish-green at the highest layer thickness. This is clear from the spectral reflectance curves shown in Fig. 2.48. In the cases of bronze, copper, and red, the reddish-brown absorption color is even amplified by the corresponding, nearly equal interference color like yellow, copper, and red. We return to the hue dependence of the layer thickness in Section 3.5.3. The superposition of interference and absorption leads to brilliant colors in the vicinity of the specular angle. Near this angle, the purple and reddish-green pearlescent pigments have a violet or green interference color caused by the relative high layer thickness of the hematite. With regard to comparable outer

96

2 Light Sources, Types of Colorants, Observer 150

R (%)

100

μas = 15°

50

25°

45°

0 400

75°

500

600

110°

λ

nm

Fig. 2.47 Spectral reflectance of the green interference pigment from Fig. 2.46 at five aspecular measuring angles μas ; measurement over black background, illumination angle β = 45◦

layer thickness, the TiO2 -mica pigments produce a broader variety of interference colors, whereas the iron-III-oxide produces diverse types of red hues. The distinct brilliance of the α-Fe2 O3 pigments comes from the slightly higher refractive index of the hematite in comparison to that of titanium dioxide. The increased covering capacity is again a consequence of the typically increased scattering of the inorganic oxide.

2.3

Effect Pigments

97

R (%)

a: Bronce b: Red-golden c: Copper d: Purple e: Red-green

80

Fe2O3Layer thickness

60

a b c

40

d

e 20

0

400

500

600

λ nm

Fig. 2.48 Diffuse spectral reflectance of mica pigments coated with Fe2 O3 of different layer thicknesses; measurement over black background, measuring geometry de:8

2.3.6.3 Combination Pigments Interference with more brilliant colors and higher absorption is generated by coating the substrate with two or more different metal oxides. A multitude of combination pigments are based on titanium dioxide-mica flakes, where the flakes are additionally vapor coated with Fe2 O3 or Cr2 O3 . These sorts of combination pigments show, in comparison to the original single-coated mica pigments, higher brilliance and a more distinct color flop. In the following, we consider the reflection behavior of mica combination pigments composed of layers of titanium dioxide in rutil modification coated with iron-III-oxide. The optical interactions between both metal oxides lead to many brilliant golden colors reaching from pale yellow-gold to rich red-gold to green-gold. With the addition of an oxide layer, a clear color extension with regard to the simple Fe2 O3 -mica particles is achieved. Furthermore, the change of the layer thicknesses of the two metal oxides generates a wealth of superimposed interference and absorption colors. The thicknesses of all three layers can be tuned with each other in such a way that the interaction of the interference and absorption components leads to a demanded coloristical effect, within limits. Similar features result from the substitution of the outer Fe2 O3 layer by Cr2 O3 . The combination of Cr2 O3 with TiO2 in anastas modification results in

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2 Light Sources, Types of Colorants, Observer

R (%) Fe2O3 TiO2 Rutil

80

60

Fe2O3 TiO2 Rutil

40 Cr2O3 TiO2 Anastas Cr2O3

20

TiO2 Anastas

0

400

500

600

λ nm

Fig. 2.49 Diffuse spectral reflectance of two combination pigments Fe2 O3 /TiO2 (rutil), Cr2 O3 /TiO2 (anastas) on mica substrate of different layer thicknesses; measurement over black background, measuring geometry de:8; the curves of Fe2 O3 /TiO2 are shifted 20 units toward higher reflectance values for better overview

high-luster blue-greenish to moss-green colors. Figure 2.49 shows the spectral reflectance curves measured under diffuse illumination of Fe2 O3 /TiO2 pigment and Cr2 O3 /TiO2 -mica pigment. Each has two different outer layer thicknesses, but the total coating thickness is constant in each case. The greater thickness of the outer oxide layer reduces the reflection. This is clearly caused by increased absorption due to greater layer thickness. In the case of a higher layer thickness of Cr2 O3 , this pigment additionally develops a clear peak in comparison to the quite broad peak belonging to the thinner outer layer. The increased influence of the Cr2 O3 layer on the interference development can be ascertained from the reflectance curve (f) in Fig. 2.46 for the green pearlescent pigment together with the two lower curves in Fig. 2.49. Altogether, titanium dioxide is responsible for the brilliance of the colors and the outer layer thickness for the interference color. The included absorption colors of iron oxide or chromium oxide exist otherwise under all observation angles. Nearly equal-colored oxide layers create a pearl luster effect of intensive colors which, at nearly all observation angles, show a noticeable color shift. However, if the color of an absorption colorant corresponds to that of the complementary color of the interference pigment, then a distinct two-colored effect is achieved, especially in the oxide combination of Cr2 O3 /TiO2 . Further inorganic absorption pigments such as Prussian blue, cobalt blue, Fe3 O4 , or carbon

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99

black generate, in combination with titanium oxide on mica substrate, unusual color effects as well. 2.3.6.4 Liquid Crystal Pigments The color flop of liquid crystal pigments consisting of polysiloxane depends on the pitch height p and causes, for example, the color pairs violet/blue, blue/turquoise, blue/green, green-blue/gold, and green/copper-red. Flakes with blue/green-flop are shown in Color plate 10 at bright and dark field illumination over black background. The noticeable sparkle particle there is already green from the top view because of the higher pitch height compared to the other flakes. To the present day, mixtures of the transparent particles with absorption or other effect pigments produce unequaled and extremely brilliant colors. As mentioned at the beginning of the section, further shades can be obtained with a suitable background color. With regard to liquid crystal pigments, this concept is underlined by the example in Fig. 2.50. The reflectance curves correspond to background colors of white, red, and black as well as each top coated with a transparent film of the same liquid crystal pigment. The reflectance peak over black at 512 nm corresponds to the green color impression from the top view. The red background leads to a minimal blue-tinged red and the

R (%)

Background alone

White

80

Layer over background

Red

60

40

20

Black 0 400

500

600

λ nm

Fig. 2.50 Spectral reflectance of a transparent liquid crystal pigment with d50 = 30 μm over white, red, and black background, as well as backgrounds alone; wavelengths of maximum reflectance over background: 512 nm (red), 531 nm (black); measuring geometry de:8

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2 Light Sources, Types of Colorants, Observer

white background creates a very light pink. The flat maximum of the white background at a wavelength of 490 nm is caused by fluorescence emission of the contained optical brightening agent (Section 4.2.6). Figure 2.51 shows the directional reflectance over a black background for illumination angle β = 45◦ and aspecular observation angles μas = 15◦ , 25◦ , 45◦ , 75◦ , 110◦ . With increasing measuring angle, the wavelength of the peak shifts from 483 to 520 nm. Simultaneously, the peak height decreases from R = 108% to about R = 2%. The angle dependence of the interference wavelength corresponds to the cosine function in Equation (2.1.20). R (%)

μas = 15°

β = 45° d50 = 30 μm

100

25° 50

45° 75° 0 400

500

λ nm

110° 600

Fig. 2.51 Spectral reflectance of a liquid crystal pigment with d50 = 30 μm for five aspecular measuring angles; measurements over black background, illumination angle β = 45◦

The brilliance of liquid crystalline pigments depends also on the average lateral dimension of the flake particles. This can be seen from the reflectance maxima in Fig. 2.52. The reflectance curves are measured for three different particle size distributions over the same black background and at constant illumination and aspecular angle of β = 45◦ and μas = 15◦ , respectively. The flakes have size distributions which correspond to the ratios of d50 /d99 = 30/90, 23/60, and 18/50 μm/μm. The reflectance maxima in Fig. 2.52 correspond to the ratios 30:23:18 of accompanying d50 values. In addition, this result is confirmed by measurements over red and blue background. The height of the reflectance maximum of liquid crystal pigments is, therefore, directly proportional to the mean flake size. Clearly, large-sized LCPs orient better to the background surface, in analogy to metallic flakes. These larger particles then show increased brilliance

2.3

Effect Pigments

101

R (%)

β = 45° μas = 15°

d50 = 30 μm 100

23 μm

50

0 400

18 μm

500

λ

600

nm

Fig. 2.52 Directional reflectance at constant aspecular measuring angle μas = 15◦ of a liquid crystal pigment in dependence on particle size d50 ; measurements over black background, illumination angle β = 45◦

compared to smaller sized flakes. The colorimetric properties of the most important pearl luster and interference pigments are outlined in Sections 3.5.3 and 3.5.4.

2.3.7 Opaque Films Containing Absorbing and Effect Pigments From discussions up until now, it is clear that the creation of new colorations by mixtures or films consisting of modern effect and classical absorption pigments represents quite a challenge. Especially needed to meet this challenge are hiding layers which consist of transparent effect pigments and covering absorption pigments. A coating is called covering or hiding, if any colored background can no longer be observed from above the film (Section 3.4.3). Covering layers with coarse metallic, transparent pearlescent, interference, or even diffraction pigments are produced by mixtures of colorants which generate suited scattering or absorption. It is also important to consider the change of the originally desired color effect with the addition of further colorants and also the compatibility with the pure effect pigment. Experience shows that covering coatings with effect pigments can, in the most cases, be manufactured with three different colorants: carbon black, suitable absorption pigments, and fine metallic pigments [54].

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2 Light Sources, Types of Colorants, Observer

A small amount of carbon black turns out to be quite effective for this. The resulting dark and sometimes even intense colors are caused by absorption of the complementary color with carbon black already inside the layer. Only a small volume amount < 1.0% of carbon black is normally sufficient. For larger concentrations, the interference color often appears too dark and the color travel (respectively, color flop) is too small or suppressed. The selective absorption and scattering of mixed in absorption pigments also result in covering layers of additionally impressive colors. Small amounts cause mostly brilliant interference colors. Larger amounts, however, can obliterate the pearl luster and interference effect: on the one hand, the interference intensity is increasingly masked and on the other hand, the complementary and part of the interference color are absorbed depending on the chroma of the absorption pigment. Finally, in the third method, the transparent pearl or interference pigment is mixed with a small amount of a special fine and, therefore, strongly scattering metallic pigment. In this case, the covering formulation is, however, accompanied by reduced brilliance, color saturation, and gloss. In all three mentioned cases, it is important to pay attention to the compatibility of the added components. Furthermore, the added amounts should be as small as possible in order to result in a coloristic effect changed only slightly. Altogether, the level of pigmentation of effect pigments in recipes depends on the desired color effect, the spreading rate of the pigment, and the molecular surroundings. For a few applications the pigment volume content is shown in Table 2.16. Apart from the application, the processability, the light, temperature, or weathering resistance, etc., additionally determine the total pigment amount of a coloration; see also Section 3.4.6. Table 2.16 Pigmentation level of effect pigments in various fields Pigment level % Employment

Metallic pigments

Pearlescent, interference pigments

Lacquers, emulsion paints Thermoplastics, thermosets Printing inks Cosmetics Toiletries

0.5–2 0.5–3 1–45 1–15 0.05–1

0.5–20 0.5–2 1–30 1–50 0.05–1

From color physical point of view, mixtures of effect pigments together or with absorption colorants obey the laws of additive or subtractive color mixing (Section 2.4.3). Mixtures or layers of effect pigments show additive color mixing if the colors of the single pigments are based on the superposition of wavelengths; see Table 2.17. But, as soon as the light interacts only with one

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Effect Pigments

103

Table 2.17 Additive and subtractive color mixing of effect and absorption pigments Pigment combination

Color mixing

Cause

Interference pigment mixed with interference pigment Interference pigment mixed with diffraction pigment

Additive

Interference pigment on absorption pigment

Additive and subtractive

Interference pigment mixed with absorption pigment

Subtractive

Absorption pigment mixed with absorption pigment

Subtractive

Superposition of reflected interference and transmitted complementary color Superposition of interference, diffraction, and transmitted complementary color Superposition interference color, absorption of complementary color and selective absorption of the colored pigment Colored pigment absorbs parts of the complementary and natural color of the interference pigment Each colored pigment absorbs parts of the influx light

Additive

absorption pigment together with any further pigment sort, the resulting color impression is based on subtractive color mixing; see also the literature [54, 55]. Additive color mixing is fundamental in the human color sense and the based on colorimetry of versatile industrial applications. Finally, we return to optical properties of interference pigments which depend on the particle size and morphology. Smaller particles show a reduced pearl luster effect in comparison with those of larger sized. This is because the higher edge scattering reduces the luster. In the case of dominant scattering, the pearl luster and gloss of interference pigments can even disappear completely. An increasing lateral size dimension up to 200 μm improves not only the brilliance but also the sparkle of the pearlescent and interference pigments; furthermore the color travel passes more wavelengths. However, with increasing flake geometry, the hiding power and DOI are reduced. The pigment properties shown in Table 2.18 correlate with related properties of metallic pigments listed in Table 2.9. In other words, comparable color properties of metallic, pearlescent, and interference pigments change in the same direction depending on the flake size. We have, thus far, still neglected opaque multi-layered pigments (Table 2.13). A considerable part of their color physical properties correspond to those of transparent effect pigments. An advantage of these pigments is the fact that the color effect is not distorted by additional colorants, but the absorption and selfscattering of the particles softens the original color. Further similarities between different effect pigment sorts are uncovered in Section 3.5.3.

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Table 2.18 Assessment of some properties of pearlescent and interference pigments in dependence of the particle size Particle size Φ/μm Property

Φ<5

5 ≤ Φ ≤ 25

10 ≤ Φ ≤ 60

30 ≤ Φ ≤ 200

Brilliance Color flop Covering capacity DOI

Low Reduced Excellent Very good

Silky Low Very good Good

Very well Striking Good Low

Sparkling Distinct Low Very low

2.3.8 Colors of Diffraction Pigments Although the diffraction of light waves has been known since the beginning of the 19th century, this phenomenon has only been used in color physics for a few years [56]. Waves are diffracted, analogous to interference, when they interact with structures of dimensions on the same order of magnitude as the wavelengths. Polychromatic light is split up by diffraction into the spectral components. Reflective diffraction pigments are the sorts that is of interest in industrial color physics. These consist of a metal substrate with a grating of embossed periodic grooves. In addition, the substrate is coated with inorganic substances; see Table 2.12. Each particle works as a reflection grating. The nano-engineering method of interference lithography can be used to produce periodic grating structures on surfaces. This method is based on the interference of two laser beams; see Fig. 2.53. With the two coherent waves from the same source, each of wavelength λL , an interference fringe pattern of parallel fringes of distance d results (Michelson configuration). This distance is given by the relation d = λL · sin ϑ,

λL

(2.3.1)

d

2ϑ

λL

Fig. 2.53 Generation of parallel structures with two coherent light rays using the Michelson configuration (schematically)

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105

where ϑ denotes the half of the divergence angle. The periodic laser fringe pattern is directed onto a metallized thermoplastic film or photoresist film so that this pattern is transferred into the film as an exposure. After removing the organic backing layer by thermal or chemical procedures, the remaining metal foil of projected groove geometry is evaporated in vacuum with further substances which are also used in nanotechniques. The substrate is coated symmetrically so that the diffraction always develops at the illuminated surface of the reflection grating. The films of up to 1 μm in thickness can be broken up into particles with lateral dimensions between 10 and 300 μm using ultrasound. The resulting flakes can be sifted by pigment size. Flakes with mean diameters of d50 = 20±2 μm are preferably used for print media and coatings [56]. The regular grating patterns consist not only of parallel but also rectangular, diagonal, or hexagonal lines. Presently, diffraction pigments are manufactured with 100 up to 5,000 equidistant parallel lines per millimeter (l/mm). The cross section of the grooves can be chosen in such a way that the first diffraction order corresponds to the geometrical direction of reflection (blaze techniques, Section 2.1.7). The diffraction particles shown in Fig. 2.54 have a grating constant of 1,000 l/mm. The substrate consists of highly reflective aluminum of about 80 nm thickness and periodic folding. Using PVD techniques, the metal film is coated on both sides with MgF2 of about 450 nm thickness; see cross section at fracture in Fig. 2.55. Diffraction pigments have been produced industrially since 2002, although symmetric grating cross sections have been used in technical optics since about 1980. The spectral reflectance of diffraction pigments with three different grating constants measured by diffuse illumination and de:8 geometry is shown in Fig. 2.56. For pigments with 1,400, 2,000, or 3,000 l/mm, the reflectance

Fig. 2.54 Diffraction pigments with periodic grating dividing of parallel lines; grating constant 1,000 l/mm (source: Flex Products Inc, Santa Rosa, CA, USA)

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Fig. 2.55 Cross section at fracture of a diffraction particle with periodic grating dividing; the surfaces of the aluminum substrate are vapor coated with MgF2 (source: Merck KGaA, Darmstadt, Germany) R (%) 3,000 l/mm

80

70

60

2,000 l/mm

50

1,400 l/mm

40

30 400

500

600

λ

nm

Fig. 2.56 Spectral reflectance of diffraction pigments of magnesium fluoride/aluminum of three different grating constants; the middle and upper curves are shifted 5, respectively 10 units toward higher scale values

is between R = 0.4 and R = 0.7. This quantity depends on the reflectivity of the metal layer, the number and thickness of the non-metallic vapor-deposited films, the accompanying refractive indices, as well as the mean particle size. The pigment with 1,400 l/mm produces a silver metallic color impression at observation perpendicular to the coating surface. The nearly flat spectral reflectance

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107

shown in Fig. 2.56 is known from silver-colored metallic or pearlescent pigments. Under diffuse illumination, particles of grating constant 2,000 l/mm appear bluish and those of 3,000 l/mm produce a yellow-orange color impression. The almost equal form of the reflectance curves in Fig. 2.57 indicates that the thickness of the aluminum substrate between 80 and 240 nm is, as to expect, of no influence with regard to the color impression. For a fixed grating constant of 1,400 l/mm, the height of the reflectance – and, therefore, the brightness – is only slightly increased with aluminum thickness. With substrate thicknesses greater than 240 nm, the surface roughness increases and, therefore, also the interface scattering. This scattering reduces, however, the intensity of diffraction. R (%)

240 nm

70 160 nm

80 nm 60

50 400

500

600

λ

nm

Fig. 2.57 Spectral reflectance of MgF2 /Al pigments with aluminum layer thicknesses of 80, 160, and 240 nm; grating constant 1,400 l/mm; the middle and upper curves are shifted 10 units toward higher scale values

Reflective diffraction pigments consisting of a ferromagnetic substrate allow for the orientation of the particles according to an external magnetic field. Preferred ferromagnetic materials are nickel, iron, cobalt, but also elements of the lanthanide series. For better understanding of the orientation of ferromagnetic particles in an external magnetic field, it is useful to make a short excursion into magnetostatics. The ferromagnetic state in solids is caused by non-compensated electron spins of the unsaturated 3d- and 4f-electron shells of metal ions. The unbalanced

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angular momentums of the electrons behave, therefore, like magnetic elementary dipoles which, in the crystal state, respond to two different forces. The first, the anisotropic force causes an alignment of the dipoles parallel to an external bounding face and the second, the exchange force forces the dipoles to parallelize to each other in the volume. In a magnetic material, anisotropic and exchange forces produce zones of equal magnetization direction. These zones have linear extensions between 1 μm and 1 mm which are called domains or Weiss areas, after their discoverer P.E. Weiss. An external magnetic field can drive two geometrical processes in the domains: a change of the threedimensional extension as well as a rotation from the original position at constant volume. Both effects are represented schematically in Figs. 2.58a–c. Figure 2.58a represents the original position of four domains in a ferromagnetic substance. A weak external magnetic field causes a shift of the domain walls; see Fig. 2.58b. An even higher external magnetic field strength turns the molecular magnets more into the direction of the external field, accompanied by a further extension of the domain volume, Fig. 2.58c. In general, domain wall shifts occur at moderate magnetic field strengths and domain rotations only at higher strengths. The basic driving mechanism for the orientation of ferromagnetic domains or particles parallel to an external field is the tendency of such systems to seek a state of minimal total energy. This is achieved by an internal demagnetization field which counteracts the external field. A sufficiently high external magnetic field strength causes a total rotation of the domains. If the lateral dimensions of free particles are on the order of magnitude of the domains, then the rotation into magnetic field direction is performed by the entire particle. This effect is

H a)

b)

H c)

Fig. 2.58 Schematic representation of Weiss domains in a ferromagnetic material: (a) without external magnetic field, (b) with weak magnetic field, and (c) with strong external magnetic field H

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known from iron filings, which align along the lines of an external magnetic field. Also ferromagnetic diffraction pigments are subject to such rotations. With the help of an external magnetic field, a nearly uniform orientation of diffraction pigments of ferromagnetic substrate can, therefore, be fixed before and during crosslinking of a binder. The both diffraction pigments given in the last line of Table 2.12 are ferromagnetic. With an additional Ni layer for the substrate, their composition corresponds to the two- and three-layered diffraction pigments one line higher in the table. The ferromagnetic particles consisting of the three different layers MgF2 /Al/Ni and grating constants of 1,400, 2,000, or 3,000 l/mm produce, under diffuse illumination, a nearly uniform silver-colored impression. A further PVD coating with chromium causes a gold-yellow color. The spectral reflectance curves of ferromagnetic diffraction pigments are of the same shape as the curves of non-ferromagnetic particles with grating constants of 1,400 and 2,000 l/mm; see Fig. 2.56. Certainly, on account of the particle orientation in a magnetic field, an angle-dependent colorimetrical behavior results, one which had not been observed before (Section 3.5.5). After a detailed description of the optical operation modes of industrially used colorants, we can conclude: between the rock and cave paintings of natural colors and the modern produced and applied colors there was, without doubt, an impressive historical development. This is accompanied by broad research as well as process and application technology in color industry. In spite of this, all presently known and also newly developed or modified colorants are merely orientated toward the color perception of humans. The human color sense is, therefore, discussed in some detail in the next section.

2.4 Observer In addition to the importance of the light source and the color pattern, the most important and final crucial factor for the emergence of color perception is the observer. His/her subjective color sensation comes from the remaining light waves that reach the retina of the eye after source light interacts with the colorants. Nearly all efforts in processing and applying natural, modified, and synthesized colorants are oriented toward the capability of human color sensation. Long ago, Newton emphasized the fact that color production is ultimately based on neural processes. This means that outside the visual cortex, colors do not exist, but only electromagnetic waves. In the following, we give a brief overview of the modern physiological and neurological understanding of color sensation [57, 58]. These connections lead directly to the fundamental Grassmann laws of additive mixing of colors. The standardized CIE color values of the years 1931 and 1976 as well as the broad color physical and colorimetrical applications are based on these laws.

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2.4.1 Color Perception and Color Theories As already mentioned in the introduction of this text, color perception is produced by psychophysical and neural processes. Color sensation is, therefore, not directly measurable with normal physical methods. The first important element contributing to overall color perception is the eye. The essential optical components of the human eye for image formation are shown in Fig. 2.59. The cornea is only 0.5 mm thick and of fixed focal length of 25 mm, the iris has a lightness-dependent aperture ratio of 28:1, and the crystalline lens is of variable focal length. This focal length amounts about 50 mm for the resting eye when relaxed for distant vision. For an eye of normal vision, these optical elements provide a clear and focused image on the retina in the area of the fovea. In the retina are located two types of photosensitive receptors: first, the rods responsible for dusk vision and achromatic colors for luminance less than 0.01 cd/m2 (scotopic vision); second, the cones for color vision for luminance higher than 10 cd/m2 (photopic vision).17 If the eye is fully adapted to darkness, just nine photons are required before a light stimulus is detected. In cases of middle illumination for luminance in the range of 0.01 and 10 cd/m2 , rods and cones are simultaneously active (mesopic vision). The retina of the human eye contains altogether about 125 million visual cells, but only 5% consist of cones. In the small area of the fovea for visual angles of about ±0.5◦ , there are only cones present. Here they are of maximum density and enable focused vision only at this spot of the retina. The majority of

lris

Crystalline lens

Cornea

Fovea Macula Optic axis

Blind Spot

{ Optic nerves

Retina

Fig. 2.59 Horizontal section of the human eye

17 For

definition of unit cd (candela) see footnote 1 in Section 2.1.3.

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111

the rods are, however, distributed outside of the macula in an area of visual angle greater than ±5◦ (for visual angle, see Fig. 2.65). At the place of the blind spot of missing visual cells, the cord of the optic nerves continues to the brain.18 The different density distributions of rods and cones in the human retina are shown in Fig. 2.60. Four different photosensitive pigments are contained within the outer zones of the visual cells: one pigment in the rods and three pigments in the cones. The photosensitive pigments consist of a protein molecule denoted as opsin to which a derivative of vitamin A1 molecule is bound. This chromophoric group is common to the four pigments. The absorption spectrum of a photosensitive pigment depends on the kind of protein as shown in the normalized representation in Fig. 2.61. The pigment in the rods, called rhodopsin, shows an absorption maximum at the wavelength of 498 nm, the tails of the maximum project into the spectral absorption ranges of the cones. The absorption maxima of the three cone pigments occur at wavelengths 437, 533, and 564 nm. These wavelengths are perceived as blue, green, and red and the corresponding cones are, therefore, called blue, green, and red receptors. In the English-speaking medical literature they are denoted as S-, M-, and L-cones, indicating the selective sensitivity at short, middle, and long wavelengths of the visible spectrum. The three cone types are not equally distributed throughout the retina. The relative distribution for blue:green:red is about 1:20:40. From the similar shapes of absorption

Number of retinal receptors × 103/mm2

Blind spot

Rods Cones

150

100

50

0 –80°

–40°

0° Angle from fovea center

40°

Fig. 2.60 Density distribution of rod and cone receptors across the human retina

18 In modern neurology,

the retina is interpreted as a part of the brain.

80°

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λ max 437

498

533

564

nm

Relative absorption

1.0

0.5

Cones Rods 0

400

500

600

λ

nm

Fig. 2.61 Normalized spectral absorption of the four different photo receptor types in the human retina [57]

curves in Fig. 2.61, it is possible to conclude that the spectral absorption mechanisms of the three cone types are basically identical, although the curves seem to be shifted along the wavelength axis. Up until now, we have assumed normal color perception, called trichromacy on account of the three different cone receptors. Some sort of color vision deficiency with at least one defective color receptor occurs in about 0.5% females and about 7% males [58–60]. Although some animal species show an evolutionary caused tetrachromacy [61], this property has almost disappeared completely of human beings. In only extremely rare cases of women a fourth color receptor was proven; such a receptor shows a maximum sensitivity in the range of yellow wavelengths [62]. The unusually varying description of defective color vision in the literature is a consequence of the fact that previous generations of researchers used the constrained color perception for indications of the neural color sensation. The following kinds of anomalous color vision can be distinguished: – color asthenopy: fast onset of tiredness during color vision; – color amblyopy: selectively reduced ability to distinguish colors; – color anomaly: caused of defective or missing color receptors. The color anomaly can be divided further into: – anomalous trichromasy: the characteristic feature is a reduced perception of one color during simultaneous observation of several varied colors. In this

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case, there are three color receptors available but one does not work normally. More concretely, there are red-, green-, and blue-yellow weakness; – dichromasy: is present if a certain color is not perceived at all; dichromasies are the most frequent anomalies in humans and can be passed on gender specifically. Dichromates have only two color receptors, leaving red-, green-, and blue-yellow blindness; – achromatopsy: this term is a synonym for complete color blindness. Affected people do not possess any of the three well-operating color receptors; such people can only differentiate lightness levels. Although the after retinal occurring neural processes are only vaguely understood for normal color vision – and surely will remain for a long time at this level understanding – we follow at least roughly the path of the neural signals in the brain. This discussion is also carried out here to indicate to the reader how difficult the search of an adequate color vision theory is. About 50 ms after photon absorption in the photo pigments, retinal signals reach the optic nerve behind the eye. Each left and right half of the retina of both eyes form a nerve fiber; see Fig. 2.62. At the position of the optic nerves crossing over, the so-called chiasma or optic chiasma, each nerve fiber coming from the left and right half of both retinas is shared to the related brain lope. In the optic chiasma, however, there is no mixing of the incoming neural signals from both eyes. The subsequent optic tract ends at the region of the so-called lateral geniculate nucleus, where Lateral geniculate nucleus

Optic radiations Optic tract

Visual cortex

Chiasma Optic nerve

Fig. 2.62 Top view of the head to the human brain with indicated path of neural signals between the two retinas and the visual cortex; the visual cortex is in no way a sharp restricted region of both brain lopes, but individually different expanded

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the separate incoming neural impulses from both eyes are united into groups. Along several optic radiators, the transformed signals directly reach the region at the back of the brain lopes. This is the location of the center of our visual perception, called the visual cortex. In one special zone of the visual cortex, the actual color perception is produced, just about 100 until 150 ms after photon absorption in the retina. On the basis of this incomplete knowledge, it is necessary to have at least an adequate theory of human color perception. Although the trichromatic color theory of Young–Helmholtz and the opponent color theory of Hering have been shown to be true on physiological basis, neither theory can concretely explain the mechanism in the visual cortex for producing color perception (cf. Chapter 1). Likewise the modern retinex theory of Land, based on studies of color constancy, delivers no usable ideas about the emergence of color impression in the visual cortex. According to the zone theory, proposed by Müller [63] and Judd [64], the first zone contains the three independent S-, M-, and L-color receptors of the trichromatic theory. In the second zone, the generated nerve impulses are transformed according to Hering’s theory into an achromatic signal and two opponent chromatic signals. The visual cortex represents the third zone. The arriving neural signals initiate the color impression including the memory. Accordingly, the zone theory merely brings together the verified and accepted basics of both theories mentioned above. On the basis of current knowledge, it is possible to distinguish 30 different zones of the brain responsible for vision. Merely five of them are assigned to the color sense, of these only one dominates for color perception. The neural processes between photon absorption in the retina and induced color impression in the visual cortex remain unclear. It is not surprising, therefore, that only the physiological confirmed theories of Young–Helmholtz and Hering lead to scientific color applications such as colorimetry. The lacking of a theory which accurately describes the human color sense turns out to be disadvantageous, in particular for quantification of visually perceived color differences. As shown in Sections 3.1 and 3.2, only empirical formulas are given to express a color difference numerically. This unsatisfactory situation has existed since the beginnings of colorimetry.

2.4.2 Color Perception Phenomenon In the previous section, we have roughly outlined how humans perceive color. Now we direct our attention at some remarkable color phenomena of the complex processes of color sensation. Among these are the so-called simultaneous contrast, negative, or positive after-images or phosphene perception. These effects are important in different color applications and need to be taken into

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115

consideration during color assessment. These are, therefore, considered briefly here. The term simultaneous contrast describes the phenomenon that the lightness and the color of the surroundings influence the color impression. We should differentiate between simultaneous lightness contrast and simultaneous color contrast. A middle gray, for example, appears darker in a white neighborhood than in a black one; this is lightness contrast. In a similar manner, differences between dark chromatic colors are resolved worse with bright-surround field than with dark-surround field; this is color contrast. For visual assessment of color differences, it is, therefore, necessary to always use the same achromatic surrounding color (Section 3.2.2). Phenomenon of simultaneous color contrast is more varied than simultaneous lightness contrast. In both cases, the contrast is amplified just at the edges of color areas. The so-called lateral inhibition is responsible for the altered color perception. This term means that the neural signals in adjacent receptors within the retina interact with one another. Changes that are caused by lightness or color alone are, therefore, felt lesser strongly; contrasts, on the other hand, are perceived amplified. An additional quite remarkable ability of human color sense is the fact that many object colors are unchanged in spite of illuminant change. For example, a blue color is still perceived as the same blue, although the spectral energy distribution of the illuminant is varied. Another property of the visual sense is combined with the so-called afterimage. This after-effect occurs, if a single-colored area of high color saturation is seen with fixed eyes during about 1 min, and without movement of the eyes or head. If the eyes are afterward adapted to a white area, the color impression remains for some time. This negative after-image is produced in the complementary color. The reason for this is that the most of the corresponding receptors are desensitized during the fixing of the chromatic surface; the remaining sensitized receptors continue to have a reinforced effect during the white impression. The contrary phenomenon becomes apparent if the eyes are closed for some minutes, and then for a short time directed toward a contrasting object and then closed again. The resulting positive after-image shows that the color sensation actually lasts longer than the original action of the light. A longer continuing involuntary persistence occurs on chromatic adaption. After-images are in particular of importance for colorists if color shades are to be judged after a bright lighting phase with monochromatic sources. Therefore, before assessment of critical hues, for example, is important to look at a large, plane, and achromatic area for several minutes in order to avoid influences of after-images. A further time-dependent characteristic of color perception is associated with a short viewing of a color stimulus. For a viewing time of less than 0.1 s, most hues appear progressively desaturated. In the case of very short stimuli of about 3 ms duration, monochromatic light in the range from 490 to 520 nm appears

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achromatic white or gray. These effects are attributed to the property that the response time of the receptors is clearly shorter for achromatic colors than for chromatic colors. Colors can be perceived, quite astonishingly, also without an outside stimulus; this phenomenon is called phosphene. After the eyes are accommodated to darkness for a sufficient time, it is normal to observe light or colored spots. The effect of outside pressure on one or both eyes results in time-dependent changing light areas or colored schlieren. Phosphene occurs in addition without dark adaption by sudden pressure or electric current. In these cases, the influence from outside the closed eyes evokes disturbances in circulation and neural transmission. There are further phenomena of color perception which depend on spatial conditions or movement of the colored object. These are not of interest to color physics and are beyond the scope of this book. With exception of phosphene, the above outlined effects enable us to perceive a world of stable colors and lightness, although we are surrounded by an even greater variety of electromagnetic waves of different frequencies and changing intensities. But these electromagnetic waves are responsible for further color effects such as additive color mixing, which is effective especially in the human eye, or subtractive color mixing which, for example, is present in absorption colorants. Both phenomena are subjects of the following section.

2.4.3 Subtractive and Additive Mixing of Colors A deeper insight into the human color sense has been obscured as a consequence of subjective color judgment and, up to now, the inability to detect directly the chromatic signals at each position from the retina to the visual cortex by objective measurements. Therefore, Grassmann tried an indirect way to come closer to the human color sense [65]. In order to describe color impressions quantitatively, he avoided some of the briefly mentioned problems by using three monochromatic light sources of different wavelengths; see Fig. 2.63. Consequently, color stimuli are generated under defined conditions and color effects can be observed and registered. Non-self-luminescent colors are unsuited for such investigations, because the colors of these light sources are required to be produced quite simply, reproducible, and additionally, they need to be mixed trouble-free with one another. For better understanding of the results achieved by Grassmann, it is useful to go into details of subtractive and additive color mixing, both of which are basic properties of color physics. Subtractive color mixing is present, for example, after polychromatic light has passed through an optical selective filter: one part of the incident light is absorbed by the filter, and, therefore, denoted as subtracted. In Table 2.19, some

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117

Red Green

Blue White screen

Observer

Black partition

Incandescent lamp

Fig. 2.63 Configuration for producing light colors for additive color mixing

examples of subtractive color mixing are listed, cf. also Color plate 1, overlapping upper colors. In the case of several simultaneously used filters, the resulting transmitted color is independent of the order of the filter. Subtractive color mixing dominates in all sorts of absorption colorants and their mixtures; the color result of a colorant mixture is, therefore, independent of addition sequence. Subtraction of light waves occurs also in special cases of mixtures with effect pigments, as well as during developing of color pictures, among other things. In contrast, additive color mixing occurs by superposition of colored lights. The likely most important example is the human color perception. Additive color mixing is subject to photons of visible light, provided they enter the retina at the same time. Superimposed, the waves or photons are denoted as added. If the retinal photo pigments are stimulated simultaneously by polychromatic light, merely one single color stimulus is induced. In this context the synonymous term “additive color mixing of light” is correct. Further examples of additive color mixture are computer or television screens. For such screens, addressable groups of closely packed red, green, and blue pixels of diameter less than 0.1 mm (of phosphorus, liquid crystals, or inert gas cells) are stimulated simultaneously to produce color points. Color pixels of such small lateral dimension cannot be perceived separately by the eye at sufficient distance; therefore, only a single color stimulus is produced. A macroscopic example is shown in Color plate 1, lower overlapping colors, cf. Table 2.19.

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Table 2.19 Examples of subtractive and additive mixing of colors; cf. Color plate 1, overlapping color segments Subtractive mixing of colors

Additive mixing of colors

Initial colors

Mixed color

Initial colors

Mixed color

Magenta, yellow Yellow, cyan Cyan, magenta Magenta, yellow, cyan

Red Green Blue Black

Green, blue Blue, red Red, green Green, blue, red

Cyan Magenta Yellow White

From the position and shape of the curves in Fig. 2.61, it is clear that the light absorption of the red, green, and blue color receptors occurs in different and partially overlapping spectral regions, although only one color stimulus is evoked in the retina. The color stimulus Φ clearly depends on wavelength λ. If the photons of the source directly reach the retina, the color stimulus Φ(λ) equals the spectral energy distribution S(λ) of the source. For non-self-luminous colors, Φ(λ) is given by the product of S(λ) and the reflection R(λ) or transmission T(λ) of the illuminated layer: Φ(λ) = S(λ) · T(λ),

(2.4.1)

Φ(λ) = S(λ) · T(λ).

(2.4.2)

Using several illumination sources simultaneously, each individual color stimulus is summed up to form an aggregate stimulus, exactly the same procedure as in the retina or in screens. All further color physical themes dealt within this book, such as color vision, colorimetry, color measurement, or color recipe prediction, are based on the fundamental empirical laws of additive color mixing formulated by Grassmann in 1853. As briefly mentioned at the beginning of this section, for his investigations, Grassmann used three monochromatic light sources emitting constant red, green, and blue wavelengths, respectively. With a similar configuration to that shown in Fig. 2.63, but without the incandescent lamp, a new light color is mixed on a white field by superposition of these three wavelengths. The experimental results show that any additive color mixing is characterized by three arbitrarily chosen primaries R, G, B, in this case lights, which are given by the corresponding color values R, G, B. They correspond to the relative luminance of the light sources used. Of importance is that each of the three color values R, G, B is not necessarily restricted to a fixed wavelength in the red, green, or blue range; they are optional. The three color values R, G, B are combined in the so-called color

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stimulus specification C, which is written as a vector in three dimensions19 C = (R, G, B)T .

(2.4.3)

Furthermore, the primaries R, G, B can be interpreted as the unit vectors of a three-dimensional space R = (1, 0, 0)T ,

G = (0, 1, 0)T ,

B = (0, 0, 1)T .

(2.4.4)

Using terms such as color values, color stimulus specification, and primaries, Grassmann summarized his results in the following three empirical laws of additive color mixture: 1. For identification of a color stimulus specification, three independent primaries, which cannot be matched by additive mixture of the other stimuli, are necessary and sufficient. 2. The result of an additive color mixture is influenced only by the color stimulus specifications, not by their spectral compositions. 3. All producible color mixing series change continuously. Now, we consider the consequences of the Grassmann laws. According to his first law, each color stimulus specification C is represented by an equation of the form C = RR + GG + BB.

(2.4.5)

It is called the color equation. Equation (2.4.5) is of central significance to all of color physics. It is valid for arbitrarily selected primaries R, G, B. The three color values R, G, B provide the contributions of the three primaries to the color stimulus specification C. In other words, the color values describe the entire color impression. This is synonymous with the statement that every color impression is given in quantitative form by three color values R, G, B. The numerical quantities of R, G, B are, therefore, representatives of a color impression. The second law of additive color mixing turns out, for example, to be applicable to all cases of color matching; it is the basis of color recipe prediction. For any color impression, it is irrelevant of which individual colorant components the corresponding coloration is composed. The same holds for the mixture of lights of different spectral distributions. If, for example, two arbitrarily given color stimulus specifications C1 and C2 are matched by three primaries R, G, B, then from the first Grassmann law, it follows 19 The exponent

T symbolizes a transposed vector.

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C1 = R1 R + G1 G + B1 B ,

(2.4.6)

C2 = R2 R + G2 G + B2 B.

(2.4.7)

According to the second law, the additive mixture of both color stimulus specifications is given by the sum of the corresponding individual color values

C = C1 + C2 = (R1 + R2 )R + (G1 + G2 )G + (B1 + B2 )B.

(2.4.8)

The additive color mixing produces a new color stimulus specification C, from which is not possible to discern whether or not it consists of several individual color components (i.e., color values). The third law of additive color mixing implies that if one or more components of the mixture are gradually changed, then the resulting color values also change gradually, that is, as opposed to changing in a discontinuous fashion. Therefore, if we assign the color stimulus specification to a point in three-dimensional space, all additively mixed color stimuli generate an own continuous and coherent color solid, the color space. Grassmann has formulated originally four laws, which in modern literature are summarized to the three given above. In the next section, we pursue the industrially important question, how we can use color values to treat color impressions objectively?

2.4.4 Tristimulus Color-Matching Experiments According to the laws of additive color mixing, a given color is characterized by three color values R, G, B. For handling of these laws, it is useful to establish the optional primaries R, G, B in such a way that at least the requirements that follow are fulfilled. The primaries should – be definite, reproducible, and computable from color measurements; – have always a positive sign; – correspond to photopic vision; – achieve numerical color differences corresponding to the visually perceived color differences. The system introduced from the CIE in 1931 meets, to a large extent, the listed first three criteria [7]. The final listed requirement is as yet not satisfactorily solved (see, e.g., Section 3.2). The CIE 1931 system rests on the results of tristimulus color-matching investigations of Wright [66] and Guild [67], which used ten and seven people, respectively. For these experiments, the complete configuration used is shown in Fig. 2.63. With a source of nearly monochromatic

2.4

Observer

121

light, a test stimulus of ±2.5 nm wavelength accuracy was generated by illuminating a bipartite white screen which was shielded against the other half. On this second field, the additive mixture of three primaries was projected. These were the matching stimuli of the three monochromatic lights. By using adjustable light controllers, the observer subjectively adjusted the light flux of the three primaries to obtain a color match between the two separated fields. Wright and Guild used three monochromatic light sources of wavelengths 700.0, 564.1, and 435.8 nm. In the case of a match, the test stimulus was characterized by the three luminance values of the primaries. Before the actual matching experiments were performed, the intensities of the three primaries were adjusted to obtain an additive mixture of achromatic white. This procedure is termed as white balance; an example is shown in Color plate 1, see lower overlapping color circles. The color values are, therefore, normalized with regard to the radiant energy density of the primaries for white balance: the accompanying color equations are valid for the so-called equienergy spectrum. Colors differing only in their lightness belong to the same chromaticity. The color stimulus specification of the test stimulus at wavelength λi is termed as Ci . The matching color values of the three primaries are called colormatching values and are assigned by ri , gi , bi ; they are valid especially for the chosen primaries R, G, B [68, 69]. Due to the subjectively different color sensation of the observers, the ri , gi , bi values fluctuate. Their mean values are denoted by r¯i , g¯ i , b¯ i . In the case of match, the color equation Ci = r¯i R + g¯ i G + b¯ i B

(2.4.9)

is fulfilled according to the first Grassmann law. However, the matching procedures certainly show that not every given spectral color can be matched with the preset primaries. For example, a blue-green color of wavelength 490 nm does not match by mixing the blue or green primaries alone. In this case, the color match of equally saturated color fields is only achieved if an amount r¯io of the primary R is directly added to the given test stimulus specification Ci . This procedure is called outer mixture and is represented by the color equation Ci + r¯io R = g¯ i G + b¯ i B.

(2.4.10)

By adding the quantity −¯rio R to each side of color equation (2.4.10), an equation analogous to Equation (2.4.9), but with negative amount −¯rio on the right-hand side, results. This is a less than satisfactory situation to which we return in the next section. For continuous wavelengths λ, the mean color-matching values r¯i , g¯ i , b¯ i turn into the color-matching functions (CMFs), which are assigned to r¯ (λ),

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2 Light Sources, Types of Colorants, Observer

Color matching function

0.4

– r (λ)

– b (λ)

0.3

g–(λ)

0.2

0.1

0 546.1

435.8

700.0

–0.1 400

500

600

λ nm

¯ Fig. 2.64 Color-matching functions r¯ (λ), g¯ (λ), b(λ) of the equienergy spectrum using wavelengths of 700.0, 546.1, and 435.8 nm for primaries R, G, B

2°

10°

50

1.8 9.1

Fig. 2.65 Visual angles of 2◦ and 10◦

¯ g¯ (λ), b(λ), and shown in Fig. 2.64. The color match on both color fields is carried out with a visual angle of 2◦ by the observer; see Fig. 2.65. Under this observation angle and at normal viewing distance of 50 cm, a circle area of 1.8 cm in diameter is perceived. In the same angular range, the fovea of the retina contains only cones (Fig. 2.60). An observer, which fulfills these conditions, is called 2◦ standard observer, abbreviated 2◦ observer, and is represented ¯ by the three CMFs r¯ (λ), g¯ (λ), b(λ). As shown in Fig. 2.64, the CMFs have unequal peak heights. This represents the different sensitivities of the corresponding three retinal color receptors. Among the CMFs, only the function r¯ (λ) has a negative amount with the dip minimum at about 522 nm. The negative proportion is a consequence of the outer mixture. If the reference color stimulus specification corresponds to the

2.4

Observer

123

wavelength of a primary, the other two primaries are not required and their corresponding CMFs are zero. This can be taken from Fig. 2.64 for wavelengths of 435.8, 546.1, and 700.0 nm. Finally, it needs to be pointed out that the trichromatic matching experiments were achieved only with few people, under photopic vision with an observation angle of 2◦ and dark surroundings. Insights for scotopic or mesopic vision, bright surroundings, or large colored fields have, up to now, not been performed.

2.4.5 Determination of Tristimulus Values In view of color measuring methods and colorimetric applications, we will use discrete wavelengths λi in the following. A trichromatic stimulus is induced, according to the explanations in Section 2.4.3, by additive mixture of all monochromatic color stimuli Φ i = Φ(λi ). In the case of non-self-luminous colors, the color stimulus Φ i follows, according to Equations (2.4.1) and (2.4.2), from Φi = S(λi ) · R(λi )

(2.4.11)

Φi = S(λi ) · T(λi ),

(2.4.12)

or

because a non-self-luminous color is to illuminate with a source of spectral power distribution S(λi ). Each retinal color stimulus Φ i is represented by a corresponding color stimulus specification Ci . If there are simultaneously i = 1, 2, . . . , N stimuli Ci , the total color stimulus specification is given by additive mixture: C=

N 

Ci .

(2.4.13)

i=1

Applying Equation (2.4.8) with N ≥ 2, C=

N  i=1

Ci =

N  i=1





Ri R +

N 

Gi G +

i=1



N 

Bi B

(2.4.14)

i=1

follows. The color values Ri , Gi , Bi correspond, on the other hand, to the product of the color stimulus Φ i , one of the respective CMFs r¯i , g¯ i , b¯ i , and the technically given wavelength interval width Δλ of the used illuminant. From this, it is possible to write the equations for the color values: Ri = Φi r¯i Δλ,

(2.4.15)

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2 Light Sources, Types of Colorants, Observer

Gi = Φi g¯ i Δλ,

(2.4.16)

Bi = Φi b¯ i Δλ .

(2.4.17)

If we insert these relations into Equation (2.4.14) and compare the new equation with Equation (2.4.5), finally the color values R, G, B of a non-self-luminous color follow from expressions:

R=

N 

Φi r¯i Δλ,

(2.4.18)

Φi g¯ i Δλ,

(2.4.19)

Φi b¯ i Δλ.

(2.4.20)

i=1

G=

N  i=1

B=

N  i=1

If the emitted wavelengths are quite closely spaced together so that infinitesimally small wavelength intervals dλ can be assumed, the discrete color stimuli Φ i can be written as the color stimulus function Φ(λ). Then, the color-matching ¯ Accordingly, the sums values r¯i , g¯ i , b¯ i also become the CMFs r¯ (λ), g¯ (λ), b(λ). in Equations (2.4.13), (2.4.14) and (2.4.18), (2.4.19), (2.4.20) should be substituted by integrals and the integration limits correspond to the lower and upper cutoff wavelengths of the visible range. In this text, the sum notation is generally preferred. Two straightforward reasons for this preference are as follows: first, the spectral power distribution S(λi ) of the most used illuminants in color industry and the CMFs of the 2◦ standard observer are tabulated for interval widths Δλ of 5, 10, and 20 nm [70–72] and second, all industrial spectrophotometers are designed to measure in one of these wavelength interval widths. Therefore, the color values R, G, B of a given color sample can be calculated from Equations (2.4.18), (2.4.19), and (2.4.20) if the accompanying reflection or transmission is known from measurement. To summarize, we emphasize that owing to additive color mixture, a given color can be described by three corresponding numerical color values R, G, B. These are merely representatives of the corresponding color and are by no means absolute values. On the basis of the experimental connections, they can be interpreted as the red, green, and blue components of a color shade. In the following section these results are finally modified for practical applications in colorimetry.

2.4

Observer

125

2.4.6 CIE 1931 and CIE 1964 Standard Colorimetric Observers Among the indicated color values, the red component R of a color shade has a region of negative values of the CMF r¯ (λ) shown in Fig. 2.64: this is extremely unreasonable. Depending on color stimulus Φ i , the color value R can show, according to Equation (2.4.18), a positive or negative sign and, therefore, cannot be clearly interpreted in a coloristic sense. More confusing is, for example, a red coloration with a color value of R = 0. The occurrence of negative color values is due to the choice of the three real primaries R, G, B. Negative color values cannot be altogether avoided with the spectral energy distribution of technical light sources. In order to achieve only positive or zero CMFs and color values, respectively, the CIE introduced in the year 1931 – after the conclusions of Wright [66] and Guild [67] – the so-called virtual primaries X, Y, Z. These follow from a suitable numerical transformation of the previous real primaries R, G, B [7]. The virtual primaries X, Y, Z are chosen in such a way that the following criteria are fulfilled:

– X, Y, Z are by definition independent of each another, and are, therefore, not producible by mixing (the same requirement as for the real primaries R, G, B); – the new CMFs x¯ (λ), y¯ (λ), z¯(λ) always take values ≥ 0 and follow from the previous CMFs by an appropriate transformation (see below); – the virtual primaries X, Y, Z are adjusted in such a way that the corresponding tristimulus values X, Y, Z are equally valued for the chromaticity of the equienergy spectrum: X = Y = Z; – the two virtual primaries X and Z are chosen in such a way that only the color value Y is proportional to the lightness of a color.

The new color quantities X, Y, Z are called standard color values. The new CMFs x¯ (λ), y¯ (λ), z¯(λ) are named standard color-matching functions (SCMFs) ¯ and are given with the previous CMFs r¯ (λ), g¯ (λ), b(λ) by the empirical matrix equation ⎛

⎞ ⎛ ⎞ ⎛ ⎞ r¯ (λ) x¯ (λ) 2.768892 1.751748 1.130160 ⎝ y¯ (λ) ⎠ = ⎝ 1 4.590700 0.060100 ⎠ · ⎝ g¯ (λ) ⎠ . ¯ z¯(λ) 0 0.056508 5.594292 b(λ)

(2.4.21)

These SCMFs belong to the 2◦ observer, called CIE 1931 observer; the corresponding graphs are shown in Fig. 2.66 [7]. Discrete values of these functions are used to calculate the standard color values X, Y, Z, see below.

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2 Light Sources, Types of Colorants, Observer

In practice, all color samples for visual assessment have a larger sized colored area compared with the 2◦ field at the normal vision range of 50 cm (Fig. 2.65). A considerable reason for such small-sized samples is that an observation angle of 2◦ covers only the area of the fovea. Although there is the highest cone density, the induced color stimulus from this spot is insufficient for an adequate color assessment. Because of that, the CIE introduced in 1964 the so-called 10◦ observer, also termed as CIE 1964 observer, based on investigations of Stiles and Burch as well as Speranskaja with 49 and 27 people, respectively [73–75]. Recent tabular values of the SCMFs for the 10◦ observer have been given in the literature [70–72]. Whereas the primaries of the 2◦ observer are related to the wavelengths 700.0, 546.1, and 435.8 nm and the primaries of the 10◦ observer have the corresponding wavelengths 645.2, 526.3, and 444.4 nm. For the normal vision distance of 50 cm, the visual angle of 10◦ covers a circle area of 8.8 cm in diameter, cf. Fig. 2.65. In this area, the retina certainly contains rods. For this reason, the SCMFs were corrected to eliminate the influence of the rods [7]. The CIE recommended the 2◦ observer for visual angles between 1◦ and 4◦ , and the 10◦ observer for visual angles > 4 ◦ . The 4◦ limit was arbitrarily established although there exists no discontinuity of color perception at this angle. The SCMFs of the 10◦ observer x¯ 10 (λ), y¯ 10 (λ), z¯10 (λ) are distinguished from those of the 2◦ observer by the subscript 10. The values follow from a transformation similar to Equation (2.4.21) using the corresponding CMFs r¯10 (λ), g¯ 10 (λ), b¯ 10 (λ) according to the equation ⎞ ⎛ ⎞ ⎞ ⎛ r¯10 (λ) 0.341080 0.189145 0.387529 x¯ 10 (λ) ⎝ y¯ 10 (λ) ⎠ = ⎝ 0.139058 0.837460 0.073160 ⎠ · ⎝ g¯ 10 (λ) ⎠ . 0 0.039553 1.026200 z¯10 (λ) b¯ 10 (λ) ⎛

(2.4.22)

The numerical values of the components of the matrices in Equations (2.4.21) and (2.4.22) are unequal because they belong to different primaries and different observer fields. The SCMFs of both observers are together represented in Fig. 2.66. The maxima of the functions y¯ (λ) and y¯ 10 (λ) are normalized at λ = 555 nm to 1.0. Due to the different visual field of the observers, the corresponding curves do not match perfectly. For one and the same color, the standard color values corresponding to each observer are also not identical. From the nonlinear wavelength dependence of both CMFs follows that the corresponding standard color values cannot be converted into one another. Although the 10◦ observer is used to an increasing extent in color industry, in order to avoid ambiguity, the observer must be clearly indicated with the results. For the last adjustment criterion of the four listed above, it should be stated that the graph of the function y¯ 10 (λ) agrees with the measured luminous efficiency V(λ) of the human eye

2.4

Observer

127

2° 10°

Standard color matching function

2.0 z (λ) 1.5

x (λ)

y (λ) 1.0

x (λ)

0.5

0 400

500

600

700

λ

nm

Fig. 2.66 Color-matching functions of the 2◦ and 10◦ standard observers (CIE 1931 and CIE 1964 observers, respectively)

y¯ 10 (λ) = V(λ),

(2.4.23)

see Fig. 2.67.20 The standard color value Y is, on account of this adjustment, a measure for the lightness of a coloration. The curves of V (λ) and V(λ) in Fig. 2.67 correspond to the sensitivity of the eye for scotopic and photopic adaption, respectively. The curves are shifted with respect one another by about 40 nm. This is described by the so-called Purkinje effect. From the half-widths of 150 nm, it follows that the middle wavelengths of the visible spectrum are perceived with particular sensitivity under both adaption conditions. Now, the CIE standard color values X, Y, Z are simply given from the previous quantities R, G, B by substituting the discrete color-matching values r¯i , g¯ i , b¯ i of the 2◦ observer in Equations (2.4.18), (2.4.19), and (2.4.20) by the corresponding quantities x¯ i , y¯ i , z¯i :

20 Since

V(λ).

2005, the CIE’s altered recommendation [71]; since 1931 it was accepted y¯ (λ) =

128

2 Light Sources, Types of Colorants, Observer

V (λ)

Relative luminous efficiency

V '(λ) 1.0

0.5

0

400

500

600

700

λ

nm

Fig. 2.67 Relative luminous efficiency of the human eye in scotopic adaption V (λ) and photopic adaption V(λ)

X=

N 

Φi x¯ i Δλ,

(2.4.24)

Φi y¯ i Δλ,

(2.4.25)

Φi z¯i Δλ.

(2.4.26)

i=1

Y=

N  i=1

Z=

N  i=1

The standard color values for the 10◦ observer follow from the same considerations using x¯ 10, i , y¯ 10, i , z¯10, i instead of x¯ i , y¯ i , z¯i and are denoted by X10 , Y10 , Z10 . According to the last three relations, the standard color values of a coloration can be explicitly calculated if the retinal color stimuli Φ i are known from Equation (2.4.11) or (2.4.12). The spectral power distribution S(λi ) of the light source used is known from CIE tables. This means that the spectral power distribution of the real source is substituted by a corresponding artificial source for the determination of color values. The same is the case with regard to the observer: the individual observer is substituted by one of the standard observers. In addition, the measured reflection or transmission of a given color sample can differ in different color measuring devices. The retinal color stimulus values above are, therefore, idealized quantities and it is, therefore, not astonishing that color values can differ from the individual visual assessment. Nevertheless, a given coloration is unambiguously characterized by three corresponding numerical standard color values. Colors of equal standard values are

References

129

perceived as equal, even if they consist of different sorts of colorants (e.g., dyes or absorption pigments). Because of this, it is not possible to infer the constituent colorants from the numerics of the standard color values. At best, with experience, one can estimate the lightness, chroma, or hue of the color. On the other hand, a single color value alone is not of main interest in color physical applications. Of much greater importance is the color difference between two or more similar color shades. Such questions and answers to them for nearly all sorts of colors are discussed in the sections of the following chapter.

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24. Benzing, G, Ed: “Pigmente und Farbstoffe fuer die Lackindustrie”, 2nd ed, expert, Ehningen (1992) 25. Hunger, K, Ed: “Industrial Dyes”, Wiley-VCH, Weinheim (2003) 26. Valeur, B: “Molecular Fluorescence”, Wiley-VCH, Weinheim (2006) 27. Hunt, RWG: “The Reproduction of Colour”, 6th ed, Wiley, Chichester (2004) 28. Smith, KJ: “Colour order systems, colour spaces, colour difference and colour scales”; in McDonald, R, Ed: “Colour Physics for Industry”, 2nd ed, Soc of Dyers and Colourists, Bradford (1997) 29. Kuehni, RG, Schwarz, A: “Color Ordered: A Survey of Color Order Systems from Antiquity to the Present”, Oxford University Press, Oxford (2008) 30. Munsell, AH: “Atlas of the Munsell Color System”, Wadsworth-Howland & Company, Malden, MA (1915); “Munsell Book of Color”, Munsell Color Co, Baltimore, MD (1929) until now 31. Hard, A, Sivik, L, Tonquist, G: “NCS natural colour system from concept to research and applications”, Col Res Appl 21 (1996) 129 32. Colour Index International: “Pigment and solvent Dyes”, Soc Dyers Col, Bradford, England; and American Association of Textile Chemists and Colorists, Research Triangle Park, NC (1998) 33. ASTM E 308 – 08: “Standard Practice for Computing the Colors of Objects by Using the CIE-System”, American Society for Testing and Materials, West Conshohocken, PA (2008); DIN 6164: “DIN-Farbenkarte”, Part 1–3, Deutsches Institut fuer Normung eV, Berlin (1980–1981) 34. Nickerson, D: “OSA uniform color scale samples: a unique set”, Col Res Appl 6 (1981) 7 35. RAL: “RAL-Design-System”, “RAL Effect”; RAL Deutsches Institut fuer Guetesicherung und Kennzeichnung, St Augustin, Germany (2008) 36. ASTM D 16 – 08: “Standard Terminology for Paint, Related Coatings, Materials, and Applications”, American Society for Testing and Materials, West Conshohocken, PA (2008); CIE No 124/1: “Colour notations and colour order systems”, CIE, Bureau de la CIE, Wien (1997); DIN 55943: “Farbmittel – Begriffe”, Deutsches Institut fuer Normung eV, Berlin (2001) 37. Landolt-Boernstein: “Zahlenwerte und Funktionen aus Naturwissenschaften und Technik”, new series II, Vol 15b: “Metalle: Elektronische Transportphaenomene”, Springer, Berlin (1985) 38. Wheeler, I: “Metallic Pigments in Polymers”, Rapra Techn Ltd, Shawbury, UK (2003) 39. Wissling, P et al: “Metallic effect pigments”, Vincentz Network, Hannover (2007) 40. Pfaff, G: “Spezielle Effektpigmente”, 2nd ed, Vincentz Network, Hannover (2007) 41. ASTM D 480 – 88: “Standard Test Methods for Sampling and Testing of Flaked Aluminum Powders and Pastes”, American Society for Testing and Materials, West Conshohocken, PA (2008); DIN 55923: “Pigmente: Aluminiumpigmente und Aluminiumpigmentpasten fuer Anstrichfarben”, Deutsches Institut fuer Normung eV, Berlin (1983) 42. Roesler, G: “Colorimetric characterization and comparison of metallic paints”, Polymers Paint Colour J 181 (1991) 230 43. Rodriguez, ABJ: “Color and appearance measurement of metallic and pearlescent finishes”, ASTM Standardization News 10 (1995) 44. Panush, S: “Opalescent Automotive Paint Compositions Containing Microtitaniumdioxide Pigment”, Patent US 4753829 (1986) 45. Pfaff, G, Franz, KD, Emmert, R, Nitta, K, in: “Ullmann’s Encyclopedia of Industrial Chemistry: Pigments, Inorganic”, Section 43; 6th ed, Wiley-VCH, Weinheim (1998) 46. Teany, S, Pfaff, G, Nitta, K: “New effect pigments using innovative substrates”, Eur Coatings J 4 (1999) 434 47. Sharrock, SR, Schuel, N: “New effect pigments based on SiO2 and Al2 O3 flakes”, Eur Coatings J 1–2 (2000) 105 48. Maile, FJ, Pfaff, G, Reynders, P: “Effect pigments: past, present, future”, Progr Org Coat 54/3 (2005) 150

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Chapter 3

Systems of Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

Because the trichromatic color sense of the observer is based on additive color mixture, every color impression can be assigned three numerical color values. These color stimulus specifications are, so to speak, representatives of a color. In this chapter, the previous tristimulus values are extended, specifically in order to better meet some coloristical requirements of color industry. The empirical three color values can additionally be interpreted as points in a color space. We restrict the discussion first to the standardized systems that have been proposed for application by the CIE. The most important empirical color difference formulas in colorimetry follow from these systems. These formulas enable to calculate only the color differences of similar or nearly equal colors. In addition, we discuss the standardized DIN99o color space and the newest color appearance model which offer further methods to solve problems in industrial color physics. These formalisms allow one to better detect numerically such color deviations, which inevitably occur during production, reproduction, repair, or aging. The main focus of the effort relates to properties of absorption colors, such as color constancy, metamerism, color strength, covering power, or transparency, which are described by characteristic values. Experience shows that the related colorimetric procedures are quite efficient; however, they do have limits in their applicability. The limits in applicability of these methods are shown especially with different sorts of colorations containing effect pigments.

3.1 Systems of Standardized Tristimulus Values After introduction of the tristimulus values X, Y, Z by the CIE [1] in 1931, more than 13 different color-value systems were worked out. Each of the accompanying color spaces followed from the intention of broadening the basis of the subjective description for the increasing industrial color requirements of absorption colors. This, in no way arbitrary, variety is an expression of the fact that all of these empirical systems were incapable of sufficiently answering the most G.A. Klein, Industrial Color Physics, Springer Series in Optical Sciences 154, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1197-1_3, 

133

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important questions in the color industry. In the following, we restrict the discussion to the three most applied contemporary color value systems. Although the color physical results with these systems are up to now quite promising, the numerical simulation of the human color sense needs further research.

3.1.1 CIE 1931 Tristimulus Values The standard color values X, Y, Z, also called CIE 1931 tristimulus values, given with Equations (2.4.24), (2.4.25), and (2.4.26), are applicable for opaque, translucent, as well as transparent non-self-luminous colors. However, the following modifications need to be taken into consideration depending on the type of light transmittance. For opaque colors, the color stimulus Φ(λ) from Equation (2.4.1) for each wavelength λi should be inserted in the above-mentioned three equations, which results in X=k Y=k

N  i=1 N 

S(λi )R(λi )¯x(λi )Δλ ,

(3.1.1)

S(λi )R(λi )¯y(λi )Δλ ,

(3.1.2)

S(λi )R(λi )¯z(λi )Δλ .

(3.1.3)

i=1

Z=k

N  i=1

In the case of translucent colors, the additional color stimulus Φ(λ) from Equation (2.4.2) is valid so that three further tristimulus values result. These are simply given by substitution of the spectral reflectance R(λi ) in Equations (3.1.1), (3.1.2), and (3.1.3) by the spectral transmittance T(λi ). For ideal transparent colors, only the transmittance values actually measured should be taken into consideration. The spectral power distribution of the source S(λi ) as well as the SCMFs x¯ (λi ), y¯ (λi ), z¯(λi ) of the chosen standard observer should be taken from standard tables for the relevant Δλ value, which is given by the measuring conditions [2,3].1 The constant k in the above equations depends on the chosen illuminant, the SMCF y¯ (λi ), and the wavelength interval Δλ. This comes from the following consideration. Independent of the wavelength λi in the visible range, an ideally scattering achromatic white field has a constant reflectance R(λi ) = 1.000. For an ideal transparent material, the set value is T(λi ) = 1.000. As explained in Section 2.4.6, the standard color-matching function y¯ (λi ) is accepted as identical to the 1 In addition to previous

recommendations, since the year 2001, the CIE also edits standards, which are designated by CIE S.

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Systems of Standardized Tristimulus Values

135

light sensitivity V(λi ) of the human eye, which is adjusted to the standard color value Y. Consequently, the standard color value is a measure for the lightness of a color. On account of this consideration, the CIE has arbitrarily assigned the ideal scattering field the value Y = 100.000 for every illuminant. Equation (3.2.1), therefore, leads to k=

100.000 N 

.

(3.1.4)

S(λi ) · y¯ (λi ) · Δλ

i=1

The choice of the constant k in this equation has a far-reaching consequence: the three standardized color values (and all further ones) are dimensionless quantities, particularly considering that the reflectance and transmittance are dimensionless; see Equations (2.1.14a) and (2.1.14b). Similar considerations are valid in view of the 10◦ observer. The corresponding formalism follows by insertion of the SCMFs x¯ 10 (λi ), y¯ 10 (λi ), z¯10 (λi ) into Equations (3.1.1), (3.1.2), and (3.1.3). In this case the constant k in Equation (3.1.4) simply results from using y¯ 10 (λi ) instead of y¯ (λi ). The accompanying standard color values are denoted as X10 , Y10 , Z10 . It should be pointed out again that for clear interpretation, all color values are to be specified with the underlying illuminant and standard observer. The standard color values X and Z are not subject to further adjustment conditions. Both quantities can, therefore, differ from 100.000. This is shown in Table 3.1 for ideal white (or ideal transparency) and common illuminants. The tabular values of Xn and Zn follow from Equations (3.1.1) and (3.1.3) by insertion of the ideal reflectance or transmittance R(λi ) = T(λi ) = 1.000. For all illuminants and standard observers, the quantity Yn = 100.000 remains. The quantities Xn and Zn are to be consequently interpreted as standard color values of the corresponding illuminant. A careful view of the sums in Equations (3.1.1), (3.1.2), and (3.1.3) confirms the conclusion which was already stated at the beginning of the previous chapter. Table 3.1 CIE standardized color values of different illuminants 2◦ standard observer

10◦ standard observer

Illuminant

Xn

Zn

Xn

Zn

D 65 A C

95.047 109.850 98.074

108.883 35.585 118.232

94.811 111.144 97.285

107.304 35.200 116.145

FL 2 FL 7 FL 11

99.186 95.041 100.962

67.393 108.747 64.350

103.279 95.792 103.863

69.027 107.686 65.607

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Each of the three standard color values X, Y, Z follows from the three external factors, which are responsible for the color impression of non-self-luminous colors: – the light source, approximated by the chosen illuminant of spectral power distribution S(λi ); – the color sample, characterized by the measured spectral reflectance R(λi ) or transmittance T(λi ); – the observer, represented by the standard observer of SCMFs x¯ (λi ), y¯ (λi ), z¯(λi ) or x¯ 10 (λi ), y¯ 10 (λi ), z¯10 (λi ). At this stage, it is necessary to point out a critical fact. Among the three factors, the reflectance and transmittance by the colorants, binders, and additives of the color pattern are clearly given. However, the spectral power distribution and the standard observer are idealized quantities: in most cases, the radiant energy of the light source used deviates more or less from the tabulated values S(λi ). In the same sense, the SCMFs x¯ (λi ), y¯ (λi ), z¯(λi ) or x¯ 10 (λi ), y¯ 10 (λi ), z¯10 (λi ) of the chosen standard observer do not necessarily agree with those of the actual observer. Consequently, the standard color values only represent the color impression of the special standard observer in combination with the chosen tabulated illuminant. In addition, it is important to remember the fact that the experimental quantities of reflectance and transmittance always have measurement errors. These critical connections should always be taken into consideration in order to avoid, for example, confusion during assessment of color values. It was already mentioned at the end of Section 2.4.5 that the sums in Equations (3.1.1), (3.1.2), and (3.1.3) are actually integrals. In reality, the optic nerves integrate over all simultaneously incoming spectral stimuli at the retina. This is independent of the width of the corresponding wavelength intervals. Nevertheless, we maintain the sum notation for the moment because the quantities of reflectance R(λi ) or transmittance T(λi ) have to be measured at a fine number N of discrete wavelengths λi . These are to be combined with corresponding tabular values S(λi ) and x¯ 10 (λi ), y¯ 10 (λi ), z¯10 (λi ), for example. The integrals are ipso facto approximated by N addends. Although the psychophysical knowledge with regard to colors is only based on empirical considerations, the color values enable an approximate quantification of the color impression, of special color properties, or of color differences at least of similar absorption colors. Such color quantities are no longer limited merely to subjective verbal descriptions. The reader should realize that every standard color value is only a representative of three special color components of a color pattern. These are exclusively valid for the chosen illuminant and the underlying standard observer.

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137

3.1.2 Chromaticity Coordinates and Chromaticity Diagram In spite of the achieved breakthrough, the CIE 1931 system lacks a level of clarity. However, before the more clear CIE 1976 and DIN99o systems are discussed, it is useful to glance over the so-called chromaticity coordinates x, y, z, which were also introduced by the CIE in 1931. These quantities are based on the standard color values X, Y, Z, upon which the above-mentioned two further systems are also based. The chromaticity coordinates of coloration are defined by the equations x=

X , X+Y +Z

(3.1.5)

y=

Y , X+Y +Z

(3.1.6)

z=

Z . X+Y +Z

(3.1.7)

Each coordinate is given by the ratio of one standard color value over the sum of all three color values of a color pattern. A unity sum of these coordinates follows: x + y + z = 1. This is why two given chromaticity coordinates determine the third quantity. Analogous formulas to Equations (3.1.5) and (3.1.6) follow for the 10◦ standard observer; the chromaticity coordinates are indicated by x10 , y10 , z10 . On the other hand, it is impossible to calculate the three standard color values of a color from two chromaticity coordinates such as x, y. For this calculation, one of the three values X, Y, Z would additionally have to be known. The chromaticity coordinates x and y represent the so-called chromaticity, which combines both terms hue and chroma of a color. The CIE has decided to use the x, y, Y system which is applicable to the 2◦ and 10◦ standard observers. Because the Y value is a measure for the lightness of a color and the primaries X and Z contain no lightness component, the quantities x, y can be taken, for example, as coordinates of a plane triangle. If these coordinates are chosen rectangular, we obtain the CIE chromaticity diagram [1]. In Color plate 2, the chromaticity values x, y of absorption colors with equal lightness are approximately represented.2 The limiting equal-sided triangle of (x, y) corner point coordinates (1, 0), (0, 1), (0, 0) is called the color triangle. Non-self-luminous colors of equal lightness are represented in the chromaticity diagram by the x, y coordinates located inside the sole-shaped area. This area is bordered by the arched spectrum locus and the straight purple boundary. The x, y chromaticity coordinates of visible spectral colors are represented by the spectrum locus. In Color plate 2, some of the corresponding 2

Color plates are inserted between Chapters 3 and 4.

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Table 3.2 Color areas of equal lightness Y in the CIE chromaticity diagram [4] 2◦ standard observer

10◦ standard observer

Illuminant

xn

yn

xn

yn

D 65 A C

0.31272 0.44758 0.31006

0.32903 0.40745 0.31617

0.31382 0.45117 0.31002

0.33100 0.40594 0.31867

FL 2 FL 7 FL 11

0.37207 0.31285 0.38053

0.37512 0.32918 0.37691

0.37928 0.31565 0.38543

0.36723 0.32951 0.37110

monochromatic wavelengths between 380 and 780 nm are indicated. The purple boundary joins the end points of the spectrum locus. The colors represented by the purple boundary are not contained in the visible spectrum, but they can be produced spectrally by additive mixture of monochromatic lights. In the case of the equienergy spectrum, the standard color values X, Y, Z are equal to one another and the chromaticity coordinates are x = y = z = 1/3. The coordinates x = y = 1/3 determine the so-called achromatic point. The quantities of the chromaticity coordinates xn , yn for ideal white or ideal transparency are given in Table 3.2 for various illuminants as well as both standard observers. The values follow from those of Table 3.1 and result from the different spectral power distributions of the sources as well as the slightly different SCMFs of the standard observers. Every point on a straight line through the achromatic point and the spectrum locus represents colors of the same hue. The greater the distance from the achromatic point, the purer the corresponding color. Because the spectrum locus represents only spectral colors, these are the purest colors which are to produce. The third coordinate of the x, y, Y system, represented by the lightness axis, is also arbitrary chosen perpendicular to the color triangle. Its value domain is always between zero for ideal black and 100.000 for ideal white. However, for black, all standard color values vanish. Because of the as-yet not fully defined Equations (3.1.5), (3.1.6), and (3.1.7), it is not so obvious how to represent black in the chromaticity diagram. On the other hand, the area included by the spectrum locus and the purple boundary decreases with increasing lightness. The corresponding entire colors tend to become achromatic until they reach white. This is represented by the chromaticity coordinates of the achromatic point as shown in Fig. 3.1. The volume contained by the surface extending along the lightness axes is called the coloring body. Within this volume, all colors possible to be produced are represented by chromaticity coordinates with the exception of ideal black. The limiting surface, except the base area, represents all pure spectral colors including their mixtures. The chromaticity diagrams in Fig. 3.1 and Color plate 2 belong to the 2◦ observer and illuminant C. In comparison, the spectrum locus for the 10◦

3.1

Systems of Standardized Tristimulus Values

139

520

y

10

540

20 30 40 50

60

560 70

0.6

80 90

580

95

0.4

600 100

620 650 770 nm

0.2 480 470

0

380

450

0

0.2

0.4

0.6

x

Fig. 3.1 Color areas of equal lightness Y in the CIE chromaticity diagram [4]

observer is only marginally shorter. This is caused by the higher and broader maxima of the corresponding SCMFs (Fig. 2.67). The chromaticity points of the illuminants D65, A, and C with correlated color temperatures and of the blackbody radiator are shown in Fig. 3.2. The solid curve section is termed as the Planckian locus. It follows, from Equations (3.1.1), (3.1.2), (3.1.3), (3.1.4), (3.1.5), (3.1.6), and (3.1.7), that this locus has the temperature-dependent Planck law of radiation (2.1.1). The chromaticity coordinates of real temperature or luminescence radiators, however, tend to be located adjacent to the Planckian locus. Although the x, y chromaticity diagram supplies an important contribution to the illustration of color values, it also reveals, however, a serious property of the human color sense which turns out to be a disadvantage in discerning the differences in equal or similar colors. According to the investigations of McAdam, similar colors, which are perceived as equal, lie inside the so-called

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520

y 540

560

0.6 500 580 A 2,000 K 2,856 K 3,500 K D65 4,500 K C 6,500 K 10,000 K

0.4

600

650 770 nm

∞

0.2 480 380

0

450

0

0.2

0.4

0.6

x

Fig. 3.2 Chromaticity coordinates of temperature radiators: standard illuminants A and D65 as well as illuminant C, and blackbody radiator

tolerance ellipses in the chromaticity diagram [4]. Some of these ellipses are shown magnified 10 times in Color plate 3. The length of the ellipse axis and the orientation are not constant; these parameters depend rather on the color locus and lightness. In the area of yellow to green colors, larger ellipses dominate in comparison to those for dark-blue colors. For dark-blue, violet, or dark red colors, the major ellipse axes are oriented nearly parallel to the purple boundary. If this line rotates counterclockwise around the endpoint corresponding to 380 nm, we nearly obtain the new alignment of the major axis in other areas of the chromaticity diagram. Color differences among dark colors are, therefore, more sensibly perceived by the human eye and with improved tolerance than color deviations for brilliant and light colors such as green, yellow, or orange. From colorimetric point of view, the chromaticity diagram gives no consistency in color differences

3.1

Systems of Standardized Tristimulus Values

141

in different color areas. Not only the color sense but even color differences behave in a nonlinear fashion and depend on the color locus in the chromaticity diagram. Consequently, also various color tolerances need to be arranged: a nonnegligible hindrance for industrial color judgment or finding out accurate color tolerance agreement. Brown and McAdam [5] showed, in addition, that Color plate 3 renders only projections of three-dimensional ellipsoids with major axes which are not necessarily located in the x, y plane. The observable lightness differences are also irregularly distributed in the color plane. In a further work, Wyszecki [6] found, however, that the semi-axes of the ellipses are smaller than those given by Brown and McAdam. The shortcomings of the x, y, Y system would really be eliminated, if one could succeed in transforming the X, Y, Z standard color values in such a way that the various ellipsoids change into spheres of equal radii. This would correspond to the uniform chromaticity scale triangle (UCS triangle and corresponding UCS color space). This goal has, until now, not been achieved. As long as it is impossible to quantitatively follow the neural processes of color perception, only the empirical approach of tracking down the so-called visually equidistant color space is possible. This empirical method forms the basis of the modern color value systems and the color appearance model in the following sections.

3.1.3 CIE 1976 Color Spaces The CIE revised several times the x, y, Y system and, in 1978, recommended the so-called CIELAB system, established 1976, for application to non-selfluminous colors [7]. The corresponding three-dimensional color space is based on the Munsell color atlas. These color patterns were characterized by spectrophotometric methods and evaluated visually to be basically equidistant [8]. At the same time, the CIELUV system was published. It is recommended for use with colored light sources and displays. Both systems are united by the term CIE 1976 color spaces. We restrict the discussion to the CIELAB system, which is applied to non-self-luminous colors and to some areas of digital image processing as well. Instead of the Munsell color attributes value V, chroma C, and hue H, the CIELAB color space is based on the following three color values: L∗ : a∗ : b∗ :

lightness or black to white contribution; red to green contribution; yellow to blue contribution.

These correspond to the three color pairs of the color opponent theory of Hering. With this, it should also be taken into account that experience shows that there

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exist no hues of absorption colors such as greenish red, reddish green, or bluish yellow. The term CIELAB is a synonym for CIE and the color values L∗ , a∗ , b∗ . The new color values follow from the empirical transformation equations L∗ = 116 · f (Y/Yn ) − 16,

(3.1.8)

∗

a = 500 {f (X/Xn ) − f (Y/Yn )} ,

(3.1.9)

∗

(3.1.10)

b = 200 {f (Y/Yn ) − f (Z/Zn )} .

These equations contain the previous standard color values X, Y, Z and those of the chosen illuminant Xn , Yn , Zn and standard observer. The values Xn , Yn , Zn are given in Table 3.1 for some illuminants. The three functions f(q) are given by  f (q) =

for q > (24/116)3 q1/3 , (841/108) · q + 16/116 for q ≤ (24/116)3

(3.1.11)

where q represents each of the ratios X/Xn , Y/Yn , Z/Zn in Equations (3.1.8), (3.1.9), and (3.1.10) [9]. The existing cubic root in Equation (3.1.11) was determined following the logarithmic change of a physiological stimulus in the Weber–Fechner law. The numbers of the linear term in Equation (3.1.11) are caused by the Pauli extension [10], with which the continuity of f(q) and the first derivative f (q) is achieved at position q = (24/116)3 . The CIELAB color space is based on the three orthogonal L∗ , a∗ , b∗ axes; see Fig. 3.3. The coordinates are arbitrarily chosen perpendicular to each other similar to the x, y, Y system of 1931. The positive L∗ axis represents achromatic colors of values between L∗ = 0 for ideal black and L∗ = 100.000 for ideal white. An a∗ , b∗ color plane of constant lightness represents the chromatic parts of a color stimulus specification. The greater the distance between the color locus and the lightness axis, the more chromatic and saturated the corresponding color. The lightness axis and its vicinity represent the color locus of achromatic colors. The lightness axis itself is, therefore, also named the achromatic axis. The trace point of this axis through any color plane is called the achromatic point. In Color plate 4, for example, a section of an a∗ , b∗ color plane representing absorption colors of middle lightness is shown. In Color plate 5, the central part of the CIELAB color space is depicted as a nearly spherical shape with corresponding achromatic and chromatic colors. The color locus represents the most frequently produced industrial absorption colors. The color values of absorption colors, capable of being manufactured, are by no means distributed over the entire mathematically defined CIELAB space. Such colors are rather limited to a sphere or ellipsoid-shaped volume including the entire achromatic axis. This volume is distorted differently depending on the sort of colorants and chosen color value system. When the McAdam ellipses from Color plate 3 are transformed into the a∗ , ∗ b color plane of the CIELAB system in Fig. 3.4, it is noticeable that the ellipses

3.1

Systems of Standardized Tristimulus Values

143

L∗ 100

White

C(L∗c , ac∗, bc∗)

Lc∗

∗ Cab

–a ∗

∗ Green ac

Yellow

bc∗

∗ hab

0

–b ∗

b∗

a∗ Black

Red

Blue

∗ , a∗ , b∗ ) is Fig. 3.3 L∗ , a∗ , b∗ axes of the CIELAB color space; the color locus C(LC C C ∗ additionally characterized by the chroma Cab and hue angle hab (see below)

are more uniform. The desired circles (respective spheres) of equal radius, however, fail to appear [11]. Although the x, y, Y and the CIELAB color space cannot be directly compared with each other, we get an idea of the effects of the transformation. Visually, the CIELAB and CIE 1931 color spaces are not equidistant. For chromatic colors, the color differences in the CIELAB space a∗ L∗ = 50 100

50

0

–50

–100 –200

–100

0

Fig. 3.4 McAdam ellipses in an a∗ , b∗ plane of the CIELAB system [11]

100

b∗

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have again greater tolerances than for achromatic dark colors. Nevertheless, with the widely accepted and applied CIELAB system, an important intermediate step on discovering the desired UCS is achieved. Properties of industrial colors are often characterized by attributes such as lightness, chroma, hue, or hue angle. The CIELAB system offers now the advantage that these properties can also be visualized and quantified. These quantities follow alternatively – instead of Cartesian coordinates L∗ , a∗ , b∗ – ∗ as the radius from cylindrical with chroma Cab ∗ Cab

 a∗ + b∗ ,

=

2

2

(3.1.12)

hue angle hab as the azimuth angle ◦

hab =

180 arctan (b∗ /a∗ ), π

(3.1.13)

and the lightness L∗ for third coordinate; see Fig. 3.3. However, also the cylinder coordinates describe more color points than existing real absorption colors. The ∗ of a color is represented in an a∗ , b∗ color plane by the distance chroma Cab between the achromatic point and the (a∗ , b∗ ) coordinates of a given color. The hue angle indicates the position inside a quadrant of a color plane. Because of this, it is a measure for the corresponding hue, cf. Color plate 4. According to the defining equation (3.1.13), the hue angle is the sole color value with dimensions given in degrees. The hue angle increases counterclockwise. Along the positive a∗ axis lies hab = 0. The CIELAB system has an additional fundamental feature. Because along each of the L∗ , a∗ , b∗ axes opponent colors change, the color difference between a sample color with color values LS∗ , a∗S , b∗S and a reference color with color values LR∗ , a∗R , b∗R is given by the corresponding difference contributions ΔL∗ = LS∗ − LR∗ , Δa∗ = a∗S − a∗R , Δb∗ = b∗S − b∗R .

(3.1.14)

These can now be interpreted in a coloristical sense. The amount and the sign of the so-called color differences ΔL∗ , Δa∗ , Δb∗ mean  > 0: lighter ΔL∗ = LS∗ − LR∗ = , (3.1.15) < 0: darker  > 0: redder , (3.1.16) Δa∗ = a∗S − a∗R = < 0: greener  > 0: yellower ∗ ∗ ∗ Δb = bS − bR = . (3.1.17) < 0: bluer

3.1

Systems of Standardized Tristimulus Values

145

L∗

ΔL∗

CS

∗ ΔEab

CR

Δb∗

b∗

Δa ∗ a∗ ∗ between color locus of a color sample C and reference color Fig. 3.5 Color difference ΔEab S CR in CIELAB color space

∗ follows, as shown in Fig. 3.5, The entire numerical color difference ΔEab from the three-dimensional Pythagorean theorem ∗ = ΔEab



(ΔL∗ )2 + (Δa∗ )2 + (Δb∗ )2 .

(3.1.18)

This color difference equation enables to indicate quantitatively the approximate value of the perceived color difference between two given colors. With the use of Equation (3.1.18), one must certainly take the following aspects into account: 1. The sign of the single color contributions is lost on account of the squares; ∗ alone it is not possible to draw any from the numerical value of ΔEab conclusions with regard to the three individual color differences. 2. As experience shows, Equation (3.1.18) holds – because of the present visually non-uniform color space – merely for small color differences in the range ∗ ≤ 5; the color values change nearly uniformly only in small of about ΔEab regions of the nonlinearly transformed color space. 3. According to the CIE proposal [7], the CIELAB color space should only be applied to non-self-luminous colors of the same size and shape; for visual assessment, the sample and reference color have to be kept in the same surroundings of a middle gray color (Section 3.2.2).

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It seems that some of the mentioned restrictions can be overcome with a color appearance model. A color difference can be alternatively expressed with the aid of the lightness ∗ and the hue difference ΔL∗ in combination with the chroma difference ΔCab ∗ difference ΔHab ; see below. The contribution of the chroma follows due to the defining equation (3.1.12) from ∗ ∗ ∗ ΔCab = Cab,S − Cab,R .

(3.1.19)

From a geometrical point of view, the chroma difference corresponds to the length difference of the cylinder radius between the color locus of the sample and the reference color as seen in Fig. 3.6. Similar to Equations (3.1.15), (3.1.16), and (3.1.17), the chroma difference can be interpreted using color attributes such as ∗ = ΔCab



> 0: more brillant, clearer . < 0: duller, less chromatic

(3.1.20)

b∗

S

b∗S ∗ ΔCab

∗ ΔHab

b∗R

R

Δhab

a∗R

O

a∗S

a∗

∗ , hue ΔH ∗ , and hue angle Δh of sample color S Fig. 3.6 Contributions of chroma ΔCab ab ab ___

___

∗ , OR = C∗ and reference color R: OS = Cab,S ab,R

3.1

Systems of Standardized Tristimulus Values

147

With Equation (3.1.13), the change of hue angle follows as Δhab = hab,S − hab,R .

(3.1.21)

∗ , is shown The change of hue angle Δhab , along with the hue contribution ΔHab ∗ in Fig. 3.6. The quantity ΔHab is defined by

 ∗ ∗ C∗ = 2 Cab,S ΔHab ab,R · sin (Δhab /2)

(3.1.22a)

and is identical with expression ∗ = ΔHab



∗ )2 − (ΔL∗ )2 − (ΔC∗ )2 (ΔEab ab

(3.1.22b)

∗ corresponds to the arithmetic mean of [29]. Geometrically, the quantity ΔHab the chord lengths belonging to the chroma of the sample and reference color. ∗ leads to Solving Equation (3.1.22b) for ΔEab ∗ = ΔEab

 ∗ )2 + (ΔH ∗ )2 . (ΔL∗ )2 + (ΔCab ab

(3.1.23)

This is an additional relation for determining a color difference. Equations (3.1.18), (3.1.19), (3.1.20), (3.1.21), (3.1.22), and (3.1.23) give also the hue contribution by ∗ ΔHab =

 ∗ C∗ ∗ ∗ ∗ ∗ 2(Cab,S ab,R − aS aR − bR bS ).

(3.1.24)

∗ is positive, if the inequality a∗ b∗ − a∗ b∗ ≥ 0 holds; it is The sign of ΔHab R S S R ∗ follow directly otherwise negative [12]. Both the value and the sign of ΔHab from the equation ∗ = ΔHab

a∗R b∗S − a∗S b∗R ∗ C∗ ∗ ∗ ∗ ∗ 0.5 · (Cab,S ab,R + aS aR + bS bR )

,

(3.1.25)

∗ provided the sign of the root is positive. The contributions Δhab and ΔHab have the same sign. Therefore, Equation (3.1.25) can be used for the determination of the correct sign of Δhab , for example, if the color locus of the sample and reference color are located in adjoining quadrants of an a∗ , b∗ color plane. The CIELAB system does not define which of the equivalent color values L∗ , ∗ , h ∗ a , b∗ or respective L∗ , Cab ab are to be used. It is, at least, advisable to

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indicate the L∗ , a∗ , b∗ values from which the further ones immediately follow. It is, furthermore, not specified which of the Equations (3.1.18), (3.1.19), (3.1.20), (3.1.21), (3.1.22), and (3.1.23) is to be used to calculate the color ∗ [7,13]. In any case, it is useful to list the contributions difference ΔEab ∗ , ΔH ∗ , and Δh , which state the color difference ∗ ∗ ∗ ΔL , Δa , Δb , ΔCab ab ab precisely. Although this system does not represent the ideally desirable visually uniform color space, the progressive CIELAB formalism permits, however, to approximately answer further color physical questions which has been impossible in the past.

3.1.4 DIN99o Color Space After the introduction of the CIELAB system, since about the year 1976, international teams have, to a considerable degree, fallen back on the matching of absorption colors, which were selected and disposed of, to get closer to the desired uniform color space. The endeavors lead to the establishment of the DIN99 color space [14, 15], which is more extravagant than the CIELAB system, among other things. The color values of the DIN99 space are indicated by L99 , a99 , b99 or L99 , C99 , h99 and result from a transformation of the already known L∗ , a∗ , b∗ quantities. The revised and actually valid version of the DIN99 space is called DIN99o color space. This system delivers an improved metric, which is expressed by a more effective color difference formula. The color values of the DIN99o system L99o , a99o , b99o or C99o , h99o , likewise, follow from a nonlinear transformation of the L∗ , a∗ , b∗ quantities [14 –16]. The lightness L99o varies in logarithmic dependence on L∗ in the domain 0 ≤ L99o ≤ 100: L99o =

303.67 ln (1.0000 + 0.0039L∗ ). kE

(3.1.26)

The a99o , b99o color values are determined using the following four auxiliary functions: eo = a∗ cos (26◦ ) + b∗ sin (26◦ ),

(3.1.27)

fo = −0.83 · a∗ sin (26◦ ) + 0.83 · b∗ cos (26◦ ),

(3.1.28)

 heofo = arctan (fo eo),

(3.1.29)

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Systems of Standardized Tristimulus Values

149

 Go = + eo2 + fo2 .

(3.1.30)

For the variable heofo , the following value domains

heofo

⎧ 0 ⎪  ⎪ ⎪ ⎪ arctan (fo eo) ⎪ ⎪  ⎨ π 2  = π+ arctan (fo eo) ⎪ ⎪  ⎪ ⎪ ⎪ 3π 2  ⎪ ⎩ arctan (fo eo)

for for for for for for

eo = fo = 0 eo > 0 and fo ≥ 0 eo = 0 and fo > 0 eo < 0 eo = 0 and fo < 0 eo > 0 and fo ≤ 0

(3.1.31)

should be differentiated. The chroma C99o follows from formula

C99o =

ln (1.000 + 0.075 · Go) . 0.0435 · kCH kE

(3.1.32)

The correction factors kE and kCH in Equations (3.1.26) and (3.1.32) depend on the observation conditions of the color patterns. These are set to 1.0, provided that the CIE reference conditions are fulfilled (Section 3.2.2). The new values of the red to green and yellow to blue proportions follow from relations a99o = C99o cos (h99o ),

(3.1.33)

b99o = C99o sin (h99o ),

(3.1.34)

with the hue angle h99o given by h99o =

180◦ heofo + 26◦ π

(3.1.35)

using the condition in (3.1.31). The original color locus of constant chroma in a CIELAB color plane is, in the new a99o , b99o color plane, rotated and compressed to concentric “ellipses” nearly in the direction of the yellow-blue axis. This is similar to the transformation by the DIN99 system, cf. Figs. 3.7a, b and 3.8. With increasing chroma, the ellipses narrow. The transformed a99o , b99o quantities are smaller than the original values in the domains for a∗ ≥ 20 and b∗ ≥ 20, but larger than comparable a99 , b99 pairs. The up until now visually tolerated higher color differences with the CIELAB system for yellow, orange, and red colors of high chroma have been mostly reduced in the DIN99 and DIN99o color spaces. In the CIELAB as well as the DIN99o space, the metric relations are structured in an analog fashion. The color difference relation

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40

a)

0

–40

–80

–80

–40

0

40

80

a∗

b99 40

20

b)

0

–20

–40 –40

–20

0

20

40 a99

Fig. 3.7 Transformation of color locus of equal chroma from (a) a CIELAB a∗ , b∗ color plane into (b) the accompanying DIN99 color plane

ΔE99o =

 (ΔL99o )2 + (Δa99o )2 + (Δb99o )2

(3.1.36)

to apply in the case of DIN99o color values corresponds again to the threedimensional Pythagorean theorem, cf. formula (3.1.18).

3.1

Systems of Standardized Tristimulus Values

151

b99o

40

20

0

–20

–40

–60 –60

–40

–20

0

20

40

a99o

Fig. 3.8 Transformation of color locus of equal chroma from the CIELAB a∗ , b∗ color plane in Fig. 3.7a into the accompanying DIN99o color plane

The differences contained in Equation (3.1.36) between a sample and reference color of indices S and R of lightness ΔL99o , red to green Δa99o , and yellow to blue Δb99o are ΔL99o = L99o,S − L99o,R ,

(3.1.37)

Δa99o = a99o,S − a99o,R ,

(3.1.38)

Δb99o = b99o,S − b99o,R .

(3.1.39)

These are of the same coloristical meaning as in Equations (3.1.15), (3.1.16), and (3.1.17). The chroma difference ΔC99o and the change in hue angle follow from ΔC99o = C99o,S − C99o,R ,

(3.1.40)

Δh99o = h99o, S − h99o, R .

(3.1.41)

The total color difference ΔE99o can be alternatively calculated using the formula ΔE99o =



(ΔL99o )2 + (ΔC99o )2 + (ΔH99o )2 .

(3.1.42)

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The hue contribution ΔH99o and its sign are most simply determined, analogous to Equation (3.1.25), from the relation ΔH99o = 

a99o,R b99o,S − a99o,S b99o,R 0.5 · (C99o,S C99o,R + a99o,S a99o,R + b99o,S b99o,R )

,

(3.1.43)

provided that the positive sign of the square root is used as well. As already mentioned for the CIELAB space, Equations (3.1.36), (3.1.37), (3.1.38), (3.1.39), (3.1.40), (3.1.41), (3.1.42), and (3.1.43) are only applicable to small color differences ΔE99o ≤ 5. The new values of the color differences are mostly smaller than those corresponding values of the CIELAB system. The DIN99o color space actually shows a better metric than the CIELAB space, but it is, nevertheless, not visually equidistantly structured [17]. Similar to the CIELAB formalism, the DIN99o color space is also applicable to non-selfluminous colors of colorants, lacquers, plastics, textiles, printing inks, as well as colors of hygiene and cosmetic products. The question remains: Which color space should be applied in colorimetry? This question will be answered in Section 3.2.4. First of all, for simplicity in the following, we leave out the specifying indices and exponents of color values, color differences, and color difference contributions, provided that no confusion is introduced.

3.2 Color Difference Metrics and Color Tolerances The effort of empirically discovering a colorimetrical system, which matches the visually perceived color differences as closely as possible, has created more than 20 contrasting color difference formulas for absorption colorations. Only some of them have survived the constantly rising industrial requirements. In addition ∗ and ΔE to the ΔEab 99o relations, there are three key color difference equations. These are applied in different color fields: the so-termed CMC, CIE94, and CIEDE2000 color difference formulas. These three relations and also the ΔE99o equation are based on the CIELAB formula containing differently weighted color differences. Moreover, the color appearance model CIECAM02 delivers, in addition to new chromaticities, further Euclidian color spaces and color difference relations, in which the observing and surround field conditions are taken better into account than before. Independent of these, the color deviations of effect colorations need especially careful considerations and assessments. Among the color values given until now, because of the lack of suited effect models, few are appropriate for approximate colorimetrical characterization of some angle-dependent effect properties. This is also because the original color spaces were developed only for colored lights and absorption colors. The color differences of effect colors are more complex and are additionally not simply characterized by a

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Color Difference Metrics and Color Tolerances

153

numerical color difference value. Other properties such as the flop, brilliance, distinctiveness, or sparkle, for example, often also deviate. Such kinds of differences should be assessed rather with additional methods that are described in Section 3.5.

3.2.1 CMC(l:c) Color Difference Formula In 1963, the Society of Dyers and Colourists in Britain arranged the Colour Measurement Committee (CMC) for working out a color difference relation for absorption colorations that matches the visually perceived color differences better than the CIELAB formula. Investigations with different color collections confirmed the fundamental insight: perceived color tolerances are represented by ellipses of different axis instead of spheres of equal radii as is expected from color difference Equations (3.1.18), (3.1.23), and (3.1.42). The final results of the committee were published in 1984 and are condensed into the so-called CMC(l:c) color difference formula   ΔECMC(l:c) =

ΔL∗ lSL

2

 +

∗ ΔCab cSC

2

 +

∗ ΔHab SH )

2 ,

(3.2.1)

[18]. This corresponds to the equation of a three-dimensional ellipsoid with axes ∗ , ΔH ∗ . The tolin the direction of the color difference contributions ΔL∗ , ΔCab ab erance parameters l and c in (3.2.1) serve for weighting of the lightness and hue differences. For textile colorations, the values are normally set to l = 2 and c = 1. The corresponding color difference value is specified with the index CMC(2:1). For lacquers and plastic materials, the weighting factors l = 1.3 and c = 1 are used. The empirical functions SL , SC , SH are given by SL =

0.040975LS∗ for (1 + 0.01765LS∗ ) SC =

LS∗ ≥ 16,

∗ 0.0638 · Cab,S

∗ ) (1 + 0.0131 · Cab,S

else SL = 0.511,

(3.2.2)

+ 0.638,

(3.2.3)

SH = SC (f T + 1 − f ),

(3.2.4)

   f =

(3.2.5)

where f and T are given by ∗ )4 (Cab,S

∗ )4 + 1900 (Cab,S

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and T = 0.38 + |0.4 cos (35◦ + hab,S )|.

(3.2.6a)

In the hue angle domain 164◦ ≤ hab,S ≤ 345◦ , the equation T = 0.56 + |0.2 cos (168◦ + hab,S )|

(3.2.6b)

∗ , h is valid. The quantities LS∗ , Cab,S ab,S belong to the sample color. If the sample color is not known, then the arithmetic mean of both colors which are ∗ , ΔH ∗ of the color being compared should be used. The differences ΔL∗ , ΔCab ab contributions follow from Equations (3.1.15), (3.1.19), and (3.1.25). The length of each semi axis of the ellipsoid defined by the CMC(l:c) formula in (3.2.1) is given by the correction functions in (3.2.2), (3.2.3), and (3.2.4). In Color plate 6, the sizes and orientations of projected ellipsoids are shown in the range of ΔECMC(1:1) ≤ 2.0 in an a∗ , b∗ color plane for ΔL∗ = 0. The major axis of the ellipses grows with increasing chroma for all colors; the quantity of the minor axis, however, depends on the hue angle hab . The ellipses are smallest in the field of orange near hab = 55◦ and widest in the region of green at ∗ takes extreme values at each hab = 192◦ . The correction function SH for ΔHab ◦ ◦ ◦ ◦ of the hue angles 145 , 164 , 282 and 342 . Since its introduction, the CMC(l:c) formula has been paving the way for the discovery of improved color difference relations. This is underlined in the following three sections.

3.2.2 CIE94 Color Difference Expression Essentially, the CMC(l:c) formula has shown itself to be correct only for textile dyes and synthesized polymeric materials. The CIE, therefore, suggested matching of further absorbent colors in order to find an improved color difference equation. For that, previous data records were used. Those records were selected in view of statistical confidence limits as well as equal measurement and evaluation conditions. Consequently in 1994, a color difference equation was presented. It was termed the CIE94 color difference formula [17, 18]:      ∗ 2 ∗ 2 ΔCab ΔHab ΔL∗ 2 ∗ + + . (3.2.7) ΔE94 = kL SL kC SC kH SH With regard to the CMC(l:c) equation, this new relation shows modified correction functions SL , SC , SH . Now, these follow from more simply designed expressions: SL = 1,

(3.2.8)

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Color Difference Metrics and Color Tolerances

155

∗ SC = 1 + 0.045 · Cab ,

(3.2.9)

∗ SH = 1 + 0.015 · Cab ,

(3.2.10)

kL = kC = kH = 1.0

for reference conditions, ∗ ∗ Cab = Cab,R .

(3.2.11) (3.2.12)

These are provided that the reference color is known, otherwise the geometrical  ∗ = ∗ C∗ mean Cab Cab,1 ab,2 of colors 1 and 2 to be compared should be used in Equations (3.2.9) and (3.2.10). The coefficients kL , kC , kH depend on the previously mentioned observation conditions, which were introduced at that time. These are set to 1.0 in Equation (3.2.7), as long as the so-called CIE reference conditions are fulfilled [19]. It is suitable to modify the mentioned three constants if one or more criteria deviate from the preset conditions. The CIE reference conditions, together with the experimental realizations, are listed in Table 3.3. These are to be regarded as minimum requirements and are primarily founded on the specific properties of the three factors determining color impression outlined in Sections 2.1 and 2.4. The conditions in Table 3.3 are intended to be used for the comparison and assessment of colors. They follow from these connections: 1. The standard illuminant D65 to simulate middle daylight shows a nearly constant spectral power distribution in the visible range (Fig. 2.4); colors are, to some degree, rendered neutral with this illuminant in comparison to other temperature or luminescence radiators. Change in illuminant possibly exposes nonlinear color changes such as color inconstancy, for example. Table 3.3 CIE reference conditions for assessment of absorption colors No.

Criterion

Realization of CIE reference condition

1 2 3 4 5 6 7

Illumination Illuminance Background field Viewing mode Viewing angle Sample separation Magnitude of color difference Sample structure

D65 simulator 1,000 lx Uniform, neutral gray of L ∗ = 50 Object mode > 4◦ Minimal, with direct edge contact ∗ ≤5 ΔEab

8

Homogeneous, without visible patterns or uniformities

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Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

2. The illuminance should be higher than 500 lx so that all cones of the retina are stimulated for color vision and the rods are switched off. This requirement is definitely achieved above an illuminance level of 1,000 lx; levels higher than 5,000 lx, however, cause blinding glare. 3. According to the color constancy investigations of Land and other groups, the surrounding color changes the color impression (cf. simultaneous contrast, Section 2.4.2). In order to avoid color shifts caused by peripheral colors, in the immediate surroundings of the color pattern (locating surface), a middle neutral gray of lightness L∗ = 50 should be used. It is known that differences between dark colors are resolved to a much lesser degree in a light surrounding than in a dark one. Therefore, a middle gray is a useful compromise. 4. In object modus, by definition, the reflected light of a free field non-selfluminous color is directly observed. In contrast, the assessment can also be carried out in illumination modus (direct light) or in aperture modus (observed field limited by an aperture), so that uncontrolled scattered light hardly contributes to the color impression. 5. The influence of the yellow pigment in the macula of the retina (Fig. 2.60) on color impression is nearly excluded for visual angles greater than 4◦ . Therefore, the need for the application of the 10◦ standard observer is clear. 6. Light or brilliant colors appear darker or duller with edge separation as in direct edge contact; conversely, dark and dull colors are perceived lighter or more brilliant with edge separation. Furthermore, the color assessment is increasingly influenced by the color of the gap between the color patterns; because of this “distraction effect,” higher color differences are tolerated as with perfect formation. These considerations follow directly from the third criterion above. 7. The color values of the color spaces discussed in Section 3.1 change in a nonlinear fashion with regard to color perception. Local changes, however, are assumed to be linear; this is generally valid for color differences ∗ ≤ 5. ΔEab 8. To avoid distorting the color impression from the beginning, only homogeneous, uniform, and optically defect-free color patterns should be employed. A structured surface increases the lightness and non-uniform colored samples distort the chromaticity values. All in all, the CIE reference conditions recommend that the mentioned observation criteria should be met for all sorts of color assessments and are independent of specially used color difference formulas as well. Deviations from the reference conditions or additional premises are to indicate for clarity. In a color appearance model outlined in Section 3.2.4, the color values are calculated directly considering observation criteria.

3.2

Color Difference Metrics and Color Tolerances

157

3.2.3 CIEDE2000 Color Difference Equation The color difference equation given in the previous section is only roughly appropriate for the varying expectations of different color technological industries. For this reason, the CIE recommended in 2001 an additional color difference formula which was worked out by several groups based on more than 3,600 color patterns [20–23]. This new color difference relation is denoted as CIEDE2000 and delivers the value ΔE00 :   ΔE00 =

ΔL kL SL



2 +

ΔCab kC SC



2 +

ΔHab kH SH

2

 + RT

ΔCab kC SC



 ΔHab . kH SH (3.2.13)

This expression is again founded on the CIELAB color values and has two innovations in comparison to the previous color difference formulas: and of hue ΔH are transformed by a∗ – the contributions of chroma ΔCab ab ∗ and Cab , instead of the unchanged lightness difference ΔL = ΔL∗ ; – there is a fourth summand which consists of the product of the chroma and hue contributions; the ellipsoid axes are, therefore, not oriented parallel to the , H . coordinates of Cab ab

The original CIELAB chroma and hue differences given in Section 3.1.3 are to modify in the manner that follows. The indices S and R stand for the sample and the reference color: L = L∗ ,

a = a∗ (1 + G),

b = b∗

(3.2.14)

with G=

  1  ∗ ∗ ∗ )7 /[(C∗ )7 + 257 ] , C∗ · 1 − (Cab,m ab,m = (Cab,S + Cab,R )/2, ab,m 2 (3.2.15) Cab =

h ab =



(a )2 + (b )2 ,

180◦ · arctan (b /a ), π

(3.2.16) (3.2.17)

ΔCab = Cab,S − Cab,R ,

(3.2.18)

Δh ab = h ab,S − h ab,R ,

(3.2.19)

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a R b∗S − a S b∗R . ΔHab = 0.5 · (Cab,S Cab,R + a S a R + b∗S b∗R )

(3.2.20)

Equations (3.2.14) and (3.2.15) change ellipses located near the lightness axes into circles of the a , b color plane. If the CIE reference conditions of the previous section are met, then the correction parameters in Equation (3.2.13) are kL = kC = kH = 1. The correction functions for lightness, chroma, and hue differences, SL , SC , SH , follow from the relations: − 50)2 1 + 0.15 · (Lm , SL =  − 50)2 20 + (Lm

Lm = (LS + LR )/2,

(3.2.21)

, Cm = (CS + CR )/2, SC = 1 + 0.045 · Cm

(3.2.22)

· T. SH = 1 + 0.015 · Cm

(3.2.23)

For the determination of the function values of T in Equation (3.2.23) and RT in Equation (3.2.13), the expressions T = 1 − 0.17 · cos (h m − 30◦ ) + 0.24 · cos (2h m ) + 0.32 · cos (3h m + 6◦ ) − 0.20 · cos (4h m − 63◦ ),

h m = 12 (h S + h R ) (3.2.24)

are used along with   RT = (2G − 1) · sin (ΔΘ), ΔΘ = 60 · exp −[(h m − 275◦ )/25◦ ]2 . (3.2.25) , C∗ , C , and h are the arithmetic mean of the correspondThe quantities Lm m m ab,m ing sample and reference values. Equation (3.2.22) is formally equivalent to that of the CIE94 formula in Equation (3.2.9). The function T in Equation (3.2.24) on the hue angle. The takes into account the additional dependence of ΔHab rotation function RT appears in a color difference equation for the first time. It corrects the direction of the major axis of the discrimination ellipses in the blue region of the color plane. To sum up, two different methods have been adopted until now to achieve a better match between numerical color differences and visually perceived color deviations:

3.2

Color Difference Metrics and Color Tolerances

159

– establishing Euclidian color spaces (CIELAB, DIN99o); the color difference expressions correspond to sphere equations; – creation of single color difference formulas which contain weighted and transformed contributions of CIELAB color differences (CMC(l:c), CIE94, CIEDE2000) and follow from assessment of color collections; the corresponding equations describe ellipsoids. Among these approaches, the Euclidian color space is generally preferred: every color locus of the space presents the same metric; furthermore, the color difference follows directly from uncorrected color contributions such as ΔL, Δa, Δb. These five color difference relations discussed here are standardized [24]. Out of these newest formulas at least two further questions show themselves. First, how effectively the individual color difference relations work? And second, are further improvements possible using other empirical methods? Both aspects are discussed in the next section.

3.2.4 Efficiency of Color Difference Formulas, CIE Color Appearance Models The determination of a suitable color difference formula for a special color application requires extensive coloristical experience. Ultimately, it is necessary to decide which of the discussed color difference expressions is at the most accurate. One principal reason for the lack of a generally applicable relation is the incomplete knowledge of the complicated neural sequences which cause the trichromatic color impression in the visual cortex. Furthermore it is not clear whether the postulated visually uniform color space is actually reproducible with mathematical elements. Already from the x, y chromaticity diagram or the CIELAB system, it follows that the visually accepted color difference depends on both the color locus and the direction of the color difference. As is known, the tolerated color deviations of light and brilliant colors are numerically higher than those of achromatic dark ∗ of brilliant coloration is as much colors. The tolerated chroma difference ΔCab ∗ . as four times higher than the corresponding hue difference ΔHab ∗ In comparison to the formulas for ΔEab and ΔE99o , the modified color difference relations such as CMC(l:c), CIE94, and CIEDE2000 result from the assessment of color collections of different chromatic colorant and binder sorts. Consequently, each of these relations is to be applied to the corresponding material as follows: – CMC(2:1) for textile fibers, – CMC(1.3:1) for lacquers and plastic materials, – CIE94 for textile fibers, lacquers, and plastics, ∗ , ΔE – ΔEab 99o , CIEDE2000 for every known base material.

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It goes without saying that the mentioned difference expressions cannot be compared with one another in a direct manner. The corresponding transformations are different, nonlinear, and, furthermore, based on different color collections. The workability or efficiency of such an expression can be estimated by testing additional color patterns followed by a strict statistical evaluation of the results. From this point of view, the five color difference formulas mentioned above were tested with more than 3,600 additional color samples of small color differences [16, 23]. Each spectrometrically determined color difference was divided into its contributions ΔL, Δa, Δb and assigned to an ellipsoid of volume ΔVex ; see Section 4.3.4. Afterward, the visually perceived color differences were evaluated and assigned to a separate ellipsoid of volume ΔVvis . From all of the experimental and visual color differences, a statistical dimension value PF/3 results. This characteristic is an approximate measure of the absolute difference between the ellipsoid volumes: PF/3 ≈ |ΔVex − ΔVvis |.

(3.2.26)

For complete correspondence of the visual and measured color differences, one should expect PF/3 = 0. A value of PF/3 = 30, for example, indicates a statistical deviation between both investigations of about 30%. The results of the above-mentioned five formulas for small differences ∗ ≤ 2.5 correspond to the PF/3 list in the upper half of Table 3.4. From ΔEab these, the advantage of the CIEDE2000 difference equation is clear. This result suggests that the CIEDE2000 or at best the DIN99o color difference formula should be applied for the assessment of small color differences. However, the statistic quantity PF/3 does not indicate if there are systematic deviations in one or more quadrants of the color plane or not. Table 3.4 Effectiveness of color differences from color spaces and color appearance models with regard to the absolute statistical dimension value PF/3 [16, 26] Color collections of small color differences

PF/3 value

Color collections of high color differences

PF/3 value

CIEDE2000 DIN99o CMC(1:1) CIE94 CIELAB

33 35 38 38 52

CIELAB

26

CAM02-SCD CAM02-UCS CAM02-LCD CIECAM02

34 35 47 47

CAM02-LCD CAM02-UCS CIECAM02 CAM02-SCD

23 25 25 27

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161

The other statistical dimension values PF/3 in Table 3.4 each follow from a so-called color appearance model (CAM). The corresponding color difference formula directly contains the surrounding and observation conditions, which are of significance in the color industry and also in image processing. Such a model delivers values of color difference as well as color impression and, additionally, specific color characteristics [25, 26]. The CIE recommends the special model CAM02 for specific applications [27]. As input values, the standard color values X, Y, Z are applied to the test illuminant. With this formalism, three attributes are introduced. These differ in relation to the chromatic content: chroma C, colorfulness M, and saturation s. These color attributes together with lightness J and hue angle h can create three different spaces of color coordinates (J, aC , bC ), (J, aM , bM ), and (J, aS , bS ). The derived color space of color values J, aM , bM shows the most uniform results when tested with the color collection with small color differences as mentioned ∗ ≤ 2.5 and ΔE ∗ ≈ above and a second collection with large differences (ΔEab ab 10). The color collection of small color differences results in a value PF/3 = 47, which is slightly smaller than the corresponding CIELAB value PF/3 = 52; see Table 3.4. Based on CIECAM02, three further systems (CAM02-SCD, -LCD, and UCS) for visually small (SCD), large (LCD), and combined small and large (UCS) differences have been derived [26]. These three formalisms produced the other PF/3 values listed in Table 3.4. Considering these, the CAM02-UCS is suitable for evaluation of both small and large color differences. Because improvements in CAMs are still in progress, we avoid describing the extensive formalisms of the four mentioned models in this text and simply refer to the literature [26, 27]. All in all, the CIECAM02 enables one to determine the above-mentioned parameters and further characteristics such as the color rendering index of light sources, color inconstancy index, or index of metamerism. This CAM is suited for the three major fields in colorimetry, namely color specification, color difference evaluation, and prediction of color appearance. It remains to be seen, whether the CAM02-UCS formalism is really a universal CAM or not.

3.2.5 Color Tolerances The immense effort put into the search for the visually uniform color space follows from the decades-long insight that, roughly speaking, the color differences between absorption colors are of greater significance in color technology than the sign and absolute values of the accompanying color values. Color differences almost inevitably result in production or recipe prediction, among other things. An accurate colorimetry is, however, an essential requirement for arranging an acceptable color tolerance agreement between manufacturer and customer. In

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most cases, the color tolerance agreement is supplemented by further restrictions. In order to obtain an acceptable color tolerance for non-self-luminous colors, we have, in particular, taken into account the different error influences introduced by the three outer factors – light source, color sample, and observer – producing color impression. From the term color difference, we should differentiate between the smallest visually resolvable off shade and the magnitude of the allowed numerical color tolerance [24, 29–33]. For comparison of colors, inevitably, one must consider the sort of binder, surface structures, parameters of manufacture, application, or processing, etc. In addition, it is important to realize that the smallest visually resolvable off shade is non-uniform and color locus dependent: the smallest color difference is about ∗ = 0.3 for achromatic dark absorption colors and about ΔE ∗ = 0.7 for ΔEab ab brilliant green, yellow to red colors; see Table 3.5. In comparison, the short-term ∗ | ≤ 0.02 repetition accuracy of modern spectrophotometers is at least |ΔEab and is, therefore, more than 10 times better than the visual capacity to resolve off shade differences. For series production of coatings with plain lacquers, higher tolerance values compared to the smallest visually discernable color shades are usually applied. In the case of plain colorations of automotive finishes, for example, ∗ ≤ 0.9. The corresponding color tolerances are allowed in the range 0.3 ≤ ΔEab dependence on color locus is standardized [29] and is indicated in the chromaticity diagram of Fig. 3.9. Because the tolerance areas in the diagram are discontinuously separated from each other, it is useful to meet further requirements for the vicinity of the thresholds. A continuous transition between adjoining sectors is realistic (cf. Section 3.5.2). The color tolerances for paint refinishing are twice as high as series coating of automotives; see Table 3.6. Color tolerances for absorption colorants are by no means transferable to metallic or other effect colorations; this is clear by simply considering the angle-dependent color impression. For effect colors, the color tolerances ΔE(μ) should rather be arranged as they depend on the angle of illumination β and of observation μ. As shown in Table 3.6, the tolerance value is ΔED (μ) ≤ 1.0 for delivery and first paint coating; for paint refinishing, a range Table 3.5 Smallest visually perceivable and measurable color differences Coloration, measuring equipment Absorption colorants

Spectrophotometer

Criterion

Minimal color difference value

Smallest visually perceivable color difference Achromatic dark colors Brilliant yellow until orange-red colors

∗ ∼ 0.3 ΔEab = ∗ ∼ 0.7 ΔEab =

Short-term repeatability

∗ ≤ 0.02 ΔEab

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Color Difference Metrics and Color Tolerances

163

y

0.6

0.7

0.9

0.4

0.5 0.3

0.2

0

0

0.2

0.4

0.6

x

Fig. 3.9 CIE diagram for the 10◦ standard observer showing arbitrary chosen color areas of ∗ for automotive solid paints [29] different color tolerances ΔEab

Table 3.6 Standardized tolerance ranges according to DIN 6175 [29, 30] Kind of paint

Processing

Standard tolerance range

Plain paint

First paint coating Paint refinishing, low-bake paint

∗ ≤ 0.9 0.3 ≤ ΔEab ∗ ≤ 1.8 0.6 ≤ ΔEab

Metallic paint

Delivery and first paint coating Paint refinishing, low-bake paint

ΔED (μ) ≤ 1.0 1.5 ≤ ΔEP (μ) ≤ 3.0

of 1.5 ≤ ΔEp (μ) ≤ 3.0 is permitted. Both tolerance intervals are based on a relatively small number of color samples and, therefore, need additional requirements. Apart from the necessity of color tolerance requirements, it is essential that those involved in the evaluations continuously undertake coloristical and colorimetrical training [34]. Subjective color differences, which have repeated results greater than numerical ones due to insufficient coloristical training, are unsuited for tolerance agreement. Further considerations with regard to color tolerances and their detection are discussed in Sections 4.1.5 and 4.1.6.

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3.3 Color Constancy and Metamerism Color inconstancy and metamerism are color phenomena which are mainly depending on the constituent absorption colorants. These phenomena appear exclusively by illuminant change. Both properties are rarely avoided during manufacturing or matching of colors, but the degree that these appear can be characterized numerically by colorimetrical methods. For this, each of the already mentioned color difference equations is suitable, provided that special restrictions are considered. The resulting color inconstancy value is a measure of the appearance, how much the color impression of one and the same coloration changes due to change in illumination. The fundamental color physical processes are, in fact, only vaguely understood. They can, however, be phenomenologically evaluated with a color difference formula. In contrast to color inconstancy, metamerism is related to a pair of color patterns. Such a pair of same hue at one illumination shows a color change occurring with change of illumination. Also, the extent of this special kind of metamerism is expressed by an additional color difference value. The use of colorimetric methods in this context fundamentally simplifies the manufacture of colors with required high color constancy or low metamerism.

3.3.1 Chromatic Adaptation and Color Constancy Already in Section 2.4.2, we have become familiar with some perplexing phenomena of color perception which should be considered with regard to the assessment of colors. Two further properties of human color sense also belong to this group. These are termed chromatic adaptation and color constancy. The description of both effects in this section can now provide a clearer understanding of color inconstant colorations. Apart from the capability for distinguishing as many as 10 million colors, the human color sense has another astonishing ability: the sense of vision accommodates itself, within certain limits, continuously a previous change of illumination. The corresponding adjustment is generally called adaptation. The accommodation to the altered lighting condition needs a certain time interval. This adaptation phase is nearly independent of the preceding change of light, but – like also of the most other criteria of color perception – depends on the individual constitution and other conditions of the observer. Before any conclusive visual assessment of colors, one should, therefore, wait for a relatively long-time adaptation phase of at least 30 min. This amount of time is generally sufficient to overcome differences in individual delays. We differentiate between four kinds of adaptation: those of absolute sensitivity, discrimination threshold, visual acuity, and chromatic adaptation.

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Color Constancy and Metamerism

165

The adaptation of absolute sensitivity occurs if the eye which had been sensitized to a constant illumination then registers a change of lightness. The transition from daylight vision to twilight vision is called dark adaptation; the opposite case is light adaptation. In the case of light adaptation, the visual sensitivity of the eye reduces with increasing luminance, whereas for dark adaptation, the sensitivity increases with lower illumination. Light adaptation takes place at photopic vision for luminance values higher than 10 cd/m2 , dark adaptation at scotopic vision for luminance lower than 0.01 cd/m2 . Both light and dark adaptation are caused exclusively by lightness changes of illumination and not by color differences of sources. Characteristic for adaptation of discrimination threshold is the fact that light adaptation needs shorter time than dark adaptation. Complete dark adaptation needs at least 30 min for the normal eye. In this case, the sensitivity is increased to 105 times that of the daylight sensitivity. The longer adaptation time for dark adaption comes down to the slower-acting rods in comparison to the cones. During light and dark adaptation, the diameter of the pupil changes; this is called adaptation of the visual acuity. The contraction of the pupil during light adaption increases the visual acuity and aberrations are simultaneously reduced. On the other hand, dark adaptation increases the diameter of the pupil and, therefore, the visual acuity worsens and aberrations become more noticeable. The chromatic adaptation is of special importance in our context of color assessment. This phenomenon takes place in photopic vision because the cones in the retina are capable of color vision only in this situation. It is a characteristic of this phenomenon that color sensation is kept nearly constant in spite of illumination change. The explanation for this is that in visual assessment also the illuminated surroundings are included which also change in the same way. The process of chromatic adaption can be divided into two steps: chromatic shift and adaptive shift. The chromatic shift occurs immediately, for example, by the change of a surrounding of daylight to that of a room with fluorescence lighting of illuminant FL 11. Immediately upon entering the room, all colors appear shifted somewhat reddish. This color shift is a consequence of the different spectral power distributions of natural daylight and artificial fluorescence light. During the “short” chromatic adaptation phase of some seconds, the perceived color shift disappears. This is partially a consequence of the effect that the most chromatic colored objects are in reality more or less color constant (see below). After chromatic adaption follows the adaptive shift. The exact process of this phase is not yet completely clear. It is, however, clear from observations that the perceived color impression is controlled by changes in the retina together with the color memory of the observer. In other words, the spectral differences of the sources are nearly balanced by a shift of the spectral sensitivity of the eye. Simultaneously, the perceived color is unconsciously compared to color

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impressions of similar color surroundings stored in the memory. The complete processes of chromatic adaptation needs, as mentioned above, a total time of about 30 min. The phenomenon of color constancy is closely associated with the chromatic shift. A color is considered color constant if it always produces the same color impression, independent of spectrally different illuminations. There is a comparable term, lightness constancy. This term characterizes the situation where the assessment of lightness is nearly independent of the energy level of the source. In reality, this judgment is based on a comparison of diffuse reflecting surfaces. A white sheet of paper is always perceived as white despite light changes, although the color stimulus caused by the sheet is different. It seems that the brain is trained to see colors as if they are always illuminated by white light. But color stimulus is – as discussed in Section 2.4.3 – dependent on the spectral power distribution of the light source. Color constancy is, therefore, an essential reason to characterize colors, color differences, and color appearances by applicable dimensional values rather than simply leaving it entirely up to individual visual judgment. In addition to color constant colorations, there exist a whole host of chromatic absorption colors which change considerably their color under varying illuminants (except monochromatic light sources). Such colors are called color inconstant. A quite unsettling example is a color of reddish hue in daylight which appears greenish under fluorescent illumination. Colors in applications which are subject to frequent illumination changes, therefore, should be generally of low color inconstancy. Dependent on the application field, new colors should be tested with regard to this property. The necessary approach for such tests is described in the next section.

3.3.2 Index of Color Inconstancy A high color constancy is by no means an unusual quality criterion. Safety colors, for example, must always be recognizable as such in daylight, twilight, and night light. Colors of consumer items should also certainly not change under an illumination change. The determination of color inconstancy under a test illuminant is carried out with regard to a reference illuminant, normally D65. The degree of color change, which a single color shows for both illuminants, is given by the index of color inconstancy CIC . Initially, it seems self-evident that the color inconstancy index should be identified as the color difference which appears between both illuminants. Detailed studies show, however, that the change of the corresponding color values is not suitable for an adequate description of color inconstancy. It is particularly important to know the relationship between the color values for colors which look precisely the same to an observer when they are adapted

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Color Constancy and Metamerism

167

to different illuminants. Such kinds of colors are called corresponding colors. Given color values R, G, B of a color sample under one illuminant, the color values RC , GC , BC for the corresponding color under another illuminant can be determined by calculation. The second illuminant is the so-called corresponding illuminant. The calculation of the corresponding color values is achieved by a chromatic adaptation transform (CAT). The numerical color difference ΔE between the color values of the corresponding color and the color under the original illuminant is an adequate measure for the degree of color inconstancy CCI (ΔE) = ΔE. For determination of the color values of the corresponding color, von Kries assumed that the three color receptors in the retina are differently affected by chromatic adaptation [35]. Accordingly, chromatic adaptation causes a reduction in sensitivity by constant factors for each of the three cones. The magnitude of each of the corresponding coefficients α, β, γ depends on the stimulus to which the observer is adapted. The von Kries coefficient law can be written as Rc = α · R, Gc = β · G, Bc = γ · B.

(3.3.1)

This law describes the transformation of the cone responses due to chromatic adaptation. Most of the various CATs developed have been founded on the von Kries law (3.3.1). The specific CAT to be used for absorption colors is CAT02. It was improved with a recent color collection of representative absorption colorations in 2002 and forms the basis of the sketched color appearance model CIECAM02 [36, 37]. The relevant transformation equations are given in Appendix A.2. The calculation of the color inconstancy index of a given color using CAT02 is performed in several steps. The input data are the standard color values of the sample, X, Y, Z, in the test illuminant and, XR , YR , ZR , in the reference illuminant. From these, the standard color values Xc , Yc , Zc of the corresponding illuminant are determined. The numerical degree of adaptation of the observer is also included in the process. Finally, the color difference ΔE is calculated with the triplets Xc , Yc , Zc and XR , YR , ZR . From the differences of the three color value pairs, the amount and the direction of the color shift can be seen. A value of CIC (ΔE) = 0 represents a completely color constant coloration for the chosen reference and test illuminant. The application of CAT02 is only for absorption colorants because the corresponding parameters and transformation relations were derived from the above-mentioned color collection. It is advantageous that the value CIC (ΔE) can be calculated in advance for every illuminant change without actually implementing the corresponding lighting. For clearness and comparability, along with the value of the color inconstancy index, the reference and test illuminants, the

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standard observer, the degree of adaption, the observation conditions, and the used color difference equation should also be indicated. As an example, we consider the color values of the reference colors given in Table 3.7. The standard color values X, Y, Z and L∗ , a∗ , b∗ belong to the illuminant/observer combinations D65/10◦ and A/10◦ . In the calculation, the field of the reference and sample is assumed to have a luminance of LA = 200 cd/m2 and presuppose normal observation conditions. In Table 3.8, the corresponding standard color values for the corresponding test illuminant Ac using CAT02 are listed. In the last column, the resulting quantities of the index of color inconstancy for three different color difference formulas are given. The first color sample has values between 6.27 and 6.72 and the second between 2.11 and 2.61. The second color is, therefore, much more color constant than the first. From the difference between the color values, one can estimate the direction and the amount of the color shift. In the special case of the chosen examples, the color inconstancy index is nearly independent of the used color difference formula – which is not typical for other absorption colors.

Table 3.7 Examples of starting values for determination of color inconstancy index in Table 3.8; 10◦ standard observer Sample number

Illuminants 10◦ observer

X

Y

Z

L∗

a∗

b∗

1

D65 A

20.51 24.08

24.09 22.91

26.14 8.99

56.18 54.98

–10.96 –5.64

–0.45 –4.51

2

D65 A

20.29 22.48

23.91 22.74

25.47 8.54

56.00 54.80

–11.26 –11.69

0.30 –2.66

Table 3.8 Color inconstancy index for combinations of reference/test illuminants of Table 3.7 [37] Reference/test illuminant corresponding Sample color; 10◦ observer 1 2

D65/Ac D65/Ac

Normal observation conditions

Color inconstancy index CIC (ΔE)

X

Y

Z

∗ ΔEab

ΔE(CMC1:1) ΔE00(1:1:1)

20.51 20.88

24.09 23.20

26.14 27.30

6.72

6.27

6.56

20.29 19.35

23.91 23.13

25.47 25.95

2.61

2.48

2.11

3.3

Color Constancy and Metamerism

169

3.3.3 Kinds of Metamerism The color property called metamerism is rather undesirable in many areas of color technology; it is, however, in many cases unavoidable. According to Ostwald [38], two colors are called metameric to each other (conditional equal), if they match for one illuminant but show a color difference under a change of illuminant. The metameric behavior is a consequence of additive color mixture of human color sense. It should be pointed out that color inconstancy is a property of one color, whereas metamerism is a property of the comparison of two colors. This effect is of particular importance, for example, in the production of colors in different charges, substitution of one or more colorants in recipes, or recipe prediction of a given color of unknown colorant composition. The degree of metamerism is – apart from color inconstancy and further properties – often the decisive factor for the choice of a desired coloration or suited color recipes among several alternative formulations. As an example, Fig. 3.10 shows the spectral reflectance of a pair of metameric colors which are perceived as nearly equal at midday light. The curve with a relative low maximum at λ = 510 nm for the sample color repeatedly intersects the curve of the reference color. Metameric pairs show generally one further characteristic: the spectral curves have at least three intersection points at different wavelengths in the visible range [20, 39, 40]. If there were fewer intersection points, either the lightness or the hue of the colors under reference illumination do not match. R(λ) R1(λ) 0.3 R2(λ) 0.2

0.1

0

400

500

600

Fig. 3.10 Spectral reflectance R1 (λ), R2 (λ) of a pair of metameric colors

700

λ nm

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Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

The occurrence of at least three intersections touches the heart of the methods of color recipe prediction. More than a hundred years of collective experience shows that to match a color of unknown composition requires at least three different colorants. In the simplest case, the accompanying colorant concentrations follow from the condition that the spectral curves of the recipe and reference color intersect for exactly three different wavelengths. Nearly equal spectral curves can only be achieved with greater number of colorations. A completely congruent pair of curves is only obtained with identical colorants and concentrations (Sections 6.3.3, 6.3.4, and 6.3.5). The spectral values of the color pair are only one factor for producing metamerism. The second is the spectral power distribution of the illuminant used. Both are combined in the color stimulus function Φ(λi ) which is interpreted as the original cause of the appearance of metamerism. The combination of this function and the chosen standard observer can lead – despite different spectral values – to equal standard color values X1 = X2 , Y1 = Y2 , Z1 = Z2 for the metameric color pair. For example, with the spectral reflectance R1 (λi ) and R2 (λi ) of each color, the spectral power distribution S(λi ) of the reference illuminant and, for simplified notation, the 2◦ standard observer follows from Equation (3.1.1): X1 = k

N 

S(λi )¯x(λi )R1 (λi )Δλ

(3.3.2)

S(λi )¯x(λi )R2 (λi )Δλ.

(3.3.3)

i=1

and X2 = k

N  i=1

Although the reflectance values of both colors are unequal at most wavelengths and the individual summands in Equations (3.3.2) and (3.3.3) to the same wavelength are mostly different: for metameric color pairs, equal totals always result and, therefore, identical standard color values X1 = X2 for the reference illuminant. The same holds for the remaining two color values. The existence of equal standard color values, here X1 = X2 , Y1 = Y2 , Z1 = Z2 , signifies always equal color sensation. Because of this, both colors produce the same color impression. The same holds for additional color values, especially of CIELAB or DIN99o color spaces. If the reference illuminant is substituted by another one of new spectral power distribution S0 (λi ), then the individual summands and totals of Equations (3.3.2) and (3.3.3) are different. The summands show now other weighting factors in comparison to those in the reference illuminant. Even all three color values differ, therefore, from one another, the change of illuminant results in two different color impressions.

3.3

Color Constancy and Metamerism

171

The individual contributions of different power distributions S(λ) in dependence of wavelength have already been seen, for example, from Fig. 2.4 for standard illuminants D65 and A. In the range of blue wavelengths, the distribution of D65 is essentially higher than A. In contrast, in the red range, the radiation intensity of A dominates compared to that of D65. Because unequal color values mean different color sensations, metameric pairs typically produce a color difference for a change of illuminant. Although the concept of metamerism was originally introduced by Ostwald for illuminant changes, it has also been extended to include states without illuminant changes. These are more accurately denoted as “conditional equal prerequisites”. Five metamerism kinds are known to this day. In the following survey, the first and second metamerism kinds are directly influenced by the different summand contributions. The last three are caused by the individual conditions of the observer and the measuring equipment used: 1. Illumination metamerism occurs under change of illuminant as described above; this is the most important sort of metamerism. 2. Observer geometry metamerism results from change between the 2◦ and 10◦ standard observers on the basis of the different CMFs x¯ (λi ), y¯ (λi ), z¯(λi ) and x¯ 10 (λi ), y¯ 10 (λi ), z¯10 (λi ), respectively. The consequences are by no means negligible, as is shown in the next section. 3. Observer metamerism is caused by the subjective color sense of different observers and the individual deviations from the underlying standard observer. Such deviations should be clarified between the involved partners in advance of the color tolerance agreement. 4. Geometry metamerism can occur under otherwise equal conditions and equal visual angle by a change in the line of sight. For effect pigments, for example, the color match by change of the angle of illumination or observation can be lost. A special case of geometry metamerism is the so-called silking effect. This effect means that the reflectance changes simply by rotation of an individual sample around an axis perpendicular to its surface. This phenomenon is caused by the nearly parallel needle-, flake-, or rod-shaped pigment particles. 5. Equipment metamerism can be considered due to the general lack of consistency among color measuring instruments of the same or different manufacturers. The variations in measured values are primarily caused by small but largely unavoidable production differences of the component parts or the equipment design. Illumination metamerism is, as already mentioned, practically unavoidable. It occurs during reproduction of a reference color of unknown composition by applying colorants of different batches, other colorants, binders, additives, and a different number of colored components. The chemical analysis is much

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too complex and priority is generally given to the use of available colorants. There should also be sufficient experience related to color inconstancy, color strength, color fastness, compatibility – only to list some of relevant criteria for reproduction of absorption colors. From each sketched type of metamerism above, an indication for its reduction is at the same time revealed. For lowering illumination metamerism, one can carry out a careful selection and tuning of the colorants during visual or recipe prediction. For a recipe with absorption colors, at minimum three and at best six different colorants should be used. This optimum comes because with increasing number, the risk of an “overshoot” instead of an approach of the spectral values is greater. Because of the substantially lower concentrations of the additional colorants relative to the already dominant ones, a minor admixture is often difficult to meter out or to homogenize. A reproduced color often shows an increased metamerism if for the dominant colorant the coloristical “most natural” one is chosen. To achieve recipes with minimal metamerism, each step of the spectral curve trace should be approached with a separate colorant (cf. Figs. 6.15 and 6.16). This aim in most cases, however, is not achieved with the “first best” colorant. The curve trace of the maximum as well as the rise and slope at the start or end of a curve step should be approximated as well as possible. This goes especially for violet, yellow, orange, or red and brown colors which show increased spectral values at wavelengths longer than 500 nm. For blue and green hues, however, the position of the maximum and the slope are well approximated by only few colorant components. One strategy of recipe prediction is based on the criteria that the color difference between the reproducing color and the target color is minimal for a given reference illuminant. This color difference can be calculated in advance without the necessity of implementing the corresponding recipe. A similar consideration holds for the color inconstancy index. It can be determined in advance for every change of illumination without the need of actually performing the lighting change. As a reference illuminant in both cases, D65 is mostly used, although other illuminants are possible. This strategy allows for the possibility of changing the types and numbers of colorants for a recipe in order to pursue the desired effect on the degree of metamerism. The amount of illumination or geometry observer metamerism is characterized by the index of metamerism. This is discussed in the next section.

3.3.4 Special Metamerism Indices The degree of metamerism of a color pair is indicated by the so-called special metamerism index for change of illumination. This index corresponds to the color difference which results from the change between reference and test illuminant for the given color pair [40]. The choice of the color difference formula

3.3

Color Constancy and Metamerism

173

is an option. In the cases of the CIELAB expressions (3.1.18) or (3.1.23), the color contributions are determined from equations ΔL∗ = LT∗ − LR∗ ,

Δa∗ = a∗T − a∗R ,

Δb∗ = b∗T − b∗R

(3.3.4)

or additionally from ∗ ∗ ∗ = Cab,T − Cab,R , ΔCab

(3.3.5)

a∗R b∗T − a∗T b∗R ∗ . = ΔHab ∗ C∗ ∗ ∗ ∗ ∗ 0.5(Cab,T ab,R + aT aR + bT bR )

(3.3.6)

The indices T and R belong to the test and reference illuminant, respectively. In most cases, D65 is arranged as the reference illuminant and the test illuminant should greatly differ from the reference. The index of metamerism MXilm needs to be determined for approximate illuminants in which the color pairs are subjected to during application. The degree of metamerism remains unchanged under the exchange of the reference and test illuminants, which is a consequence of the squared color contributions (3.3.4), (3.3.5), and (3.3.6) in a color difference equation. The visually perceived and the numeric color differences can definitely be different because, as already explained, the stored values of the spectral power distribution of the illuminants and the standard observers are not identical with those of the sources used or the individual observer. For a clear interpretation and comparability of the metamerism index for illumination change MXilm , in addition to the numerical value, the following must also be specified in the result: the test and reference illuminants, the standard observer, and the color difference formula used. According to a recommendation of the CIE, the special metamerism index for change of observer can also be calculated under the same illuminant [41]. This is the analog of determining the color contributions (3.3.4), (3.3.5), and (3.3.6) of the CMFs of a test person with regard to one standard observer under the same illuminant. Also in this case, the calculated numerical value of the index MX 0 obs can deviate from that for another observer. For correct determination of the special index of metamerism, a peculiarity needs to be considered. In a number of cases, the standard color values of the color pair are not exactly the same as for the reference illuminant, that is, X1 = X2 , Y1 = Y2 , Z1 = Z2 . In such cases, the standard color values X2,T , Y2,T , Z2,T of the sample in the test illuminant need to be corrected using the respective factors:    fX = X1 X2 , fY = Y1 Y2 , fZ = Z1 Z2 .

(3.3.7)

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Otherwise, incorrect metamerism values result. The color difference is, therefore, determined between the color values X1 , Y1 , Z1 of the reference illuminant and  Y2 = fY Y2,T ,  Z2 = fZ Z2,T X2 = fX X2,T , 

(3.3.8)

for test illuminant. As an example, in order to determine the special indices for change in illuminant and observer geometry, we use the color pair of reflectance shown in Fig. 3.10. The accompanying standard and CIELAB color values for the combinations D65/10◦ and FL 2/10◦ as well as D65/2◦ and FL 2/2◦ are listed in Table 3.9. The standard color values of the test illuminant are to be corrected using Equation (3.3.8) because the X, Y, Z or L∗ , a∗ , b∗ values already differ for reference illuminant. The resulting special indices of metamerism for change of illuminant MXilm ∗ , are given in Table 3.10. In and MX 0 obs , each based on color difference ΔEab the second row of this table, the respectively used combinations which follow from Table 3.9 are indicated. The description 6 (1/5), for example, means that the color values of number 6 are multiplied by the ratio of the corresponding color values in lines 1 and 5. The results in the last two rows in Table 3.10 show that the index of metamerism for illuminant change is typically higher than that for the change of observer geometry. From comparison of the CIELAB values for reference and test illuminants for the same observer in Table 3.9, it should be clear that metamerism is here mainly caused by differing red to green amounts Δa∗ between both colors. The combinations 1, 6 with 2, 5 as well as 3, 8 with 4, 7 in Table 3.9 are equivalent because test and reference illuminants are exchanged with each other. The same holds for the change of standard observer with regard to the combinations 1, 7 with 3, 5 as well as 2, 8 with 4, 6. The slight deviations of the corresponding index values are caused by measuring and rounding errors. Table 3.9 CIE 1931 and 1976 standardized values of the metameric color pair in Fig. 3.10

No.

Illuminant/standard observer

X

Y

Z

L∗

a∗

b∗

1 2 3 4

Reference

D65/10◦ FL 2/10◦ D65/2◦ FL 2/2◦

16.862 19.096 17.444 18.753

13.577 15.210 14.068 15.491

7.356 4.735 7.473 4.637

43.62 45.92 44.33 46.30

24.20 17.95 24.10 18.44

20.94 24.89 22.13 25.46

5 6 7 8

Test

D65/10◦ FL 2/10◦ D65/2◦ FL 2/2◦

16.573 17.390 17.305 17.232

13.536 13.680 13.894 13.804

7.140 4.543 7.345 4.486

43.56 43.77 44.08 43.95

22.84 18.47 24.42 20.59

21.63 22.30 22.17 22.31

3.4

Specific Qualities of Colorants

175

Table 3.10 Indices of illumination and observer-geometry metamerism following from color values in Table 3.9 Kind of metamerism Illumination metamerism

Observergeometry metamerism

Number in Color Table 3.9 pattern

Illuminant/ standard observer

1 6 (1/5) 3 8 (3/7) 2 5 (2/6) 4 7 (4/8)

Reference Test Reference Test Reference Test Reference Test

D65/10◦ FL 2/10◦ D65/2◦ FL 2/2◦ FL 2/10◦ D65/10◦ FL 2/2◦ D65/2◦

1 7 (1/5) 2 8 (2/6) 3 5 (3/7) 4 6 (4/8)

Reference Test Reference Test Reference Test Reference Test

D65/10◦ D65/2◦ FL 2/10◦ FL 2/2◦ D65/2◦ D65/10◦ FL 2/2◦ FL 2/10◦

Special metamerism index ∗ MXilm = ΔEab

∗ MX 0 obs = ΔEab

4.65 3.84 4.57 3.94 1.77 2.19 1.75 2.18

In this section, it has been shown that two essential properties of absorption colors can be approximately described and numerically compared by colorimetrical methods using color difference formulas. The following sections deal with further colorimetrical procedures, which have been developed especially for numerically characterizing other qualities of absorption colors.

3.4 Specific Qualities of Colorants The color spaces CIE 1976 and DIN99o described as well as the color difference formulas – which were, in particular, intended for the assessment of collections with absorption colors – are designed for application to additional color properties. In the following, quantities such as color strength, spreading rate, covering capacity, transparency, or color fastness of formulations with regard to colorimetric characterization are discussed. Such properties are united under the term color qualities. With these accounts, the fact that the whole of colorimetric methods are only valid for small color differences needs to be considered. In order to have reasonable comparability of results, the same color space and chosen color difference formula should be maintained throughout.

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3.4.1 Build-up and Coloring Potential All colorants can be dispersed in different concentrations in a dye or binder dependent on the desired color. The change of the color values of a single colorant in dependence on concentration is called build-up. This behavior needs to be differentiated with regard to a dye in the same solvent and a colored absorption pigment in white mixtures for constant total pigment loading in the same binder. In the case of a chromatic absorption pigment in white/black mixtures, the dependence of lightness and chroma on the concentration of all three colorants is given by the so-called coloring potential. The build-up is graphically represented by a three-dimensional curve section in a color space. In comparison, the coloring potential is represented by a three-dimensional area in color space. Its shape depends on the concentration of the chromatic pigment in white/black mixtures. From the geometrical representation of the build-up and coloring potential, characteristic features of the corresponding colorants can be obtained. These are discussed in the following. Figure 3.11 shows the projection of the build-up of different dyes in solution in an a∗ , b∗ color plane. The simultaneous change in lightness here is of secondary importance. From this simplified representation, the color development in dependence of dye content directly follows. Each curve starts for b∗ 1 2

80 60

3 40 20

4

0 7 –20 –40

–40

5

6 –20

0

20

40

a∗

Fig. 3.11 Typical build-up of different dyes: 1 light-yellow, 2 yellow, 3 orange, 4 red, 5 blue, 6 dark-blue, 7 blue-green

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Specific Qualities of Colorants

177

minor dye concentration in the immediate vicinity of the achromatic locus. It ends with the coordinates of the mass tone, which is the natural color of the absorption colorant. At first, the chroma changes directly proportional to the dye content. But at certain high dye concentrations, a particularly noticeable change in hue occurs and the chroma lowers. The build-up describes, dependent on color locus, a left-hand or right-hand turn. The amount of the hue angle |hab | changes as shown between about 20◦ and 60◦ . Chromatic pigments in white mixtures show similar shaped projected buildups such as dyes in solution. However, it has not been clarified as to why a build-up describes systematically a right or left-hand curve in dependence on the quadrant of a color plane. For visualization of a coloring potential, Fig. 3.12 shows the change in ∗ of an organic colored pigment. The lightness L∗ dependent on chroma Cab parameters P/W and P/B indicate the blending ratio of pigment to white and black colorants, respectively, at constant pigment loading. On the envelope curve, three points P, W, and B are plotted. These represent the mass tone of the colored, white, and black pigments, respectively. The intermediately located curve sections have the following meaning: – The section from point W to point P represents the build-up of the colored ∗ plane. It extends from the pigment in white mixtures projected in the L∗ , Cab L∗ W 10P/90W 75 50P/50W

1B/99W 50 3B/97W

P

25 B

99P/1B 95P/3B

0 0

25

50

75

C ∗ab

Fig. 3.12 Coloring potential of an organic absorption pigment in white and black mixtures of constant pigmentation

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mass tone of white to the mass tone of the colored pigment. The downward curved slope of this section shows the decrease in lightness with increasing chromatic pigment content, as well as the simultaneous reduction of chroma. This is an analogous behavior to dyes, Fig. 3.11. – The curve element from P to B represents the color loci for mixtures of the colored pigment and the achromatic black pigment. – The last section of the envelope curve WB is a part of the lightness axis L∗ , representing achromatic mixtures between the mass tone of the white and black colorants without a colored pigment. The area enclosed by these three curve sections represents the coloring potential of an absorption pigment. In other words, the coloring potential forms a curved three-dimensional area in color space similar to a blade of a marine propeller. This area is flat in the surroundings of the lightness axes. The coloring potential includes, therefore, all color loci which can be entirely generated with a given chromatic pigment and white/black mixtures for fixed total pigment loading. The coloring potential is an additional fingerprint of a colored pigment. As can be seen in Fig. 3.12, the distances between the plotted points on curve sections WP, PB, and BW change non-uniformly with regard to the indicated mixing ratio values. This behavior signifies that the lightness, chroma, and hue of the colorant change nonlinearly in dependence on concentration. This nonlinearity is typical for all absorption colorants but, moreover, it develops differently. Altogether, this characteristic complicates the entire color physical and coloristical handling of all colorant sorts. It forces, for example, one to carry out the dosing or weighing of colorants with the upmost care and accuracy and to perform all manufacturing steps with greatest possible reproducibility. Recipe prediction needs, therefore, nonlinear approximation methods. Generally, the chroma, hue, and lightness respond considerably more sensibly to a fraction of black than white of same amount. This behavior can already be inferred for merely 1 part black from Fig. 3.12. The combination 99P/1B ∗ = 75 by about reduces the chroma of the color-intensive mass tone in P of Cab ∗ ∗ ΔCab = −34. The ratio 1B/99W decreases the lightness L = 92 in W for the mass tone of white, however, by about ΔL∗ = –25. Dark chromatic colorants such as violet or blue behave in a similar manner. This most striking nonlinearity turns out to have a particularly detrimental effect on the assessment, manufacture, and reproduction of colors. The strong change of color for a small fraction of a dark colorant is related to the sensitivity and correctability of color recipes. In comparison to organic colorants, inorganic pigments typically show a less developed coloring potential. This is a result of the characteristically higher scattering and lower chroma of the mass tone of these sort of colorants. For inorganic pigments, the arched curve sections in Fig. 3.12 turn into nearly rectilinear line segments of only slight curvature. The nonlinearity of inorganic pigments,

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179

in dependence on concentration, is in general similarly strongly developed as for organic pigments. The term coloring potential – in its original meaning – does not apply to effect pigments. This is because for these colorants, the angle-dependent color impression is of primary interest. Slight amounts of carbon black in formulations with effect pigments serve principally to absorb the complementary color of interference pigments, reduce chroma, and occasionally for correction of lightness flop. The natural interference colors of interference pigments emerge more clearly with only a minimal amount of carbon black. A borderline case is represented by mixtures of interference and colored pigments. With such mixtures, a new type of build-up in comparison to that of absorption colorants is combined (Section 3.5.4).

3.4.2 Strength and Depth of Color The concepts strength of color and depth of color are closely linked and are of coloristical as well as economic significance. Both distinguishing features were already established at the beginning of the 20th century for assessment of dyes and in 1930 began to be applied to absorption pigments. Whereas the coloration potential and build-up describe the development of a color in dependence on colorant concentration, the strength and depth of color characterize the coloring efficiency of a colorant. Color strength is defined as that property of a colorant, which acts by coloring upon other materials [42]. In contrast, color depth describes that feature of a colorant, which produces a more intense coloration or rather higher chroma with increasing concentration. It increases, therefore, up to the saturation of the colorant but decreases with rising lightness. The color depth correlates consequently with the color build-up. It is important to point out that color strength is a relative quantity. The color strength of an absorption colorant is a quantity in relation to another (similar) absorption colorant. In Fig. 3.13, the depth of color D for a pair of similar colorants in dependence on concentration c is plotted. For small and moderate values of c, the depth of color D changes linearly for the reference and sample colorant. At higher concentrations, the so-called saturation depth of the reference DS,R and sample colorant DS,S is attained. These depths are not necessary at the same level. The two straight lines through the origin have different slopes. The slope, however, is directly proportional to color strength. From this plot, it can be inferred that the color strength of the sample is related to the same level of color depth DSD as that of the reference color. Apart from a constant k, the slope of the straight line through the origin for moderate concentrations is a direct measure for the color strength P of a colorant. Using the notation in Fig. 3.13, the general equation of a straight line through the origin is

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Reference

DS,R DS,S Depth of color D

Sample

DSD = const.

0 0

cR

cS

Concentration c

Fig. 3.13 Schematic representation of color depth D in dependence on concentration c of a pair of similar absorption colorants

D = k · P · c,

(3.4.1)

where P is the absolute color strength. However, it should be noted that it is the relative color strength rather than the absolute that is relevant in practice. Under the condition of equal color depth level DSD for the sample and reference colorants, the relations DSD = kPS cS ,

(3.4.2)

DSD = kPR cR ,

(3.4.3)

follow. If the ratio of PS and PR is defined as the degree of relative color strength Prp of an absorption pigment, it follows that Prp =

PS cR = . PR cS

(3.4.4)

The relative color strength of an absorption pigment follows consequently from the quotient of the concentration cR of the reference pigment to the concentration cS of the sample pigment at equal color depth level DSD . The quantities Prp and D are dimensionless. In the case shown in Fig. 3.13, the initial slope of the reference colorant is twice that of the slope of the sample colorant. Following the definition in Equation (3.4.4), the sample pigment has a color strength of 1/2 related to the reference pigment, and consequently half that of the reference. On the other hand, both pigments are only at equal color depth if the condition cS = 2cR is met. Procedures for determining equal level DSD are described below.

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Specific Qualities of Colorants

181

For dyes in solution, the same considerations hold, but with the difference that the relative color strength Prd is then determined from the quotient of the extinction coefficients ε S and εR at the absorption maximums of the reference and sample dyes Prd =

εS εR

(3.4.5)

at equal level of DSD [43]. The determination of relative color strength for pigments follows from brightening mixtures with a white pigment [42]. Instead of the concentration ratio, the mass ratio mR /mS of the reference and sample pigments is now inserted into Equation (3.4.4). First of all, the weighed portion mR is chosen so that the previously fixed color strength level DSD is achieved. At this step, it is necessary to pay attention to insure that the weight fraction of the white pigment mW is the same in the mixture of reference and sample pigments. Afterward the mass mS of the sample pigment is altered as long as both mixtures show nearly no difference with regard to the previously chosen adjustment criterion FAC , so ΔFAC ∼ = 0 (see below). If these conditions are fulfilled, the relative color strength Prp of the sample pigment follows from relation Prp =

mR mS

(3.4.6)

for secondary conditions: DSD = const., mW = const., and ΔFAC ∼ = 0. In addition to the value of relative color strength Prp , it is always necessary to indicate the reference colorant, the level SD of color depth, the sort of white pigment, respectively solvent, as well as the accompanying fraction, and the chosen matching criterion FAC . For comparability of the resulting relative color strength values with those of other colorants, the so-called standard color depth (SD) first for dyes and later geared to pigments was established. The standardized SDs for textile dyes are denoted as 1/25 SD, 1/12 SD, 1/6 SD, 1/3 SD, 1/1 SD, and 2/1 SD. For pigments, there are four standard depths: 1/25 SD, 1/9 SD, 1/3 SD, and 1/1 SD [44]. The production and testing of standard deep colors are described in [45]. Above and beyond the mentioned parameters DSD = const., mW = const., and ΔFAC ∼ = 0, it needs to be considered that the color strength of pigments additionally depends on the chosen reference pigment, the mean particle size, the sort and concentration of the white pigment, and a possible added black pigment. For dyes, the relations are simpler because only the reference dye, the solvent, and the concentration need to be indicated in addition to the chosen standard color depth and matching criteria. For absorption colorants, various effective matching criteria FAC exist. The perhaps obvious idea of using the visual assessment for determination of color

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Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

strength certainly must be ruled out. In particular, this approach is overtaxed if similar reflection or transmission spectra are present. The color matching needs to be carried out exclusively with measuring methods. Some widespread criteria founded on either spectral or colorimetrical quantities follow. The spectral-matching criteria are based on the term Kubelka–Munk function F(λ) = K(λ)/S(λ). It corresponds to the ratio of the absorption and scattering coefficient K(λ) and S(λ), respectively, in dependence on the wavelength λ, cf. Equation (5.3.14). Both coefficients have to be known with regard to the sample and reference colorants at the same wavelength. This quotient is, however, useless if the scattering coefficient tends to zero – especially for highly transparent or dark colorants. In such cases, it is necessary to switch to other criteria. The four most commonly used spectral-matching criteria are as follows: 1. The difference of the Kubelka–Munk functions between the sample and the reference colorant FS , FR , respectively, each at the accompanying wavelength λmax,R of the function maximum of the reference colorant: ΔFmax, R = FS (λmax, R ) − FR (λmax, R ) ∼ = 0.

(3.4.7)

2. The difference of the reflectance between the sample and the reference colorant, each at the accompanying wavelength λmin,R of the reflection minimum of the reference colorant: ΔRmin, R = RS (λmin, R ) − RR (λmin, R ) ∼ = 0;

(3.4.8)

this criterion is equivalent to the previous one. 3. The minimum of the sum, which results from the weighted differences of the Kubelka–Munk function values between sample and reference colorant, each determined at i = 1, 2, . . . , N measured wavelengths λi : N  i=1

ΔFi =

N 

αi [FS (λi ) − FR (λi )] ≡ minimum;

(3.4.9)

i=1

the weighting factors α i correspond to the sum of the CMFs αi = x¯ (λi ) + y¯ (λi ) + z¯(λi )

(3.4.10)

for the chosen 2◦ or 10◦ standard observer; this criterion leads to unsatisfactory results if the summands in Equation (3.4.9) are simultaneously of high and low values. In this case, it is opportune to apply the next criterion. 4. The minimum of the sum which follows from the squared deviation of the Kubelka–Munk function of the sample and reference colorant at N measured wavelengths λi

3.4

Specific Qualities of Colorants N  i=1

(ΔFi ) = 2

N 

183

[FS (λi ) − FR (λi )]2 ≡ minimum,

(3.4.11)

i=1

probably with other weighting factors as in the third criterion. The spectrophotometric matching of the sample to the reference colorant should certainly be performed with the same illuminant, standard observer, and spectrophotometer for each. These conditions need to be indicated along with the value of the relative color strength in addition to the employed reference colorant. Relative color strengths are only to be compared with each other if the same reference colorant and the mentioned three conditions had been used. For colorimetrical-matching criteria, one should alternatively take into consideration: 1. One of the differences ΔX = XS – XR , or ΔY = YS – YR , or ΔZ = ZS – ZR between the standard color values of the sample XS , YS , ZS and reference XR , YR , ZR , which contains the smallest value min (XR , YR , ZR ) of the reference colorant: ⎧ ⎫ ⎨ ΔX if XR = min(XR , YR , ZR ) ⎬ ΔY if YR = min(XR , YR , ZR ) ∼ (3.4.12) = 0. ⎩ ⎭ ΔZ if ZR = min(XR , YR , ZR ) 2. The difference of the standard color values YS , YR or lightness values LS , LR of the sample and reference colorant ΔY = |YS − YR | ∼ =0

(3.4.13)

ΔL = |LS − LR | ∼ = 0.

(3.4.14)

or

3. The smallest color difference ΔE with regard to the sample and reference colorant ΔE ≡ minimum;

(3.4.15)

the color difference formula is optional. 4. The depth of color Di at standard color depth SD; this matching method should be applied to dyes as well as pigments.

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Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

There exist empirical formulas, especially for matching of color strength with pigments. We restrict ourselves to these in the following discussion. Historically, these have been found in a similar manner as for dyes. A relation in widespread use is given by ΔDi =

√ Y[s · ai (ϕ) − 10] + bi ∼ = 0.

(3.4.16)

The quantities in this expression are as follows: ΔDi : difference in depth of color for standard color depth level i; i: level of standard color depth, one of 1/1 SD to 1/25 SD as mentioned above; Y: standard color value of the CIE 1931 system, a measure for lightness; s: measure of saturation, it corresponds to 10 times the distance in the CIE chromaticity diagram between color locus of the sample pigment (x, y) and the achromatic locus (xn , yn ) s = 10 ·



(x − xn )2 + (y − yn )2

(3.4.17)

for the chosen illuminant and standard observer, cf. Table 3.2 and Color plate 2; ai (ϕ): empirical function for weighing of s; it depends on hue angle ϕ = arctan [(y − yn )/(x − xn )] and the chosen level i of standard depth SD; empirically determined constant corresponding to the level i; it takes bi : values of b1/3 = 29, b1/9 = 41, b1/25 = 56. The added mass of the sample pigment mk+1 on the kth iteration step, k = 1, 2, 3, . . . , follows from the additional empirical relation mk+1 = mk · 0.9ΔDi ,

(3.4.18)

the exponent ΔDi following from Equation (3.4.16). Often the numerical value of the relative color strength Prp is emphasized alone for the argument for colorant substitution in color reproduction or color recipe prediction. But it needs to be taken into consideration that the replacement usually brings in changes in hue or lightness or other coloristic features with consequences for color constancy, metamerism, among other things. Provided that the new pigments are transparent or translucent, also the light transmission or the covering potential can change. In any case, such consequences must be thoroughly checked before the colorant exchange. Also, as normal in material sciences, it must be ensured that the replacement of one component has no negative effect on other components or system properties.

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Specific Qualities of Colorants

185

In summary, we point out that both spectrometric and colorimetric procedures are quite efficiently applied to determine and characterize the relative color strength of absorption colorants. The productive potential of colorimetry up until now is used, in the following, for a quantitative description of further color properties of absorbing colorants.

3.4.3 Covering Capacity Whereas the color strength, in some sense, indicates the coloristical effectiveness of an absorbing colorant, the covering capacity is a measure of the capability of a coating to hide a colored background [46]. This property is called hiding power in ASTM standards [47– 49]. For constant pigment loading, there exists a minimum coating thickness dC for which the color of a substrate can no longer be visually discerned. On the other hand, given a layer thickness d, there is a minimal colorant concentration cmin for which the background color is also not visible. In both cases, the inverse of the minimal covering thickness dC is a measure for the covering capacity of a scattering and absorbing coating, as in  D = 1 dC .

(3.4.19)

The quantity D is called the covering capacity value and is in units of reciprocal length. This dimension can be modified in such a way that it has an economical meaning. If the unit of film thickness dC is given in mm and multiplied by the ratio m2 /m2 , then for covering capacity value D, the quotient m2 /l follows; 1 l ≡ 103 cm3 is the volume unit liter. More precise, the quantity Dv = D relates to the volume. The corresponding numerical value of Dv indicates the area in square meters, which 1 l of color can totally cover at film thickness dC . On the other hand, if the mass density ρ m is given in dimensions g/cm3 , then the unit liter l can be substituted by the expression ρm−1 kg. If the numerical value D is given in the units mm–1 or m2 /l, then the multiplication with the value of mass density ρ m in unit g/cm3 results in dimension m2 /kg for Dm = ρm · Dv . The quantity Dm now relates to the mass, it indicates the area in square meters, which one kg of color can totally hide at film thickness dC . Therefore, the value of Dv or Dm is a realistic measure for the covering spreading rate of a formulation with absorption pigments. The quantity of the covering capacity can be simply verified from the dry film thickness. Now, the question needs to be answered as to which method can be used to find out the minimal covering film thickness dC . Just as for the color strength, the covering capacity value of absorption colorations is experimentally determined with the help of a colorimetrically defined criterion. For this purpose, there exist two methods of operation [46 – 50]. The first of them is based on the so-called

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3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

contrast ratio; it follows from the standard color values Yw and Yb which result from the measured reflectance of the film over white and black backgrounds (contrast substrate). According to this criterion, a minimal covering layer is achieved if the condition Yb ≥ 0.98 Yw

(3.4.20)

is fulfilled, for example, for the standard illuminant/observer combination D65/10◦ . The contrast ratio should, accordingly, attain a minimum value of 98% to indicate a covering film thickness. The second procedure uses the color difference ΔEbw which results also from the spectrophotometric measurement of the film over black and white backgrounds of known reflectance. A covering coating satisfies the condition

ΔEbw =

 (ΔLbw )2 + (ΔCbw )2 + (ΔH bw )2 ≤ 1,

(3.4.21)

where ΔLbw , ΔCbw , ΔHbw are the difference in lightness, chroma, and hue, respectively, each over both backgrounds. For opaque colors, the values are (Yb /Yw ) = 1 or ΔEbw = 0. Due to the color locus-dependent visual discrimination of color differences, it may be necessary to fix, in dependence of hue, a color difference in the value domain 0.3 ≤ |ΔEbw | ≤ 0.9. In the case of achromatic or nearly achromatic colors, Equation (3.4.21) reduces to |ΔLbw | ≤ 1 because the distribution of chroma and hue is now of secondary importance. In all cases, the spectral reflectance of the black and white backgrounds Rbg,b (λ) and Rbg,w (λ), respectively, should be constant and meet the conditions Rbg,b (λ) ≤ 0.2 and Rbg,w (λ) ≥ 0.8. The actual determination of the volume- or mass-related covering capacity value D from the color difference method can be achieved either graphically or numerically (for simplicity, in the following we leave out the corresponding indices v and m of D). The graphic method requires at least three colorations of different film thicknesses d, each over the black and white substrate. The layer thickness should be known to an uncertainty of ±1%. From the graph of ΔEbw in dependence of 1/d, the value D follows by extrapolation. It is represented by the slope point of coordinates (ΔEbw , 1/dC ) to which the color difference ΔEbw = 1 belongs; see Fig. 3.14. In contrast, the numerical procedure is based on the calculation of the reflectance spectra over both backgrounds. It has the advantage that only one control coloration needs to be manufactured. On the other hand, this method requires various known spectral quantities as input data: the optical constants of the pigments, binder, perhaps additives, and the reflectance of both backgrounds; in addition, the pigment loading and the single concentration ratio of

3.4

Specific Qualities of Colorants

187

ΔEbw

3.0

2.0

1.0 D ≈ 18 m2/l D 0 0

10

20

1/d/mm–1

Fig. 3.14 Determination of covering capacity value D of a pigmented coloration by extrapolation

the inserted colorants have to be known. This set of quantities is used to calculate the spectral reflectance over both backgrounds with the aid of a suitable radiative transfer approximation (see Chapter 5). Afterward, the covering capacity value D follows from iteration as given in Fig. 3.15. The process starts with an estimate for a non-covering film thickness dk = 1/Dk , which leads to the color difference value ΔEk (for simplicity, the exponent ws is left out in the following). In the next step, a suited value Dk+1 < Dk of corresponding color difference ΔEk+1 is determined. If for the sought after covering capacity value (3.4.19), the given color difference obeys ΔEC ≤ 1, then an improved value Dk+2 follows from equation Dk+2 = Dk+1 −

Dk+1 − Dk (ΔEk+1 − ΔEC ). ΔEk+1 − ΔEk

(3.4.22)

This relation results from the method regula falsi. According to this, at each iteration step, a curve section is approximated by a secant line. After a few iteration steps the covering capacity value D = 1/dC is obtained. The graphical and numerical methods do not necessarily result in same values. Covering capacity values can, consequently, only be compared with each other if they are determined using the same method. The determination of the value D runs, however, more elegantly, if this procedure is directly included in color recipe prediction. This is because the optical

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3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments ΔE bw k=0

6.0

4.0

k=1 2.0

k=2 k=3 D ≈ 22.9 m2/l

0 10

20

30

40

1/d/mm–1

Fig. 3.15 Iterative determination of the covering capacity value D

constants of the inserted colorants and the concentration ratios are already known as well as the calculated spectral reflectance values (Section 6.2.3). The color recipe prediction even allows for the possibility of fixing a covering film thickness dC = 1/D and determining the corresponding colorant fractions. In fact, this method needs intensive computation, but it requires only one coloration in order to check the calculated covering thickness dC . If another covering thickness dC, x = 1/Dx is required, and this thickness differs not more than about ±10% of dC , then the new pigment loading cx follows from the previous one by the simple approximation cx = c ·

dC Dx =c· . dC, x D

(3.4.23)

For use of the maximal pigment loading cmax (in so-called high solids), the even covering and minimal thickness dC, min = 1/Dmax is determined from dC, min = dC ·

c cmax

;

(3.4.24)

the accompanying quantity Dmax follows from formula

Dmax = D ·

cmax . c

(3.4.25)

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Specific Qualities of Colorants

189

To complete the picture, we point out that the change of film thickness or pigment loading can additionally influence the volume or surface properties, for example, gloss, microhardness, abrasive resistance, color fastness, or weathering resistance. Such features, consequently, should also be checked. Moreover, we mention that the discussed methods will produce incorrect results if measured input quantities are inaccurate (“garbage in, garbage out”). On the basis of this fact, the quantities that we are interested in D, dC , or c should be determined in view of a load factor lf > 1 in order to compensate the consequences of unavoidable production fluctuations.

3.4.4 Transparency and Coloring Power According to the previous section, the transition from translucent to opaque of a colored layer is unambiguously characterized by either the covering capacity value, contrast ratio, or color difference over black and white contrast backgrounds. However, reverse has not yet been clarified as to which characteristic value describes the changeover between nearly transparent and slightly translucent materials. Before we discuss this question, we first have to characterize the real state of nearly transparent colorations. Ideal transparency is characterized by pure absorption and completely absent scattering. The greater the transparency of a real layer, the less the light is absorbed or scattered. In addition, transparency of a real material depends on refractive index, the layer thickness, contained colorants, binder, superstructure, as well as aging processes. Clearly, the spectral absorption influences both the light transmittance and color of the material; normally, color differences as small ∗ ≈ 0.02 are just provable. The color shift of nearly colorless transparas ΔEab ent materials is called the tinting power. This color shade is perceptible over a white background, the scattering component, on the other hand, over a black background. A measure for light transmittance is, apart from spectral transmission, the so-called transparency index. It characterizes the scattering component of a transparent medium. It is consequently determined from the color difference ΔEb which follows from reflectance measurements of the layer over a black background and the background itself. Because transparency depends non-linearly on layer thickness d, samples of various thicknesses are to be manufactured for determining the transparency index. In order to attain a transparency index independent of geometrical influence, the color difference value ΔEb for zero thickness d→0 by extrapolation should be determined. The plot of the difference ΔEb in dependence on d results in a nearly positive parabolic curve through the origin, as shown in Fig. 3.16. The transparency index TN is now defined by the reciprocal curve gradient at infinitesimal thickness difference Δd→0:

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3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments ΔE b

ΔE2b

ΔE1b

0 0

d1

d2

d

Fig. 3.16 Determination of transparency index from the slope at the origin

1/TN = lim

Δd→0

Δ(ΔEb ) . Δd

(3.4.26)

The quantity of TN has the unit of d. The greater the transparency of a layer, the smaller the gradient at the origin. Consequently, the transparency index is a quantity which is a real characteristic of an optical material [51, 52]. The slope at the origin can be approximately determined by curve fitting. If we assign the measured value pairs di , ΔEib for i = 1, 2, 3. . . to a polynomial of second order through the origin ΔEb = a1 d + a2 d2

(3.4.27)

with coefficients a1 , a2 , then the transparency index follows from TN = 1/a1 .

(3.4.28)

Also for polynomials of higher order, the transparency index results from the inverse of the coefficient at d. For a rough estimate, two layers of thicknesses d1 and d2 > 1.5 d1 are sufficient. With the accompanying color differences ΔE1b , ΔE2b and using Equations (3.4.27) and (3.4.28), the relation TN =

q · (d2 − d1 ) q2 ΔE1b − ΔE2b

, q = d2 /d1 ,

(3.4.29)

can be derived. The quantity TN is quite sensitive to inaccuracies in the values of d1 and d2 . On the other hand, the relative uncertainty of the transparency index can be estimated by summing up and doubling the relative errors of color

3.4

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191

difference and thickness measurement. The computation of adjustment naturally delivers an enhanced error estimate for TN . The coloring power value CP of a layer is determined in a similar way, but using a white background rather than a black one. The quantity represents a dimensional value for the color of a nearly colorless transparent material. For the determination of the coloring power value, the color difference ΔEw between the layer separately over white background and the background is used, this again in dependence of layer thickness d. The color difference ΔEw behaves similar to ΔEb in dependence on d. Therefore, the formalism for TN can be nearly applied to derive a formula for determination of CP . From similar considerations as above, the quantity of CP follows directly from the slope in the origin CP = lim

Δd→0

Δ(ΔEw ) = a1 , Δd

(3.4.30)

provided that Equation (3.4.27) is applied with ΔEw instead of ΔEb . The unit of CP is the same as of d. The smaller the slope at the origin, the lower the value of CP and the less perceptible the natural color of the transparent material. Both quantities CP and TN can even be simultaneously determined from the same layer thicknesses if the transparent films are each measured over a black/white contrast substrate. With the thickness values d1 , d2 , and the accompanying color differences ΔE1w and ΔE2w ,

CP =

q2 ΔE1w − ΔE2w , q · (d2 − d1 )

q = d2 /d1 ,

(3.4.31)

follows from Equations (3.4.27) and (3.4.30). The error estimates of CP and TN are nearly the same, provided that in both cases the same measuring equipment, layer thicknesses, and contrast substrate are used. For more than three thicknesses, the values are to determine by computation of adjustment. The multiplication of transparency index and coloring power value of the same material leads to the dimensionless varnish power value Γ = TN · CP .

(3.4.32)

As stated at the beginning of this section, with the covering power criteria, it is necessary to differentiate between an opaque and a translucent coating. There does not, however, exist a colorimetrical method for distinguishing a nearly ideal transparent medium from a slightly translucent one. This problem is solved at the end of the next section after the discussion of aging stability.

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3.4.5 Color Fastness and Turbidity Despite experience indicating otherwise, until now we have disregarded the fact that components – especially colorants, binders, and additives – are subject to aging effects. The term color fastness is to be considered the resistance of a given coloration to physical or chemical influences during manufacturing, processing, storing, application, or aging [53, 54]. Color changes during such periods, however, are of prime importance for acceptance, quality, profitability, and conventional working life of a colored product. A high color fastness is necessary during manufacturing or processing because of the presence of the prevailing temperatures, IR radiation, pressures, shear rates, and processing auxiliaries [55]. Consequently, the short- or long-term aging stability of colorants and formulations should be known or at least approximated from results of simulation experiments. Aging processes depend on the inner and outer conditions of a material. The inner conditions consist of thermodynamic instable molecular states, which lead (especially in plastic materials) to after effects such as secondary crystallization, embrittlement, diffusion of colorants, plasticizers, or colorant auxiliaries by relief of residual stresses. Aging from the surface is primarily caused by radiation energy, from cosmic and radioactive radiation to radiation in the near-UV and far-IR range (cf. Fig. 2.1). The effects of humidity, liquids like brine solutions, acids, leaches, organic solvents, detergents, and also gases such as oxygen, ozone, carbon dioxide, and exhaust pollutants should not be underestimated. The inner and outer effects can damage both the individual colorants and the net color effect. The examinations for color aging can be divided into practice equivalent and time-lapsed methods. Of special interest among them is the long-term fastness to light under different climates. Economic reasons allow only in rare cases a long continual outdoor weathering in a temperate climate (Central Europe), warm dry (Arizona), or warm humid (Bombay) with solar energy influx between 127 and 150 W/m2 (about 10% of the solar constant). From material science, timelapsed simulation experiments are known and widely applied, from which the real aging stability should follow in a shorter exposure time. During an artificial weathering, the color samples are exposed to a continuous radiant flux and additionally loaded with thermal energy or agents [56]. For radiation sources are used independently of each other or combined: xenon radiators, luminescence and mercury vapor lamps, as well as carbon-arc lamps [57, 58]. The emitted spectra are irradiated directly without or with selective filters. In single cases, a spectral power distribution which corresponds to the standardized middle daylight D65 simulator is generally sought after. Because the higher radiant energy densities also cause an accelerated self-aging of the sources, the artificial radiators and the inserted UV filters should be exchanged no later than after 1,500 operating hours, and for multiple instrumentation this

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is in cyclic order. The overall influence of the radiant power is a measure of the radiation-invoked aging process of the relevant material. The artificially induced degradation should approximately correspond to that which will probably occur during working life of the colored product. During natural or artificial aging of colorants or color formulations, the corresponding color values change constantly. Periodically time-dependent color alterations are quite rare in practice. Only in some cases, the aging caused chalking or efflorescence of colorants and additives are pursued with colorimetrical methods. Also changes of wood substrates coated with varnishes need to be interpreted with some restriction and are at best judged qualitatively. Changes of gloss, microhardness, abrasion resistance, blistering, cracking or crazing as well as edge peeling, etc., are difficult or even impossible to characterize with colorimetrical methods; such kinds of damage are often judged subjectively according to standardized guidelines. Among the multitude of damage influences, we only describe the effects that are addressable colorimetrically, such as turbidity and yellowing of transparent and translucent materials. The color changes compared with the original condition can be established, for example, by the color difference contributions ΔL, ΔC, ΔH. Because the majority of the damage mechanisms proceed in an exponential time-dependent fashion, the color difference measurements need, at first, to be carried out in shorter, logarithmic time intervals. If the aged specimen field is larger than the aperture of the color measuring equipment, then the changes should be taken each time at the same marked measuring area and in the same orientation. In cases of fogging, striations, or grains present in the measuring field, at least three representative spots should be chosen and the arithmetic mean of the measured variables taken. Each measurement should be carried out with the same color measuring instrument and evaluated with the same method, the same color differences, and the same illuminant/observer combination. All of the aging and evaluation conditions need to be documented with the results. For performance evaluation with regard to color changes were developed, apart from color difference quantities, further empirical characteristics especially to use for different colorations or materials. Among them are the gray scale rating (principally of textiles), whiteness index (white pigments, paper), yellowness index (paper, foils), and turbidity or haze (foils, inorganic or organic glasses). The mentioned quantities are now discussed. The gray scale rating GS can be determined for achromatic and chromatic colorations. This quantity follows after the time-dependent degradation action with the special color difference formula ΔEC =



(ΔL∗ )2 + (ΔCC )2 + (ΔHC )2 .

(3.4.33)

The special chroma and hue differences ΔCC and ΔHC are empirical functions ∗ , Δh , ΔH ∗ , which are given by a standard [59]. of CIELAB differences ΔCab ab ab

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The value ΔEC corresponds to a gray scale rating value GS , which is a natural number in the range 1 ≤ GS ≤ 5. The GS value ranking in dependence on ΔEC is somewhat arbitrary and not clearly defined for intermediate ΔEC values outside given tolerance intervals; see Table 3.11. For example, a color difference in the value domain 12.6 ≤ ΔEC ≤ 14.6 corresponds to the lowest possible value GS = 1, the highest GS = 5 corresponds to the color difference interval ΔEC ≤ +0.2, a value in the domain 10.3 ≤ ΔEC ≤ 12.6, on the other hand, cannot be clearly assigned a concrete gray scale rating value. The whiteness index and whitening index were introduced for evaluation of nearly achromatic light colorations and accompanying color changes. The determination of the whiteness index W is founded on the x, y, Y system and follows from relation  W=

Yn + 800(xn − x) + 1700(yn − y) for D65/2◦ . (3.4.34) Yn + 800(xn, 10 − x10 ) + 1700(yn, 10 − y10 ) for D65/10◦

It comes from extensive color matching [60, 61]. In Equation (3.4.34), xn , yn and xn,10 , yn,10 denote the coordinates of the achromatic point in the chromatic diagram to the 2◦ and 10◦ standard observers, respectively (see Table 3.2). According to Equation (3.4.34), ideal white has a value W = Yn = 100.000 for each of the standard observers. Colorations containing fluorescent brighteners can exceed this value and need a suitable corrected formula or an entirely different one. For description of near-white shades of paper or textiles, on the other hand, the so-called whitening index Tw is used [62]. In comparison to W, its determination follows from a similar relation:  Tw =

for D65/2◦ 1000(xn − x) − 700(yn − y) , 900(xn, 10 − x10 ) − 800(yn, 10 − y10 ) for D65/10◦

(3.4.35)

Table 3.11 Gray scale value, corresponding color difference, and color-difference tolerance of gray scale rating Gray-scale value GS

Color difference ΔEC

Tolerance ±δ(ΔEC )

1 1–2 2 2–3 3 3–4 4 4–5 5

13.6 9.6 6.8 4.8 3.4 2.5 1.7 0.8 0

1.0 0.7 0.6 0.5 0.4 0.35 0.3 0.2 +0.2

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where the indexed chromaticity coordinates have the same meaning as in Equation (3.4.34). The test color appears greenish for Tw > 0 and reddish in the case of Tw < 0. Ideal white is represented by the value Tw = 0 for both standard observers. For equal color coordinate values of x, y, Equations (3.4.34) and (3.4.35) show numerically different results. Clearly, the whiteness index is a measure for the lightness including the weighed relative red and green fractions of the sample. In contrast, the whitening index represents the difference between the relative red and green fractions. Both quantities, consequently, cannot be compared with one another, neither colorimetrically nor numerically. Greater color changes of white or brightened materials are characterized by the yellowness index Yness or yellowing index Ying . The yellowing is caused by energy absorption at wavelengths < 490 nm. The determination of the yellowness index Yness is based on a relative equation of type Yness = (α · X − β · Z)/Y,

(3.4.36)

this again comes from extensive color pattern matching. The empirical weighting factors α and β depend on the chosen illuminant and standard observer [61, 62]. The yellowness index is to calculate from  Yness =

(1.298 · X − 1.133 · Z)/Y for D65/2◦ . (1.301 · X10 − 1.149 · Z10 )/Y10 for D65/10◦

(3.4.37)

Each of the expressions in Equation (3.4.37) corresponds to the difference of the weighted red and blue fractions relative to the lightness value of the yellowish material. The yellowing index Ying is simply defined as the difference in a o : and at the original condition Yness yellowness index after aging Yness a o Ying = Yness − Yness .

(3.4.38)

Yellowing is accompanied by changes of chroma ΔC and hue ΔH, which characterize the color shift more closely. The yellowing of transparent organic materials like coatings, foils, inorganic or organic glasses is also often accompanied by increasing turbidity or haze. The turbidity of transparent finishes or haze of plastic materials transforms a directional light intensity into an attenuated directed and a diffuse component, as shown in Fig. 3.17. A measure for the diffuse scattering effect in the volume of the layer is the haze index HI [63]. This quantity is defined as the ratio of the diffuse transmissivity Td to the entire transmissivity Te of a layer multiplied by hundred HI = (Td /Te ) · 100.

(3.4.39)

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Fig. 3.17 Directional and diffuse intensities determine the haze index of a transparent turbid material

Id Io Ie

jdTe

The entire transmissivity equals the transmission of the material. The diffuse component Td is given by the ratio of the transmitted diffuse intensity Id to the irradiated intensity I0 : Td = Id /I0 .

(3.4.40)

The entire transmission Te follows from an analog consideration by the ratio Te = Ie /I0 ,

(3.4.41)

where Ie stands for the entire transmitted intensity. The value of Id is normally determined with angle dependence. For simplicity, the scattered radiation Jd is determined integrally at an axial angle of ±80◦ with respect to the direction of the incident intensity I0 . The diffuse flux Jd is composed of the transmitted component Id and the scattering jd Te caused by the physical elements of the measurement setup: Jd = Id + jd Te .

(3.4.42)

The diffuse flux jd is determined from an empty measurement in normal operating conditions. The haze index HI is consequently calculated from the relation   Jd jd HI = − · 100. (3.4.43) Ie I0

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This measuring principle responds quite sensitively to defects at the surfaces such as dust or scratches. In order to observe the haze coming from scattering centers in the volume, the samples need to be handled quite carefully. Transparent materials with structured surfaces are, therefore, also unusable for this measuring method. This is because turbidity depends on discontinuity of refractive index at the interface only to a small extent. The haze index of transparent real materials is in the domain HI ≤ 0.05 and of translucent mediums in the interval 0.05 ≤ HI ≤ 0.30. The haze index value HI ≤ 0.05 is a suitable criterion for discerning transparent layers from translucent ones. The haze index HI is, according to definition equation (3.4.39), a dimensionless quantity, whereas the transparency index TN , from Equation (3.4.29), has the units of length. Also this comparison shows clearly that both quantities represent different optical properties. The discussed evaluation relations in this chapter have been outlined generally for the representation of typical properties of colorations with absorption colorants. Certainly, colors containing flake-shaped effect pigments need separate considerations with regard to color fastness because some color properties of this pigment sort cannot be characterized by colorimetrical methods. More detailed information about this is given in the next section for colorations especially with metallic pigments.

3.4.6 Stability of Effect Pigments Similar to that which was discussed for absorption colors, colorations with effect pigments lose their initial color physical properties due to physical or chemical influences. Because of the composition, morphology, and greater size of such particles compared with absorption pigments, the effect flakes present – in a real and figurative sense – a bigger target. The effect pigments should have high resistance against shear load, light or heat load and to a large degree against aggressive chemical environments. Formulations with effect colorants generally require more extensive fastness tests than for absorption colorants. From the catalog of possibilities for metallic and interference pigments, we restrict the discussion to some fundamental degradation mechanisms. The mechanical loads which flake-shaped effect pigments undergo during manufacturing or processing should, by no means, be underestimated. The mechanical ring pumps of painting plants or fluid plastic melts through narrow nozzles and spiral channels cause enormous pressure gradients – shear rates up to 105 s–1 . Incorporated effect particles should survive a process with a maximal shear stress of 102 MN/m2 undamaged. Because of shear stress load, preferential thin metal particles are uncontrollably deformed and scratched, outer sections of the flakes fold down or fracture: the metallic brilliance grays. In a similar way, the brilliance of pearl luster and interference pigments is also reduced.

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The degree of shear stability of effect particles is characterized either by the gray scale rating GS or colorimetrical lightness L. For determination of the decrease in lightness, test effect pigments are agitated in a cooled mixer machine of a suited agitator at a speed of 20,000 rpm for 15 min. Afterward a coating with such shear-loaded particles is manufactured in a clear binder and the CIELAB lightness difference ΔL∗ is determined between the damaged flakes LS∗ and the lightness of the reference coloration LR∗ . The measurements are each performed at an illumination angle of 45◦ and fixed aspecular measuring angle of μas = 20◦ . In dependence of the amount of lightness difference value ΔL∗ (20◦as ) = |LS∗ (20◦as ) − LR∗ (20◦as )|,

(3.4.44)

three empirically defined groups can be differentiated. These are listed in Table 3.12: shear stable, low shear stable, and non-shear stable platelets. In order to further characterize the degradation, a higher measuring angle is used to determine the difference of flop index (Section 3.5.1), or the damage time is to increase. The highest shear stability show clearly thick glitter particles, the lowest metallic corn flakes. The stability of pearlescent and interference pigments can be increased by coating with suited interface-active substances. Although metallic pigments show a higher light resistance in comparison to absorption pigments, high-energy radiation, thermal energy, or an unfavorable polymer surrounding can lead to immediate degradation of the flakes. The metallic character is further lowered by such influences. Particles of copper/zinc alloys (brass) turn out to be particularly susceptible. Because of the lower melting point, the flakes become thermally unstable at temperatures higher than about 200◦ C; the flakes are, therefore, suitably stabilized for application in baking finishes or plastic materials of high melting and glass transition temperature, respectively. Water-borne systems or hygroscopic polymers are able to activate decomposing effective zinc and copper ions from brass particles. These can cause considerable color shift and luster decrease. Both ions lead to accelerated disintegration of PVC molecules; already a thermal load at a temperature as low as 80◦ C leads to separation of hydrogen chloride. The combined coloristical variTable 3.12 Connection between change of lightness and shear stability of metallic pigments Lightness difference ΔL∗ (20◦as )

Degree of shear stability

<5

Shear stable

5–10 > 10

Low shear stable Non-shear stable

Examples of metallic pigments Glitter particles of metal foils Silver dollar flakes Cornflakes

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199

ations can be quantified by the previously discussed colorimetrical methods for determination of yellowing. The damaging effects of the copper and zinc ions can be reduced, however, by appropriate polymer coatings of the flakes. In such a way, stabilized particles can have elevated processing temperatures of up to 260◦ C. An additional property of effect pigments, which especially need to be considered for aluminum flakes, is the leafing stability. This particle formation can be unintentionally reduced or altogether time-dependent decay by false additives. The standardized testing value for leafing stability indicates the time during which a leafing pigment is flooded and finally shows visible wetting [64]. Leafing stability can also be determined relative to a reference formulation. It is possible to additionally pursue the leafing decay with the colorimetrical lightness L. The chemically caused influences can induce particularly enormous degradations of the metal particles. For example, the brilliance of aluminum flakes is reduced in the presence of water as in water-based paint systems or in the form of water vapor. The particles are oxidized with the release of hydrogen and lose, therefore, their initial brilliance. The dehydration of polymeric materials with comparably high humidity content such as polyamides, polycarbonates, or polyethylene terephthalates should be performed separately and actually directly before incorporation of the aluminum pigments. Otherwise the particles become partially oxidized from the surface by released water vapor under the formation of Al2 O3 (Appendix A.1.1). This unwanted influence causes a distinct reduction of lightness. Moreover, the surface-oxidized aluminum platelets tend to conglomerate during incorporation in binders or melts of high polymeric materials. The conglomerations in paints lead to local restricted mottlings, which are caused by turbulent solvent evaporation. Such heterogeneities are detected as local lightness fluctuations with a small aperture color measuring instrument and applying the method described above in connection with Equation (3.4.44). Coatings with effect and absorption pigments, which are exposed to high humidity, are indispensable for the examination of the so-called condensation water resistance. This test is performed with colorations containing single or various combined pigment types. For this, effect pigments can be coated with various covering organic substances. In such a way, modified particles are subjected to a saturated water vapor atmosphere of 40◦ C for 240 h [65, 66]. The colorimetrical changes to detect are again followed in logarithmic time steps. In general, these actions produce stronger damages as the standardized salt spray testing [67]. This was originally introduced to control recipes for absorption pigments. With this method, conventional effect paints and water-based systems are controlled with regard to swelling, blistering, adhesion, reduction of brilliance, and distinctiveness of image DOI. For outdoor use, metallic, pearl luster, and interference pigments have to meet the greatest demands concerning color constancy or weather stability. In

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addition to the testing procedures mentioned until now, the exterior weathering is also assessed with the non-standardized Florida test. This test is regarded as passed, if after a 2-year natural weathering – under daytime-dependant strong fluctuations of air humidity and temperature – the color difference of a chromatic color regarding the undamaged reference coloration is less than unity: ∗ <1. This color difference holds also for effect colorations for illumination ΔEab and measuring angle of β = 45◦ and μas = 15◦ , respectively. For control, it is sensible to perform a standardized outdoor exposure [68]. The weather durability of effect pigments can be greatly enhanced by coating the flakes with a clear colorless UV-absorbing paint, for example, with a silane. Also inorganic layers can improve the climate resistance of effect pigments. If, for example, the mica substrate of a pearl luster pigment is first coated with SnO2 , then the light fastness is improved. A topcoat of the muscovite consisting of TiO2 nanoparticles causes an enhanced UV stability. This is due to the reduced photo activity of the TiO2 layer by UV absorption. Combination pigments with Fe2 O3 or Cr2 O3 layers show – compared to those with only TiO2 -evaporated pearlescent pigments – a refined adhesion between the particle surface and the binder of the formulation. To summarize, the consequences of a natural or artificial degradation of effect colorations can be partially pursued with specially geared colorimetric evaluation methods. In some cases, these procedures are even overtaken from tests for absorption colorants. Since we discussed essential spectrometric properties of effect pigments already in Section 2.3, it is absolutely consistent to represent the primary angle-dependent color physical properties of such pigments with colorimetrical elements in the following.

3.5 Chroma of Effect Pigments Because, until present, for effect colors there is no specific color space, therefore, conventional standard color values are of limited use for characterizing properties of such colorations. As a few examples, we already discussed the changes in brilliance and lightness difference of metallic pigment coatings in connection with the assessment of shear and weathering resistance. In this section, we first become acquainted with specific characteristics of metal effect colorations which are partly based on colorimetric determination procedures including metal effect value, lightness-flop index, distinctiveness of image grade DOI, sparkle degree, graininess, as well as an angle-dependent color difference formula. Afterward, the projection method of color build-up into a color area, already known from absorption colorants in dependence of colorant concentration, is transferred to effect pigments. These curve elements, now taken in dependence of illumination or measuring angle at constant colorant concentration, are denoted as chroma curves for differentiation. From such

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Chroma of Effect Pigments

201

kinds of curves, the color changes of pearl luster and interference pigments in dependence of the evaporated overlay thickness on the particle’s substrate are immediately observed, for example. Moreover, chroma curves of different interference pigments show common characteristics for similar thicknesses and dependences of the illumination and measuring angle. Chroma curves of interference pigments mixed with absorption and additional effect pigments are particularly instructive. Even more impressive forms show angle-dependent chroma curves of diffraction pigments with different grating constants or for those with ferromagnetic substrates. Consequently, chroma curves enable a direct overview of almost the entire angle-dependent color development of effect pigments.

3.5.1 Coloristic Quantities of Effect Pigments As already seen in Section 2.3, the platelet-shaped geometry of effect pigments needs an almost completely different color physical characterization as that for absorption pigments. For the moment, it is most suitable to consider properties of effect pigments which are quantified with as simple methods as possible. Although these efforts are by no means complete, some optical characteristics can be addressed, for example: – metallic value MV, this quantity is a measure of metallic brilliance; – flop index FI, this can quantify the lightness flop; – distinctiveness of image DOI, this property characterizes the covering capacity; – sparkle effect (microbrightness, glints, diamonds) describes the lightness impression caused by dominant reflecting particles; – graininess (coarseness, texture, structure, “salt and pepper”) is a quantity which is independent of sparkling. The degree of reflection caused by metallic particles in formulations is described by the so-called metallic value MV. This characteristic is determined by reflection measurements at two different angles: a plane sample, consisting of metallic pigments in a clear coat, is directionally illuminated at angle β = 45◦ with white light and reflectance R is measured at aspecular angles μas = 7◦ and μas = 45◦ [69].3 The illuminant type is actually unimportant; however, an LED is suitable with regard to emission constancy and long lifetime. The metallic value is calculated with the formula MV = 3 For angle

R(μas = 7◦ ) · 100. R(μas = 45◦ )

counting, see Section 4.1.2, Figs. 4.5 and 4.6.

(3.5.1)

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Because the specular reflectance is usually high for small aspecular angles μas , the quantity MV generally has values higher than 100. Experience shows that pigments of low metallic effect have metallic values of about MV = 150. The reasonable range extends to roughly MV = 5,000 for especially distinct metallic reflection behavior. The second property to characterize is the lightness flop. The reflection of metallic pigments typically decreases with increasing aspecular measuring angle μas , as already shown in Fig. 2.39, and correlating with CIELAB lightness L∗ . This behavior can be taken from Fig. 3.18. Accordingly, the lightness difference ΔL∗ between two extreme aspecular measuring angles can be regarded as a measure for the amount of lightness flop. For determination of the so-called flop index FI, the coating containing metallic pigments is illuminated under β = 45◦ and the lightness L∗ (μas ) is measured at three aspecular angles μas = 15◦ , 45◦ , and 110◦ . The value of the flop index follows from the empirical expression

FI = 2.69 ·

[L∗ (μas = 15◦ ) − L∗ (μas = 110◦ )] 1.11 [L∗ (μas = 45◦ )] 0.85

(3.5.2)

[70]. The flop index corresponds to the lightness difference at small and large aspecular angles L∗ (15◦as ) and L∗ (110◦as ), respectively, related to the lightness L∗ (45◦as ) at observation perpendicular to the surface of the coating. The numerator and denominator in Equation (3.5.2) are weighted with different exponents. Because the measured reflectance values are related to an absorbing and scattering white standard with reflectance of almost R = 1.0, metallics can show reflectance values of R > 1.0 not only at near specular angles; this also leads to lightness values higher than 100. By definition, a dark lightness flop corresponds to a high lightness difference between small and large aspecular observation direction angles. A light lightness flop is, however, characterized by a small lightness difference (Section 2.3.2). Standard metallic flakes of middle light flop show FI values in the domain 10 ≤ FI ≤ 20, high-quality brilliant formulations with metallic pigments of dark flop attain, however, FI values up to 60. As can be taken from the values listed in Fig. 3.18, in comparison to the change of lightness curves, the FI value is directly proportional to the lightness difference between small and large aspecular measuring angles. This corresponds to the numerator in Equation (3.5.2). From Fig. 3.18, it is to infer that the lightness value L∗ (25◦as ) at angle μas = 25◦ plays a central role – it shows values in the range 95 ≤ L∗ (25◦as ) ≤ 110. The curves seem to be slightly rotated at a point with rough coordinates (μas ≈ 25◦ , L∗ ≈ 100). The steeper the lightness curve in the vicinity of this point, the higher the corresponding FI value and the darker the lightness flop. Metallic pigments of high lightness flop index reach lightness values of L∗ (15◦as ) > 300 at small aspecular angles.

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103

L∗

β = 45°

Flop index 11.7 15.5 19.8 37.5 55.4

102

101 0

20

40

60

80

100

μas / o

Fig. 3.18 Lightness L∗ in dependence of aspecular measuring angle μas of metallic pigments with different flop indices

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It seems obvious to use similar considerations for characterization of the color flop index. For that, however, it is necessary to verify if there is a suitable additional formula structured like Equation (3.5.2) with chroma instead of lightness, but at the same aspecular angles and other weighting parameters. Additionally, we need to know if it is necessary to vary the angle of illumination. In each case, both extreme flop colors should be indicated together with the color flop index value(s). An additional property of metallic pigments which can be expressed numerically is the distinctiveness of image DOI, cf. Section 2.3.2. The DOI value more or less represents the uniformity of the visually perceived metallic effect and is correlated with the covering capacity of the metallic formulation. The determination of the DOI value is again carried out under an illumination angle β = 45◦ , the reflection measurement at an aspecular angle μas = 15.0◦ , and slightly varied angles μas = (15.0 ± 0.3)◦ . The source used is normally white light, but illuminations of other spectral composition in the visible range are also possible. The DOI value is given by one of the following relations: ! R(μas = 14.7◦ ) · 100, R(μas = 15.0◦ )

(3.5.3a)

! R(μas = 15.3◦ ) DOI = 1 − · 100 R(μas = 15.0◦ )

(3.5.3b)

DOI = 1 −

[71, 72]. For uniformly parallelized flakes, each of the reflection values in the numerator of both equations should be approximately the same. Both formulas (3.5.3a) and (3.5.3b) limit the DOI value domain to the interval 0 < DOI < 100. The higher the distinctiveness of image, the smaller the reflectance value in the numerator compared to the relative value R(μas = 15.0◦ ) in the denominator. Large-sized flakes show low DOI values, on the other hand, fine particles have high DOI values. These higher values indicate that minimal contrast differences are better reproduced. This property comes down to the smaller edge circumference and the smaller surface area of these flakes. Also high-contrast objects are better imaged against the surroundings by the reflections from the small-sized particles. Moreover, this method can also be applied for DOI determination of pearlescent and interference pigments. The DOI value is related to the covering capacity of a formulation with metallic and interference pigments. Both the sparkle effect and graininess of effect colorations are determined using directional light. Actually, sparkling depends on illumination and measuring angle, but in order to reduce the effort of measuring, three illumination angles of β v = 15◦ , – 45◦ , and –75◦ with respect to the normal to the surface are generally sufficient (the negative sign means that the angle is, as regards the normal, opposite to positive ones). Each of the measurements is performed at an

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205

angle μv = 0◦ with a digital CCD camera.4 The registered gray levels are processed as lightness values of the relevant effect coloration. For evaluation of the sparkle effect, two quantities are used: sparkling area Ask and sparkling intensity Isk . The sparkling area corresponds to the total area, which follows from the summation of each registered particle. But for agreement of the result with visual impression, the only reflecting particles taken into consideration are those lighter than the middle basic lightness of the actual coloration. On the other hand, the sparkling intensity Isk follows from the summation of all individually registered intensities. The sparkle effect is determined separately for each of the three measuring angles. For characterization of this phenomenon, the sparkle grade SG and sparkling intensity Isk are used. The sparkle grade is defined as the geometric mean of Isk and sparkling area Ask :  (3.5.4) SG = Isk · Ask . For an effect coating without any sparkle particles with Ask = 0, a sparkle grade of SG = 0 follows. Normal sparkling effect colorations of size 34 μm ≤ d50 ≤ 54 μm show illumination angle-dependent values in the interval 0 ≤ SG ≤ 10. In extreme cases, values up to SG = 30 at β v = 15◦ are attained. On the other hand, for a constant sparkle degree, if Isk is plotted against Ask , then variations in manufacturing or processing with empirical tolerance ellipses follow, those have limits of permissible variations ΔIsk and ΔAsk (for tolerance specifications, see Section 4.3.3). The sparkle grade of a fine and coarse gray effect coating at three illumination angles is listed in Table 3.13. Both colorations are visually perceived as identical. As expected, the sparkle grade decreases with higher value of illumination angle for both colors. This is because smaller particles reflect light in the direction of the sensor at more flat angles. On the other hand, the lower sparkle grade of the fine coloration continues to higher illumination angles. This behavior comes from the orientation distribution of the flakes, although the majority of the particles are orientated parallel to the coating surface. Because the appearance of graininess of a coating can be recognized only at a small distance from the coating’s surface, this property is independent of observation angle. The quantitative determination of this phenomenon is performed by measurement with diffuse illumination. Graininess is mainly seen out of direct daylight. In order to ascertain the graininess, the non-uniformity of the light/dark fields is evaluated. These fields are registered by the CCD chip as gray scales. The more non-uniform the fields appear, the higher the value of graininess Gr ≥ 0. A value Gr = 0 represents a coated area without discernible graininess. The value domain of real colorations extends to about Gr = 30. The 4 CCD: charge

coupled device

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Table 3.13 Sparkle grade and graininess value of fine and coarse, dark gray visual and colorimetric equal effect colorations [73] Sparkle grade SG at illumination angle β v Grain quality

15◦

– 45◦

–75◦

Graininess Gr

Fine Coarse

1.3 6.9

1.2 2.9

0.8 1.6

6.0 9.5

graininess values of the above fine and coarse effect colors are also indicated in Table 3.13. Altogether, it is important to emphasize that only recently introduced experimental methods of sparkle grade and graininess enable the accurate description of additional optical properties of effect colorations. The results are consistent and agree well with visual impressions. On the other hand, the entirety of measuring methods discussed in this section is only to be regarded as complementary to the usual angle-dependent color measurements of effect pigments.

3.5.2 Color Difference Equation for Metallics As already pointed out, the color difference formulas given in Sections 3.1.3, 3.1.4, 3.2.1, 3.2.2, 3.2.3, and 3.2.4 are only applicable to absorption colorants. For formulations with metallic or other flake-shaped pigments, it is necessary to additionally consider the angle-dependent, anisotropic reflection of such coatings. A color difference formula only for use in automotive coatings which contain metallic pigments is subject to a standard [30]. This standard is principally designed for refinishing paintwork and low-bake paints. Comprehensive color difference relations for other effect pigments have not been developed up until now. The need of such relations really depends on further use of those colorants. On the other hand, the change of the unusual color physical properties of such effect pigments can generally not be simply reduced to a color difference expression. For determination of color differences occurring with totally covering metallic colorations, the customary illumination angle of β = 45◦ and aspecular measuring angles μas = 15◦ , 25◦ , 45◦ , 75◦ , 110◦ are normally used. For simplicity, the index of the measuring angle is left out below. The measuring results should only be compared with each other if the sample and reference colors are of similar surface gloss and other colorimetrical requirements are the same for both. The reflectance measurement of a metallic coating is performed at about five different points of the plane patterns of the sample and reference coloration. From each of the resulting CIELAB color values of the sample color

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Chroma of Effect Pigments

207

LS∗ (μ), a∗S (μ), b∗S (μ) and the reference color LR∗ (μ), a∗R (μ), b∗R (μ), the arithmetic mean for each of the five measuring angles is taken. The calculated color difference contributions ΔL∗ (μ), Δa∗ (μ), Δb∗ (μ), ΔC∗ (μ), ΔH∗ (μ) lead to five angle-dependent color differences, denoted by ΔEeff (μ). If the coloration consists of both metallic and absorption pigments, then the color flop can be represented by two color loci which are possibly positioned in different areas of a color plane. In such cases, the local color differences are calculated with different formulas. In the mentioned standard, two extended color difference expressions for uncolored and colored colorations are given: ∗ (μ) < 10 as well as pas1. For achromatic colors with chroma values 0 ≤ Cab ∗ < 18 and lightness tel colors (of high “silver” or white content) of 0 ≤ Cab L∗ (μ) > 27, relation

(μ) ΔEab

 =

ΔL∗ (μ) gL SL (μ)



2 +

Δa∗ (μ) ga Sa



2 +

Δb∗ (μ) gb Sb

2 (3.5.5)

is applied. ∗ (μ) ≥ 10, but not including pastel colors, the 2. For chromatic colors with Cab formula (μ) ΔECH

 =

ΔL∗ (μ) gL SL (μ)

2

 +

ΔC∗ (μ) gC SC (μ)

2

 +

ΔH ∗ (μ) gH SH (μ)

2 (3.5.6)

is valid. The visually tolerated color difference for chromatic metallic formulations is non-uniform as well. This is already known for absorption colors or colored lights. This conclusion is particularly clear for dark and pastel shades. The values of the empirical weighting factors gL , ga , . . . , gH in Equation (3.5.6) are given in Table 3.14 for different manufacturing steps. The correction functions SL , Sa , . . . , SH in Equations (3.5.5) and (3.5.6) follow also from the assessment of color collections:   SL (μ) = 0.15 L(μ) + (31.5◦ μ), (3.5.7) Sa = Sb = 0.7,    SC (μ) = max [0.7; 0.48 C(μ) − 0.35 L(μ) + (42◦ μ)],    SH (μ) = max [0.7; 0.14 C(μ) − 0.2 L(μ) + (21◦ μ) + 0.7],

(3.5.8) (3.5.9) (3.5.10)

where L(μ) is the geometric mean of the lightness values of the sample color LS∗ (μ) and reference color LR∗ (μ), each at the same measuring angle μ:

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Table 3.14 Empirical weighting factors for color differences in Equations (3.5.5) and (3.5.6) Empirical weighting factors Condition

gL

ga

gb

gC

gH

Delivery First paint coating, refinishing paint, low-bake paint

1.0 2.0

1.0 1.2

1.0 1.2

1.0 1.8

1.0 1.2

L(μ) =

 LS∗ (μ) · LR∗ (μ).

(3.5.11)

Also the function C(μ) is given by a geometric average, now of the chroma of the sample CS∗ (μ) and the reference color CR∗ (μ): C(μ) =

 CS∗ (μ) · CR∗ (μ).

(3.5.12)

Concerning the correction functions (3.5.9) and (3.5.10), it is important to point out that, at first, the calculation is based on the algebraic expression. If the calculated value is less than 0.7, then it is set to 0.7, otherwise the relevant value ≥ 0.7 is inserted into Equation (3.5.6). At the switchover between the above-mentioned color areas, discontinuous color differences follow from Equations (3.5.5) to (3.5.6). This discontinuity issue is rectified using the Fermi function σ (μ). It is based on Equation (6.3.21) and given as σ (μ) =

1 1 + e[C(μ)−C0 (μ)]

,

(3.5.13)

where C0 (γ ) = 10 +

8 27−L(γ )] [ 1+e

(3.5.14)

is another angle-dependent correction function. The value domain of σ (γ ) is limited to the interval [0,1], cf. Fig. 6.18. The single valuedness of σ (μ) depends ∗ (μ) on the chroma value Cab  σ (μ) =

∗ (μ) >> 10 0 for Cab ∗ (μ) << 10 . 1 for Cab

(3.5.15)

The finite color difference ΔEeff (μ) for each measuring angle μ is calculated using Equations (3.5.4) and (3.5.5) from the function

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Chroma of Effect Pigments

209

ΔEeff (μ) = σ (μ) · ΔEab (μ) + [1 − σ (μ)] · ΔECH (μ).

(3.5.16)

With this adjustment, the color difference ΔEeff (μ) changes continuously between quantities ΔEab (μ) and ΔECH (μ). This follows immediately from Equation (3.5.16), given the relation  ΔEeff (μ) =

(μ) for σ (μ) = 1 ΔEab . ΔECH (μ) for σ (μ) = 0

(3.5.17)

Delivered and also first paint effect coatings should have a color difference of ΔED (μ) ≤ 1.0 in comparison to a reference coating, cf. Table 3.6. For paint refinishing or low-bake paints, the color difference between adjoining body panels should not exceed the value of ΔEP (μ) = 1.5. In cases of body shell apertures separated by body flanges, slats, or gaps, the standard permits a color difference of ΔEP (μ) ≤ 3.0, but this tolerance is generally much too high and it should be kept to ≤ 2.0 and for dark metallic paints down to ≤ 1.0. The angle-dependent color difference of matched metallic colors is, however, still not a comprehensive criterion for the quality of the matching. The color difference value of ΔE(μ) is rather an evaluation measure for a discernable color difference. It leaves additional specific metallic properties unaddressed, which are often the most characteristic for the actual coloration. The working out of effect colorations succeeds, as experience shows, with coloristical feeling; see Section 6.3.5, among other things. The optical behavior of pearlescent, interference, and diffraction pigments results in a host of novel color appearances. These can still be increased and improved by mixtures with absorption colorants. The change of the chroma of natural effect pigments and of different mixtures in dependence on layer thickness, illumination, and observation angle are topics of the next sections.

3.5.3 Chroma of Pearlesence and Interference Pigments Whereas some typical properties of metallic pigments can be described by the colorimetrical lightness, this quality is of minor importance for the description of the color appearance of pearl luster, interference, or diffraction pigments. The dominant color properties of such effect colorants are rather described by the colorimetrical quantities of chroma, hue, or hue angle. As already discussed in Sections 2.1 and 2.3, interference and diffraction colors depend on the refractive index, especially of each single layer thickness, as well as the illumination and observation angle. The dependence of interference colors on layer thickness is simply demonstrated with pearlescent pigments of muscovite substrate evaporated with TiO2

210

3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

or Fe2 O3 . The similar color physical behavior of both pearl luster types follow not only from Figs. 2.46 and 2.48 but also from corresponding curves in an a∗ , b∗ color plane. The meaning of such, generally called chroma curves, is explained in the following. Figure 3.19 represents the chroma curves for different layer thickness of the above-mentioned pearl luster pigments determined by diffuse illumination and directional measurement. The layer thickness increases in the clockwise direction. In the case of TiO2 layers, this corresponds to the given quantities and colors in Table 2.15. The chroma curve of the TiO2 pigment sweeps almost over all four quadrants, whereas the chroma curve of flakes with hematite layers nearly remains in the first quadrant. This second curve represents colors which change from yellow to “golden” to rust colored. To manufacture a needed intermediate color, the accompanying thickness of the metal-oxide layer can be estimated from the corresponding chroma curve. Each of the chroma curves represents a special color build-up. This build-up describes the interference colors in dependence on the evaporated layer thickness. For further increases in the layer thickness, each curve extends as a spiral which ends, due to increasing absorption, in the vicinity of the achromatic point. For constant layer thickness, directional illumination, and directional measuring, however, two sorts of chroma curves result. Those are due to a change of illumination or measuring angle β and μ, respectively [74, 75]. The measuring conditions are characterized by the standardized angle direction as indicated b∗ 40 TiO2

Fe2O3

20

0

–20

–40

–40

–20

0

20

∗ 40 a

Fig. 3.19 Chroma curves of mica pigments coated with rutile TiO2 or α-Fe2 O3 ; the layer thickness increases in the clockwise direction; the color values a∗ , b∗ are determined from spectral values shown in Figs. 2.46 and 2.48; measuring geometry de:8, standard illuminant D65, 10◦ standard observer

3.5

Chroma of Effect Pigments

211

in Fig. 4.4: the illumination angle β v is counted from the vertical to the illuminated surface and the measuring angle μas from the direction of the specular reflected beam. Measuring angles are positive signed in direction of the source – the aspecular direction – negative otherwise. According to the nomenclature in Section 4.1.2, this corresponds to the measuring geometry indicated by β v /μas . Both of the above-mentioned types of chroma curves are determined by the following measuring conditions: 1. β v is a variable and μas is kept constant; the interference colors change in dependence on β v along a so-called interference line; 2. β v is kept constant and μas is varied; the accompanying interference colors vary in dependence on measuring μas along a so-called effect line or aspecular line. The following chroma curves result from reflectance measurements over a black background using standard illuminant D65 and 10◦ standard observer. The first example in Fig. 3.20, this one for a blue pearlescent pigment, shows one interference line for constant measuring angle μas = 15◦ and two aspecular lines at constant illumination angles β v = 45◦ and β v = 65◦ , respectively. The starting points of the effect lines are located on the interference line, the respective end points for β v = 45◦ /μas = 65◦ and β v = 65◦ /μas = 70◦ are in the vicinity of the achromatic point. The effect line corresponding to β v = 65◦ represents a change ∗ ≈ 100. The interference line represents a color change in chroma of about ΔCab from violet blue at 65◦ /15◦ to turquoise at 20◦ /15◦ . The configuration of the three chroma curves in Fig. 3.20 is reminiscent of the projection from the side of a surfboard (interference line) with a stretched sail (restricted by both effect lines); the head seems to “hang in” in the neighborhood of the achromatic point. The greatest possible surface patch of the color area enclosed by the interference and aspecular lines can be interpreted as a kind of chroma potential of the corresponding interference colorant. The chroma curves in Fig. 3.20 are limited only by special construction of the illumination and measuring unit. Similarly shaped interference and aspecular lines are shown in Fig. 3.21 for an interference pigment consisting of a SiO2 substrate coated with TiO2 . A typical cross section of such particles is shown in the SEM picture in Fig. 2.44b. For the aspecular line at constant measuring angle μas = 15◦ , the color of this pigment changes from purple to weak red to green. In the end, the interference line traverses three quadrants of a color plane. Each effect line begins for small illumination angle at the interference line and ends again in the vicinity of the achromatic point. Compared to that of Fig. 3.20, the entire graphical plot seems rotated around the achromatic axis and simultaneously reduced in size. The situation is the same for transparent liquid crystal pigments as depicted in Fig. 3.22: the interference line is simply more curved and the “surfboard” stands on sail top.

212

3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments b∗

45°/65°

65°/70°

Effect line 45° = const. –40

Effect line 65° = const.

45°/20° 20°/15° 45°/15°

65°/20°

–80

Interference line 15° = const. 65°/15° –120 –40

0

a∗

40

Fig. 3.20 Interference line with β v = 15◦ and two effect lines with μas = 45◦ , 65◦ of the blue pearl luster pigment in Fig. 2.46, curve (e) b∗ 40

20

65°/15°

0

45°/65° 45°/15° 65°/70°

–20

–40

20°/15° –40

–20

0

20

40 a∗

Fig. 3.21 Interference line and two effect lines of a combination pigment with titaniumdioxide/silicon-dioxide/mica layers; see Fig. 2.40

The mutual configuration of the chroma curves is maintained for opaque multi-layered pigments as can be seen in the next two plots. The interference line for the Al pigment coated with SiO2 and Fe2 O3 with β v = 15◦ in Fig. 3.23

3.5

Chroma of Effect Pigments

213

b∗ 100

80

45°/15°

60 40

20°/15° 20

65°/15° 65°/70° 45°/65°

0 –20 –80

–60

–40

–20

0

20

40

a∗

Fig. 3.22 Interference line and two effect lines of a liquid crystalline interference pigment b∗ 80

65°/15°

60

40

45°/15°

65°/70°

20

45°/65° 0

20°/15°

–20 –40 –40

–20

0

20

40

60

80

a∗

Fig. 3.23 Interference line and two effect lines of an iron-III-oxide/silicon dioxide multilayered interference pigment

actually covers only two quadrants, the interference color changes from deep red to light yellow. As seen in Fig. 3.24, the interference line produced from the Al pigment – now evaporated with MgF2 and Cr – takes a big swing and travels

214

3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

b∗ 80 65°/15° 40 20°/15°

45°/65°

65°/70°

0

–40

45°/15°

–80

–120 –120

–80

–40

0

40

80

a∗

Fig. 3.24 Interference line and two effect lines of an optical variable green/purple multilayered interference pigment; see Fig. 2.41 and Color plate 9

through all four quadrants! It represents a color travel from green over blue and red to orange (cf. the corresponding light microscopic pictures in Color plate 9). The layer thicknesses of such multi-layer particles, which vary merely some ∗ ≈ 115 combined nanometers, produce a remarkably high chroma of about Cab with an extreme color flop. In summary, it should be emphasized that, from topological point of view, equal chroma curves are produced from pearl luster and different sorts of interference pigments. The interference and effect lines are rotated in a characteristic manner around the achromatic axis and contracted or stretched. Clearly, the color travel of such interference pigments includes the more colors, the greater the number of quadrants which are traversed by the interference line. Altogether, it is to conclude that not only absorption pigments can be described by chroma curves but also the impressive color travel of chromatic effect pigments. The chroma results are in agreement with visual perception.

3.5.4 Mixtures of Effect Pigments From mixtures of effect colorants with other effect colorants or with absorption pigments, it is possible to obtain additional new color shades. For two colorants, this results in hues somewhere between those of the natural colors of the involved colorants. The types of the resulting color mixing of some pigment combinations are listed in Table 2.17. If pearlescent or interference pigments of complementary natural colors are mixed, then – at perpendicular observation –

3.5

Chroma of Effect Pigments

215

no white is produced. Instead, a gray results; this is caused by edge scattering of the particles. This achromatic color depends on the d50 value and the particle size distribution. The smaller the particles and broader the size distribution the darker the gray. To begin with, we discuss homologous multi-layered interference pigments of aluminum substrate and layer composition Cr/MgF2 /Al/ . . . /Cr, which differ only with regard to the magnesium fluoride layer thickness. Consequently, each pigment of this series shows a color travel with systematically altered extreme colors. Such a type of interference particles is shown in Color plate 9. The colored flop of this pigment travels between the extreme colors of green to purple. In bright-field illumination, the pigments aligned parallel to the image plane appear green or nearly turquoise, the purple ones are tilted. The same image detail shows only the purple flop color at higher observation angles in darkfield illumination. This color then dominates, because the particles are laterally illuminated but observed perpendicularly. For dark-field illumination, the morphology and the edges of the particles are much easier to distinguish than in bright-field observation. The colorimetric measurements for seven natural interference pigments of the above-mentioned layer composition were performed over a black background with diffuse geometry de:8. The evaluation was performed as described in the previous section. The color locus of each of the seven pigments is represented in Fig. 3.25 with an open circle. The indicated color pairs correspond to the extreme flop colors of the respective pigment. Starting at the color locus of the silver/green in direction of the green/purple pigment, the thickness of the MgF2 layer evaporated on the Al substrate increases in the clockwise direction through all other points. This curve again traces out a clockwise spiral and seems to be the contour of a nautilus swimming to the right. On the basis of only slight thickness fluctuations of the relevant evaporated layer thickness, the chroma values of the present interference pigments in Fig. 3.25 are always higher than those of pearlescent pigments with mica substrate, cf. Fig. 3.19. The color locus of mixtures produced by two neighboring natural pigments always traces out a straight line between both initial points as seen in Fig. 3.26 [77]. This indicates additive color mixture. The spiral shown in Fig. 3.25 is, for better overview, not included in Fig. 3.26. It is not surprising that mixtures of homologue pearl luster pigments show an analogous behavior under the same measuring conditions, as we will see below. On the other hand, mixtures of the cyan/purple interference and a red absorption pigment produce under directional illumination and measuring conditions the results represented in Fig. 3.27. The original interference line, already known from Fig. 3.24, decreases with increasing red pigment content and converges to the color locus of the absorption pigment (subtractive color mixture). Similarly mixtures of pearlescent and absorption pigments behave as discussed below.

216

3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments b∗ 80

Gold/Silver

40 Silver/Green

Green/Purple 0

Red/ Gold –40

Cyan/ Purple Purple/ Orange

Blue/ Red –80

–40

–20

0

40 a∗

20

Fig. 3.25 Color locus of seven interference pigments composed of Cr/MgF2 /Al/MgF2 /Cr with different MgF2 -layer thicknesses; the thickness increases in the clockwise direction b∗ 80 Gold/Silver 40 Silver/Green Red/ Gold

0 Green/ Purple Cyan/ Purple –40 Blue/ Red

Magenta/ Gold Purple/ Orange

–80

–40

–20

0

20

40 a∗

Fig. 3.26 Color locus of mixtures with eight interference pigments composed of Cr/MgF2 /Al/MgF2 /Cr layer of different MgF2 -layer thicknesses

Although the interference and effect lines are suitable for the visualization of the angle-dependent color development of effect pigments, the measurements are actually quite extravagant to perform. For a relevant and colorimetric

3.5

Chroma of Effect Pigments

217

b∗

80

100/0

Interference/ Absorption Pigment

90/10 40 70/30 0

–40

–80

–120 –120

–80

–40

0

40

80

a∗

Fig. 3.27 Interference lines of the optical variable multi-layered interference pigment of Fig. 3.24 with three different amounts of a red quinacridone pigment

identification of such colorants in mixtures, it is not advisable to determine the total slope of the relevant interference line and effect lines. It is more reasonable to simply apply an abbreviated measuring procedure. From extensive investigations, it follows that the eight measuring geometries listed in Table 3.15 are sufficient for this. The angle combinations are already subject to standards [78–81]. As an example of the reduced measuring method, we discuss now mixtures of the blue pearlescent pigment characterized in Fig. 3.20, a blue absorption

Table 3.15 Standardized measuring geometries for characterization of interference pigments or mixtures with other sorts of colorants [78–81] Illumination angle β v /degree

Aspecular measuring angle μas /degree

Interference line

15 15 45

15 –15 –15

Effect line

45

15 25 45 75 110

Chroma curve

218

3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

pigment, and a silver dollar aluminum pigment. The content of the blue absorption pigment is kept constant at a fraction of 60% in three of the four following mixtures. The shortened interference and effect line forms a so-called anchor curve [82].

3.5.4.1 Pearl Luster with Absorption Pigment In Fig. 3.28, five anchor curves are plotted. The curve parameters indicate the mixing ratio of pearl luster/absorption pigment for constant pigment loading. For simplicity, the measurement for μas = –15◦ has been left out. One endpoint of each effect line tends to the color point of the blue absorption pigment. At these coordinates, the effect and the interference lines vanish. On the other hand, both line lengths increase with increasing content of the pearl pigment. The interference line section of the natural pearl luster pigment (100/0) falls out from the family of the parallel interference lines. This is caused by the measurement of the transparent natural pearlescent pigment over a black background (cf. Fig. 3.20). The endpoint of the accompanying effect line at an angle μas = 110◦ , therefore, does not agree with the color point of the natural colored pigment. Already for a slight fraction of absorption pigment, the coloration becomes opaque. This is the reason why the anchor curves then change systematically. From such mixtures of pearl and absorption pigments, these are the following colorimetric properties:

b∗

–20

0/100

–40 100/0 20/80

–60

40/60 –80

60/40 80/20

–100 –80

–60

–40

–20

0

a∗

Fig. 3.28 Anchor curves of mixtures with a blue pearl luster and blue absorption pigment; the ratio of pearlescent to absorption pigment content is indicated

3.5

Chroma of Effect Pigments

219

a. the sections of the interference line behave nearly parallel to one another in dependence on concentration of the chromatic pigment; b. for low concentration of the absorption colorant, the effect line of β = 45◦ tends to reach the color locus of the absorption colorant; c. the lengths of the interference and effect lines decrease simultaneously with increasing colored pigment content; both lines vanish in the color point of this pigment; d. the endpoints of the effect lines for a measuring angle of μas = 110◦ are located on a spiral segment between the color points of both natural pigments. 3.5.4.2 Aluminum Flakes with Absorption Pigment Colorant mixtures of a silver dollar aluminum pigment with the same blue absorption pigment as above result in an altered colorimetrical behavior. The results in Fig. 3.29 essentially come down to the presence of the stacked metallic flakes. The anchor curves can be characterized as follows: a. the sections of the reduced interference lines have almost the same slope as the adjacent effect line; b. the distance between effect lines increases with increasing content of aluminum pigment; c. in addition, for both lines the conclusions in Section 3.5.4.1 (b) – (d) hold. b∗

100/0

–20

0/100 80/20

–40

60/40 –60

20/80

40/60

–80

–100 –80

–60

–40

–20

0

a∗

Fig. 3.29 Anchor curves of mixtures with a silver dollar blue pearl luster and blue absorption pigment; the ratio of metallic to chromatic pigment content is indicated

220

3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

Such distinguishing colorimetrical features enable one to discern mixtures of aluminum with colored pigments from mixtures with other effect pigments. 3.5.4.3 Interference with Metallic Pigment Some of the previously mentioned colorimetric features become more clearly evident from mixtures of the blue pearlescent and the silver dollar aluminum pigment. From the family of anchor curves in Fig. 3.30, which are now characterized by the fractions of the pearl and metallic pigment, it is possible to list the following characteristics: a. the mean slope of the interference lines changes in dependence on the metallic pigment concentration; b. independent of the concentration of the metallic pigment, the initial slope of the effect lines remains constant; c. for both lines, the properties mentioned in Section 3.5.4.1 (c), (d) remain valid, but where the term colored pigment is substituted by metallic pigment. b∗

0/100

20/80 40/60

–20

60/40 80/20 –40

100/0 –60

–80

–100 –80

–60

–40

–20

0

a∗

Fig. 3.30 Anchor curves of mixtures with a blue interference and an aluminum pigment; the ratio of interference to aluminum pigment content is indicated

3.5.4.4 Interference, Metallic, and Absorption Pigments Each of the measured mixtures now contains 60 proportions of the blue absorption pigment. As seen from the form of the anchor curves in Fig. 3.31 and from comparison with the results of the previous three mixtures, the colorimetrical

3.5

Chroma of Effect Pigments

221

b∗

–20

–40 0/40 20/20

–60

30/10 38/2

40/0

–80

–100 –80

–60

–40

–20

0

a∗

Fig. 3.31 Anchor curves of mixtures with three colorants: blue pearl luster, aluminum cornflake, and blue absorption pigment; the ratio of pearlescent to effect pigment content is given; each mixture contains 60 parts of the same blue absorption pigment

behavior of the mixtures is predominantly controlled by the aluminum and blue absorption pigments. On the one hand, again the endpoints of the effect lines for μas = 110◦ tend toward the color point of the colored pigment; on the other hand, the interference lines turn with increasing aluminum content into that direction, which corresponds to the initial slope of the adjacent effect line for μas = 15◦ , cf. Fig. 3.29. In addition, one can observe in Fig. 3.31 that the anchor hooks come close to this initial slope with increasing concentration of the pearlescent pigment. From investigations with other effect and absorption pigments, similar shaped chroma curves follow [82]. To summarize, the abbreviated method of anchor curves produces new insight into the colorimetrical behavior of various pigment mixtures. This procedure is not, however, suitable for quantitative or qualitative analysis of real mixtures with different pigment sorts as demanded. The light microscopic procedure as part of expert system remains more practical for this purpose (Section 6.3.5).

3.5.5 Color Development of Diffraction Pigments Diffraction pigments based on grating structures and groove geometries are the most complex to produce effect pigments (Section 2.3.8). The grating geometry establishes the number of observable diffraction orders and the degree of

222

3

Standardized Tristimulus Values, Color Qualities, Chroma of Effect Pigments

optical efficiency or brilliance. Now, we consider the angle-dependent chroma of diffraction pigments with a linear sine-shaped groove cross section and grating constants of 1,000 and 1,400 l/mm (l/mm meaning lines/mm) [82, 83]. As we will see, the chroma curves of diffraction pigments differ considerably from those of interference pigments regarding the effect line. The angle-dependent chroma curves in the following are determined with a special gonio spectrophotometer. This device allows for an illumination angle of β = 45◦ and adjustment of the measuring aspecular angle in steps as small as Δμ = 1◦ in the interval –35◦ ≤ μas ≤ 78◦ . Each of the samples is in optical contact with the same black background. The resulting CIELAB color values a∗ and b∗ belong to standard illuminant D65 and 10◦ standard observer. The change of lightness in this case – similar to interference pigments – is not being considered here, but rather the color development. First, we follow the chroma change of simple structured particles with symmetrical permutation of layers of MgF2 /Al/MgF2 and grating constant of 1,000 l/mm. The corresponding angle-dependent effect line is shown in Fig. 3.32. It runs clockwise from μas = –35◦ in the vicinity of the achromatic point to μas = +78◦ the diffraction orders z = 0, 1, 2. Between angles μas = –35◦ and μas = +15◦ , the chroma has a kind of hairpin bend; the sharply shaped chroma maximum corresponds to a specular angle μas = 0◦ . The angle range 15◦ ≤ μas ≤ 50◦ belongs to the diffraction order z = 1. The accompanying section of the chroma curve traces out a kind of ellipse centered near the achromatic point. A following “ellipse” corresponds to the second diffraction order z = 2; it ends for a measuring angle μas = 78◦ in the neighborhood of the achromatic point. Higher orders, z > 2, are of quite low luminosity and, due to the technically limited angle adjustment, are not measured. The curve shape in Fig. 3.32 looks altogether like an “unfinished treble clef.” Each diffraction order contains all wavelengths emitted by the applied source, which for a reflecting grating are given by Equation (2.1.24) in dependence of diffraction angle. Correspondingly, in the case of the xenon discharge lamp used, the chroma curve section of each diffraction order z > 0 extends over all four quadrants of the color plane. Because the intensity of the higher diffraction orders decreases quickly (Fig. 2.21), the accompanying chroma becomes also smaller. The spectra of higher orders tend, therefore, to achromatic colors. With increased diffraction order, the chroma curve always traces out smaller “ellipses” until these vanish in the achromatic point. If the illumination angle is increased with respect to the normal of the sample, then the chroma maximum at the specular angle shifts nearer to the start of the curve for μas = –35◦ . The corresponding height of the chroma maximum reduces due to increasing scattering at the grooves and the edges of the particles. However, the chroma curve in Fig. 3.32 includes the majority of the second diffraction order. This indicates that the grating geometry is not optimized because the intensities of the first two orders are each lowered by the higher orders.

3.5

Chroma of Effect Pigments

223

∗ b

0°

40 10° –10° 20

40°

– 20°

78°

60°

0 30°

– 35°

20°

50° –20 –20

–10

0

10

a

∗

Fig. 3.32 Chroma curve respective effect line representing diffraction orders z = 0, 1, 2 of a two-layered pigment; illumination angle β = 45◦ ; parameter is the measuring angle with regard to the specular angle; grating constant 1,000 l/mm

However, in the range until the limiting measuring angle μas = 78◦ , the optimized diffraction particles with linear grating constant of 1,400 l/mm and sine-shaped groove cross section produce a chroma curve, which does not contain the order z = 2, cf. Fig. 3.33. This conclusion is also true for extrapolation to higher measuring angles on account of the curve sections always being smaller for higher angle decades to a maximum of μas = 135◦ . In comparison to the pigment with 1,000 l/mm, the curve section for z = 1 in Fig. 3.33 shows a chroma about double: the intensities of the order z = 0 and z ≥ 2 are mostly transferred to the diffraction spectrum for z = 1. This is achieved by the realized blaze technique for the present diffraction particles (Section 2.1.7). Particles consisting of right-angled crossed groove structure, each of 1,400 l/mm with sine-shaped groove cross section, produce the chroma curve shown in Fig. 3.34. The undeflected (zeroth) order and first diffracted order have lower chroma values than the linear grating of same grating constant compared to Fig. 3.33. This behavior is caused by mutually perpendicular crossed √ grooves, with which an additional grating constant of g˜ = (1,400/ 2) l/mm = 990 l/mm is introduced. Whereas the chroma of diffraction order z = 0 is nearly

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b∗ Linear

40

0°

50°

–10°

10°

60°

20 –20°

0

–35° 20°

–20

70°

40° 78°

30°

–40

–20

–10

0

10

20

a∗

Fig. 3.33 Chroma curve of an optimized linear sine-shaped diffraction pigment; grating constant 1,400 l/mm

the same as in Fig. 3.33, the new grating constant reduces the chroma of the higher orders in magnitude comparable to Fig. 3.32 for grating constant of 1,000 l/mm. Moreover, the combined perpendicular and diagonal grating structures produce an azimuthal anisotropy of chroma, which shows a periodicity of 45◦ for rotation of the sample vertical to its surface [83]. An increased linear grating constant of 3,000 l/mm leads, however, to a reduced chroma; in the chroma curves (not shown), the different diffraction orders are barely distinguishable. The reason for this is the more closely crowded grooves: at the edges and corners the light is increasingly scattered so that the remaining light intensity for diffraction is weakened. In addition, small-sized grating particles reduce the chroma caused by increased light scattering at the particle edges. Though it is inappropriate to arbitrarily extend the dimensions for the given pigments of d50 = 20 ± 2 μm, because the smaller groove distances lead to a stronger notch effect of the brittle particles. Such pigments are, depending on application method or processing, subject to increased mechanical degradation. Mixtures of diffraction pigments with other effect or absorption colorants have not been described in literature until now. On the basis of the correspond-

3.5

Chroma of Effect Pigments

225

b∗ Crossed 40

0° 10°

–10°

20 –20°

50° 20° –35° 0 70°

78° 40°

–20

30°

–40

–20

–10

0

10

20

a∗

Fig. 3.34 Chroma curve of a linear diffraction pigment with perpendicular crossed grooves; grating constant in both directions 1,400 l/mm

ing insight with interference pigments it is, however, sufficient to regard only the first-order diffraction and to estimate the chroma changes in dependence on the further added colorant fractions. With increased content of the additional pigment, the “diffraction ellipse” decreases and is pulled back into the color locus of the contained absorption or metallic pigment. As already discussed in Section 2.3.5, the optical anisotropy of diffraction pigments is additionally influenced by a substrate consisting of a ferromagnetic material such as nickel, for example. In a magnetic field, the grooves of elongated grating particles orient parallel to the lines of the field. If the particles are simultaneously subject to a shear gradient parallel to the direction of the magnetic field, then this alignment is preferentially fixed during crosslinking time of the binder. The particles are orientated similar to a smectic texture (Fig. 2.42). Ferromagnetic diffraction pigments of symmetrical permutation of layers MgF2 /Al/Ni/Al/MgF2 with a 400 nm thick coating of magnesium fluoride and linear grating constant of 1,400 l/mm appear brilliant metallic under diffuse illumination. With directional illumination however, these particles develop an

226

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impressive chroma. This chroma depends on the orientation of the grooves perpendicular or parallel to the optical measuring plane. The optical measuring plane is built from the illuminating and measuring beam. In the case of grooves preferentially oriented normal to the measuring plane, ∗ ≈ the first diffraction order shows an increased maximal chroma of about Cab 85 at an orange color of high brilliance; see outer curve in Fig. 3.35. In this alignment, the particles enhance the diffraction effect. This conclusion is clear by comparing Fig. 3.33: for the ferromagnetic pigment, the chroma values of the first diffraction order are in the domain 30◦ ≤ μas ≤ 78◦ . This is more than two times greater than that of the previously optimized non-ferromagnetic pigment. The chroma scale between both pigment orientations can be influenced by magnet field strength and exposure time. However, if the grooves are aligned parallel to the measuring plane, then the previous brilliant chroma development is almost lost. This is represented by the inner curve in Fig. 3.35. This chroma curve is shrunk. Only the specular color can be identified and the chroma of the first diffraction order degenerates to achromatic. Because the influx is oriented parallel to the grooves and not in the normal direction, nearly no real diffraction develops. In this case, the b∗ Measuring plane normal parallel

80

60°

0° 40

10° 50°

–10° –20°

20°

–35°

0

70°

30° –40 78°

–80 40°

–120 –60

–40

–20

0

20

40

a∗

Fig. 3.35 Chroma curves of a three-layered ferromagnetic diffraction pigment; grating grooves perpendicular and parallel to the measuring plane; grating constant 1,400 l/mm

3.5

Chroma of Effect Pigments

227

incident light is attenuated by the covering MgF2 and essentially reflected off the aluminum layer. For diffuse illumination and right angle observation, ferromagnetic diffraction pigments with an additional cover layer of chromium Cr/MgF2 /Al/Ni/. . ./. . ./Cr produce a yellow-golden color impression. For directional illumination and measuring, however, the accompanying chroma curve changes dramatically, if the grooves are normal to the measuring plane, as can be seen in Fig. 3.36. Both particle alignments produce no “ellipses” around the achromatic point; furthermore, the chroma curves travel through at least two adjacent quadrants. This means that the zero and first diffraction orders are missing corresponding colors such as yellow-green, green, or blue. In the case where the grooves are normal to the measuring plane, the color impression oscillates with increased measuring angle only between yellow to orange-red to red-violet with increasing chroma. This particle alignment acts as an interference filter for blue to yellow-green wavelengths caused by the reflecting chromium and aluminum layers spaced by dielectric MgF2 , cf. Fig. 2.18. With the alignment parallel to the measuring plane, the first diffraction order is degenerated again near the achromatic point. This particle orientation behaves b∗

Measuring plane normal parallel

60

0° 60°

0° –10°

40

10°

–20°

–30° 20

–35°

50° 20°

–35°

70° 30°

78°

0

30°

78°

–20 40° –20

0

20

40

60

a∗

Fig. 3.36 Chroma curves of a four-layered ferromagnetic diffraction pigment; grating grooves perpendicular and parallel to the measuring plane; grating constant 1,400 l/mm

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like a dielectric mirror with a specular color of orange red. The brilliance and the diffraction colors as well as the filtered or reflected wavelengths are influenced by the thickness of the nonmetallic layer and the grating structure. To complete this picture, diffraction and interference are superimposed for these pigments and this gives rise to the extreme chroma curves restricted to the first quadrant of the color plane in Fig. 3.36. With the above description of ferromagnetic diffraction pigments, we finish the colorimetrical characterization of some special color physical properties of both absorption and effect colorants. The fact that the color development of effect colorants, which changes with illumination and observation angle, is mainly reproduced by different chroma curves is actually quite astonishing. However, the necessarily large extent of measuring brings a degree of impracticality. On the other hand, the methods for detailed coloristical description of colorations with flake-shaped effect pigments are by no means completed.

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19. CIE No 101: “Parametric Effects in Colour-Difference Evaluation”, CIE, Bureau Central de la CIE, Wien (1993) 20. Luo, MR, Cui, G, Rigg, B: “The development of the CIE 2000 colour difference formula: CIEDE2000”, Col Res Appl 26 (2001) 340 21. Chou, W, Lin, H, Luo, MR, Westland, S, Rigg, B, Nobbs, J: “Performance of lightness difference formulae”, Coloration Techn 117 (2001) 19 22. CIE No 142: “Improvement to Industrial Colour-Difference Evaluation”, Commission Internationale de L’Éclairage (CIE), Bureau Central de la CIE, Wien (2001) 23. Witt, K: “CIE color difference metrics”, in: Schanda, J, ed.: “Colorimetry: Understanding the CIE System”, Wiley, Hoboken, NJ (2007) 24. ASTM D 2244: “Standard Practice for Calculation of Color Tolerances and Color Differences from Instrumentally Measured Color Coordinates”, American Society for Testing and Materials, West Conshohocken, PA (2005) 25. Fairchild, MD: “Colour Appearance Models”, 2nd ed., repr, Wiley, Chichester (2006) 26. Luo, MR, Li, CJ: “CIE Colour Appearance Models and associated Colour Spaces”, in: Schanda, J, ed.: “Colorimetry: Understandig the CIE System”, Wiley, Hoboken, NJ (2007) 27. CIE No 159: “A colour appearance model for colour management systems: CIECAM02”, Commission Internationale de L’Éclairage (CIE), Bureau Central de la CIE, Wien (2004) 28. Luo, MR, Cui, GH, Li, CJ, Rigg, B: “Uniform color spaces based on CIECAM02 colour appearance model”, Col Res Appl 31 (2006) 320 29. DIN 6175-1: “Farbtoleranzen fuer Automobillackierungen”, part 1: “Toleranzen fuer Unilackierungen”, Deutsches Institut fuer Normung eV, Berlin (1986) 30. DIN 6175-2: “Farbtoleranzen fuer Automobillackierungen”, part 2: “Toleranzen fuer Effektlackierungen”, Deutsches Institut fuer Normung eV, Berlin (2001) 31. McDonald, R: “A Review of the Relationship between Visual and Instrumental Assessments of Colour Difference, Part I, II”, J Oil Col Chem Assoc 65 (1982) 43–53, 93–106 32. DIN Fachbericht 49: “Verfahren zur Vereinbarung von Farbtoleranzen”, Deutsches Institut fuer Normung eV, Berlin (1995) 33. ASTM D 3134–97(2008)e1: “Standard Practice for Establishing Color and Gloss Tolerances”, American Society for Testing and Materials, West Conshohocken, PA (2008) 34. ASTM E 1499–97(2003): “Standard Guide for Selection, Evaluation, and Training of Observers”, American Society for Testing and Materials, West Conshohocken, PA (2003) 35. von Kries, J: “Chromatische Adaption”, Festschrift der Albrecht-Ludwigs-Universitaet, Freiburg (1902) 145–158; reprint in: “Sources of Color Science”, MIT Press, Cambridge MA (1970) 109 36. CIE No 159: “A color appearance model for color management systems: CIECAM02”, CIE, Commission Internationale de L’Éclairage, Bureau Central de la CIE, Wien (2004) 37. Luo, MR, Li, CJ, “CIE Color Appearance Models and associated Color Spaces”, in: Shanda, J, ed.: “Colorimetry”, Wiley-Interscience, New York (2007) 38. Ostwald, W: “Farbenlehre”: Vol 1: “Mathematische Farbenlehre”, 2nd ed., Unesma, Leipzig (1930); Vol 2: “Physikalische Farbenlehre”, Unesma, Leipzig (1923) 39. Kang, HR: “Computational color technology”, SPIE Press, Bellingham (2006) 40. DIN 6172: “Metamerie-Index von Probenpaaren bei Lichtartwechsel”, Deutsches Institut fuer Normung eV, Berlin (1993) 41. CIE No 80: “Special Metamerism Index: Change in Observer”, Commission Internationale de L’Éclairage (CIE), Bureau Central de la CIE, Wien (1989) 42. DIN 55978: “Bestimmung der relativen Farbstaerke in Loesungen, Spektralphotometrisches Verfahren”, Deutsches Institut fuer Normung eV, Berlin (1981) 43. DIN 55603: “Pruefung von Pigmenten – Bestimmung der relativen Farbstaerke und des Restfarbenabstandes von anorganischen Pigmenten in Weissaufhellungen nach dem Helligkeitsverfahren”, Deutsches Institut fuer Normung eV, Berlin (2003)

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44. DIN 53235-1: “Pruefung von Pigmenten – Pruefungen an standardfarbtiefen Proben – Part 1: Standardfarbtiefen”, “Pruefung von Pigmenten – Pruefungen an standardfarbtiefen Proben – Part 2: Einstellen von Proben auf Standardfarbtiefe”, (2005); Deutsches Institut fuer Normung eV, Berlin 45. DIN 53235-2: “Pruefung von Pigmenten – Pruefungen an standardfarbtiefen Proben – Part 2: Einstellen von Proben auf Standardfarbtiefe”, (2005); Deutsches Institut fuer Normung eV, Berlin 46. DIN 55987: “Bestimmung des Deckvermoegenswertes pigmentierter Medien”, Deutsches Institut fuer Normung eV, Berlin (1981) 47. ASTM D 2805 – 96a(2003): “Standard Test Method for Hiding Power of Paints by Reflectometry”, American Society for Testing and Materials, West Conshohocken, PA (2003) 48. ASTM D 6441–05: “Standard Test Methods for Measuring Hiding Power of Powder Coatings”, American Society for Testing and Materials, West Conshohocken, PA (2005) 49. ISO 6504-3:2006: “Paints and varnishes – Determination of hiding-Power – Part 3: Determination of contrast ratio of light-coloured paints at a fixed spreading rate”, International Organization of Standardization, Genf (2006) 50. DIN 55984: “Bestimmung des Deckvermoegenswertes von weissen und hellgrauen Medien”, Deutsches Institut fuer Normung eV, Berlin (1986) 51. DIN 55988: “Bestimmung der Transparenzzahl (Lasur) von pigmentierten und unpigmentierten Systemen”, Deutsches Institut fuer Normung eV, Berlin (1989) 52. ASTM D 1746–03: “Standard Test Method of Transparency of Plastic Sheeting”, American Society for Testing and Materials, West Conshohocken, PA (2003) 53. DIN EN ISO 105 – A01: “Textilien – Farbechtheitspruefungen – Part A01: Allgemeine Pruefgrundlagen (ISO 105-A01:2008)”, Deutsches Institut fuer Normung eV, Berlin (2008) 54. DIN EN ISO 105, A06: “Textilien – Farbechtheitspruefungen – Part A06: Farbmetrische Bestimmung der 1/1 Richttype (ISO 105-A06:1995)”, Deutsches Institut fuer Normung eV, Berlin (1997) 55. Graystone, J: “Journeys into Colorspace”, Surf Coat Int, Part B Coat Trans, 87 (B3) (2004) 221 56. ISO 4582:2007: “Plastics – Determination of changes in colour and variation in properties after exposure to daylight under glass, natural weathering and laboratory sources”, International Organization of Standardization, Genf (2007) 57. ISO 11507:2007: “Paints and varnishes – Exposure of coatings to artificial weathering – Exposure to fluorescent UV lamps and water”, International Organization of Standardization, Genf (2007) 58. ISO 4892-1/4: “Plastics – Methods of exposure to laboratory sources; Part 1: General Guidance; Part 2: Xenon-arclamps; Part 3: Fluorescent and UV-lamps; Part 4: Open-flame carbon-arc lamps”, International Organization of Standardization, Genf (1999–2006); identical with: DIN EN ISO 4892-1/4: “Kunststoffe – Kuenstliches Bestrahlen oder Bewittern in Geraeten”, Deutsches Institut fuer Normung eV, Berlin (2000–2001) 59. ISO 105-J05: “Textiles – Test for colour fastness – Part J05: Instrumental assessment of change in colour for determination of grey scale rating”, International Organization of Standardization, Genf (1997); identical with: DIN EN ISO 105-J05: “Textilien – Farbechtheitspruefungen – Part J05: Instrumentelle Bewertung der Aenderung der Farbe zur Bestimmung der Graumassstabszahl”, Deutsches Institut fuer Normung eV, Berlin (1997) 60. ISO 105-J02: “Textiles – Test for colour fastness – Part J02: Instrumental assessment of whiteness”, International Organization of Standardization, Genf (1998) 61. ASTM E 313–05: “Standard Practice for Calculating Yellowness and Whiteness-Indices from instrumentally measured Color Coordinates”, American Society for Testing and Materials, West Conshohocken, PA (2005)

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62. DIN 6167: “Beschreibung der Vergilbung von nahezu weissen oder nahezu farblosen Materialien”, Deutsches Institut fuer Normung eV, Berlin (1980) 63. ASTM D 1003 – 07e1: “Standard Test Method for Haze and Luminous Transmittance of Transparent Plastics”, American Society for Testing and Materials, West Conshohocken, PA (2007) 64. DIN 55923: “Pigmente: Aluminiumpigmente und Aluminiumpigmentpasten fuer Anstrichfarben”, Deutsches Institut fuer Normung eV, Berlin (1983) 65. DIN 50017: “Kondenswasser-Pruefklimate”, Deutsches Institut fuer Normung eV, Berlin (1982) 66. DIN EN ISO 6270-1/4: “Beschichtungsstoffe – Bestimmung der Bestaendigkeit gegen Feuchtigkeit (ISO 6270)”, Deutsches Institut fuer Normung eV; Part 1–4 (2002 – 2007) 67. DIN EN ISO 9227: “Korrosionspruefung in kuenstlichen Atmosphaeren – Salzspruehnebelpruefungen (ISO 9227:2006)”, Deutsches Institut fuer Normung eV, Berlin (2006) 68. DIN EN ISO 2810: “Beschichtungsstoffe – Freibewitterung von Beschichtungen – Bewitterung und Bewertung (ISO 2810:2004)”, Deutsches Institut fuer Normung eV, Berlin (2004) 69. Wissling, P et al.: “Metallic Pigments”, Vincentz, Hannover (2007) 70. Kelly, RJ: “Process for matching color of paint to a colored surface”, US Patent No 4692481, DuPont Nemours and Company, Wilmington, Del (1987) 71. ASTM D 5767 – 95(2004): “Standard Test Methods for Instrumental Measurement of Distinctness-of-Image Gloss of Coating Surfaces”, American Society for Testing and Materials, West Conshohocken, PA (2004) 72. ASTM E 430 – 05: “Standard Test Methods for measurement of Gloss of High Gloss Surfaces by Abridged Goniophotometry”, American Society for Testing and Materials, West Conshohocken, PA (2005) 73. Kigl-Boeckler, G: “Total Color Measurement of Effect Finishes”, ECS Nuernberg, Nuernberg (2007) 74. Gabel, PW, Hofmeister, F, Pieper, H: “Interference Pigments as Focal Point of Color Measurement”, Kontakte (Darmstadt) 2 (1992) 25 75. Cramer, WR, Gabel, PW: “Effektvolles Messen”, Farbe Lack 107(1) (2001) 42 76. Nadal, ME, Early, EA: “Color measurements for pearlescent coatings”, Col Res Appl 29 (2004) 38 77. Droll, FJ: “Stunning Views”, PPCJ 191(4441) (2001) 78. ASTM E 2194–03: “Standard Practice for Multiangle Color Measurement of Metal Flake Pigmented Materials”, American Society for Testing and Materials, West Conshohocken, PA (2003) 79. ASTM WK 1164: “New Standard Practice for Multiangle Color Measurement of Interference Pigmented Materials”, American Society for Testing and Materials, West Conshohocken, PA (2006) 80. ASTM E 2539–08: “Standard Practice for Multiangle Color Measurement of Interference Pigments”, American Society for Testing and Materials, West Conshohocken, PA (2008) 81. DIN 6175-3: “Farbtoleranzen fuer Automobillackierungen”, Part 3: “Messgeometrien fuer Interferenzpigmente”, draft, Deutsches Institut fuer Normung eV, Berlin (2006) 82. Cramer, WR, Gabel, PW: “Dreiecksbeziehungen aus Bunt-, Aluminium- und Interferenz-Pigmenten”, Farbe u Lack 109(10) (2003) 78 83. Argoitia, A, Witzman, M: “Pigments exhibiting diffractive effects”, Soc Vacuum Coaters, 45th Ann Techn Conf Proc (2002) 84. Argoitia, A, Chu, S: “The concept of printable holograms through the alignment of diffraction pigments”, Flex Prods Inc, Santa Rosa, CA (2002)

Yellow

Hue

Green

Red

Blue Color plate 4. A color plane of CIELAB space (source: Konica-Minolta, Langenhagen/ Hannover, Germany)

Color plate 7. Light microscopic images of pearlescent pigments with mica substrate in bright-field illumination: top: coated with titanium-dioxide (rutile), the particles appear yellow from the top view; bottom: coated with iron-III-oxide (α-hematite), the particles appear deep red from the top view (source: Clariant Produkte GmbH, Frankfurt am Main, Germany)

Color plate 8. Light microscopic images of a red pearl luster pigment with mica substrate coated with titanium-dioxide (rutile) of greater thickness than those in color plate 7; top: in bright-field, bottom: in dark-field illumination of the same frame (source: Clariant Produkte GmbH, Frankfurt am Main, Germany)

Color plate 9. Light microscopic images of an optically variable green/purple interference pigment of aluminum substrate coated with chromium and magnesium-fluoride; top: brightfield, bottom: dark-field of the same frame (source: Clariant Produkte GmbH, Frankfurt am Main, Germany)

Color plate 10. Light microscopic images of a blue liquid crystalline interference pigment of polysiloxane, both over black background; top: bright-field, bottom: dark-field of the same frame (source: Clariant Produkte GmbH, Frankfurt am Main, Germany)

Color plate 11. Light microscopic bright- and dark-field image of an effect coloration of yellow-gold color impression from the top view; the images show mixed metallic, pearlescent and absorption colorants; top: bright-field, bottom: dark-field of the same frame (source: Clariant Produkte GmbH, Frankfurt am Main, Germany)

Color plate 12. Light microscopic bright- and dark-field image of a brown effect color coating from the top view; the images show mixed pearl luster and absorption pigments; top: bright-field, bottom: dark-field of the same frame (source: Clariant Produkte GmbH, Frankfurt am Main, Germany)

Chapter 4

Measuring Colors

At the beginning of the previous chapter, empirical color value systems suitable for numerical quantification of color impressions were described. A fundamental requirement for such systems is an accurate and precise measurement of the spectral reflectance or transmittance of the relevant color patterns. This necessity comes down to the fact that the retinal color stimulus is caused by the spectral reflection or transmission of the colored object, among other things. Accordingly, the measuring result should agree with the visual impression as well as possible. Color measuring instruments have been established and used since about 1960. Today, they are indispensable pieces of equipment for industrial color physics. Color measuring and especially the clear interpretation of the results require extensive knowledge and experience with regard to the underlying visual, color physical, and metrological interrelations. The measurement quantities are of importance for answering questions in research and development, quality control, and process control or application techniques. Moreover, the measurement results offer an objective base for realistic color tolerance agreements. For such agreements, the people involved should be familiarized with the measuring conditions, performance features of the instruments used, the meaningfulness, and trustworthiness of the measuring quantities. The technological progress attained with regard to color measuring instruments has, however, often been overrated, especially concerning the measuring uncertainty. Undoubtedly, the improvement of component parts has contributed to an increased reliability of the measured quantities and extent of information. Such components include high-resolution reflection gratings or photodiodes with half-width of about 5 nm, cleverly designed interconnect elements for optical wave guides, high-speed microprocessors, and storage mediums of several terabytes at the time of this writing. On the other hand, for measurement of colors, one must take into consideration three fundamental points. These are perhaps self-evident, but nevertheless pointed out explicitly at this stage:

G.A. Klein, Industrial Color Physics, Springer Series in Optical Sciences 154, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1197-1_4, 

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1. The measured values obtained from a color pattern represent the actual physical state at the moment of measurement; even if the measurement is of the highest quality, it is useless, if the color sample does not maintain the required color physical state. 2. The sensitivity of color measuring instruments is already about an order of magnitude better than the visual color sensitivity of even a practiced colorist! Further reductions in measurement uncertainty below this level are, therefore, unnecessary for color industry. 3. Although the result of a color measurement is at present accurate to at least three digits, the final human visual impression determines if the actually measured color pattern is acceptable or not. Experience shows that the combination of subjective visual judgment and objectively obtained color characteristics can be quite successful. These considerations are the subject to a more detailed discussion in the following. At first, we clarify the double meaning of the term color measurement and describe useful requirements for visual and metrological execution of color measurement. Moreover, the measurement geometries proposed by the CIE and the most important color measuring procedures are discussed [1–8]. In addition, we give suggestions for preparation and conditioning of color patterns. With regard to effect pigments, additional measurement methods are given. Finally, we discuss in detail the assessment of measurement uncertainty and statistics of chromaticities. These are indispensable for visual color matching, tolerance agreement, and quality control, among other things.

4.1 Measurement of Reflecting and Transmitting Materials Among the three factors contributing to the color impression of non-selfluminous colors, the final decision maker, as mentioned previously, is undoubtedly the observer. Simply put, in color measuring techniques, the observer is substituted by the color measuring instrument and computer. This substitution follows, e.g., also by comparison of Figs. 2.29a and 4.1, for example. More precisely, the optical image formation by the eye and the trichromatic elements of the retina are simulated by the optical components and photodiodes of the measuring instrument. The transfer of the trichromatic stimulus to the visual cortex and its assessment by the brain are simulated by analog–digital converters interfaced with a computer using suitable software. These days, this software includes the majority of the colorimetrical relations given in Chapter 3 as well as right designed graphic diagrams. Furthermore, it is important to stress that the characteristics and quantities which result from color measurements are also merely representatives of the special spectral or colorimetric properties of the relevant color pattern. The

4.1

Measurement of Reflecting and Transmitting Materials

Fig. 4.1 Principle sketch of a color measuring equipment; the observer is substituted by the measuring instrument and computer, cf. Fig. 2.29a

235 Computer X, Y, Z, L, a, b

Monochromator, Analog-digital converter Light source

Sample

interpretation and judgment of the numerical or graphical results should be carried out by experienced colorists. Often, the subjective color sensation is the decisive factor, because the colorimetrical formalisms only approximately imitate the nonlinear peculiarities of human color sense.

4.1.1 Measurement of Colors and Visual Judgment To measure means to compare. In the context of colors, to compare has a double meaning, namely metrological and visual. In the first case, a property which we are interested in is compared quantitatively with a suitably chosen reference color or standard; in the second case, the comparison is performed using the subjective color sense. First, we consider the fundamental aspects of quantitative measuring of color characteristics. Then, we go into the peculiarities which need to be taken into account for visual comparisons. Already from the law of energy conservation, it is known that light interactions in optical materials can cause effects such as reflection and transmission. The reflection of a layer is comprised of at least two superimposed principle components: backscattering from the volume and reflection from the interfaces. Reflection can be not only directional or partly directional but also diffuse or mixed such as partly diffuse, depending on the main directed or diffuse component (cf. Fig. 2.28). In some cases, the angle-dependent diffuse reflection of a matt surface obeys the Lambert law I(ϑ) = A · B · cos ϑ,

(4.1.1)

where I(ϑ) is the luminous intensity which is reflected into angle ϑ with respect to the face normal of a completely diffuse reflecting surface of size A and luminance B. A diffuse reflecting field satisfying this law is called Lambert radiator.

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On the other hand, even the specular reflection of a highly glossy color pattern or one with metallic pigments is superimposed with a quite small, but detectable diffuse component. Reflectance measurements are performed with regard to two so-called standards which serve as fixed points on the photometer scale. With the blackbody at absolute zero, the theoretical zero point of the photometer scale is established. At this temperature, the blackbody radiator emits no light but absorbs all entering light (Section 2.1.2). In practice, however, a hollow body of napped black interior called blackbody trap is used. The second fixed point of the photometer scale is given by a white standard, which – ideally – reflects, respectively scatters, the incident light completely. The spectral values obtained from the white and black standards are corrected in a suited manner and interpreted as 0.00 and 100.00% reflectance, respectively. The scale between both points is linearly subdivided and called photometer scale. This scale is applied to all sorts of colorants, the reason why colorations containing effect or fluorescence colorants attain apparent reflectance values of R > 100% or lightness values of L > 100. However, to this day, an ideal white standard does not exist. Such a standard is only realized approximately, e.g., by a freshly pressed layer of powdered BaSO4 or polytetrafluorethylene (PTFE). Both materials show a wavelength-dependent reflectance of about 98.5% in the visible range. This value is determined from the ratio of the reflected spectral energy of the white standard to the emitted spectral energy of the illuminating source. Other materials of high spectral reflectance such as MgO or ZnO are, on account of process reasons, very rarely used as white standards these days [1]. Altogether, there exist three different sorts of standards for measuring colors. The relevant quality levels arise from the following rank order: 1. Primary standard is manufactured and certified by a Bureau of Standards.1 The remarkable features of such a primary standard are the highest possible technical precision and greatest dependability. 2. Transfer standard is calibrated using a primary standard; it is preferentially employed by instrument manufacturers for calibration of reference (master) instruments. 3. Secondary standard is calibrated in turn on the basis of a transfer standard; this type of standard is enclosed within the measuring instrument; its photometric scale is previously calibrated with such black and white standards. Secondary standards are only really designed to use for scaling of the accompanying instrument during operation.

1 For

example: National Institute of Standards and Technology (NIST), Gaithersburg, MD, USA; Community Reference Bureau (CRB) of the European Union, Bruxelles, Belgium; Physikalisch-Technische Bundesanstalt (PTB), Braunschweig, Germany.

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Measurement of Reflecting and Transmitting Materials

237

In the case of transparent materials, the determination of transmission seems to turn out easier. Transmission is a measure for transparency of a material; it is reduced by interface reflection, absorption, and scattering. Clearly, the transmission can occur not only directionally or partly directionally but also diffusely or mixed such as partly diffusely. There does not exist a specific standard for the determination of the transmittance of a material. Rather the spectral intensity of the light source – without any sample – receiving the detector for closed and opened aperture diaphragm is simply interpreted as 0.00 and 100.00% transmittance, respectively. However, the measurement of transmittance is often influenced by heterogeneities at the interfaces of the relevant material. For non-self-luminous colors, there also does not exist unambiguously defined color standards. Certainly it is useful within itself to check the linearity of the gray scale of color measuring instruments with regard to the reflective factors in the domain 0 ≤ R(λ) ≤ 1.0 with tiles of so-called BCRA II ceramic.2 This consists of an entire set of 12 colored and hueless tiles which, for years ahead, have turned out to be almost color constant in appropriate storage conditions. Actually, the reflection of these patterns depends slightly on azimuthal angle. This influence can be excluded by repeated measurements in which the same measuring spot at the same orientation is always used. Otherwise gonio spectrophotometers can fake a falsely calibrated angle scale. Known and reproducible wavelengths of illuminant sources are used for calibration of the wavelength scale of a color measuring equipment. Such sources are also used to characterize light colors of sources, LEDs, screens, or panels. In order to cover the range of interest, devices such as the gas discharge of helium, mercury, and sodium vapor lamps are used. These sources emit different numbers of sharp spectral lines of known wavelengths in the visible range: He emits 18 lines in the interval 365.0 nm ≤ λHe ≤ 690.7 nm, Hg shows 12 lines in the domain 370.5 nm ≤ λHg ≤ 728.1 nm, and Na merely the pair of lines 589.0 and 589.6 nm. For inspection of the angle scale of gonio spectrophotometers, there exist no obligatory standards until now. As experience shows, each scaling depends on temperature and humidity of the surroundings, the instrument construction, and the standards used. The calibration should, therefore, be performed in operating conditions. Furthermore, the instrumentation and the secondary standards should not be exposed to higher fluctuations of temperature and humidity than specified by the manufacturer. Generally, the best situation is an installation of the instrument and secondary standards in a constant standard climate at temperature ϑ = 20◦ C and atmospheric humidity of 50%.

2 BCRA Series II Ceramic Standards: previously manufactured by British Ceramic Research Association, nowadays by Ceram Research, Penkhull, UK.

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The visual assessment of colors turns out to be the something more problematic; this can by no means be ignored. After all, the entire coloring efforts are oriented toward the human color sense. Clearly, only people with normally developed color perception should be involved in the assessment of colors. The capability of the individual color perception should be checked at regular intervals, and the test should be passed with the Ishihara color panels [9], even better with the Nagel anomaloscope [10], or with another standard [11]. In addition, also the ability to distinguish colors can be selectively reduced or unequally developed (color amblyopy). This also needs to be checked with a visual color test. Moreover, possibly the CMFs of joined observers differ from each other and deviate from those of both CIE standard observers. This observer metamerism needs to be considered by the involved people. The visual judgment is performed not only between similar colorations but can also include a great variety of special colorations. The systematic comparison of sample colors (sample, test, pattern) with reference colors (reference, standard, target) is called equality method. A sample color can be compared with special own standards of the manufacturer, reference colors, a systematic color collection, a pattern collection selected from manufacturing, or with patterns of a color-order system, among other things. A disadvantage of the equality method is the subjective judgment of the colorist – this often turns out to be too strict. For more objective methods, however, the tristimulus and the spectral measuring methods are available (Sections 4.2.1 and 4.2.2). In case of routine examinations under daylight conditions, it is sufficient to expose the colors to diffuse north skylight in the northern hemisphere and to diffuse south skylight in the southern hemisphere. In each of these cases, the skylight should be incident through windows which are as high as possible. During the visual comparison under artificial illumination, of course, it is necessary to meet at least the CIE reference conditions. In a light booth for color judgment, the illuminants that correspond to the colorations to which the color will presumably be exposed in future are to be activated. The basic equipment of a standardized booth includes simulators of illuminants D65, A, FL 2, FL 11 as well as two UV sources of defined bandwidth for stimulation of fluorescence emission [12, 13]. Due to a general lack of time in practice, often the sources are switched in fractions of a second among the various illuminants which means within the short-termed chromatic adaption phase. This procedure is certainly left for only experienced colorists; otherwise, it is better to wait for the longer adaption phase. Special diligence and experience are necessary, if equal colorations are to be compared with each other in different binders or of different surface structures. Further critical color aspects may arise with colorations containing effect and absorption pigments. In most cases, problems emerge by comparing the angledependent lightness or color flop, brilliance, or covering capacity. These require intensive training in color perception in unusual domains of a color space as well

4.1

Measurement of Reflecting and Transmitting Materials

239

as a sensitive interpretation of spectral reflectance and transmittance or rather of the relevant color values.

4.1.2 Measurement Geometries Instrumental color measurements are performed with different light sources and detectors for wavelength-dependent measurement of the reflected or transmitted light of a color pattern. The term measuring geometry refers to the kind of illumination of the color sample and the placement of the measuring detector. The choice of the measuring geometry depends on the consistency of the color pattern, the surface structure, and in particular, the sorts of colorants. From the accompanying measurement results, it is expected that they agree with the visual impression. Among the whole host of possible combinations, the CIE proposed four measuring geometries for measurements of opaque materials [14]. These can be gathered into two groups: – directional geometries 45:0, 0:45, – spherical geometries d:0, 0:d. The indication 45:0 stands for influx at 45◦ and measurement at 0◦ , e.g., perpendicular to the sample surface; d:0 means diffuse illumination and measurement at 0◦ . The inverse geometries produce identical results because the reversal of a light path has no influence of the effect – this is a fundamental principle of geometrical optics [15]. For absorption and effect pigments, in addition, there are – variable directional geometries β:μ, μ:β or β/μ, μ/β available, which were not proposed by the CIE until now. Before we discuss further details of these geometries, it is necessary to comment on the nomenclature. The measured quantity of opaque color patterns is assigned to a standardized term which depends on the chosen measurement geometry [16]: – reflection factor ρ: beam focused to conical illumination and semi-spatial observation (0:d, 8:d); – reflectance R: semi-spatial illumination and conical observation (d:0, d:8); – radiance factor β: illumination and observation are directional (45:0, 0:45; β/μ, μ/β).

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Be aware of the fact that β stands for either the angle of illumination or the radiance factor. For measurements of transmittance of transparent materials, corresponding terms such as transmittance factor τ and transmittance T have been introduced (see also end of this section). In the following, we discuss the directional geometries, followed by the spherical geometries, and those proposed for measuring transmittance. 4.1.2.1 Directional Measuring Geometries In the simple case of directional geometry, a light beam is incident at an angle β v = 45◦ to the surface of a colored sample; the measurement is carried out normal to this surface, corresponding to an angle μv = 0◦ . This configuration is denoted as the 45:0 measurement geometry and is shown schematically in Fig. 4.2a. The detector takes on the function of the observer. Permutation of source and detector is indicated by 0:45, as shown in Fig. 4.2b. The designation and way of counting of angles are described below and can be taken from Figs. 4.5 and 4.6.

Fig. 4.2 Directional measuring geometries: (a) 45:0 and (b) 0:45

Instrument manufacturers these days prefer the installation of the 45:0 geometry. This configuration has the advantage that the specular beam or ray cone reflected by an even and glossy sample is excluded from the measurement. Primarily, the reflection coming from the volume is registered. The 45:0 geometry corresponds to the observation method of colorists. Therefore, the measuring result agrees best with visual judgment. This conclusion is also correct for minor glossy, matt, or structured surfaces because this measuring arrangement only registers the outward flux at an angle μv = 0◦ . For a color sample of an incompletely homogeneous surface, illumination from opposite sides, each at an angle of β v = 45◦ , is suitable. This configuration is shown in Fig. 4.3a. Instead of two or more lamps, alternatively only a single source can be used. In this case, the source illuminates the color sample in combination with a circular 360◦ mirror at an angle β v = 45◦ . The sample

4.1

Measurement of Reflecting and Transmitting Materials

241

Detector

Detector Source

Source

45°

45°

45°

Sample

Sample

a)

b)

Fig. 4.3 Directional measuring geometries: (a) bidirectional 45bi:0 geometry and (b) circular 45c:0 geometry

is, therefore, illuminated circularly. For distinction, this measuring geometry is denoted as 45c:0 and shown in Fig. 4.3b. When applying this geometry, inhomogeneous surfaces have nearly no influence on the spectrometric or colorimetric results. With the 45c:0 configuration, it cannot be determined through the measurement whether the color producing mechanism of the sample is caused by a volume or surface effect or both. However, this differentiation is of particular importance for colorations with optically anisotropic pearlescent, interference, or diffraction pigments. The characterization of the angle-dependent interference and diffraction orders needs a so-called gonio spectrophotometer with variable directional geometry β/μ. In the simplest version, such a color measuring device has an illumination at a fixed angle β v = 45◦ and a detector of variable measuring angle μas with respect to the specular angle. In convenient instruments, the angles of the illumination and measuring are adjustable to a large extent and independent of one another. In Fig. 4.4, the principle of such a gonio spectrophotometer is represented. This configuration has an illumination angle of β v = 45◦ and, for simplicity, only four measuring angles μas = 25◦ , 45◦ , 75◦ , and 110◦ . This device is exclusively used for measurements of colors containing metallic pigments [17, 18]. Recently, standardized reflectance measurements have begun to be carried out at two additional angles of μas = 15◦ and –15◦ , also out of plane. However, these four to six measuring angles are not sufficient for clear colorimetrical characterization of effect colorations. At this stage, it is necessary to explain a clear method for angle counting of measurements with goniospectrophotometers. In ASTM and DIN standards for

242

4 Measuring Colors Detector Source

Specular direction

45°

25°

75°

110° 45°

45°

Fig. 4.4 Principle of a gonio spectrophotometer of variable directional geometry with only four standardized measuring angles following DIN or ASTM [17, 18]

measuring interference pigments [19, 20], the measuring geometry is denoted for given illumination and measuring angle in the form β v /μas ; see Fig. 4.5. Illumination angles β v are with respect to the normal to the sample surface; however, the measuring angle μas is with respect to the specular angle. The sign of μas is positive in direction toward the source. The aspecular angle is sometimes termed the effect angle in the literature, though there exist two nonstandardized ways of angle counting with regard to the horizontal (β h /μh ) and vertical (β v /μv ) for both angles; cf. Fig. 4.6. The instrument industry supplies gonio spectrometers of fixed illumination angles in the range 0◦ < β v < 90◦ , and also variable illumination up to β v = 360◦ . Measurements can be carried out normally at six standardized angles μas = –15◦ , 15◦ , 25◦ , 45◦ , 75◦ , 110◦ , or at variable angles which, in an extreme case, are adjustable with steps of 1◦ in the angle domain –35◦ ≤ μas ≤ 78◦ . Such a large number of independent angles of illumination and measurement are especially needed for pearlescent, interference, and diffraction pigments

μas = 0 + μas

Fig. 4.5 Fixing of the sign for aspecular measuring angle μas with respect to the specular beam following ASTM and DIN [19, 20]

βv

βv

– μas

4.1

Measurement of Reflecting and Transmitting Materials

Fig. 4.6 Non-standardized illumination and measuring angle pairs for angledependent measurements, with respect to the horizontal β h /μh and to the vertical β v /μv of the sample surface

243

βv

μv μh

βh

(Sections 3.5.3, 3.5.4, and 3.5.5 and [21–25]). For approximate recipe prediction and adjustment of color recipes with effect colorants, measurements at one illumination angle and at least five measuring angles are needed. Such multiple angle measuring geometries are also suitable for investigations of azimuthally dependent effects, e.g., orientation of colored particles or flakes in flow direction especially in high viscous states of high solid lacquers or melts of high polymers. This is valid for marbleizing effects with absorption pigments as well as homogenized sparkling flakes. Such investigations by measurement are performed at defined azimuthal angles and at different points of the sample surface. Altogether, the reflection values resulting from measurements with a gonio spectrophotometer are relative quantities provided that the investigations are performed with respect to a BaSO4 secondary standard. This standard shows an angle-dependent radiance factor and, in comparison to the 45:0 geometry, leads to large deviations in the vicinity of the specular direction. The color values following with this standard can be interpreted as dependent quantities of the spectrophotometer used. Color values of different effect pigments can, therefore, only be compared with one another if the measurements are also based on the same reflection standard and the same instrument. In Table 4.1, available devices with directional or diffuse measuring geometries are listed in order of the type of illumination as well as respective application situations.

4.1.2.2 Diffuse Measuring Geometries The measuring configurations d:0 and 0:d, which are recommended by the CIE, are not explicitly itemized in Table 4.1. These are, however, approximated by the measuring geometries d:8 and 8:d. The notation d:0 means diffuse illumination and directional measurement at an angle μv = 0◦ ; see Fig. 4.7. The emitted light of a source is reflected in different directions at the interior surface of a suitable coated Ulbricht sphere (e.g., powder coated with PTFE). The sample

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Table 4.1 Measuring geometries with directional or diffuse illumination and proposed applications

Illumination

Measuring geometry

Directional

45:0 45:0 bidirectional 45c:0 circular β v /μas gonio spectrophotometer

Diffuse

di:8

de:8

Suitable applications or advantages Smooth, slightly brilliant to high glossy surfaces; specular beam excluded; fluorescent materials Non-uniform, structured, dull to high glossy surfaces; Transparent to opaque materials, dependent on measuring method; agreement of colorimetrical with visual impression Anisotropic, transparent to opaque samples with absorption or effect pigments, all sorts of colorant mixtures; determination of reflection indicatrix Smooth, slightly brilliant surfaces of transparent to opaque materials; agreement of colorimetrical with visual impression; numerical subtraction of specular gloss; color recipe prediction Smooth to glossy, transparent to opaque color samples; specular beam excluded

is consequently nearly diffusely illuminated. In modern instruments, the measuring receiver is not positioned at μv = 0◦ , but rather at an angle μv = 8◦ ; the CIE even allows a deviation of up to 10◦ [14]. As can be seen below, this displacement has some metrological advantages. The diffuse measuring geometry d:8 is versatile enough to be used for absorption colorants and also to survey effect colorants. This geometry is schematically represented in Fig. 4.8. An advantage of the d:8 configuration is that light incident with an angle 8◦ regarding the normal of the sample surface can be included in or excluded from the measurement. In other words, the specular component can be incorporated in or eliminated from the result. Therefore, it is necessary to differentiate between the diffuse geometry with included specular reflection di:8 and excluded specular component de:8. With an absorbing specular light trap installed, the specular component can be excluded from the measurement. Without a light trap, the specular reflection at angle μv = 8◦ is included in the result. Consequently, with sphere geometry has always to be indicated the applied measuring modus. The specular component, which is included in the di:8 measurement modus, can be later numerically subtracted from the result. The resulting spectral values of even or glossy color patterns are nearly identical to those of the de:8 modus. The following color values agree well with the visual impression, analogous to

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Measurement of Reflecting and Transmitting Materials

245

Detector d:0

Source

Sample

Fig. 4.7 Sphere geometry d:0

d:8

Detector

Light trap

Source

8°

Sample

Fig. 4.8 Sphere geometry d:8 with specular light trap

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results with the 45c:0 geometry. However, from semi-matt or matt colors, only a fraction of the specular reflected component at the angle 8◦ reaches the light trap. The measurement results lead to color values of higher lightness in comparison to the visual impression. In this case, the numerical subtraction of the specular reflection leads to false color values. This means that the de:8 geometry is only suitable for measuring of even or glossy color patterns. On account of the included specular component of the di:8 geometry, it is impossible to discern different surface structures, that is, different textures result in nearly equal color values. The measurement results of matt and structured color samples deviate somewhat from the visual impression. In this case, the 45c:0 geometry is more suitable. The spherical geometry with di:8 modus is preferentially applied for color recipe prediction, if matt or structured color patterns of unknown colorant composition are manufactured with smooth or glossy surfaces. In any case, the choice of the measuring geometry should be considered carefully (cf. Section 4.1.5). When in doubt, the possibilities should be examined experimentally beforehand to decide which of the mentioned geometries is at the best suited for the actual problem. The survey given in Table 4.2 can be helpful. It is divided up with regard to the surface or volume properties of the color pattern of interest. Fluctuations occurring during manufacture of optical and microelectronic component parts are a reason that the results obtained with various color measuring instruments often differ from each other. For clear comparison of results which were achieved by different instruments from the same manufacturer or by different manufacturers, every manufacturing company should generally insure that the actual instrument generation fulfills at least the CIE recommendation [14]. Moreover, possibly several of the updated standards [26 –31] should be met. 4.1.2.3 Geometries for Transmission Measurements The reflectance and transmittance of translucent materials are best determined using one of the above-mentioned geometries and applying the measuring method over two different backgrounds (Section 4.2.4). This procedure is not standardized. For measurement of transparent or nearly transparent materials, the CIE proposed altogether one directional and five spherical geometries as follows [14, 32]: 1. directional normal–normal geometry (0:0): each of the incident and outgoing fluxes is perpendicular to the plane parallel sample surface; 2. diffuse–normal geometry (di:0): the color pattern is illuminated diffusely, the measurement is performed normal to the sample surface with specular reflection included;

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Measurement of Reflecting and Transmitting Materials

247

Table 4.2 Survey over suitable measuring geometries for color measurements of non-selfluminous colors

Optical surface or volume property Surface

Matt

Slightly glossy, structured

Glossy, high glossy

Volume

Transparent

Translucent

Opaque

Suitable measuring geometries and colored materials Three geometries are suitable: di:8 with subtraction of specular reflection, de:8, 45c:0; colors, lacquers, plastic materials, textile fibers, glass, ceramics, fluorescent colorations Smooth and slightly bright color patterns: di:8 with gloss subtraction, otherwise gonio spectrophotometer; more rarely 45c:0; lacquers, embossed or injection molded surfaces of polymeric materials, textile fibers, paper, leather, glass Also with diffuse component: alternative di:8 with gloss subtraction, 45c:0; lacquers, high polymeric materials, foils, backed paper, ceramic Directional transmission: measurement with transillumination: di:8 with gloss subtraction, de:8; directional and diffuse transmission: reflection over two different backgrounds: di:8 with gloss subtraction, 45c:0; dyes, colored and effect pigments, inorganic and organic glass Homogeneous, inhomogeneous, cloudy, or flow structures: as given for transparent sample under directional and diffuse transmission, see above; comparison of measuring geometry Isotropic scattering: di:8 with gloss subtraction, de:8, 45c:0; colors, lacquers, plastic materials, ceramics; anisotropic reflecting or scattering: gonio spectrophotometer; absorption and effect pigments, fibers, high synthetic polymers

3. diffuse–normal geometry (de:0): unlike to 2., the specular reflection is excluded from measurement; 4. normal–diffuse geometry (0:di): this geometry is the inverse of that in 2.; 5. normal–diffuse geometry (0:de): geometry inverse to that in 3.; 6. diffuse–diffuse geometry (d:d): illumination and measuring are each performed with diffuse light.

The third and fourth geometries with included specular component result in transmittance factor τ , the other listed geometries in transmittance T [32]. The

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construction of the Ulbricht sphere corresponds to that for reflectance measurements. Further details for color measurement of optically different materials follow in Sections 4.1.4 and 4.2.4.

4.1.3 Sample Requirements As already mentioned, the measurement result represents an aspect of the actual physical state of the sample. From there, it is essential that the experimental handling of the color pattern is performed with extreme care. All influences, which are able to distort the measurement result, need to be keep away from the colored sample. Apart from this, the result is influenced by the manufacturing, preparation, selection, and storage of the coloration, among other things. Such factors are often underestimated or disregarded. Among the varied aspects to be considered, we restrict ourselves to the most important influences which are caused by the surface, shape, homogeneity, and manufacturing parameters that possibly change the optical properties of the color pattern [33]. Issues of quality control are not considered in this text. During the manufacture of colored samples, the components are mixed, dispersed, or kneaded in such a way that process variables such as shear velocities, temperatures, pressures, and action times are maintained similar to the future production. Loose powders, granulates, or pellets are completely unsuitable for color measurements. From these, indefinable shadowing is superimposed on the backscattering from the volume; this causes, under both directional and diffuse illumination, false color values: in fact, the colors are lighter. Pigmented powders of required particle size distribution should be homogenized in a transparent binder. Powdered materials are otherwise manufactured in stampings with defined smooth, diffuse scattering surface, and without cracking of crystallites. In this case, the natural chroma and brilliance are maintained to the greatest possible extent. Granulates or pellets of plastic materials are transformed by melting into a sample of required size, shape, and surface structure. From the applied procedure for homogenizing absorption pigments in a binder should result optically isotropic color patterns; therefore, spectral values are independent of the sample orientation. The measurement of optical anisotropic materials – like some textile fibers and fabrics, effect pigments in lacquers, injection molded or extruded plastic materials – is performed by systematical rotation of the sample with regard to the aperture. This is in order to investigate whether the spectral values are independent of azimuthal angle. Modern instruments for this are equipped with a sample objective. On the other hand, naturally or artificially aged materials can develop anisotropic properties such as color streaking, chalking, micro- or macro-cracks and other near-surface or volume variations.

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Measurement of Reflecting and Transmitting Materials

249

For all sorts of colorants or binders, defect-free sample surfaces without any scratches, bubbles, blisters, inclusions, fingerprints, etc., should always be used. Otherwise unwanted reflections are created. These uncontrollably falsify the color impression and the color values. In addition, the electrostatic charging of organic materials which often occurs needs to be avoided. Dust particles cause surface scattering: dark hues appear lighter, light hues darker. Rough and coarse surfaces, e.g., of leather, textiles, ceramics, lacquers or plastic materials, behave in a similar way. On the other hand, in some cases, the brightening effect is directly created with suitable additives in lacquers; in the case of plastics, pearl-, conic-, prismatic-, or linen-shaped surface structures are applied. Saturated and dark hues are extremely difficult to handle in this context. For such colorations, and also for matt surfaces of plastic materials, coatings, and lacquer systems, it is recommended to apply a colorless and transparent topcoat. In this case, not only the applicable color impression is attained but also untainted color values. For spectrometric measurement, the di:8 modus with numerical subtraction of specular reflection, alternatively the de:8, or the 45c:0 measuring geometry should be applied. The entire color measurements and visual color difference assessments should be performed exclusively with smooth and plane samples. Curved or undulated surfaces are unusable. Arched surfaces, however, can be suitable for visual judgment of (a) the gloss or roughness of colorations with absorption pigments or (b) the travel of lightness or color flop of effect pigments. Additionally, inhomogeneities such as flat spots or sink marks in the sample surfaces are to be avoided. These occur preferentially in thin layers of lacquers or at the surfaces of unfavorably constructed and manufactured plastic components. The color samples should be dimensioned in such a way that metrological and visual conditions are satisfied. Irrespective of the colorant sort, the measurement aperture is often adjusted to the largest possible diameter with the goal of achieving an averaging of the spectral values over the measurement field. But with regard to the aperture of a measuring instrument, the need can be laid out more precisely: – effect pigments need always the greatest possible aperture; – absorption and phototropic colorants need a middle-sized aperture; – fluorescent and thermochromic colorants need the smallest possible aperture. There is a rule of thumb that the sample diameter should be at least three times that of the adjusted aperture diameter. With that, the influences of edge reflection and edge loss are somewhat reduced. An applicable theoretical description of such edge phenomenon is, however, as yet unpublished. Both the multiple reflection at the side wall and the emerging light fraction there cause false measurement values. This systematic error exists in the entire previous color measurements up to that point and is withhold in most cases in literature.

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Color patterns of rectilinear, orthogonal edges, at which the color samples are aligned without any gap, are indispensable for visual judgment of color differences (simultaneous contrast). Furthermore, the other CIE reference conditions are met during the evaluation of colors. Also, the different color perception phenomena which can affect the judgment need to be considered. The storage of the manufactured color patterns is normally carried out pressurelessly between pH neutral separating paper in a dark surrounding, and possibly in standard climate or at low temperature and middle humidity. The storage of the calibration patterns for color recipe prediction should be considered with special prudence. This also goes for color collections, long-term internal or external standards, or all sorts of calibration standards of the color measuring instruments used.

4.1.4 Transparent, Translucent, Opaque Colors Optical materials can be distinguished depending on the sort of light transmission. As already mentioned, they are termed as transparent, translucent, or opaque. In the color industry, continuous transition between such different optical states is often desired, for example, a given shade may be manufactured with varying degrees of transparency. For precise characterization of these kinds of colorations, colorimetric terms such as transparency index and contrast ratio are used. An optical transparent medium absorbs a fraction of the incident light in dependence on the wavelength. The residual fraction is ideally transmitted without any scattering. Such a material appears colored not only in transmitted illumination but also in incident illumination. The reason for color impression at incident illumination is the difference between the refractive index of the binder (n2 ≈ 1.5) and of air (n1 ≈ 1.0). From directional light perpendicularly incident on the interface, 4% of the incident light are immediately reflected, from diffuse incident light about 9.2%, cf. Equations (2.1.8) and (2.1.12). For constant colorant concentration, the measured quantities and the visual judgment of transparent colors also depend on the layer thickness (Sections 3.4.3 and 5.1.2). The spectrophotometric measurement of a transparent color pattern is carried out either with transmitted light or reflected light in one of the measuring conditions and geometries listed in Table 4.2. These decisions depend on the question, as to whether only the directional or both the directional and diffuse components of transmission should be recorded. By definition, a transparent material shows no reflection from the volume due to lack of scatter. With the d:8 measuring geometry, the transmittance can be determined with three different procedures: (a) the colored layer is positioned between source and Ulbricht sphere, (b) between the sphere and aperture of the detector,

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Measurement of Reflecting and Transmitting Materials

251

To detector

Position b)

8° Position a)

Fig. 4.9 Two alternative placements for measuring transparent materials: (a) in front of the illumination aperture and (b) at the measuring aperture

or (c) over two different backgrounds; methods (a) and (b) are shown in Fig. 4.9, method (c) in Fig. 4.14. Both sample placements (a) and (b) result in slightly different measurement values, provided that a scattered component from the interfaces appears in position (b). The measurement of transparent materials is often complicated by inhomogeneous surfaces and multiple reflections at the interfaces of a plane parallel layer. In order to exclude such reflections from the measurement result, it is generally not appropriate to simply tilt the layer a few degrees from normal to the beam. Refraction causes not only dispersion but also a shift of the sample beam. A better method for quantitative characterization of transparent or translucent layers, which avoids such disadvantages, is given by the method of reflection measurements over two different backgrounds. From a physical point of view, the color measurement of translucent materials is more complex than for transparent or opaque systems: in this case, it is necessary to determine both reflectance and transmittance, which are already listed in Table 2.6. Both quantities of colored layers can be measured with spectrophotometers by two different procedures: either by separate determination of reflection and transmission or by two reflection measurements over two different backgrounds of known spectral reflectance; see Section 4.2.4.

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The measurement of reflection over two different backgrounds can be performed with the 45c:0 or di:8 geometry. Diffuse illumination, however, is preferentially applied. The accompanying geometry is shown in Fig. 4.8. The measuring procedure over the two backgrounds is complicated by a basic physical requirement: the translucent layer and the background have to be in optical contact with each other. The light background of known spectral reflectance reflects the majority of the light which is not absorbed by the particles of the layer. The measured reflectance, therefore, corresponds to the color of the translucent system. Because the dark background absorbs the majority of the transmitted light, then the reflected fraction is caused by the scattering of the translucent material. Opaque colorations of inorganic colored and opaque interference pigments completely cover a colored background. The measured spectral values over different backgrounds are equal within measurement uncertainty of the instrument used. Nevertheless, the two-background procedure is also applied to opaquecolored materials. The reasons for this are either for the determination of the contrast ratio of a covering layer or the precise separate determination of the optical constants of an incorporated colorant. Layers containing effect pigments need more extensive color measurements than translucent samples. The most commonly used diffuse measuring geometry for these sorts of samples, de:8, shows results which agree well for pearl, interference, and diffraction pigments with visual impression perpendicular to the pattern surface. On the other hand, from directional illumination at fixed angle β and directional measurement for at least five aspecular angles μas , for example, chroma, color travel respective flop, or the interference color of the coloration can be observed. In contrast, apart from at short visible wavelengths, metallic pigments show a spectral reflection nearly independent of wavelength which increases with specular angle (cf. Fig. 2.39). Different shear rates and consequently altered structural viscosities of binders or plastic melts additionally superimpose all sorts of optically anisotropic effects on flake-shaped particles. A possible anisotropy with respect to azimuthal angles, often caused by finishing or processing, can be shown by systematic rotating of the sample perpendicular to the surface of the color pattern.

4.1.5 Color Matching The visual comparison of two or more sample and reference colorations of similar shade and their assessment is generally called color matching. Normally, this method is accompanied by colorimetrical measurements. This procedure of drawing comparisons can be applied to all sorts of absorption and effect colorations in research, development, or quality control. The criteria and suitable tolerances to which the results need to comply are selected beforehand. For

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253

visual judgment, at least the CIE reference conditions need to be realized. A light booth should be equipped with standard illuminant D65. Additional illuminants such as A, C, FL 2, or FL 11 are necessary in cases for visual assessment of color inconstancy, metamerism, or fluorescence. The evaluation of the computed color values is carried out with the same illuminants as in the visual method. For competitive color measurements, the involved colorists have to employ the same optical instruments, possibly with the same equipment and capacity. This means the use of the same color-matching booths, (gonio) spectrophotometers of the same measuring geometry, gloss meters, etc. Both sides have to agree and meet the measuring geometry, the angles of illumination and measurement, as well as the illuminants and standard observer. Only in such a situation it is possible to clearly compare and interpret the distinct angle-dependent measurement values of effect colorations. Moreover, the spectral power distribution of the sources used needs to be determined beforehand and compared. Deviations should be eliminated by exchanging lamps with those of the same type. The UV or IR range should be included in the comparison only in cases where fluorescent and phototropic or thermotropic colors are to be matched [33]. In addition, the color capability of the involved colorists is to test and systematically to train. Different color capabilities between the colorists should be known and taken into account. The required color measurements should be performed immediately before or after the visual color matching. Producing two measurement series, one direct before matching and one check series afterward, has proven to be effective. A possible time-dependent change of sample and reference colors during the procedure time can, by this method, be nearly excluded from the results. The mean from the results for each color pattern of both measuring series should be taken. The deviations should not exceed twice the measuring uncertainty of the instrument. As mentioned previously, the color impression of an object comes from the superposition of light interactions from the volume and reflection on account of the conditions of the interfaces. Most of misinterpretations of color measuring results come down to the fact that it is unknown whether the surface reflection is included in the measurement result or not. Applying the diffuse measuring geometry di:8 or 8:di, the specular reflection is included in the result. Before determination of color values and color differences, therefore, from each spectral value, the quantity of the directional reflection coefficient r needs to be subtracted. Afterward, the resulting color values and color differences can be calculated. The color differences attained from different measuring instruments should now agree with each other within the measuring uncertainty of the instruments. With the di:8 modus or reverse, the color differences coming from the colorants homogenized in the volume of the sample are essentially measured. Surface structures have no effect on

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the resulting color difference. These are the reasons why measurements with included specular reflection are preferentially used for colorant tests. On the other hand, subtraction of the directional reflection coefficient of a highly glossy color pattern shows a result which agrees with the visual impression. For a matt sample, the numerical subtraction of the specular component leads to a fictitious color. In this case, an unaffected and realistic color impression is achieved by a highly glossy clear coat of the sample and aspecular observation. The corresponding measurement results are, therefore, valid especially for highly glossy and unstructured color patterns. In addition, the results agree with the visual color impression. The interface reflection of distinctly structured matt to glossy colors such as emulsion paints, plastic materials, and textile fibers needs to be determined by empirical methods. In all other cases, the spectral values are taken without subtraction of the specular component. In the second measurement modus with the diffuse geometries de:0, de:8 and the directional geometries 45:0, 45c:0, the specular reflection is excluded from the result. In reality, the specular reflection is excluded from the measurement only if plane, ideally glossy samples are used. However, fractions of the specular reflection can unintentionally be included in the result. This depends on the construction of the light trap, optical shielding, or the conical divergent light beams with half angles > 5◦ . Measurement results from different instruments are, therefore, not necessarily comparable with each other. The specular excluding modus leads to measurement results which are due to light interactions in the volume of the colored pattern. These results agree with the visual impression excluding specular reflection. However, if the specular component is reduced by color fastness tests, the diffuse surface reflection increases and also, consequently, the entire reflection. Changes in the surface and volume are, therefore, registered together. Consequently, in color fastness tests, it is useful to follow the surface influence in dependence on time separately with a gloss meter. On the other hand, in cases of extreme changes of surface reflection with the observation angle, the measuring geometry de:8 produces differences with regard to the visual color impression. Except of effect colorations, this influence is especially well developed not only for semi-glossy and structured samples but also for patterns showing coat blooming. Such colorations result in even higher deviation of the measurement result if the 45:0 geometry is not exactly realized and the observation is unintentionally performed with partly diffuse illumination. With absorption colorations, a nearly perfect agreement between measurement and observation is only achieved with the 45:0 geometry. Diffuse and directional measurements with the specular component excluded are preferentially employed for color patterns of nearly same hue, which, however, differ with regard to the: – geometrical shape (plane parallel, undulated); – surface (even, structured);

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255

– interface reflection (partial directed, partial diffuse, diffuse); – degree of gloss (matt to high glossy). In cases of colorations of the same color, which are manufactured with quite different surfaces – e.g., for textile fibers, foils, interior, and exterior fittings of automotives –, an agreed “secondary standard” for each surface of interest is normally fabricated for assessment. In general, the present considerations are valid for opaque materials containing absorption colorants. Similar ideas can be transferred to transparent and translucent colors of plane parallel layers with glossy surfaces. For such color patterns of different light transmission, it is necessary to take into account the reflections at the outer and inner interfaces. In this context, the method of reflection measurement over two different backgrounds requires that both backgrounds are in optical contact and have the same high-glossy surface. If these conditions are not met, the spectral values will be incorrect (cf. Figs. 5.26 and 5.27). The measurements of the backgrounds alone or together with a chromatic layer should be performed with the same measurement geometry and instrument. It is possible to maintain the above-mentioned criteria for opaque materials for visual assessment of transparent and translucent colored patterns of the same color but of different surface structures.

4.1.6 Acceptability and Tolerance Agreement In the context of color matching, the verbs agreed and tolerated have already been used. The terms acceptability and tolerance agreement are closely interconnected with each other and, moreover, with the procedure of color matching. A color tolerance agreement precedes an extensive color matching, accompanied by the intension of finding the visually accepted color patterns from a given color collection. The experimental and additional numerical evaluation conditions with which the color matching and assessment are to be performed should first be clarified by the involved people. A representative collection of color patterns, which is possibly pre-selected by various independent colorists in groups such as “accepted,” “only just accepted,” or “excluded,” needs to be available for ascertaining acceptability. The sets of accepted or conditionally accepted colorations constitute an ensemble for the succeeding more strict color matching. The pattern collections are either assembled by (a) specifically produced new colorations which show systematic hue error or color deviations from the reference color (possibly in each of the three directions in a color space); (b) selected colorations of the production with random color differences with respect to the reference color. During matching, it is always necessary to distinguish between the smallest visually discernable color difference and the acceptable color difference:

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– the smallest visually discernable color difference is a psychophysical quantity; – the acceptable color difference, however, is a technically or economically reasoned value. However, if the reference threshold of acceptability is set too low and fluctuations of production, for example, always cause systematic crossing of this threshold, then an acceptable serial production is illusive to obtain. Furthermore, a threshold lower than the – color locus dependent – visual capability of discerning color differences is an excessive and unrealistic demand. The analog is true with regard to the stochastic variations in the color measurement instruments used. An acceptability threshold which is below the measurement uncertainty is not experimentally obtainable. The well-configured tolerance agreement comprises arrangements with regard to the sort and extent of deviations which are to accept or to tolerate between sample and reference color(s). In addition, the experimental, numerical, and visual methods are fixed, with which the tolerance parameters are determined and currently checked in future. Apart from that, the random and also systematic errors are taken into account, which occur by changeover from laboratory to production scale. In general, a tolerance agreement includes such parameters, which allow tolerating visual and colorimetrical differences between sample and reference colors regarding agreed thresholds. The criteria to find out are so varied that only some ideas can be given here for that intention: – the magnitude of the tolerance ranges of the different color contributions which are to determine and to fix depends on the sort of colorant, color range, or color point, surface structure of the pattern, binder, etc.; – the measuring and evaluation conditions are unambiguously to arrange; – permissible variations of color difference values are definitely to restrict; – the agreed color difference formula for evaluation use is to apply; – the statistical test of a current production is in detail to arrange and to meet: the formulation of the time frame, the parent population or sample size, test parameters, etc.; – the color tolerance values can include not only color physical criteria but also application-specific or economic aspects; – completely inappropriate is a color tolerance smaller than the visual color discrimination; – on the other hand, it is to consider that too large color tolerances are strictly to avoid, in order to achieve product acceptance by the consumer. Already this short enumeration elucidates the impossibility to formulate a universal valid tolerance agreement for each special case. The partners should rather realize that each current case requires a separate, individually tuned

4.2

Measuring Methods

257

color tolerance agreement. Finally, the concluding document with the tolerance agreement is to enclose with all measuring and evaluation protocols.

4.2 Measuring Methods Altogether, there exist three different procedures for measuring colors: the equality, the tristimulus, and the spectral methods. The equality method is identical to visual color matching described previously, whereas the tristimulus and the spectral methods are based on technical measurement principles. In the following, we describe the construction and interaction of the most important components of color measuring instruments upon which the tristimulus and spectral methods are based. Just to sum up, the tristimulus colorimeter uses three filters of light transmission with which the three standard color-matching functions (SMCFs) are simulated in the range of visible wavelengths. The more precise and versatile spectrophotometers allow for the measurement of reflectance or transmittance of a color pattern at discrete wavelengths in the visible range. Whereas the tristimulus method simply delivers color values, with the last mentioned measuring procedure, the spectral reflectance or transmittance of transparent, translucent, and opaque colorations is determined in different ways. Among these methods is an often applied technique the special measurement over two different backgrounds. This procedure is now discussed in detail. As already explained, the varied color properties of effect pigments are not fully characterized simply with angle-dependent color measurements. Therefore, we give some supplementary and industrially used optical methods. Finally, aspects for color measuring of fluorescent, thermochromic, and phototropic systems are discussed. On the basis of selective absorption mechanisms, these need modified measurement procedures.

4.2.1 Tristimulus Colorimeter The tristimulus technique is carried out with the so-called tristimulus colorimeter. Among the established color measuring instruments, this equipment has the simplest optical structure. The essential optical components consist of an illuminant and three separately positioned color filters in front of connected photocells; see Fig. 4.10. The first filter has a spectral transmittance nearly of the SCMF x¯ (λ), the second nearly of y¯ (λ), and the third of z¯(λ) with respect to the 2◦ or 10◦ standard observer. For simplicity, these colored glasses are also named R, G, B filters. In reality, however, four or more filters are usually inserted. On the one hand, the short- and long-wave fractions of the SCMF x¯ (λ) cannot only be imitated with just a single filter with excessive effort; on the other hand, the

258

4 Measuring Colors Broad band – diode

Source

X

Y Z

Sample

Grating

R, G, B Filter

Computer

Fig. 4.10 Principal configuration of a tristimulus colorimeter for determination of color values

edge steepnesses of usual filters deviate to a large extent from those of the three SCMFs. By comparison of Figs. 4.10 and 2.29a, it can be recognized that the observer in the second mentioned figure is substituted by the color measuring instrument in Fig. 4.10. In order to have the best agreement between the measurement and the visual color impression, the optical construction of a tristimulus colorimeter is kept as simple as possible and is mostly realized in the 45:0, 45bi:0, or 45c:0 geometries. The light emission of a high-pressure xenon lamp is filtered in order to simulate standard illuminant D65. The sample is illuminated at an angle of 45◦ in a linear, bidirectional, or circular manner. The spectral reflection of the color sample supplies the three R, G, B filters, which transmit only wavelengths in the red, green, or blue range with regard to the SCMFs. Afterward, the separate filtered optical signals reach each of the broadband semiconductor photodiodes. These transform the incident light into an equivalent electrical current. Each current is amplified and read in by a microprocessor which calculates the standard, CIELAB, DIN99o, or color values of a color appearance model. For color difference measurements, color differences are computed with various formulas. The color values and differences correspond to the illuminant used and to the observer which is simulated by the three filters. In most cases, the 10◦ observer is simulated. Some manufacturers offer tristimulus colorimeters of selectable illuminants and exchangeable R, G, B sets of filters for both standard observers. It is doubtless that the functional range of the tristimulus colorimeter is quite restricted. However, the clear construction constitutes an economic solution to determine the simplest colorimetrical quantities with sufficient approximation. Certainly, the three-filter photometer measures no single spectral values but

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Measuring Methods

259

produces and registers only three photoelectric currents corresponding to the installed illuminant. Therefore, color inconstant or mutually metameric colors are not detected. Moreover, this method is really restricted to colorations containing absorption colorants and is absolutely unsuited for the characterization of all optical properties of effect pigments. Finally, color recipe prediction for absorption colorations using this instrument should be carried out only with great restrictions, especially in cases of missing spectral values and those given only single color values for the reference color. The simple instrument construction causes also a low absolute accuracy and reproducibility – both properties are, for the most part, five to ten times worse than those of a spectrophotometer; values are given numerically in Table 4.4. The application of this instrument is, therefore, restricted to colorations with absorption colorants in fields of quality control and product information. Compared to this method, the spectral procedure is more complex but versatile for solutions to the majority of color physical problems.

4.2.2 Spectrophotometer The spectral method can be realized in directional, diffuse, and variable directional measuring geometries as discussed. The technical construction of a spectrophotometer of d:8 geometry is shown schematically in Fig. 4.11. The approximate D65 filtered and pulsed light of a high-pressure xenon lamp is reflected many times and scattered at the surface of coarse white inside of a

Measuring beam

Photodiode arrays

Measuring signal

Light trap

Xenonsource

8° Reference beam Reflection grating

D 65-, UV-Filter

Reference signal

Sample

Fig. 4.11 Diagram of a dual-beam spectrophotometer: d:8 geometry with measuring and reference beam (source: Datacolor AG, Dietlikon/Zürich, Switzerland)

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Ulbricht sphere. The aperture of the xenon lamp has a shielding and the light is emitted parallel to the surface of the color sample which is, therefore, diffusely illuminated. The UV fractions of the xenon spectrum are filtered in such a way that one of the four possible daylight sorts is simulated or rather the stimulation and measurement of luminescent colorants can be performed. The light reflected by the sample under an angle of 8◦ is, through an objective and optical waveguide, directed onto a concave diffraction grating. Similar to diffraction pigments, the grating diffracts the incident light and breaks it into its individual constituents. The concave curved reflection grating collimates the diffraction spectrum which is diverted onto an array of silicon photodiodes. The number of light-sensitive diodes depends on the relevant wavelength resolution and the bandwidth of the measurement spectral range. Instruments of first generation possessed 16 diodes for covering the range between 400 and 700 nm at equidistant interval width of 20 nm. Modern spectrophotometers are equipped with at least 40 diodes and register wavelengths from 360 to 750 nm with interval width of 10 nm. For security, there are sometimes two or three such arrays in the instrument. Such a collection of semiconductor elements with extremely narrow photoelectric tolerances enhances the accuracy and dependability of the spectrometer, but this comes at cost. In Table 4.3, typical characteristics and specifications of a modern reference spectrophotometer are given. The so-called abridged spectrophotometers with spectral range from 400 to 700 nm of equidistant interval width of 10 or 20 nm (31 or 16 measuring wavelengths, respectively) are common these days. Such instruments are absolutely sufficient for a wealth of colorimetric applications with regard to absorption colorants such as quality control, documentation, determination of color values, color differences, inclusive color inconstancy index, metamerism index, covering capacity, and recipe prediction. For colorations with effect pigments, gonio Table 4.3 Characteristics and specifications of a modern reference spectrophotometer Characteristic

Specification

Measuring geometry Spectral range Number of wavelengths Bandwidth Wavelength accuracy Inter-instrument agreement

de:8, di:8 360–750 nm 40 10 nm 0.05 nm ∗ ≤ 0.3 , ΔE∗ ≤ 0.15 average ΔEab ab BCRA II tiles 0–250% 0.001% ∗ ≤ 0.01 ΔEab 3s

Photometric range Photometric resolution Measurement repeatability Measurement time

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261

spectrophotometers of variable directional geometry (β/μas or μas /β) of at least five different measuring angles μ are often applied. Spectrophotometers of high-wavelength resolution of Δλ < 5 nm or extended wavelength range from 360 to 780 nm sometimes use a numerical interpolation method with the measurement values. This is for adjustment with regard to the tabular values for illuminant and standard observer of interval widths of 1 or 5 nm given by the CIE. In cases of unequal measurement and standardized spectral ranges, the tabular values are to modify in such a way that for every illuminant, the given standard color values Xn , Yn , Zn of ideal white result (cf. Table 3.1). The narrow interval widths might be necessary for color physical aspects in basic research for precise determination of narrowband extinction or transmission spectra of dyes. However, such spectrophotometers are often overkill for purely colorimetric questions, including for effect colorations. In addition to the main sample beam, spectrophotometers are often equipped with a further reference beam, as shown in Fig. 4.11. This is for enhancement of accuracy and reliability. In such a dual-beam spectrophotometer, the reference beam serves as a measure for the incident light intensity at the sample surface. The reference beam is collected from the diffuse reflected light inside the sphere and supplied to a second optical system – identical to that of the measuring beam. Either the same photoarrays are used for measurement and reference beam and the signals are appropriately time-shifted or two or more arrays are installed. The separately produced photocurrents of measurement and reference beam are amplified, digitized, and numerically edited. The ratio of the measurement and reference signal for each of the wavelengths λi corresponds to the reflectance R(λi ) or transmittance T(λi ) of the sample. In the data storage device of the integrated microprocessor, the tabular values of different illuminants and standard observers to each measured wavelength are stored. In the evaluation unit, routines are implemented for determination of spectrometric and colorimetric quantities. The required evaluation quantities are stored for reference measurements [14, 27, 30]. Additional software should have appropriate algorithms available for performing recipe prediction. Repeatedly, we have pointed out that the measured reflectance or transmittance of a color pattern is independent of the applied light source. The reason for supplying a modern tristimulus meter or spectrophotometer with a xenon lamp is, however, primarily to reduce to two random aspects: first, the spectral power distribution of xenon light is roughly the same at each wavelength in the visible range and second, it corresponds nearly to that of middle daylight with color temperature 6,500 K. Both properties together lead to the fact that standard illuminant D65 is regarded as a kind of universal reference illuminant. If, on the other hand, a tungsten filament lamp of low emission power in the violet and blue range is used, then the relative measuring error rises seriously – especially for dark color samples. This can be avoided by using a xenon

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lamp. Additionally, the nearly constant emission energy of a xenon lamp over more than 5×105 measuring flashes is of economic advantage. The output light impulse of 1 ms duration brings only a minor thermal effect to a normal colored sample (cf. Section 4.2.6). Furthermore, the emitted light energy of only few milliwatts is not a major influence on the instrument drift or aging of optical elements. A major effect contributing to the long-term drift occurring during instrument operation is thermal – due to ohmic heating of electrical components. This may lead to slight thermal variations in the elements. These elements are used for both the measurement and reference lights. Even the aging of the source, the inner reflecting surface of the Ulbricht sphere, and the optical waveguides are experienced equally by both beams. For a well-constructed spectrophotometer, such drifts and aging do not affect the measuring result, due to the fact that the measurement and reference beams experience these effects the same. Nevertheless, it is important to periodically keep an eye on the interior coating of the sphere because the absolute reflection of the coating material can be reduced by aging and, therefore, also the measuring sensitivity. The main advantage of the dual beam concept is the improved and nearly constant accuracy. For reliability, the endpoints of the gray scale of the color measuring device should be calibrated regularly. This might generally occur after the warm-up phase as well as at given regular intervals during operation. Modern instruments automatically ask for calibration if the measurement fluctuations are higher than the measurement sensitivity. If this temporary calibration is neglected, the obtained spectral values are unreliable and reproducible measurements are not ensured. For this, both secondary standards are needed. They are normally delivered with the spectrophotometer and one must be extremely careful working with and storing them. Unusable or lost standards should only be substituted by the instrument manufacturer, who should also perform the necessary new calibration. The calibration of a spectrometer with sphere geometry is carried out each time by reflection measurement with the inclusion of the specular component in the di:0 or di:8 geometry. For temporary calibration of the gray scale, it is assumed that the instrument drift between calibrations is so small that a linear correction is sufficient. This correction can be described as follows. The last obtained calibration values with the black and white standards are denoted as B0 and W0 . From a color pattern, the correct quantity Mcor with regard to B0 and W0 is obtained. In the meantime, instrument drift causes the corresponding new readings Bm , Wm , and Mm for the black and white standards as well as the same color pattern as before, respectively; cf. upper and middle scales in Fig. 4.12. Along with the basic theory of intersecting lines, the valid proportionality relation is given by Mcor − B0 W0 − B0 = . Mm − Bm Wm − Bm

(4.2.1)

4.2

Measuring Methods

263 B0

a) b)

Mcor

W0

Mm

Bm

Wm

c)

0

50

100

Fig. 4.12 Calibration of a spectrophotometer with black and white standards; the scales mean (a) last calibration, (b) before actual calibration, and (c) absolute scale

From this, the correct measurement quantity Mcor follows from the equation of a straight line Mcor = αMm + β

(4.2.2)

of positive slope W0 − B0 Wm − Bm

(4.2.3)

β = B0 − αBm .

(4.2.4)

α= and intercept

The coefficients α and β are separately calculated for each measurement wavelength λ and used for correction of the corresponding measurement value Mm . The original calibration of the manufacturer and the absolute accuracy of the spectrophotometer are not altered by such a calibration procedure. Under careful and normal operating conditions, the calibration of the wavelength or angle scale of gonio spectrophotometers is necessary only in extremely rare cases. Nevertheless as a routine matter, it is reasonable to undertake an inspection or new scaling after about 3 years of continuous operation. For calibration of wavelengths, emission lines of different sources are used (Section 4.1.1). Independent of that, it is recommended that a fundamental reconditioning of the instrument be performed before calibration of new color pattern series for recipe prediction.

4.2.3 Accuracy of Spectrophotometers In addition to the short- or long-term reproducibility issues, all measuring devices are subject to statistical measurement uncertainty. This is independent

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of the temporary calibration. Apart from this, equipment from the same or different manufacturers does not normally give exact the same measuring results. Because the spectrometric procedure is the most important color measuring method in color industry, we deal with its accuracy here in more detail and also the possibilities for enhancement. The more precisely the spectral values Ri = R(λi ) or Ti = T(λi ) are determined, the lower the measurement uncertainty. If we concentrate on the standard color values X, Y, Z given by Equations (3.1.1), (3.1.2), and (3.1.3), the corresponding error increases with increasing number of spectral values i = 1, 2, . . . , N. In some spectral range, apart from the reduction of the wavelength step width Δλ, an increase of N is only useful if the measurement uncertainty of each spectral value is simultaneously reduced. Because the color values of the CIELAB or DIN99o system and further evaluation formulas depend on X, Y, Z in a nonlinear fashion, an analytical representation of the respective error propagation is nearly impossible. The individual and overall error can be estimated with an experimentally determined dependability – which is sufficient for most practical applications. Especially for dyes in solution with erratically changing spectral values, the photometrical resolution should be precise to five valid significant figures. At worst then, this means that the fifth digit is incorrect and the corresponding relative error of one spectral value is in the thousandth range. Modern spectrophotometers meet virtually this condition, cf. Table 4.4. From the given

Table 4.4 Error limits of spectrophotometers Measuring property

Error limit

Photometrical resolution at Ri = 1.0

|ΔRi /Ri | ≤ 20 ppm

Short-term repeatability after calibration for ≤ 30 measurements with intervals of 10 s

White standard: ∗ | ≤ 0.05 |ΔRi /Ri | ≤ 50 ppm or |ΔEab Black standard: ∗ | ≤ 0.01 |ΔRi /Ri | ≤ 20 ppm or |ΔEab

Long-term reproducibility Standards and 12 BCRA II tiles

|ΔRi /Ri | < 0.1% or ∗ | ≤ 0.3 |ΔEab

Temperature drift for Δϑ = |ϑR − ϑN | ≤ 10 K , where ϑ R : room temperature, ϑ N = 20◦ C

|ΔRi /Ri | ≤ 1 × 10−3 × Δϑ/K or ∗ | ≤ 0.05 × Δϑ/K |ΔEab

Inter-instrument correspondence regarding 12 BCRA II tiles and master spectrophotometer

∗ | ≤ 0.15 instruments of same |ΔEab manufacturer ∗ | ≤ 1.0 devices of different |ΔEab manufacturers

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measuring uncertainties and the color differences in Tables 3.5 and 3.6, it follows that the accuracy of actual spectrophotometers can be well above the given color tolerances. Due to the simpler structure, however, the measuring uncertainty of a tristimulus colorimeter is about five to ten times higher. On the other hand, recall that a color difference is measured up to ten times more precisely using a spectrophotometer than by visual determination. For a color measurement device in continuous operation, the short-term repeatability needs to be checked from time to time. Note that this term is often incorrectly denoted as short-term reproducibility. Suitable internal or external standards, each with marked measuring fields, are normally used to check short-term repeatability. The measuring field is positioned in the indicated direction at the aperture; the spectral and standard color values are then determined repeatedly after time intervals of about 10 s. The resulting spectral values should have a relative error of no greater than ΔRi /Ri = ± 20 ppm for the black standard and ΔRi /Ri = ±50 ppm for the white standard. The color differences ∗ | ≤ 0.01 and |ΔE ∗ | ≤ 0.05, respectively; see Table should show values of |ΔEab ab 4.4. In addition, it is a good idea to determine whether or not the deviations of the spectral values accumulate in different wavelength bands or are uniformly distributed over the entire visible measuring range. The color difference contributions are analyzed in a similar manner with regard to systematic or color locus-dependent deviations from the reference colorations. However, the most important requirement for determination of the short-term repeatability is a stable operating state of the instrument after the warm-up phase. The full warm-up time is attained after the color values of a nonthermochromic sample show no further systematical trend after short-term repeated measurements. In any case, one should strive for constant room temperature and humidity during the measuring phase in order to avoid measuring results influenced by change of surround temperature and humidity. This phenomenon is known as temperature drift (Table 4.4). Color measurement devices are to be placed at a distance from direct heating or cooling sources and at best operated in constant standard climate. Spectrophotometers should additionally have a particularly high long-term reproducibility for several years. This also needs to be ensured for recipe prediction measurements. Verification of this follows using again the same standards as for testing the short-term repeatability as well as the chromatic and achromatic set of BCRA II tiles. The actual measurement values are compared with the previous ones and stored [34]. The long-term repeatability of constantly controlled and readjusted instruments should produce spectral relative devia∗ | ≤ 0.3. tions of ΔRi /Ri < ± 0.1% and color differences in the range |ΔEab Systematical trends need to be pursued over time and need to be eliminated. An expert readjustment is absolutely essential. In this context, for testing of short- or long-term behavior, one must also consider whether to refer to the stored scaling values or those of the actually

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measured reference standards. Experience shows that, in case of doubt, the data stored for many years are preferable over the actual obtained values. To be sure, both records should be evaluated and possibly subjected to a statistical examination (Section 4.3.4). In order to prevent unchecked data loss, a digital storage medium should be exchanged at least every 10 years (the way things are at the moment). A color difference value is used for measure of inter-instrument correspondence or rather absolute accuracy, for orientation. Because it contains no information about the direction or amount of each of the three color difference contributions, this quantity is of low meaning. Experience shows that equal or different instruments of one and the same manufacturer have relative uncer∗ | ≤ 0.15. This range is valid for mean reflection tainties in the range |ΔEab values in the domain 0.2 ≤ R (λi ) ≤ 0.6. Instruments from different manufacturers can show even higher differences, particularly for achromatic, brilliant, and especially for dark shades which frequently attain color differences up to ∗ | = 1.0. In other words, before substitution of an instrument or find|ΔEab ing of a tolerance agreement, it is necessary to perform reliable tests with suited color collections. The efficiency of the relevant instrument in different regions of a color space should be evaluated and be considered only from such measurements. The mentioned differences are somewhat higher than normal color tolerances for classical absorption colors. It is, therefore, useful that the absolute instrument accuracy is enhanced by all manufacturers. With regard to a tolerance agreement value, the visually agreed color difference alone is not sufficient, but also, apart from production fluctuations, above all the short- and long-term measuring uncertainties as well as the absolute accuracy of the measuring device need to be taken into account. Over and above that, with regard to tolerance agreement or color reproduction, one cannot go without exchange of corresponding color patterns between the interested parties: in particular it is not known to the people involved, for example, in what fashion the measured spectral values are accurately and uniformly corrected. In addition, it is not useful for color reproduction simply to convey the spectral values of a reference color pattern. On the basis of these considerations, the requirement for color toler∗ < 0.3 should be evaluated as unrealistically low: a person of normal ances ΔEab color perception ability can perceive such tiny color differences only in very rare cases.

4.2.4 Reflectance and Transmittance of Layers From color physical point of view, translucent colorations are of particular interest for at least three practical reasons: first, often a defined transmittance is required of a colored object, this goes well together with colorant loading

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267

and layer thickness; second, the degree of transmittance can be achieved with different sorts of binders or transparent plastic materials; and third, opaque colored layers are often manufactured with minimal covering layer thickness, this can be realized with aid of the two covering criteria. The determination of the reflectance and transmittance of a translucent material fundamentally needs two separate measurements. In the three mentioned cases, the reflectance and transmittance can be determined alone by two separate reflection measurements: the translucent layer is measured separately over a white and a black background of known spectral reflection. This method offers some practical advantages and is, therefore, common: – a spectrophotometer equipped for opaque colorations can additionally, without changing the measurement technology, be employed for translucent and transparent materials; – this method can be directly implemented in recipe prediction in such a manner that the required covering capacity value follows immediately together with the recipe, without the necessity of preparing any coloration in advance; – the covering of the layer over white and black backgrounds and the scattering and absorption of the coloration are directly accessible visually; – from colorations of transparent pearlescent or interference pigments over both backgrounds, the interference and the complementary color can be observed separately; – multiple reflections at various interfaces of the transparent or translucent layers are included in the evaluation formalism. This measuring procedure comes down to the mathematical structure of an accompanying algebraic relation. For derivation, we consider the structure of optically different layers shown schematically in Fig. 4.13. Adjacent interfaces are in optical contact with each other. A set of optically contacted layers means either painted over or pressed as to avoid air gaps. At each interface of different refractive indices, light is refracted, transmitted, and reflected. The relevant single processes at the interfaces of the upper and lower layers are also schematically shown in Fig. 4.13. The contributions to the reflection at the illuminated surface sum up to the quantity R13 . The total transmission T13 follows from the sum of the contributions that are transmitted by all layers. Note that the measured quantities R13 and T13 are composed of reflections and transmissions due to single processes in the volume and at the interfaces which need to be considered. The actual internal reflection and transmission follow, therefore, in an indirect way. Consequently, the evaluation formulas for the so-called external reflection and transmission follow from both reflection measurements.

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ˆ 1R2 T1T32T

R1

ˆ 1R22Rˆ 1 T1T34T

ˆT1

R1

T1T32R2

ˆ1 R

T1 T1T3

R2

ˆ

T1T32R2R1

T2

ˆ1 T1T2T33R2R

T1T2T3

Fig. 4.13 Reflection and transmission caused by multiple reflections from three layers and single reflection and transmission at both interfaces

For ascertaining of the corresponding expressions, we return to the quantities and single optical processes shown in Fig. 4.13. By summation of all reflection contributions at the upper surface of the layer, the combined reflection R13 is given by the geometric series R13 = R1 + R2 T1 Tˆ 1 T32 (1 + Rˆ 1 R2 T32 + · · · ),

(4.2.5)

which sums to the result R13 = R1 +

R2 T1 Tˆ 1 T32 . 1 − Rˆ R T 2

(4.2.6)

1 2 3

The entire transmission at the lower surface of the permutation of layers is given by the geometric series T13 = T1 T2 T3 (1 + Rˆ 1 R2 T32 + · · · ),

(4.2.7)

which sums to the result T13 =

T1 T2 T3 . 1 − Rˆ R T 2

(4.2.8)

1 2 3

Based on both Equations (4.2.7) and (4.2.8), we consider now two special cases of layer configurations. 4.2.4.1 Two Different Boundary Layers This configuration corresponds to the method of determining the reflection and transmission of a translucent layer by measurement over a black and a white background. In this special case, we assume optical contact of two different

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269

interfaces by leaving out the intermediate layer in Fig. 4.13. This leads to the evaluation expressions for determining the reflection and transmission of a layer. Because the middle layer is missing in Fig. 4.13, we set T3 = 1. In addition, we assume an isotropic upper layer, which means R1 = Rˆ 1 and T1 = Tˆ 1 . On account of Equations (4.2.6) and (4.2.8), the entire reflection R12 and transmission T12 of the layer combination are given by expressions R12 = R1 + T12 =

R2 T12 , 1 − R1 R2

(4.2.9)

T1 T2 . 1 − R1 R2

(4.2.10)

Because only reflection R1 and transmission T1 of the upper layer are of interest, we obtain R1 =

2 R12 T22 − R2 T12

(4.2.11)

2 T22 − R22 T12

and  T1 =



(R12 − R1 )

1 − R1 R2

 (4.2.12)

from Equations (4.2.9) and (4.2.10). However, the determination of R1 from Equation (4.2.11) is impossible because T2 and T12 are unknown; Equation (4.2.12), on the other hand, can only be evaluated if reflection R1 has already been determined. In order to solve this problem, as indicated in Fig. 4.14, we first denote the known reflection of the black and white backgrounds by Rbg,b and Rbg,w , respectively, and the measured layer in optical contact to each background by Rw and Rb, respectively. The spectral reflection values of both backgrounds should fulfill the conditions Rbg,b ≤ 0.2 and Rbg,w ≤ 0.8 for numerical stability. Substitution of the quantities R12 , R2 into Equation (4.2.9) by the respective corresponding pairs results in R = R1 + w

R = R1 + b

T12 Rbg,w 1 − R1 Rbg,w T12 Rbg,b 1 − R1 Rbg,b

,

(4.2.13)

,

(4.2.14)

from which the sought-after reflection R1 of the layer is given by

270

4 Measuring Colors Rb

Rw

T1

Rbg,w

T1

Tbg,w = 0

Rbg,b

Tbg,b = 0

Fig. 4.14 Procedure for determination of reflection and transmission of a layer by reflection measurements over two different backgrounds of reflections Rbg,w and Rbg,b

R1 =

Rbg,w Rb − Rbg,b Rw . Rbg,w (1 + Rbg,b Rb ) − Rbg,b (1 + Rbg,w Rw )

(4.2.15)

Performing the same substitutions for R12 , R2 in Equation (4.2.12), along with Equation (4.2.15), leads to two expressions for calculation of the layer transmission:    1 (4.2.16) − R1 T1 = (Rw − R1 ) Rbg,w and  T1 =

 (Rb − R1 )

1 Rbg,b

 − R1 .

(4.2.17)

From each result for T1 from Equations (4.2.16) and (4.2.17), the mean should be determined so that an error estimate is also possible. Clearly, the calculations of Equations (4.2.15), (4.2.16), and (4.2.17) should be repeated for each measured wavelength. The key to this procedure is the structure of Equation (4.2.9). 4.2.4.2 Transparent Liquid Between Two Equal Transparent Interfaces This case corresponds to the conditions shown in Fig. 4.13 for R1 = Rˆ 1 = R2 and T1 = Tˆ 1 = T2 . Such an optical configuration is given, for example, by a cuvette. The interface reflection R13 and the measured transmission T13 follow now from Equations (4.2.6) and (4.2.8):

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271

R13 = R1 + T13 =

T12 T32 R1 1 − T32 R21 T12 T3

1 − T32 R21

,

(4.2.18)

.

(4.2.19)

If the transmission T1 and reflection R1 of the outer layers are known from a measurement of the empty cuvette and setting T3 = 1 in Equations (4.2.18) and (4.2.19), then the transmission of the intermediate material follows from T3 =

T12 2T13 R21



 2 /T 4 1 + 4R21 T13 1

−1 .

(4.2.20)

2 /T 4 << 1, this expression can be approximated by T ≈ In cases of 4R21 T13 3 1 2 T13 /T1 . Incidentally, the measured transmission values of a static and a stationary flowing liquid are equal within measurement uncertainty, provided that cavitation (vapor-filled cavities due to low pressure of the flowing liquid) is absent. Finally, it should be stated that the procedure with two known backgrounds can be applied to further configurations in addition to the cuvette just considered.

4.2.5 Auxiliary Optical Methods for Effect Pigments As discussed in the previous chapters, the flake-shaped effect pigments have a special position among the existing sorts of colorants. Though some spectral and colorimetrical properties of effect pigments should be determined with gonio spectrophotometer measurements, other characteristics such as sparkle, distinctiveness of image, orientation, or particle morphology, which influence the brilliance, lightness, or flop behavior, are difficult or impossible to measure with this method. In order to register those properties of effect colorants, it is necessary to perform further investigations possibly with the following instruments: – a device for determination of distinctiveness of image (DOI); – an equipment for quantification of sparkle and graininess of metallic pigment coatings; – an apparatus for characterization of leafing stability; – a light microscope for identification of absorption and effect pigments, transparency, morphology; size and particle size distribution, and other properties of effect pigments and their mixtures; – a scanning electron microscope (SEM) for higher optical resolution of surface structures or fracture cross sections of effect particles.

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Presently, there exists a DOI measuring instrument which allows for the determination of distinctiveness of image values at measuring angle μas = 30◦ and variation in the range Δμ = ±3◦ . This angle interval is larger than the Δμ = ±3◦ given in Section 3.5.1. This exceeds the requirements of the corresponding standard [35]. This equipment allows for the adjustment of the measuring angle with minimal step width of Δμas = 1.6 angular minutes. Therefore, quite small differences in distinctiveness of image and reflection are resolvable. The measuring uncertainty of the DOI value is less than ±0.5%. The instrument is equally suited for characterization of the sparkling effect. Quantitative methods for the determination of sparkle and graininess of effect colorations are, however, integrated into some of the newest generation of gonio spectrophotometers (Section 3.5.1). The quantity of sparkle is determined at three illumination angles β v = 15◦ , –45◦ , and –75◦ with respect to the vertical of the sample surface. These three angles imitate the effect of solar radiation under cloudless sky. The measurements are carried out using a CCD camera fixed at an angle μv = 0◦ relative to the same vertical. For each of the three measuring angles, the light that reflects off of the sparkle particles is registered by the CCD chip, transformed into lightness steps, and evaluated with an image editing software. From the information, the characteristics of sparkle area Asp , sparkle intensity Isp , and sparkle degree SG are determined separately for each of the three measuring angles. This instrument is additionally equipped with a diffuse illumination device because under diffuse illumination and at a small distance from the coating surface, only the graininess of the flakes of a given coloration can be observed. For the determination of graininess, the number of light reflections is summed up and evaluated in dependence on lightness. The graininess Gr follows from the sum of all reflections relative to a uniformly interpreted reference coating. The graininess value is, therefore, a relative quantity. Moreover, this instrument has a directional illumination at β v = 45◦ and measurement angles μas = –15◦ , 15◦ , 25◦ , 45◦ , 75◦ , 110◦ for spectral measurements. A separate measuring instrument equipped with three fixed sensors for direct determination of lightness flop index FI of metallic pigments has thus far not been available. Rather with normal gonio spectrophotometers, the illumination angle β v = 45◦ and the three measurement angles μas = 15◦ , 45◦ , and 110◦ contained in Equation (3.5.2) should be separately adjusted [34]. FI values determined with such instruments are accurate to about ±1.5%. A standardized method for determining the leafing behavior, which influences the metallic impression, is described in [36]. A light microscope serves mainly for identification of the type, morphology, and size of effect pigment particles of a single sort or in mixtures with different pigments. In the adjusted image plane, the transparency or covering capacity is assessed over a black background. In addition, the fractions of each of the comprised absorption and effect pigments can be estimated by the numbers of

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273

accompanying particles. A system of two identical light microscopes each with an integrated transmittent and incident illumination is worthwhile. The incident illumination should be equipped with a switch over bright- and dark-field illumination. The dark field is produced by a condenser, which, in this case, consists of a circular mirror system from which only the diffracted light of first order or higher orders generated by the sample material is used for optical imaging. The undeflected light is completely masked out – the reason why the image background is dark with this sort of illumination. The linear structures of higher orders are, however, brightly reproduced. Examples of bright- and dark-field images of same picture fields are shown in Color plates 8, 9, 10, 11, and 12.3 If the pigmentation is not too high, the following properties can be directly observed or estimated from the bright field: – sorts and sizes of effect pigment particles (brilliance, size distribution); – color and morphology of the pigments (chroma, cornflakes, metallized particles); – quantitative ratio of mixed absorption and effect colorants (chroma, hue, recipe composition); – covering capacity of the particles (leafing, DOI). The dark field delivers additional information with regard to: – formation of particle surfaces and edges (brilliance, pigment sorts); – effect particles parallelized or tilted with respect to the substrate surface (flop behavior, uniformity of sparkle); – surface coating of mica, silicon dioxide, or combination pigments and evaporated metal particles (pigment analysis); – colors of chromatic pigments (hue, fraction, and substitution of pigments). The method turns out to be particularly instructive for viewing the same image area alternating in both illumination conditions. In this way, details of single particles become more apparent. The microscopes should have 50–500 times optical magnification. The resolution capability is, however, more important. This capability means the smallest distance δ, under which two adjacent objects are perceived as even separated. Microscopes for science have a resolution capacity of about δ ≥ 400 nm for visible light. This is sufficient for light microscopic analysis of single effect pigments and colorant mixtures. A planar optical system especially without field curvature is advantageous, therefore, providing the most faithful reproduction. The microscopic instrumentation can be supplied by further devices suited for automated determination of sorts, sizes, and numbers of different particles. The analysis, archiving, and 3 Color plates

are inserted between Chapter 3 and 4.

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management of the digital images taken are carried out with the assistance of a computer as a part of expert system. The mentioned resolution capability can be further improved using a scanning electron microscope (SEM). With this type of microscope, an object is scanned point by point by an electron beam; the accompanying secondary emissions are digitized, interpreted as gray levels, and synthesized into a black and white picture. Instead of light, this imaging method uses electrons which allow for imaging much smaller resolutions than that which is possible with visible light: the electrons effectively have a much smaller wavelength. The effective electron wavelength depends on the accelerating voltage; a typical voltage of 30 kV produces an electron wavelength of 7.07 pm. In comparison to a light microscope, greater than 100 times better resolution capability (δ ≥ 1 nm) and a higher optical magnification (≤ 5,000 times) are achieved. This procedure, therefore, delivers more precise optical details of single effect particles than a light microscope. For example, the size and surface structure, the layer composition, and damage mechanisms of effect particles can be analyzed. Further advantages are the extremely high depth of field and the characteristic pseudo-three-dimensional imaging. Disadvantages include the high costs, high voltage supply, the careful and complex sample preparation procedures, and uncolored pictures. Examples of SEM pictures are given in Section 2.3, Figs. 2.32–2.36.

4.2.6 Fluorescent, Thermochromic, Photochromic Colors Color measuring of fluorescent, thermochromic, or photochromic materials requires other special precautions to be taken because the testing procedure immediately interferes with the color production of such colorants and, therefore, can falsify the measurement results. Depending on the molecular structure, fluorescent, thermochromic, and photochromic colors absorb radiation in the UV, visible, or IR range. After interruption of the radiation supply, the thermochromic and photochromic color changes are reversible. At first, we discuss problems which need to be taken into consideration for measurements of fluorescent colors; the explanations show similarities for both of the other colorants titled in this section. Fluorescent colorants are widely employed in lacquers, plastic materials, textile and paper fibers, and printing inks. These so-called optical brightening agents, e.g., in textiles, paper pulps, and safety colors, result in an increased spectral reflection in the visible range. In addition, the effect of fluorescence is employed in optical waveguides, in light microscopes for enhancement of contrast, and for detection of molecular mobility or marking of biomolecules. The phenomenon of fluorescence was discovered by Stokes and is due to the fact that radiation energy of the UV or visible range is absorbed at wavelength λA and, within less than 10 ns, it is reemitted as light of a longer wavelength λE

4.2

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275

Stimulation

Energy

Emission

λA

λE

λ

Fig. 4.15 Schematic representation of normal fluorescence stimulation and emission

in the visible or IR range [37]. Examples of fluorescent pigments are given in Table 2.3. Fluorescence stimulation and emission take place mainly in continuous energy bands which generally do not overlap one another. The connection between stimulation and emission is schematically shown in Fig. 4.15. The fluorescence characterized by λA < λE is called normal-Stokes emission. In quite rare cases, anti-Stokes emission at wavelengths λE < λA can occur. Such a type of fluorescence is only achieved by simultaneous absorption of light and thermal energies, because the energy of the emitted light is higher as the absorbed, cf. Equation (2.1.2). This phenomenon has up to now not been observed in luminescence colorants. However, due to fluorescence emission, reflectance values higher than R > 1.0 can be produced. Color measurements of fluorescent colors are performed with the same measuring geometries which are already used for absorption colors. With regard to the stimulation and fluorescence emission, one should differentiate between two measuring modes from which the fluorescence component results. This procedure is termed two-mode spectroscopy and is based on the following different illumination and measuring conditions [1, 37–39]: First modus: polychromatic illumination and sequential monochromatic measurement; Second modus: sequential monochromatic illumination and polychromatic measurement. In the first modus, the fluorescent colorant is, for example, illuminated with a xenon or mercury high-pressure vapor lamp. The measurement is repeated serially using filters of narrow bandwidth (monochromator) but discrete wavelengths in the visible range. The whole of the measured spectral values correspond, therefore, to the visual impression of the color, provided that

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the measurements and observation are performed with the same illumination source. In the second modus, a monochromator produces an illumination at each given wavelength. The spectrum of fluorescence emission is measured at all wavelengths in the visible range. From the difference in the spectral values between the first and second modus, a measure of the extent of the fluorescence emission results. In Fig. 4.16, the two reflection curves obtained from the two-mode spectroscopy of a red fluorescence colorant are shown; the curves intersect at wavelength λs . From subtraction of both curves for wavelengths λ ≥ λs , a measure for the actual fluorescence emission results. The fluorescence behavior depends on some fundamental factors which make the handling of such colorants more difficult. For example, the height and width of the emission band of normal fluorescent colorants for wavelengths ≥ λs depend on: – the colorant concentration; initially, the emission intensity increases with increasing concentration, but the intensity is lowered while approaching a saturation concentration; this is caused by intrinsic absorption due to denser

ρ(λ) 1.2

Polychromatic Monochromatic

1.0

0.8

0.6

0.4

0.2 λS 0 400

500

600

λ nm

Fig. 4.16 Spectral reflectance of a red fluorescence colorant at polychromatic and monochromatic illumination; intersection wavelength λs = 600 nm

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277

molecular packing; the behavior is similar to chroma reduction of normal absorption colorants, cf. Fig. 3.12; – the sort and chemical structure of the dye or pigment; – the sort and operating mechanisms of additives; these can, by foreign absorption, even lead to total extinction of fluorescence radiation; – the chemical and physical constitution of the binder, e.g., the sort of solvent, polymer, or fibrous material.

During execution of both operation modes further aspects should be considered in order to avoid systematic measuring errors. Reliable and comparable results are only attained using the recommended measuring geometry 45c:0, therefore, the specular component is excluded. Otherwise the excitation energy is additionally – but uncontrollably – increased by the specular reflection. In order to achieve constant excitation, measurements should be performed under stationary operating conditions of the instrument and each with the same illumination and measuring aperture. On the other hand, if fluorescent materials are measured using diffuse geometries such as de:8 or d:0, then the illumination of the sphere interior, which is superimposed by fluorescence emission, can lead to lower reflection factors. This suggests the insertion of the smallest possible measuring aperture. Otherwise the sample field is additionally too intensively illuminated by the emitted fluorescence light [40]. Although the empirical skills for recipe prediction of fluorescent colorants seem to be promising for textiles [41], the experimental realization is – in view of the metrological conditions mentioned – to be judged extremely skeptically. Fluorescent colorants are, therefore, not included in the discussion of recipe prediction in Chapter 6. Similar considerations as the above can also be applied to thermochromic colorants. A reasonable fraction of absorption colorants behave thermochromic, that is, to a high degree the reflection or transmission depends on the temperature of the colorant. The change in temperature can be caused by thermal or optical radiation. Thermally or optically produced color dependence is, however, undesirable for normal absorption colorants. On the other hand, the color changes of thermochromic substances are specifically used, e.g., for determination of surface temperatures to more than 100◦ C. With special liquid crystalline polymers of cholesteric texture, temperature measurements are often accurate to Δϑ = ± 0.1 K. If the color pattern was exposed to undefined illumination or heat release of the instrument, color measurements of thermochromic materials should only be performed after temperature equilibration. For illumination of the sample, only flash lamps of maximum 0.5 ms pulse duration combined with low energy release of no more than 0.1 mW should be applied. Sequential, short-termed

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monochromatic illumination at discrete measuring wavelengths also turns out to be effective. For systematic investigations of thermochromic behavior, the color pattern is fixed in the instrument and measured in this position several times in a row. After each measurement, the time is recorded and the color differences with respect to the first measurement are determined. The initial and the end temperatures always need to be recorded as well. The specific temperature control of the sample holder is much more well defined. Consequently, the temperature variation can be precisely registered and the thermochromic temperature interval can be determined. Thermochromic behavior, however, can also be mimicked by changing humidity. During measurement and storage one must, therefore, ensure constant air humidity, even better using a standard climate. Measurements on photochromic colorations can be performed just as precisely. Photochromic substances in textile fibers or optical glass are subject to color changes by light absorption in the UV or visible range. Also this sort of color change is reversible; the absorbed energy is dissipated over time in the form of thermal radiation. Phototropy (synonymous to photochromism) can be evoked by different mechanisms such as isomerization of covalent bonds, absorption or emission of photons, tautomerism by proton transfer, cyclo-addition, and dissociation [42]. For measurements of photochromic colorations, the geometries 45c:0 or de:8 installed with smallest possible illumination and measurement aperture are at best suited. The time constant for reaching color saturation and the decay constant result from color measurement in dependence of time under different constant illuminations. The repeated measurements should be performed in the same way as for thermochromic colors. Because of the time-dependent optical behavior and rare demand, thermochromic and photochromic colorants are out of the scope of this text and not included in discussion of recipe prediction.

4.3 Uncertainties of Spectral Color Measurement It is impossible to make a statement about the trustworthiness of a result so long as we have no information about the numerical uncertainty of a measured quantity: if the accuracy of the measurement results is not available, the numerical values are of no meaning. The accuracy, reproducibility, and carefulness with which spectral or colorimetrical quantities are determined are deciding factors for color matching, dependability of covering capacity, color tolerances, acceptability of color differences, and observance of criteria for recipe prediction, among other things. First, we discuss the types and characteristics of errors. Afterward, we give insight into fundamental methods for determining and verifying measurement uncertainty and error distributions of three-dimensional

4.3

Uncertainties of Spectral Color Measurement

279

variables. Finally, we focus on statistical testing of color differences. This is of importance, for example, for color matching or quality control in color industry.

4.3.1 Qualitative Errors Since the beginnings of exact natural sciences, there has been a sobering concept: irrespective of how careful, well considered, or prepared a measurement is performed, the measurement and observational data are unavoidably combined with measurement errors. The combined errors, which occur during measurements, are called deviations. The indicated error of a measurement result is, however, denoted as measurement uncertainty [43]. Because measuring is inevitably accompanied by errors, one always attempts to keep the deviations as small as possible by suitable approaches. This is done in order to obtain the best possible approximate values under the given conditions with regard to the quantity to measure. In addition, as possible, it is useful to estimate the measurement uncertainty. For this, well-established methods of mathematical statistics can be used. These methods of operation are founded on the realization that the error-free measured value is absolutely not known: the range of possible values can only be narrowed with the methods of theory of errors [44 – 47]. Experience shows that the invariably occurring measurement errors are divided into three primary categories which can be considered to be independent of each other: gross, systematic, and statistic or random errors (stochastic uncertainty). Gross errors are to put down to falsely taken readings, confusions, inattentiveness, etc. These are often due to the mistakes of the person doing the measuring. Of course, such errors are not to be treated by the methods of error theory. This type of error is often discernible in a measurement series as socalled anomalies. The nature and way that the anomaly occurs often gives indications as to how such errors can be eliminated. In context of color patterns, gross errors are created, for example, by: – undefined, unequal, defective, or mixed up color samples, binders, backgrounds, or substrates; – employment of color patterns with the wrong surface or surface structure; – manufacturing with inappropriate colorants or incorrect parameters, aged or stored color patterns; – confusion of colorant components, additives, or binders. Systematic errors are based on incorrect conditions which, to a great extent, are due to limitations of the measuring instrument or evaluation method. Among those are:

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– incorrectly chosen measuring method(s); – incorrect measurement taking with not considering the measurement instructions; – wrongly calibrated measuring instruments; – drift of long-term reproducibility or lack of absolute accuracy of color measuring instruments; – incorrect evaluation methods or inapplicable formulas, approximations; – comparison of incorrect colors in test series.

Systematic errors usually alter the measurement result by the same amount or in the same direction. The detection of systematic errors is sometimes difficult and requires a certain sensibility and experience. This type of error is certainly controllable by supervision and selection. However, discovered systematic errors are also relevant in this context, for example, the edge loss of color patterns especially by measurements using spherical geometry. Random errors are caused by stochastic variations of the measuring conditions or other statistical influence factors. Statistical errors can change the measurement result in the positive and negative directions – in contrast to systematic errors. In accordance with theory of errors, experience shows that random errors can be reduced by either improved measuring conditions, sufficiently high number of repeated measurements, or suitable evaluation procedures. In practice, such methods can be limited by economic constraints. Despite these restrictions, the measuring conditions and extent should be structured in such a manner that the uncertainty obtained can be assessed as acceptable and trustworthy. Now that we have become acquainted with the three categories of measurement errors, we discuss in the following section the determination and reduction of purely random measurement errors by mathematical methods. It is assumed that the measuring uncertainty is influenced only by stochastic errors and all other error types are excluded. Even though the error-free value of a measured variable remains unknown, with these methods it is possible to specify the interval which contains the error-free value within a certain probability.

4.3.2 Quantitative Errors and Error Distribution The uncertainty of a measurement result is most simply indicated by estimation. This method is possible in cases where a computed uncertainty is only for verification of whether the measurement corresponds to the expected aim or not, or if it seems to be successful for orientation in similar problems. Such estimations are somewhat critical with regard to several independent or dependent

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Uncertainties of Spectral Color Measurement

281

measurement variables. The exact determination of such dependent measurement uncertainties with regard to color values is discussed in the next two sections. For the moment, it is useful to concentrate on a single measurement variable. In order to reduce the uncertainty of measurement value x, the measurement is repeated N times – each time under identical conditions. It is self-evident that the result from this series with random (statistical or stochastic) readings is always more trustworthy than a single measurement value. The N values xi , for i = 1, 2, . . . , N, are called parent population. For reduction of the uncertainty of the N measured values, the arithmetic mean value x¯ is determined N 1 xi . x¯ = N

(4.3.1)

i=1

In order to derive the uncertainty of x¯ , the deviations from the mean vi = xi − x¯ are unusable because the sum over all N differences vi vanishes N  i=1

vi =

N 

(xi − x¯ ) = 0

(4.3.2)

i=1

on account of Equation (4.3.1). Therefore, the sum of squares v2i is summed up as the squared deviations. The expression   N  1   σ = ± v2i (4.3.3) N−1 i=1

is the standard deviation of the measurement of the entire parent population of individual measurements for N ≥ 2. The denominator in Equation (4.3.3) contains the quantity N – 1, because a single measurement value has no mean. The value of  σ is a measure of the magnitude with which most of the N individual values xi fluctuate around the mean x¯ . It is also called mean deviation or scattering; the quantity  σ 2 is called variance. The middle quadratic error of the mean,  σAM , is called standard deviation, SD (or root-mean-square error, r.m.s. deviation) and follows with Equation (4.3.3) from  σ  σAM = √ , N

(4.3.4)

where the subscript AM means arithmetic mean. From Equations (4.3.3) and (4.3.4), it should be clear that the deviations are smaller for a larger number N of the measurement population. This is synonymous with the fact that the accuracy of the mean (4.3.1) increases with the number N of repeated measurements. This dependence on the number N corresponds with hundreds

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4 Measuring Colors

of years of experience and is, in general, called law of large numbers. For reasons of economy and of time, the number N is generally limited to a reasonable value given the measurement time. How much time is necessary to spend on increasing N depends, in a practical sense, on how critical the actual measurement is. In addition, if an excessively long series of N measurements is performed, then a drift can actually make the statistics worse if carried to an extreme. Therefore, there is an additional practical limit to the concept. The quantities  σ and  σAM are approximates of the real SD σ of the actual parent population. According to the theory of errors, the probable, mean, and ηAM , and  σAM , respectively, are in proportion to root-mean-square error  γAM ,  each other: ηAM :  σAM = γAM :  

2 : 0.8 : 1.0. 3

(4.3.5)

The result of the measured variable is a narrowed value and can be given in the form x = x¯ ± ΔxAM = x¯ ±  σAM .

(4.3.6)

The quantity ΔxAM is called the absolute error and has the  same unit as the mean x¯ . The relative error of x¯ is given by the quotient ΔxAM x¯ and is dimensionless. Therefore, the measurement result can also be indicated as4 x = x¯ ±

ΔxAM  σAM = x¯ ± . x¯ x¯

(4.3.7)

In addition, the probability for which the error-free value of x is in the computed interval x¯ ±ΔxAM can be determined. The probability P is a dimensionless value in the closed interval [0, 1] and it characterizes a statistical event. A value P = 0 means that the event happening is not possible; however, P = 1 represents an absolutely certain event. Although a dimensionless value of, e.g., P = 0.95 means a high probability, it is not for sure that the event actually occurs. Considerations in this regard go back to Gauss. For random errors, the single values xi typically scatter symmetrically about the arithmetic mean x¯ . Because SD becomes smaller with increasing number N according to Equation (4.3.4), the frequency of large deviations from the mean is decreased. In the case N → ∞, the discrete measuring values xi become continuous quantities x. The mean x¯ then equals the error-free value μ = x¯ and the r.m.s. deviation  σAM becomes 4 In

connection with expression (4.3.7), the extra terms introduced representing rough accuracy and precision of a measurement result should be differentiated: the fictive quantity of accuracy stands for the difference between the mean x¯ and the error-free, but unknown value; precision is simply a measure for standard deviation  σAM .

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Uncertainties of Spectral Color Measurement

283

σ = σAM . The accompanying distribution curve of the measured results is called the Gaussian distribution or normal distribution.5 The corresponding function f(x) is called the distribution function. Now the question should be addressed as to whether it is possible to predict for every experiment the limit distribution f (x). In most cases, it can be assumed that the limit distribution is the Gaussian distribution: ! (x − μ)2 1 . f (x) = √ exp − 2σ 2 σ 2π

(4.3.8)

The graph of the Gaussian distribution takes the form of a symmetrical bellshaped curve. The density function is characterized by both the quantities μ and σ : the maximum is located at the true mean value μ = x¯ and the inflection points are located at x = μ ± σ . From integration of the normal distribution (4.3.8), the normal distribution function, follows "x F(x) =

f (t) dt.

(4.3.9)

−∞

√ The normalizing factor 1/σ 2π in f(x) follows from the plausible condition that the area enclosed by the Gaussian distribution takes the value 1. Accordingly, the normal distribution function F(x) takes for x → ∞ the limit F(x → ∞) = 1. This is synonymous with the fact that the error-free quantity x in Equation (4.3.6) is with absolute certainty, or probability P(x) = 1, found in the infinite interval – ∞ ≤ x ≤ ∞. Now, we assume that an arbitrary random variable X (stochastic variable) obeys the statistics of distribution (4.3.9). In this case, the probability P(X) for X to adopt an arbitrary value within the finite interval a < X ≤ b is given by the relation "b P(a < X ≤ b) = F(b) − F(a) =

f (t) dt.

(4.3.10)

a

If for integration limits in Equation (4.3.10) are used the deviations ΔX = ± lσ from the mean μ, which are given by a = μ – lσ and b = μ + lσ for l = 1, 2, 3, . . . , then the first three probable values are

5 Other distributions like binomial distribution, hypergeometric distribution, or Poisson distribution are not considered in this book.

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4 Measuring Colors

⎧ ⎨ 0.683 for l = 1 P(μ − lσ < X ≤ μ + lσ ) = 0.955 for l = 2 . ⎩ 0.997 for l = 3

(4.3.11)

The first of these cases means that already 68.3% of the N measuring values are located between the inflection points of f(x). The probability of finding a measured value in the interval μ−3σ < x ≤ μ+3σ is already 99.7%. Consequently, nearly all measured values are contained in this extended interval. If a parent population is subject to a Gaussian distribution, then it can generally be assumed that for N ≥ 30, the actual values can be reasonably given by those statistical 2 = σ 2. ones, i.e., μ = x¯ and  σAM

4.3.3 Normal Distribution in Three and More Dimensions With regard to colorimetrical aspects, the considerations up to now for one stochastic variable in Section 4.3.2 need to be extended to at least three. The numerical effort is now greater due to two reasons: first, the color values themselves depend on each other and second, the total deviation does not follow only from the deviations that each variable has from its respective mean. For the moment, we concentrate on the general case of n Gaussian random variables xk , for k = 1, 2, . . . , n. These are written in the form of a vector x = (x1 , x2 , . . . , xn )T . Instead of the Gaussian distribution function (4.3.8), the generalized density function 1 1 Φ(x) = · exp − (x − m)T C(x − m) n 2 1 2 / / 2 (2π) ( det C)

! (4.3.12)

[48] is then used. In Equation (4.3.12), m is an n-dimensional vector which contains the means μ1 , μ2 , . . . , μn of each random variable xk , each for infinite number of repeated measurements. The quantity det C is the determinant of the quadratic (n, n) matrix C; the elements of the respective inverse matrix V = C –1 consists of the variances and covariances of the n Gaussian values of vector x. The variance σˆ (xk ) for each random variable xk simply follows from Equation σ 2 . The covariance is composed of the Gaussian (4.3.3) by relation σˆ (xk ) =  variables xk , xl , where k = l; see below the case for three variables. Now multidimensional areas with equal likelihood random variables are of interest. These are given by the constant exponent of the generalized density function (4.3.12) without the factor –1/2: (x − m)T C(x − m) = const.

(4.3.13)

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Uncertainties of Spectral Color Measurement

285

This relation written in full corresponds to the equation of an n-dimensional ellipsoid of center point M = (μ1 , μ2 , . . . , μn ), the coordinates agree with the components of vector m. In the special case for colorimetry of n = 3 Gaussian variables x1 , x2 , x3 , the vector x = (x1 , x2 , x3 ) T follows. N repeated measurements comprising (x1i , x2i , x3i ) for i = 1, 2, . . . , N have values scattering around the respective means of x¯ 1 , x¯ 2 , and x¯ 3 . The corresponding single SDs superimpose over each other along the three coordinate axes. This can be illustrated by a three-dimensional ellipsoid with center in about M ≈ (¯x1 , x¯ 2 , x¯ 3 ). In analogy with the case of one dimension, in consequence of Equation (4.3.11), it follows that a longer semi-axis of the ellipsoid results in a higher probability that the result of the kth measurement (x1k , x2k , x3k ) is represented by a point inside of the ellipsoid. The parameters of this ellipsoid follow altogether from six quantities: first, the three variances σˆ (xk ), k = 1, 2, 3, which, with regard to Equation (4.3.3), are given by the squared deviations of the mean (xki − x¯ k )2 for the N single values 1  (xki − x¯ k )2 , N−1 N

σˆ (xk ) =

k = 1, 2, 3;

(4.3.14)

i=1

second, the three covariances σˆ (xk , xl ) for k, l = 1, 2, 3 with k = l. These are given by the combined deviations of two variables xk , xl for k = l: 1  σˆ (xk , xl ) = σˆ (xl , xk ) = (xki − x¯ k )(xli − x¯ l ). N−1 N

(4.3.15)

i=1

It needs to be determined for the covariances σˆ (xk , xl ) whether or not the deviations of both variables xk , xl for k = l are really independent of one another. In cases of dependence, both terms in brackets of Equation (4.3.15) have the same sign, this is the reason why the accompanying variances σˆ (xk ), σˆ (xl ) result in high values. For independent measuring values, covariances virtually vanish. Now, we form the matrix of variances and covariances ⎞ σˆ (x1 , x2 ) σˆ (x1 , x3 ) σˆ (x1 ) σˆ (x2 , x3 ) ⎠ , V = (vkl ) = ⎝ σˆ (x2 , x1 ) σˆ ( x2 ) σˆ (x3 , x1 ) σˆ (x3 , x2 ) σˆ (x3 ) ⎛

(4.3.16)

which, due to the commutative factors in Equation (4.3.15), is symmetrical with regard to the main diagonal. From the inverse V–1 , the elements ckl of the new matrix C follow: C = (ckl ) = V −1 ,

ckl = clk , k, l = 1, 2, 3.

(4.3.17)

These elements determine the equation of the accompanying scatter ellipsoid:

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4 Measuring Colors

c11 (Δx1 )2 + c22 (Δx2 )2 + c33 (Δx3 )2 + 2(c12 Δx1 Δx2 + c13 Δx1 Δx3 (4.3.18) +c23 Δx2 Δx3 ) = P(χ 2 ,3). The differences Δxk are given by the deviations Δxk = xk − x¯ k , k = 1, 2, 3.

(4.3.19)

The inhomogeneity P(χ 2 , 3) of the ellipsoid equation in Equation (4.3.18) is, in analogy to the one-dimensional case, the distribution function of the f = 3 Gaussian random quantities x1 , x2 , x3 . The quantity f is called degree of freedom. In the case of vanishing covariances, Equation (4.3.15) and, therefore, coefficients ckl = 0 for k = l vanish; therefore, the axis of the ellipsoid are parallel to the coordinate axis x1 , x2 , x3 , and these variables change independent of each other. The distribution function P(χ 2 , f ) for natural values f is given by

P(χ 2 , f ) =

1 2f /2 Γ (f /2)

"χ

2

ν (f /2)−1 e−ν/2 dν,

(4.3.20)

0

where Γ (f /2) is Euler’s gamma function [46]. The values of P(χ 2 , f ) = (χf ; S )2 with f = 1, 2, 3 are listed in Table 4.5 in dependence on different levels of confidence S = 1 – α, whereas α is the so-called critical value (significance level). The values of P(χ 2 , 3) and, therefore, the length of each semiaxis of the ellipsoid increases exponentially with S. The case f = 1 corresponds to the one-dimensional Gaussian distribution (4.3.8). Table 4.5 Threshold values in dependence on significance level S for two different freedom degrees f of the χ 2 probability distribution [47] (χf ; 1−α )2

χf ; 1−α

S = 1–α (%)

f=1

f=2

f=3

f=1

f=2

f=3

50.0 70.0 75.0 80.0 90.0 95.0 97.5 99.0 99.5 99.9

0.455 1.07 1.32 1.64 2.71 3.84 5.02 6.63 7.88 10.83

1.39 2.41 2.77 3.22 4.61 5.99 7.38 9.21 10.60 13.82

2.37 3.66 4.11 4.64 6.25 7.81 9.35 11.34 12.84 16.27

0.675 1.03 1.15 1.28 1.65 1.96 2.24 2.57 2.81 3.29

1.18 1.55 1.66 1.79 2.15 2.45 2.72 3.03 3.26 3.72

1.54 1.91 2.03 2.15 2.50 2.79 3.06 3.37 3.58 4.03

4.3

Uncertainties of Spectral Color Measurement

287

With regard to color values such as L, a, b, the corresponding quantities of mean, SD, variance, and covariance are only indirectly applicable. This is reasoned from the mathematical point of view that color values are not independent of one another. Nevertheless, the above-mentioned considerations are by no means useless with regard to color values because over a statistical test, it is possible to decide, for example, whether or not the assumed normally distributed color deviations of a special color series are caused by measuring error. The formulation of such questions is not at all restricted to color differences but also for graininess and sparkle area, for example. Furthermore, similar problems occur in a multitude of industrial fields; these would be insurmountable without statistical methods. Such a procedure especially geared to color differences is discussed in the next section.

4.3.4 Statistical Testing of Color Differences Statistical methods are an essential aid not alone in natural and social sciences but also in economics and industrial production. One aim of statistics consists of making a likelihood statement over an interesting property. This is only based on random (stochastic) spot checks. Although the meaningfulness of statistical statements are absolutely unreliable, statistics allows, however, for the possibility of stating confidence levels and testing of a hypothesis by suitable statistical test quantities. The accompanying approach is termed as statistical testing procedure [44, 45, 47]. We now follow the assumptions for examination of color differences of such a statistical test method. With a test method, at any one time it is possible to check an alternative question as to whether or not the mean value of a sample property μS deviates from the mean of the relevant reference property μR . As a rule, the so-called null hypothesis is formulated, which means both average values are assumed to be equal μS = μR , leading to a zero difference. It is then investigated by statistical methods, if the sample property according to this hypothesis turns out to be of high or low probability. For this assumption, as discussed in the previous section, there is a statistical level of confidence S = 1 – α with significance point (critical value) α of, e.g., α = 30, 5, 1, or 0.5%. If the determined probability falls below the given value, it is then assumed that the hypothesis is not true; the hypothesis is rejected. In order to make this decision, a test quantity t is created. This represents itself again a stochastic variable – for example, depending on the color value F to test. With the aid of the distribution of the test quantity and with regard to the chosen statistical security S, a critical value tS can be derived. If the sample property produces a test value |t| ≤ tS , then the hypothesis is not disproved by the sample property and the hypothesis can be accepted. Because of this, t is termed as statistically not significant. However, if |t| > tS , then the hypothesis

288

4 Measuring Colors

is rejected, t turns out to be statistically significant:  |t| =

≤ tS : hypothesis is statistically not significant > tS : hypothesis is statistically significant

.

(4.3.21)

The rejection is synonymous with the fact that the hypothesis is declined with the probability α, although it can still actually be true (the so-called error of first kind). In contrast is if a wrong hypothesis is accepted (error of second kind). Accordingly, application of the statistical testing procedure means that in α% of all cases, the hypothesis is wrongly rejected. Because of that, the quantity α is also termed as producer risk, since in α% of all cases, e.g., a colorant fabrication is rejected as inappropriate, though the production is actually satisfactory. At the latest, the objection should be raised at this stage that chromaticities are not independent of each other and, therefore, are not random variables. After all, the color values depend on – according to Section 3.1 – the standard color values X, Y, Z, those on the overlapping SCMFs x¯ (λ), y¯ (λ), z¯(λ), and the measured spectral values. Certainly, not the absolute values of the chromaticities, but color difference contributions ΔF such as ΔL, Δa, Δb or others are true stochastic variables. For small color differences, so to speak, the mathematical transformation of the spectral values disappears: it remains only the stochastic property of the color differences. The following considerations are consequently only valid for individual, small color difference contributions. These were already assumed to have value |ΔF| ≤ 3 (meaning |ΔE| ≤ 5) considering the non-uniformly scaled color spaces in the previous chapter. This sort of statistical test only answers the question as to whether or not an experimentally determined color difference contribution ΔF is to be considered incorrect due to measurement errors. A critical test quantity tΔF belongs to ΔF. As we will see below, this follows from a coefficient, which is contained in the accompanying equation of the standard deviation ellipsoid (4.3.18). Clearly, the determined ellipsoid coefficients are only valid for the measured sample and reference color patterns, the applied measuring instrument, and the chosen colorimetrical formalism. At least these three conditions need to be indicated with each determined ellipsoid. A previously determined ellipsoid, therefore, is only applicable to a new sample series, if the same conditions were used, such as the same reference colors, manufacturing methods, same spectrophotometer, the same illuminant. The following statistical significance testing (evidence testing) can be applied to colors and colorants as well as to intermediate and final products. For performance of the statistical testing of color differences with the assistance of statistic test values tS , the following procedure has proven to be worthwhile [48 –50]. At first, N ≥ 20 sample and reference color patterns are manufactured. If only one reference color needs to be matched, then it is

4.3

Uncertainties of Spectral Color Measurement

289

assumed that the accompanying mean color differences came from N repeated measurements. The procedure is, in general, divided into four steps: 1. determination of the desired mean color difference contributions ΔF between sample and reference colors ΔL, Δa, Δb, for example; 2. computation of the coefficients of the accompanying scattering ellipsoid ckl with k, l = 1, 2 , 3; 3. calculation of the test quantities tΔF;1 –α concerning the chosen statistical level of confidence S = 1 – α; 4. comparison of the numerical values of |ΔF | and |tΔF; 1−α | . If the condition |ΔF| > |tΔF; 1−α | is satisfied, the color difference contribution ΔF is significant to a statistical level of confidence of S%. However, if it turns out that |ΔF | ≤ |tΔF; 1−α | is valid, then the quantity ΔF is not significant to the same statistical level and the null hypothesis is accepted, cf. alternatives in Equation (4.3.21). In the first step indicated above, the mean color differences between the sample and reference color patterns follow from Sections 3.1.3 and 3.1.4 and Equation (4.3.1), for example, by ΔL = LS − LR , Δa = a¯ S − a¯ R , Δb = b¯ S − b¯ R ,   ΔC = CS − CR = (¯aS )2 + (b¯ S )2 − (¯aR )2 + (b¯ R )2 ,  ΔE = (ΔL)2 + (Δa)2 + (Δb)2 .

(4.3.22) (4.3.23) (4.3.24)

In Equations (4.3.22) and (4.3.23), LS , a¯ S , b¯ S and LR , a¯ R , b¯ R stand for the means of the color values of the sample color and reference color patterns, respectively. From both measurement series, the middle difference of the hue angle can also be determined Δh =

$ % $ %& 180◦ # arctan b¯ S /¯aS − arctan b¯ R /¯aR , π

(4.3.25)

and the mean value of the hue contribution ΔH =

a¯ R · b¯ S − a¯ S · b¯ R  . + 0.5 · (CS · CR + a¯ S · a¯ R + b¯ S · b¯ R )

(4.3.26)

In the above-mentioned second step, the ellipsoid parameters with the variances σˆ (Δa), σˆ (Δb), σˆ (ΔL) and covariances σˆ (ΔL, Δa), σˆ (ΔL, Δb), σˆ (Δa, Δb) are calculated. These follow from Equations (4.3.14) and (4.3.15) by substituting xk with Δxk :

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4 Measuring Colors

1  σˆ (Δxk ) = (Δxki − Δxk )2 , xk = L, a, b, N−1 N

(4.3.27)

i=1

1  (Δxki − Δxk )(Δxli − Δxl ), xl = L, a, b, (4.3.28) N−1 N

σˆ (Δxk , Δxl ) =

i=1

where Δxki = xki,S − xki,R is the difference between the color value xki,S of the sample and the reference color xki,R and Δxk is the average value of each connected N single color values Δxk1 , Δxk2 , ... , ΔxkN , k, l = 1, 2, 3. With the elements ckl of the inverse matrix, C = (ckl ) = V–1 , the equation of the pattern of dispersion is given by c11 (ΔL)2 + c22 (Δa)2 + c33 (Δb)2 + 2 [c12 ΔL · Δa + c13 ΔL · Δb + c23 Δa · Δb] = 1.

(4.3.29)

The third step is the determination of at best six statistical test quantities tΔF; 1 – α with regard to the various chromaticity differences. These are assumed to be Gaussian and are, therefore, normal distributed. The conditional equations for the single test quantities result from the formulation of the question, what is the likelihood that a color difference contribution ΔF is represented by a point inside the corresponding standard deviation ellipsoid? The equation of the 2 ellipsoid is given by Equation (4.3.33) with inhomogeneity χ3;1−α instead of unity. The test quantity t F;1–α of the color difference contribution F follows from the generally valid expression σΔF tΔF; 1−α = √ · χ3; 1−α , N

(4.3.30)

where σΔF is the SD with regard to the color difference contribution ΔF, N the number of measurements of the parent color population, and χ3; 1−α the distribution quantity of the chosen statistical level of confidence 1 – α (see Table 4.5). The SDs of the various color values are given with the geometrical properties of the scatter ellipsoid, which are represented by the coefficients ckl of the scatter matrix C. The test quantities for the color difference distributions of ΔL, Δa, Δb follow from: χ3; 1−α , tΔL = √ Nc11

(4.3.31)

χ3; 1−α tΔa = √ , Nc22

(4.3.32)

4.3

Uncertainties of Spectral Color Measurement

291

χ3; 1−α tΔb = √ . Nc33

(4.3.33)

In other words, the value of 1/σΔF equals the square root of the corresponding coefficient ckk (k = 1, 2, 3) in the ellipsoid equation (4.3.33). The test quantity for the total mean color difference ΔE is given by χ3; 1−α , tΔE = √ NcE

(4.3.34)

 2 whereas, using Equation (4.3.29), the reciprocal squared variance cE = 1 σΔE results in cE =

1 [c11 (ΔL)2 + c22 (Δa)2 + c33 (Δb)2 + 2(c12 ΔL · Δa (ΔE)2 + c13 ΔL · Δb + c23 Δa · Δb )].

(4.3.35)

In addition, the test quantities of moderate contributions of chroma tΔC , hue tΔH , and hue angle tΔh can be derived by consideration of the geometrical context in a color plane (cf. Fig. 3.6):  tΔC = χ3; 1−α

1 2N



 1 1 + , cS cR

(4.3.36)

where cS is given by cS = c22 cos2 h¯ S + c33 sin2 h¯ S + 2c23 sin h¯ S cos h¯ S ;

(4.3.37)

the quantity cR is immediately given by substitution of the index S with R into Equation (4.3.36), which means that the sample quantities are exchanged with the reference ones. The mean values of the hue angles h¯ S and h¯ R are calculated using Equation (4.3.1). Equation (4.3.36) corresponds to an ellipse in the a, b color plane of curve parameters sin h¯ S , cos h¯ S instead of Δa, Δb. In analogy, the test quantity of the hue contribution is given by χ3; 1−α tΔH = √ , NcH

(4.3.38)

¯ cH = c22 sin2 h¯ + c33 cos2 h¯ − 2 c23 sin h¯ cos h;

(4.3.39)

with

where h¯ is the arithmetic mean of h¯ S and h¯ R . With that, the statistical test quantity for the hue angle can finally be determined from

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4 Measuring Colors

tΔh = arctan

2 · tΔH ΔH

!  · tan (Δh 2) .

(4.3.40)

In the preceding formalism, the contributions Δa, Δb can be substituted by the color distributions ΔC, ΔH right from the beginning. In this case, analogous relations to Equations (4.3.30), (4.3.31), (4.3.32), (4.3.33), and (4.3.34) result, but they contain other ellipsoid coefficients c˜ kl . From there, the test values of tΔL , tΔC , tΔH , tΔE follow. The determination of tΔh is unchanged with regard to Equation (4.3.44). Finally, the fourth step is the decisive one. Now, one of the possibilities given in Equation (4.3.21) is true. This means that the hypothesis is either significant or not. The absolute value of the test quantity gives only an indication of how well the assumptions of the null hypothesis are fulfilled. A somewhat better, but always unreliable evaluation is given with the so-called sequential analysis. With this procedure, the double-ended risk of wrongly deciding (errors of first and second kind) is more manageable. The entire parent population N is not directly used, but rather stepwise with numbers of N = 2, 3, . . . . After each step, it is decided if the tested hypothesis is accepted, rejected, or if the procedure is to continue. With that, the number of random tests can often be substantially reduced. Although the formalisms described are complex to program and implement, a large portion of formulation of questions, for example, in quality control can only be answered with the aid of mathematical statistics. At this stage, we conclude this short exposition of measuring uncertainty and statistics of color measurements with respect to color difference contributions. The seemingly trouble-free measurement of color patterns is just compounded by the typical double meaning of the term color measuring. Those are the subjective visual perception and the metrological determination of color values – which are mutually dependent. In principle, the visual non-uniformity of the applied color spaces and, moreover, the nonlinear color development of absorption and effect colorants in dependence on concentration should also be mentioned. The color physical and colorimetrical properties and measurements of colorations with effect pigments are far more complicated to measure and evaluate. It is clear that an accurately applied and correctly performed measurement technique has developed itself into an indispensable instrument in industrial color physics.

References 1. CIE No 46: “A review of publications on properties and reflections values of material and reflection standards”, CIE, Bureau de la CIE, Wien (1979) 2. Hunter, RS, Harold, RW, Eds: “The Measurement of Appearance”, 2nd ed, Wiley, New York (1987)

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293

3. Berger-Schunn, A: “Practical Color Measurement”, Wiley, New York (1994) 4. Berns, RS, Billmeyer, FW, Saltzman, M: “Billmeyer and Saltzman’s Principles of Color Technology”, 3rd ed, Wiley-Interscience, New York (2000) 5. Hunt, RGW: “Measuring Colour”, 3rd ed, Fountain Press, Kingston–upon–Thames (1998) 6. Hunt, RGW: “The Reproduction of Colour”, 6th ed, Wiley, Chichester (2004) 7. Noboru, O, Robertson, AR: “ Colorimetry”, Wiley, Chichester (2005) 8. Schanda, J, Ed: “Colorimetry: Understanding the CIE System”, Wiley, Hoboken, NJ (2007) 9. Ishihara, S: “The series of plates designed as a test for colour-blindness”, Kanehara, Tokyo (1995) 10. DIN 6160: “Anomaloskope zur Diagnose von Rot-Gruen-Farbenfehlsichtigkeiten”, Deutsches Institut fuer Normung eV, Berlin (1996) 11. ASTM E 1499–97: “Standard Guide for Selection, Evaluation, and Training of Observers”, American Society for Testing and Materials, West Conshohocken, PA (2003) 12. ASTM D 1729–96: “Standard Practice for Visual Appraisal of Colors and ColorDifferences of Diffusely-Illuminated Opaque Materials”, American Society for Testing and Materials, West Conshohocken, PA (2003) 13. ISO 3664: 2000: “Viewing conditions–Graphic technology and photography”, International Organization of Standardization, Genf (2000) 14. CIE No 130: “Practical Methods for the Measurement of Reflectance and Transmittance”, CIE, Bureau Central de la CIE, Wien (1998) 15. Ditteon, R: “Modern Geometrical Optics”, Wiley, New York (1998) 16. DIN 5036-1: “Strahlungsphysikalische und lichttechnische Eigenschaften von Materialien; Begriffe, Kennzahlen”, Deutsches Institut fuer Normung eV, Berlin (1978, 1980) 17. DIN 6175-2: “Farbtoleranzen fuer Automobillackierungen”, Teil 2: “Toleranzen fuer Effektlackierungen”, Deutsches Institut fuer Normung eV, Berlin (2001) 18. ASTM E 2194–03: “Standard Practice for Multiangle Color Measurement of Metal Flake Pigmented Materials”, American Society for Testing and Materials, West Conshohocken, PA (2003) 19. ASTM WK 1164: “Standard Practice for Multiangle Color Measurement of Interference Pigmented Materials”, American Society for Testing and Materials, West Conshohocken, PA (2006); ASTM E 2539: “Standard Practice for Multiangle Color Measurement of Interference Pigments”, American Society for Testing and Materials, West Conshohocken, PA (2008) 20. DIN 6175-3: “Farbtoleranzen fuer Automobillackierungen”, Teil 3: “Messgeometrien fuerInterferenzpigmente”, Project, Deutsches Institut fuer Normung eV, Berlin (2006) 21. Alman, DH: “Directional color measurement of metallic flake finishes”, Proc ISCC Williamsburg Conf on Appearance 53 (1987) 22. Hofmeister, F, Maisch, R, Gabel, PW: “Farbmetrische Charakterisierung und Identifizierung von Mica-Lackierungen”, Farbe and Lack 98 (1992) 593 23. Gabel, PW, Hofmeister, F, Pieper, H: “Interference pigments as focal points of colour measurement”, Kontakte (Darmstadt) 2 (1992) 25 24. Cramer, WR, Gabel, PW: “Effektvolles messen”, Farbe & Lack 107 (2001) 42 25. Nadal, ME, Early, EA: “Color measurements for pearlescent coatings”, Col Res Appl 29 (2004) 38 26. DIN 5033: “Farbmessung”, Part 1-9, Deutsches Institut fuer Normung eV, Berlin (1979–2008) 27. ISO 7724-1: “Paints and Varnishes – Colorimetry – Part 1: Principles”, International Organization of Standardization, Genf (1984) 28. JIS Z 8722/C: “Methods of Measurement of Color of Reflecting or Transmitting Objects”, Japanese Standards Association, Tokyo (1994)

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29. ASTM E 308–08: “Standard Practice for Computing the Colors of Objects by Using the CIE-System”, American Society for Testing and Materials, West Conshohocken, PA (2008) 30. ASTM E 1164–07: “Standard Practice of Obtaining Spectrometric Data for ObjectColor Evaluation”, American Society for Testing and Materials, West Conshohocken, PA (2007) 31. ASTM D 2244–07e1: “Standard Practice for Calculation of Color Tolerances and Color Differences from Instrumentally Measured Color Coordinates”, American Society for Testing and Materials, West Conshohocken, PA (2007); ASTM D 3134 – 97: “Standard Practice for Establishing Color and Gloss Tolerances”, American Society for Testing and Materials, West Conshohocken, PA (2008) 32. CIE No 15.3: “Colorimetry”, 3rd ed, CIE, Bureau Central de la CIE, Wien (2004) 33. DIN Fachbericht 49: “Verfahren zur Vereinbarungen von Farbtoleranzen”, Deutsches Institut fuer Normung eV, Berlin (1995) 34. Rich, DC, Battle, D, Malkin, F, Williamson, C, Ingleson, A: “Evaluation of the longterm repeatability of reflectance spectrophotometers”, in: Burgess C, Jones, DG, Eds: “Spectrophotometry, Luminescence and Color; Science and Compliance”, Elsevier, Amsterdam (1995); ASTM E 2214–08: “Practice for Specifying and Verifying the Performance of Color Measuring Instruments”, American Society for Testing and Materials, West Conshohocken, PA (2008) 35. ASTM E 430–05: “Standard Test Methods for measurement of Gloss of High Gloss Surfaces by Abridged Goniophotometry”, American Society for Testing and Materials, West Conshohocken, PA (2005) 36. DIN 55923: “Pigmente: Aluminiumpigmente und Aluminiumpigmentpasten fuer Anstrichfarben”, Deutsches Institut fuer Normung eV, Berlin (1983) 37. Valeur, B: “Molecular Fluorescence”, repr 1st ed, Wiley-VCH, Weinheim (2005) 38. CIE 76: “Intercomparisation of Measurement of (Total) Spectral Radiance Factor of Luminescent Specimens”, Commission Internationale de L’Éclairage (CIE), Bureau Central de la CIE, Wien (1988) 39. ASTM E 991–06: “Standard Practice for Color Measurement of Fluorescent Specimens”, American Society for Testing and Materials, West Conshohocken, PA (2006) 40. DIN 53145-1/2: “Pruefung von Papier und Pappe – Messgrundlagen zur Bestimmung des Reflexionsfaktors; Messung an fluoreszierenden Proben”, Deutsches Institut fuer Normung eV, Berlin (2000) 41. McDonald, R: “Recipe prediction for textiles”, in: McDonald, R, Ed: “Colour Physics for Industry”, 2nd ed, Soc Dyers Colourists, Bradford (1997) 42. Duerr, H, Bouas-Laurent, H: “Photochromism”, Elsevier, Amsterdam (2003) 43. Kirkup, L, Frenkel, RB: “An introduction to uncertainty measurement”, Cambridge Univ Press, Cambridge (2006) 44. Zurmuehl, R: “Praktische Mathematik fuer Ingenieure und Physiker”, 5th ed, Springer, Berlin (1984) 45. Crawley, MJ: “Statistics”, Wiley, Chichester (2008) 46. Bevingston, PR: “Data reduction and error analysis for physical sciences”, 3rd ed, McGraw-Hill, Boston, MA (2003) 47. Sawitzki, G: “Computational statistics”, Chapman & Hall, Boca Raton, FL (2009) 48. Frieden, BR: “Probability, statistical optics, and data testing”, 3rd ed, Springer, New York (2001) 49. ASTM E 1345–98: “Standard Practice for Reducing the Effect of Variability of Color Measurements by use of Multiple Measurements”, American Society for Testing and Materials, West Conshohocken, PA (2008) 50. DIN 55600: “Pruefung von Pigmenten–Bestimmung der Signifikanz von Farbabstaenden bei Koerperfarben nach der CIELAB-Formel”, Deutsches Institut fuer Normung eV, Berlin (2008)

Chapter 5

Theories of Radiative Transfer

In this chapter, we ask the question as to how closer theoretical examinations could aid the calculation of the spectral values or rather color values which one would obtain experimentally. This is necessary in order to apply them to criteria such as strength of color, covering capacity, or most importantly to numerical recipe prediction. We will describe the optical radiative transfer in layers quantitatively with two different physical approaches: simple and multiple scattering. Although both formalisms achieve nearly the same results, for practical applications multiple scattering is preferred. In particular, for multiple scattering, the formalism is quite simpler, and the measurement geometry – which forms the basis of the experimental results – can be taken into account. In the following, we consider practical methods in order to obtain the interesting spectral and color values with a procedure as simply as possible. We concentrate on the two-, three-, and multi-flux approximations of the radiative field in detail. Within these formalisms, the light at the inside and outside of a colored layer is separated into two, three, or more fluxes. In this way, the interesting parts of reflectance or transmission of the optical material can be assessed. The efficiency of each approximation can be evaluated by means of special optical cases which we illustrate with the so-called optical triangle. The origin of these models basically goes back to questions of physical chemistry, atmosphere physics, and astrophysics. They are summarized by the radiative transfer equation, first formulated by Chandrasekhar [1]. The radiative transfer equation arises out of a basic thermodynamic principle which has fundamental physical meaning and permeates the discussion of this chapter.

5.1 Fundamentals Various interactions of light with the electric charges of involved molecules can occur in optical materials. On a macroscopic level, for example, one can observe reflection, absorption, scattering. These interactions may take place alone or altogether, and they are phenomena which are described by geometric optics. In G.A. Klein, Industrial Color Physics, Springer Series in Optical Sciences 154, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1197-1_5, 

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fact, reflection and absorption are formulated on simple physical principles, but they are both included in the radiative transfer equation in a more generalized form. With this fundamental equation, we are able to describe various properties of transparent, translucent, or opaque materials. Moreover, with the phase function, the nature and degree of the light interaction in materials – from isotropic to anisotropic – are characterized. This powerful factor, also called scattering function, indicates the probability of a certain scattering process. Although the radiative transfer equation allows other configurations, the forthcoming explanations are restricted to plane parallel layers and axial symmetric illumination. This restriction is in effort to keep the formalism and results as simple as possible.

5.1.1 Basic Concepts and Definitions In Section 2.1.5, we use the Poynting vector to characterize the energy transferred by an electromagnetic wave. The term radiation energy is closely related to the term intensity. Intensity is a measure of the energy transported through a specific surface during a unit of time and given with dimension W/m2 . In order to express the intensity, we consider a radiation field of energy dEf in the frequency interval between f and f + df. We assume that this energy enters a unit area da and is transferred in the direction of vector s with solid angle element dΩ during the time interval dt; see Fig. 5.1. A vector n perpendicular to da is called the surface normal. The direction of s is defined distinct with the declination and azimuth angles ϑ and ϕ, respectively. These angles can vary independently over the domains 0◦ ≤ϑ ≤ 180◦ and 0◦ ≤ ϕ ≤360◦ . The energy I

n

dΩ

ϑ

da

ϕ

Fig. 5.1 Three-dimensional direction of intensity I

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transferred dEf and the corresponding intensity If are associated by the relation dEf = If · cos ϑ · df · da · d · dt.

(5.1.1)

We regard this expression as the definition of radiation intensity. In general, the intensity If of light is a function of frequency f, time t, position r, and direction s. For our considerations, however, it is sufficient to assume that the intensity is independent of frequency f and time t. Therefore, instead of If we write simply I, and it is given only as a function of position r and direction s I = I(r, s).

(5.1.2)

Consequently, the dimension of intensity is J/m2 . Due to the time and frequency independence of the intensity, the forthcoming formalisms prove greatly simplified. Such simplification readily facilitates the calculation, particularly of the spectral values of reflection, transmission, optical coefficients, or of the phase function separately for each interesting visible wavelength. On the other hand, the calculation of the mentioned variables has to be carried out separately for each wavelength of interest. An additional energy quantity is radiant flux or just flux. Mathematically speaking, the term flux is the product of radiant light intensity I(r, s) and the cosine of the angle between surface normal n and direction s of the light, as in the equation F(r, s, n) = I(r, s) · cos (n, s).

(5.1.3)

Accordingly, the flux is only the component of the intensity aligned perpendicular to the surface of the material. The flux corresponds to the energy which is transferred through a unit area and solid angle of unit steradian1 (sr) in the direction of s; therefore, the dimension of F(r, s, n) is J/(m2 sr). Even though the term directed radiation is simple to imagine (Sections 2.1.5 and 2.1.6), we now give an additional formulation, to which we will return in the following. The directed radiation of intensity I at position r with angles ϑ 0 , ϕ 0 , can be described by the expression I(r, ϑ, ϕ) = I0 (r) · δ( cos ϑ − cos ϑ0 ) · δ(ϕ − ϕ0 ).

1 For definition

of unit Steradian see footnote 1 in Section 2.1.4.

(5.1.4)

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Here, δ(x – a) is the Dirac delta function, which is often used to simplify mathematical expression of functions used in physics and technology.2 The following two characteristics of the Dirac delta function are of primary importance in our context: first, for x = a the function value is infinity and zero for all other x; second, the definite integral of f(x) · δ(x – a) over x has the result f(a), provided that a lies in the integration interval and f(x) is a continuous function in the neighborhood around a. We assume a three-dimensional Cartesian coordinate system with position vector r = (x, y, z), as shown in Fig. 5.2. If we further consider a directed intensity I(z) at coordinate z and angles ϑ = 0, ϕ 0 = 0, from Equation (5.1.4) follows

I(z, ϑ, ϕ) = I(z) · δ( cos ϑ − cos ϑ0 ) · δ(ϕ − ϕ0 ).

(5.1.5)

Integration of this expression over semi-space in the ranges of 0◦ ≤ ϑ ≤ 90◦ and 0◦ ≤ ϕ ≤ 360◦ simply results in the intensity I(z). With these considerations, the intensity of diffuse radiation can be formulated in a practical way. The flux consists of all those intensities included in the angle ranges of ϑ and ϕ; therefore, the entire diffuse flux at position z results from an integration over all angle-dependent intensities I(z, ϑ, ϕ) in the semi-infinite space.

z

I(z, ϑ, ϕ)

ϑ Fig. 5.2 Three-dimensional orientation of the directed intensity I(z, ϑ, ϕ) with angles of declination ϑ and azimuth ϕ in Cartesian coordinates x, y, z

y

ϕ x

2 The Dirac delta function is not a function in usual mathematical sense; it rather belongs to the class of generalized functions known as distributions.

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299

5.1.2 Absorption and Scattering The first quantitative description of directed light interacting with an optical material goes back to the observations of Beer regarding dilute solutions. The description of his results was based on the theoretical work of Lambert. Assume, for simplicity, direct radiation is incident on a plane parallel optical layer. For this case, according to the theory of Lambert, within the infinitesimal distance dz between positions z and z + dz, absorption of the medium attenuates the incident intensity I0 (z) by an amount dI0 ; mathematically, this is stated by the relation dI0 = I0 (z + dz) − I0 (z) = −K · I0 (z) dz,

(5.1.6)

as shown in Fig. 5.3a. Integration of this expression results in the Beer–Lambert law of absorption I0 (z) = I0 · e−Kz .

(5.1.7)

I0 = I(z = 0) represents the incident directed intensity at z = 0 and K is the wavelength-dependent absorption coefficient of the medium. The Lambert–Beer law is an approximation which is only valid for weak absorption, for instance, in solutions with low dye concentrations. Deviations from this law, for instance, at higher concentrations and reasons for these deviations at a molecular level are out of the scope of the discussion presented here. The Lambert–Beer law is a special case of the Bouguer–Lambert law, which describes further exponential attenuation phenomena in physics, technology, or biology. In the following, we assume that a plane parallel layer contains N pigment particles per unit volume V. The particle number density is defined as n˜ = N/V. Note that in most cases, the incident radiation is not only partially absorbed

I(z + dz, ϑ, ϕ) I0(z + dz)

I0(z)

I0(z)

dz

dz z + dz

z a)

z + dz

z b)

Fig. 5.3 Change of intensity I0 (z) as a result of particle interactions between the positions z and z+dz: (a) pure absorption and (b) absorption and scattering (schematically)

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5 Theories of Radiative Transfer

but also scattered by the colorant particles in the material. The direction of the intensity I(z + dz, ϑ, ϕ) scattered during propagation from position z to position z+dz is given as a function of angles ϑ and ϕ; see Fig. 5.3b. Now, the change of intensity dI0 due to absorption and scattering within the distance dz is given by dI0 = −(K + S) · I0 (z) · dz.

(5.1.8)

Compared to Equation (5.1.6), this relation now is expanded by the inclusion of the scattering coefficient S. In the case of three dimensions, Equation (5.1.8) changes to (s · grad)I(r, s) = −(K + S) · I(r, s).

(5.1.9)

In this expression, (s · grad)I(r, s) is the vector gradient, which describes the three-dimensional change of I(r, s) between positions r and r + sdr in the direction of s in the material; see Fig. 5.4. Due to Equations (5.1.6), (5.1.7), and (5.1.8), the differential equation (5.1.9) is called the generalized Lambert–Beer law; it represents a special case of the general equation of radiative transfer. Both the absorption coefficient K and the scattering coefficient S can be expressed in terms of the number density n˜ K = n˜ σK ,

(5.1.10)

S = n˜ σS ,

(5.1.11)

with the inclusion of the quantities σK and σS , called cross section per particle of absorption and scattering, respectively; their units are area per number of

sdr

r r + sdr

Fig. 5.4 Light beam at position r traveling the distance sdr in direction of s

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particles. Hence, K and S are expressed in the same units, namely a reciprocal length. The intensity of light I(ϑ, ϕ) scattered in the direction with angles ϑ and ϕ is the product of the incident intensity I0 (z) at position z, the number density n˜ , and the three-dimensional change of the scattering cross section I(ϑ, ϕ) = I0 (z) · n˜ ·

dσS (ϑ, ϕ) dz. d

(5.1.12)

This expression can be interpreted as the general definition of the differential scattering cross section dσS /d. If we integrate the quotient in Equation (5.1.12) over the entire space we obtain the total scattering cross section " σS =

dσS (ϑ, ϕ) d. d

(5.1.13)

Another important quantity in our considerations is the scattering function or phase function p˜ (ϑ, ϕ, ϑ0 , ϕ0 ). In the literature [1], the phase function is introduced by the expression n˜ dσS (ϑ, ϕ) 1 · p˜ (ϑ, ϕ, ϑ0 , ϕ0 ) = · . 4π S d

(5.1.14)

By integrating this expression over the entire three-dimensional space and using Equations (5.1.11), (5.1.12), and (5.1.13), the result is unity: 1 4π

" p˜ (ϑ, ϕ, ϑ0 , ϕ0 ) d = 1.

(5.1.15)

This result shows a first important property of the phase function: the value of the quotient p˜ (ϑ, ϕ, ϑ0 , ϕ0 )/4π corresponds to the probability that light of direction ϑ 0 , ϕ 0 is scattered into the direction of ϑ, ϕ. Further features of the phase function are described in Sections 5.1.5 and 5.1.6. By consideration of Equations (5.1.12), (5.1.13), and (5.1.14), the scattered intensity is given by I(ϑ, ϕ) = I0 (z)

S p˜ (ϑ, ϕ, ϑ0 , ϕ0 ) dz. 4π

(5.1.16)

If we replace ϑ 0 and ϕ 0 by ϑ and ϕ and use Equation (5.1.5), the relation I(ϑ, ϕ) =

S dz 4π

"

I(z, ϑ, ϕ ) p˜ (ϑ, ϕ, ϑ, ϕ ) d

(5.1.17)

follows. The scattered intensity I(ϑ, ϕ) is, therefore, proportional to sum over all intensities of direction ϑ , ϕ at position z multiplied by the accompanying phase

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5 Theories of Radiative Transfer

function. Finally, by integrating this equation in respect of z, we obtain the intensity J(r, s) scattered into direction s at position r; see Fig. 5.4. Consequently, the expression for the so-called source function follows: " S (5.1.18) J(r, s) = I(r, s ) p˜ (s, s ) ds . 4π The source function describes the intensity at position r, which is scattered from direction s into direction s. Both the source function and the Lambert– Beer law are included in the radiative transfer equation. Before discussion of this fundamental equation, it is useful to deal with scattering processes in more detail.

5.1.3 Single and Multiple Scattering Scattering processes can have a variety of origins; phenomenologically they can be caused, for example, by the morphology of the interacting particles, their geometry, packing, or arrangement of electrical charges. Scattering of light can be described quantitatively only if the mean particle size r and the wavelength λ of the incident light are of the same order of magnitude. In contrast to Raman or Brillouin scattering, the incident and scattered radiation normally have identical wavelengths for absorption and effect pigments that we are interested in here, as well as for binders or color additives. The intensity of the scattered light is not necessarily isotropic; it can be anisotropic with angle dependence. Phenomenologically, we can distinguish three concepts of scattering interactions: single scattering, multiple scattering, and dependent scattering. In single scattering, the intensity I(ϑ, ϕ) scattered with declination angle ϑ and azimuth angle ϕ is directly proportional to the product of the number density n˜ and the layer thickness d of the material I(ϑ, ϕ) ∝ n˜ · d.

(5.1.19)

Single scattering takes place at each particle separately and independently of the particle’s neighborhood. A strict theory of single scattering for molecules with r << λ was worked out independently by Rayleigh and Born [2]. Mie developed a theory [3] of single scattering by dielectric and absorbent spherical particles with radius r > λ. His theory was later extended to particles of other geometrical shapes. The ideas of Mie are described in detail in the literature [4, 5]. According to this theory an incident electromagnetic plane wave excites electric charge vibrations in a particle. The resulting amplitudes and phases are given by the solutions of the Maxwell equations of electrodynamics [6]. In this context, the particle charges

5.1

Fundamentals

303

2

3 1

0

Fig. 5.5 Calculated scattering functions for different particle radii: 0: isotropic scattering, particle radius r = 0 nm; 1: 10 nm; 2: 200 nm, 3: 400 nm; refractive index binder n = 1.50, refractive index pigment np = 1.74, wavelength 660 nm

behave as electrical multipoles. With increasing particle diameter, the amount of scattered radiation increases in the forward direction and decreases in the backward direction. In Fig. 5.5, the normalized scattering functions of four different particle radius are projected in a plane, for constant azimuth angle ϕ. The circle in the figure represents therefore a sphere in reality; this amounts to equal levels of isotropic scattering in all spatial directions arising from an infinite small dielectric particle at the center point of the sphere. The dumbbell-shaped diagram 1 shows equal amounts in forward and backward directions, in this case, produced by a dielectric sphere of radius 10 nm. The scattering functions 2 and 3, however, characterize purely forward scattering and are generated by particles with radii 200 and 400 nm, respectively. Forward scattering is in general dominant for particles with radius greater than about 100 nm. The scattering diagram 3 of Fig. 5.5 with dominant forward scattering is enlarged in Fig. 5.6. The angle of declination ϑ varies with respect to the direction of illumination and the azimuth angle ϕ varies perpendicular to the surface. The displayed scattering field with axial symmetry reminds one of the longitudinal cross section of an airship: the fuselage represents the predominant part of forward scattering. Moreover, the scattering field can be approximated by empirical functions. The scattering diagram in Fig. 5.6 is approximated with sufficient accuracy by the Henyey–Greenstein function (Section 5.1.6). Technically, the Mie theory of single scattering is only valid if the distance between particles is at least three times larger than their diameters.

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5 Theories of Radiative Transfer

ϑ

I (ϑ, ϕ)

ϕ Illumination direction

Fig. 5.6 Scattering diagram of a dielectric sphere with predominant forward scattering I(ϑ, ϕ); the angle of scattering ϑ varies parallel to the surface and the angle of azimuth ϕ perpendicular

Unfortunately, this requirement is not fulfilled for the most practical pigment particles fixed in a binder. Nevertheless, the results of this formalism based on electrodynamics are by no means of secondary significance in color physics. For a colored pigment, the reflection spectrum can be calculated approximately if the refractive index, concentration, and distribution of the dielectric particles are known [7, 8]. But this procedure is not pragmatic for practical questions because the calculations are too time-consuming, and, furthermore, the theoretical and experimental results are not identical. This disagreement may be caused by the fact that the geometry of the particles cannot be determined with sufficient accuracy necessary for a reliable model. In comparison to the single scattering of Mie, multiple scattering – the second method describing the scattering processes – has proven to be exceptionally effective for industrial applications. In nearly all pigmented systems single scattering changes into multiple scattering with increasing number density of the particles. Due to the closer particle packing, light is scattered not only once but also several times, provided that it is not absorbed or leaves the medium. According to the scattering of different particles with potentially different orientations, the scattered radiation is not always strictly directed, but rather diffuse and mostly anisotropic. The intensity of the scattered radiation I(ϑ, ϕ) is now a function of the product of particle density n˜ and thickness d of the layer I(ϑ, ϕ) = f (˜n · d).

(5.1.20)

To complete the picture, it is useful to briefly mention dependent scattering. Dependent scattering arises only if the packing density of the particles is so high that their mutual distance is comparable to the incident wavelength (e.g., in high solid paints); here the amount of scattered light is a function of particle density n˜ and layer thickness d, both variables are independent of each other I(ϑ, ϕ) = f (˜n, d).

(5.1.21)

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305

In the case of dependent scattering, the particles no longer scatter independently. This sort of situation means that scattering occurs from about units of particles and therefore requires a more detailed treatment which is beyond the scope of this book. First ideas about multiple scattering were formulated by Schuster [9] and Schwarzschild [10]. Both authors consider an absorptive and scattering medium and divide the internal radiation field into two anti-parallel and diffuse fluxes with intensities denoted as I+ (z) and I– (z); see Fig. 5.7. In the concept of Schuster, a system of two empirical differential equations shows the change of these intensities with z coordinate, +

dI+ = −(2 K + S)I+ + SI− , dz

(5.1.22a)

−

dI− = +SI+ − (2 K + S)I− , dz

(5.1.22b)

where K and S are the coefficients of absorption and scattering of the medium. The subdivision into two fluxes of opposite directions is carried out following the kinetic theory of gases. In this theory of Joule, to simplify matters, the statistical fluctuations of the particles are combined into three directions of movement, which are mutually anti-parallel and perpendicular; in this way, Joule reduced some gas properties such as pressure, temperature, or internal energy to well-known mechanical quantities like momentum or kinetic energy. z=0

I–(z) K, S z

I+(z) z=d

Fig. 5.7 Diffuse intensities I+ (z) and I– (z) within an absorptive and scattering medium

In this context, Kubelka and Munk [11, 12] proceed in a simple and empirical manner. Their approach differs from that of Schuster and Schwarzschild only in the weighting factor of the absorption coefficient, cf. Equations (5.1.22a), (5.1.22b), (5.3.1a), and (5.3.1b). In any case, it remains a fact that the results of this formalism are mathematically quite simple and not surprisingly of only limited applicability. Nevertheless, use of this theory is applied widely in the color industry. The experience of decades with these approximations shows that the real behavior of absorption and scattering in colors is incompletely represented by the empirical assumptions of Kubelka and Munk. Consequently their ideas have been supplemented and modified. Some of those theories, which certainly

306

5 Theories of Radiative Transfer

have no further importance for applications in color industry, are given in the literature [13–15]. In the following, we are primarily interested in the radiative transfer equation and the related approximations of the radiant flux.

5.1.4 Radiative Transfer Equation Based on modern understanding, the concept of multiple scattering offers the only real possibility in industrial applications for adequately describing optical interactions inside absorptive and scattering materials. An appropriate basis is offered by the radiative transfer equation (RTE), upon which all known phenomenological theories about light interactions are founded [1]. The RTE results directly from conservation of radiative energy in an isolated system. From another point of view, it can also be derived from the fundamental Boltzmann equation of statistical thermodynamics, which describes with time dependence the collisions of fluctuating particles in three dimensions [16]. If the particles are interpreted as photons then the time-independent Boltzmann equation reduces to the RTE [17]. In order to formulate the RTE, imagine a light ray incident in threedimensional direction s in a medium containing discrete particles such as pigments; see the illustration in Fig. 5.4 (Section 5.1.2). The ray is directed from position r to the infinitesimally distant position r + sdr. Suppose that the path sdr is much greater than the mean particle distance allowing for the assumption that the optical medium is homogeneous and isotropic. Anisotropic optical materials are treated in Section 5.1.6. Fluorescent or phosphorescent colorants are excluded from the following considerations, although it is possible to take them into account. Along the path from r to r + sdr, the beam undergoes repeated intensity alterations due to interactions with particles. Following the considerations in Section 5.1.2, we assume that all of these intensity changes have two contributions. First, the intensity I(r, s) decreases due to absorption and scattering caused by interactions with the particles along the distance sdr. The reduction of intensity can be described with the differential equation (5.1.9) – the generalized Lambert– Beer law. Second, other light incident on the direction s with intensity I(r, s ) is additionally scattered into the direction s. This causes an increase in I(r, s). We recognize that the net radiation transferred into position r + sdr corresponds to the source function J(r, s) in Equation (5.1.18). The entire three-dimensional change of intensity is, therefore, the generalized Lambert–Beer law combined with the source function, S (s · grad)I(r, s) = −(K + S)I(r, s) + 4π

"

I(r, s )˜p(s, s )ds .

(5.1.23)

5.1

Fundamentals

307

This expression is called the general radiative transfer equation (GRTE). Other transport phenomena, such as those in hydrodynamics, thermodynamics, or electrodynamics, lead to balance equations of similar structure: the entire flux is a result of the sink (absorption component) and the source (scattered contribution). The solution I(r, s) of the GRTE characterizes the optical properties of the medium considered. Even with the modified source function J(r, s), which is also called source term or scatter term, various optical materials can be characterized. In the case of pure absorption, the GRTE (5.1.23) becomes the multidimensional Lambert– Beer law (5.1.9). Due to the interaction between the pair of terms in Equation (5.1.23), further special optical cases can be characterized – those such as opaque systems or purely scattering materials. All of these systems may behave as isotropic as well as anisotropic media; this behavior is accordingly represented by the phase function p˜ (s, s ), which is contained in the integrand of the source term. The great importance of this function for various kinds of colorants is elucidated in the two sections that follow. The quantity ds = sin ϑ · dϑ · dϕ

(5.1.24)

in the GRTE (5.1.23) is the solid angle element for the direction of s , given by the angles ϑ and ϕ ; see Fig. 5.8. Integration of expression (5.1.24) over the entire space in the angular intervals 0 ≤ ϑ ≤ π and 0 ≤ ϕ ≤ 2π gives the scale factor 4π in the source function (5.1.18) as well as in the phase function (5.1.14). The three-dimensional position r and the directions s and s are not subject to restrictions. It is, however, useful to adapt these variables in order to realize I(z, ϑ⬘, ϕ⬘)

z

dΩs⬘

ϑ⬘

ϕ⬘

Fig. 5.8 Illustration of the solid angle element ds

308

5 Theories of Radiative Transfer

practical applications in color physics, among other things. In the next section, we transform the GRTE to describe the light interactions in a plane parallel layer.

5.1.5 Radiative Transfer in Plane Parallel Layers The primary concern in most situations in industrial color physics is the optical properties of colored layers. In this context, we now intend to cast the GRTE (5.1.23) in a suitable form so that it is directly applicable to coatings. Let us assume a plane parallel layer, extended to infinity in the x, y plane and of finite thickness d in z-direction. In order to simplify the equations, we employ modified polar coordinates instead of the Cartesian coordinates x, y, z, as shown in Fig. 5.2. Thus, the intensity I(r, s) is written as I(z,ϑ,ϕ), a function of the coordinate z, the declination angle ϑ with regard to the z-axis, and azimuth angle ϕ in the plane perpendicular to the z-axis. Furthermore, we replace the path differential dr by dr =

1 dz cos ϑ

(5.1.25)

based on the conditions shown in Fig. 5.9 and introduce the abbreviation μ = cos ϑ.

(5.1.26)

The solid angle element ds in Equation (5.1.24) can now be written as ds = −dμ dϕ . We additionally define the optical path length as τ = (K + S)z ,

dr

(5.1.27)

ϑ dz

z

Fig. 5.9 Relation between the differentials dr and dz in a plane parallel layer

5.1

Fundamentals

309

which is a measure of the attenuation of the incident radiation due to absorption and scattering along the path from z = 0 to z = d. Finally, we introduce the albedo which is defined by the quotient ω0 =

S . K+S

(5.1.28)

In contrast to directed reflection, the albedo indicates the relative amount of attenuation due solely to scattering at the surface or in the volume of a material. Consequently, the albedo is a measure of the scattering of the medium. For example, the surface of the Earth has a mean value of ω0 ≈ 0.3, clouds between 0.7 and 0.9. The albedo has some important properties to keep in mind. The domain is restricted to the interval 0 ≤ ω0 ≤ 1, and both domain boundaries represent special optical cases. The case ω0 = 0 characterizes pure absorption; the expressions (5.1.30), (5.1.31), (5.1.35), and (5.1.37) below turn into the generalized Lambert–Beer law with S = 0. The special condition ω0 = 1, on the other hand, is called the conservative case, which is a synonym for pure scattering. To restate, the boundary values of the albedo can be expressed by  ω0 =

0: 1:

pure absorption, Lambert−Beer law . pure scattering, conservative case

(5.1.29)

The term conservative case is derived from mechanical systems: if such a system fulfills the principle of conservation of energy, it is called a conservative system. Pure scattering means no absorption and the energy equation (2.1.14) takes the simple form R + T = 1. Based on this analogy, an optical system without absorption is called a conservative case. Finally the variables μ and ϕ in the GRTE (5.1.23) are restricted to the intervals −1 ≤ μ ≤ 1 and 0 ≤ ϕ ≤ 2π, which consequently determine the corresponding integration limits of μ and ϕ . If we take into account Equations (5.1.25), (5.1.26), (5.1.27), and (5.1.28) as well as the integration limits of μ and ϕ , the RTE (5.1.23) becomes the radiative transfer equation for plane parallel layers ω0 dI(τ ,μ,ϕ) = −I(τ ,μ,ϕ) + μ dτ 4π

"+1 "2π dμ p˜ (μ,ϕ; μ ,ϕ ) · I(τ ,μ ,ϕ ) dϕ . −1

0

(5.1.30) This formulation models all scattering and absorbing cases relevant to the physics and metrology. The optical path length τ is directly proportional to the absorption and scattering. The albedo ω0 characterizes the relative scattering and the phase function p˜ (μ,ϕ; μ ,ϕ ) indicates the degree of anisotropy together with the probability of scattering from direction μ , ϕ into direction of μ, ϕ.

310

5 Theories of Radiative Transfer

With the solution I(τ ,μ,ϕ) of Equation (5.1.30), both the reflection and the transmission of the layer can be calculated, provided that the so-called boundary conditions are known. As mentioned before, each required spectral value must be calculated separately for each wavelength of interest. On the other hand, the geometries for measurements are fully specified by the angles of declination and azimuth μ, μ and ϕ, ϕ , respectively; the quantities μ and ϕ correspond to the illumination and μ and ϕ to the measurement conditions. The calculated and correspondingly measured reflection and transmission can, therefore, be compared with one another if the constants for absorption and scattering as well as the phase function and the layer thickness are known. A variety of reasons allow for a simplification of Equation (5.1.30); the formalism and the computing time are reduced. First, we assume that the incoming light at z = 0 is independent of ϕ and ϕ, i.e., axial symmetry (see Fig. 5.10). Therefore, also a rotationally symmetric measurement has to be performed around the z-axis (e.g., 0:45c measuring geometry). In this case, the intensity I = I(τ , μ) is only a function of the optical path length τ and the direction given by μ. Furthermore, the phase function p(μ,μ ) is also independent of ϕ and ϕ; it represents now the amount of incoming light in direction μ which is scattered into direction μ. Integration of Equation (5.1.30) with regard to μ results in ω0 dI(τ ,μ) = −I(τ ,μ) + μ dτ 2

"+1 p(μ,μ ) I(τ ,μ )dμ . −1

z

I(z, ϑ, ϕ)

ϑ ϕ

ϑ⬘ x

y

Fig. 5.10 Coordinates for axial symmetry in a plane parallel layer [18]

(5.1.31)

5.1

Fundamentals

311

This special RTE is specific for plane parallel layers under axial symmetric illumination. It contains only the variables τ , ω0 , μ , and μ, which are proportional to the coordinate z and the angles ϑ, ϑ, respectively. All further explanations and calculations in the text will be based on these assumptions and therefore utilize Equation (5.1.31). Due to its mathematical structure, expressions such as these are called integro-differential equations. The phase function p(μ,μ ) in the source term of Equation (5.1.31) results from integration of the phase function p˜ (μ,ϕ; μ ,ϕ ) over ϕ . Therefore, we write this, from Equation (5.1.15), as the following p(μ,μ )

"2π

1 = 2π

p˜ (μ,ϕ; μ ,ϕ )dϕ ,

(5.1.32)

0

and require the simple normalization condition 1 2

"+1 p(μ,μ )dμ = 1.

(5.1.33)

−1

Based on this result the phase function indicates, as mentioned previously, the probability for light from direction μ is scattered in direction μ. Because Legendre polynomials Pl (μ) form a basis of orthogonal functions, it is possible to write a function such as p(μ,μ ) as a superposition of this basis set as in the equation: p(μ,μ ) =

∞ 

dl Pl (μ)Pl (μ ).

(5.1.34)

l=0

The quantity dl is the coefficient of the lth Legendre polynomial of the given expansion. Condition (5.1.33) implies d0 = 1. The series expansion of functions is extensively described in the literature; see, for instance, [19]. If we combine Equation (5.1.34) with (5.1.31), we obtain the following: " ∞ ω0  dI(τ ,μ) dl Pl (μ) Pl (μ )I(τ ,μ )dμ . = −I(τ ,μ) + μ dτ 2 +1

l=0

(5.1.35)

−1

In order to give an approximate solution I(τ ,μ) among other things, at least some of the first Legendre coefficients have to be known. For clarity, we summarize here the requirements for the validity of both the ERTs (5.1.31) and (5.1.35):

312

5 Theories of Radiative Transfer

– plane parallel layer, extending to infinity in the x, y plane, – the optical path τ depends only on the absorption and scattering coefficients K and S, and coordinate z, – the illumination is axis symmetrical with respect to the z-axis and independent of azimuth angles ϕ and ϕ, – the light is unpolarized before and after interaction, although this can be included in the calculations. In the case of isotropic scattering, all scattering directions are equally likely. From Equation (5.1.33), the phase function becomes p(μ,μ ) = 1.

(5.1.36)

This result represents a sphere with radius r = 1; see Fig. 5.5 in Section 5.1.3. Therefore, from expression (5.1.31) follows the simplest RTE ω0 dI(τ ,μ) = −I(τ ,μ) + μ dτ 2

"+1 I(τ ,μ )dμ .

(5.1.37)

−1

Beyond the requirements mentioned above, Equation (5.1.37) is simultaneously valid for absorptive and isotropic scattering layers. In the following, the twoand three-flux approximations are based on expression (5.1.37) and are mainly applied to samples containing absorptive colorants. The term two- or multi-flux approximation is based on the procedure chosen to determine the intensity I(τ ,μ) in the RTE. Among the various possible solution methods, we chose in the following the procedure to approximate the integral of the source function (5.1.18) by a sum of n ≥ 2 terms (discrete ordinate method). With this approach, the continuous radiation field inside and outside of the layer is divided into n discrete angle ranges, the so-called radiation channels or light cones. Section 5.5 deals with the general case of the multi-flux approximation with up to n = 256 fluxes, respectively, light cones. The accuracy of the approximation increases with finer subdivision of these channels; see also Fig. 5.29. Finally, it is necessary to make a remark concerning Equation (5.1.30) with respect to calculations and measurements of layers containing effect pigments. In these cases, the phase function has to be developed into a series of all four variables μ, μ , ϕ, ϕ ; the integral over ϕ , the source term, has to be approximated by a quadrature equation of 2n terms. The appropriate directed measurement geometry for the effect colorants defines the angle of illumination and several measurement angles. The number i = 1, 2, . . . , N of measurement angle pairs μi , ϕ i defines the number of differential-equation systems that has

5.1

Fundamentals

313

to be resolved with respect to I(τ , μi , μ , ϕi , ϕ ). Further work along these lines requires extensive formalism and is beyond the scope of this book.

5.1.6 Phase Function for Anisotropic Scattering The phase function p(μ,μ ) in the ERT (5.1.31) not only corresponds to the probability of light being scattered from direction μ toward direction μ but also its numerical value characterizes the type of scattering. For the simplest case with p(μ,μ ) = 0, the source term disappears; it is, therefore, equivalent to pure absorption. In contrast, the maximum p(μ,μ ) = 1 represents complete isotropic scattering, as explained in the previous section. The series expansion of the phase function can become complicated, if the Legendre coefficients dl do not converge sufficiently fast. In this case, anisotropic scattering is often characterized approximately by using different empirical phase functions [20]. In order to simulate the single scattering of Mie, the Henyey–Greenstein function [21] is used, among other things; for perpendicular illumination, i.e., corresponding to μ = −1, this function reads as

p(μ, − 1) =

1 − g2 . (1 + g2 − 2 gμ)3/2

(5.1.38)

Due to the domain of the phase function 0 ≤ p(μ, μ ) ≤ 1, the coefficient of anisotropy g is limited to values |g| ≤ 1: purely isotropic scattering corresponds to g = 0, whereas values of g > 0 represent anisotropic scattering in forward direction and g = 1 belongs to the case of “extreme anisotropy” of transparent materials. Conventional absorption pigments show frequency-dependent values of g between 0.2 and 0.8 [22]. The expansion coefficients dl of the phase function in Equation (5.1.35) can be calculated in this case by the simple formula dl = (2 l + 1)gl , l = 0, 1, 2, ...

(5.1.39)

[20]. In this case, the smaller the value of g, the faster the values of dl converge to zero with increasing l. Just as before, it follows that the modeling of highly anisotropic materials requires a large number of coefficients. In the case of solely backward scattering with g = –1, the expansion coefficients can be calculated by the equation dl = ( − 1)l (2 l + 1), l = 0, 1, 2, ...

(5.1.40)

314

5 Theories of Radiative Transfer

which produces alternating signs for dl . However, these coefficients do not represent the realistic reflection behavior of metallic or reflecting diffraction pigments appropriately. The absolute value |dl | ≤ 2 l + 1 can merely be used to estimate the size of the coefficients. Although the properties of layers containing effect colorants cannot be exactly represented by the Henyey–Greenstein function, the value of g supplies at least an idea of the extent of anisotropy in the present case. For metallic pigments, we are able to acquire an accurate phase function by another consideration. Let us assume that the directed reflection of a metallic flake can be interpreted as an extreme scattering process. This “scattering” corresponds to the effect of a microscopic mirror; see Fig. 5.11. The metallic flake reflects the incident light from direction μ = cos (ϑ/2) toward the main direction μ = −μ ; this process can be described by the Dirac delta function3 as in p(μ,μ ) = δ(μ + μ ).

(5.1.41)

However, in a real layer the metallic particles are not oriented completely parallel to the surface, but rather their alignment is statistically distributed with preference to being parallel to the surface. In that case the phase function can

ϑ 2

ϑ 2

Fig. 5.11 Schematic representation of the reflection of a metallic flake [18]

3 For some

properties of the delta function, see Section 5.1.1.

5.1

Fundamentals

315

be approximated by a suitable orientation-distribution function. If the reflecting metallic flakes are distributed rotationally symmetric with respect to the direction of illumination – i.e., independent of azimuth angle ϕ – then the phase function is correlated with an orientation-distribution function f(ϑ). Figure 5.12 shows the plot of a typical phase function in dependence on declination angle ϑ for a sample containing metallic pigments, among other things. The more evenly the flakes are oriented parallel to the substrate surface, the more the phase function approximates a delta function. The maximum of the phase function is decreased if metallic pigments are mixed with scattering absorption pigments. In general, the phase function of a colorant mixture is a linear combination of the phase functions of the respective individual colorants.

p (μ, μ′ = –1)

15

Metallic pigment Absorption pigment Mixture 50/50

10

5

1 0 0

20

40 ϑ/degree

60

80

Fig. 5.12 Typical phase functions of a metallic pigment, an isotropic scattering absorption pigment, and of a mixture of the two [22]

The various contributions of absorptive colorants to either the phase function p(μ, μ ), the optical path τ , or to the albedo ω0 are usually dependent on the wavelength λ. In contrast, the albedo of metallic pigments is nearly independent of λ, because the spectral absorption is practically constant over the entire visible range (cf. Fig. 2.39, Section 2.3.3). This also implies that the phase function of metallic flakes is nearly independent of the wavelength; certainly, due to the individual orientation distribution of the flakes, the phase function can be slightly different for each coloration of the same charge. For pearlescent, interference, and diffraction pigments, the situation is more complex. In these cases, the particles reflect or transmit the light in dependence

316

5 Theories of Radiative Transfer

of illumination and observation angles, layer thicknesses, colored background, and other contributions. Apart from the reflection of the interference or diffraction wavelengths, the transmission of the complementary color of interference pigments must also be taken into consideration. The absorption of these kinds of colorants corresponds roughly to that of metallic pigments, in fact it decreases the albedo but does not affect the phase function.

5.2 Directional Two-Flux Approximation In this section, we begin with the version of the RTE (5.1.37) which is valid for plane parallel layers of isotropic scatterers and corresponds to the simple phase function p(μ, μ ) = 1. In order to solve this integro-differential equation for the case of two-flux approximation, we separate the continuous radiation field into two anti-parallel fluxes, this, following the principle of Joule. With this simplification, the integral of the RTE reduces to a sum of two terms, each consisting only of the intensities corresponding to these fluxes. A system of two linear equations is, therefore, obtained. This system is conducive to straightforward calculation of the desired directed intensity quantities. In view of the boundary conditions, the solutions result in useful expressions for the reflection and transmission of a colored layer. In order to obtain these values for the entire visible spectrum, the calculations must be repeated for each wavelength in the desired range given by the measuring conditions chosen for the inward and outward fluxes of directed light. The calculated values of spectral reflection or transmission belong only to the chosen directional two-flux approximation; they are not transferable to any other approximation. We verify the correctness of this directional two-flux approximation with three special cases: a transparent layer, an opaque layer, and a purely scattering layer which corresponds to the conservative case. These special cases can be illustrated quite simply using the so-called optical triangle. The methods developed under the two-flux approximation are further applicable to other approaches of radiative transfer. The spectral quantities obtained from the two-flux approximation are used in three different areas of application (see Chapter 6). The first application is to compare them with the correspondingly measured spectral quantities. The second is to calculate the optical constants of a layer material, for example, the absorption and scattering coefficients or the optical path and albedo. The third is color recipe prediction, that is, to determine the type and concentrations of colorants for a given color of unknown colorant composition. This application is quite important and commonly used in the color industry.

5.2

Directional Two-Flux Approximation

317

5.2.1 Reflection and Transmission As shown in Fig. 5.13, we express the continuous radiation field inside a plane parallel optical medium as two directed intensities written as i(τ ,μ) and j(τ ,μ). These directed intensities are oriented in positive and negative directions with regard to the given z-coordinate axis. The orientation μ is related to the angle of declination ϑ by definition (5.1.26). For the two-flux approximation, the integral of the source term in Equation (5.1.37) is replaced by the sum of the intensities i(τ ,μ) and j(τ ,μ). Most admittedly, this is a quite rough approximation; we do, however, obtain a system of two linear differential equations with constant coefficients μ

ω0 di(τ ,μ) = −i(τ ,μ) + [i(τ ,μ) + j(τ ,μ)] , dτ 2

(5.2.1a)

z ≤ 0: − μ

dj(τ ,μ) ω0 = −j(τ ,μ) + [i(τ ,μ) + j(τ ,μ)]. dτ 2

(5.2.1b)

z ≥ 0:

z=0 j (τ, μ)

ω0 , τ z

i (τ, μ) z=d

Fig. 5.13 Illustration of the two-flux approximation with directed intensities i(τ ,μ) and j(τ ,μ) inside a plane parallel layer

Such a system lends itself to solution much easier than the earlier form involving an integral expression. The determination of i(τ ,μ) and j(τ ,μ) leads to the characteristic equation μ2 k2 + ω0 − 1 = 0

(5.2.2)

with roots k1/2 = ±k = ±

1 1 − ω0 . μ

(5.2.3)

The particular solutions of i(τ ,μ) and j(τ ,μ) contain integration constants which are determined from the boundary conditions i(τ = 0, μ) = 1,

(5.2.4)

j(τ = τd , μ) = 0,

(5.2.5)

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5 Theories of Radiative Transfer

with the optical path τd = (K + S)d

(5.2.6)

from Equation (5.1.27) using thickness z = d. The incident flux is normalized as given in Equation (5.2.4). Part of this incident flux is reflected by the medium, part is absorbed in the layer, and the remainder is transmitted. Now, we define the reflection R as the ratio of the reflected to the incident flux; analogously, the transmission T is defined as the ratio of the transmitted to the incident flux, cf. Equations (2.1.14a), (2.1.14b), and (2.1.14c). In every approximation, reflection and transmission have to be put in concrete terms. In the present case the influx and efflux are both directed by an angle μ. Accordingly, the directed reflection Rμμ is the proportion of intensity returning at the upper surface of the optical layer at z = 0 and is written as Rμμ = j(z = 0, μ).

(5.2.7)

The directed transmission Tμμ at position z = d is likewise given by Tμμ = i(z = d, μ).

(5.2.8)

With the solutions for i(τ ,μ) and j(τ ,μ) from Equations (5.2.1a) and (5.2.1b), we obtain expressions for these directed quantities, ω0 , 2 − ω0 + 2μk coth (kτ )

(5.2.9)

2μk . (2 − ω0 ) sinh (kτ ) + 2μk cosh (kτ )

(5.2.10)

Rμμ = Tμμ =

Both formulas have, for simplicity, τ written in place of τ d , cf. Equation (5.2.6); the quantity k corresponds to the characteristic root in Equation (5.2.3) with positive sign. It is to keep in mind that the above expressions are only valid in the relevant two-flux approximation. From the above expressions (5.2.9) and (5.2.10), the simplicity of the result becomes apparent; a mere four parameters, two geometrical and two optical, are necessary to calculate the reflection or transmission of the layer under this approximation. The required parameters, specifically, are the layer thickness d, angle of illumination μ, albedo ω0 , and optical path τ . Moreover, the optical and coloristical properties of the layer are represented only by the values ω0 and τ . These results are of fundamental importance. They follow from the concept of multiple scattering and are valid for all subsequent approximations based on multiple scattering. In the case that all four of the above required parameters are known for the calculation, the obtained theoretical quantities Rth = Rμμ and T th = Tμμ

5.2

Directional Two-Flux Approximation

319

can be directly compared with those experimentally obtained, Rex , Tex . On the other hand, given the values of the geometrical parameters μ, d, along with measured values of reflection and transmission, Rex , Tex , the optical constants ω0 and τ of the layer can be determined by Equations (5.2.9) and (5.2.10). In order to simplify the evaluation, the illumination of the layer can be carried out perpendicular to the surface, which corresponds to the condition μ = 1. It is possible to use the absorption and scattering coefficients K and S for calculation rather than ω0 and τ . These parameters follow from Equations (5.1.27) and (5.1.28) with z = d: K = (1 − ω0 )τ/d,

(5.2.11)

S = ω0 τ/d .

(5.2.12)

In any case, the above approach is applicable for all approximations within the multiple scattering concept. Equations (5.2.9) and (5.2.10) are approximations for many sorts of optical layers, chromatic or achromatic, provided that only directional rays are considered. In particular, they are considered for three special optical cases in the next section.

5.2.2 Optical Special Cases An unverifiable model is epistemologically worthless. Moreover, the results of the model must be consistent with actual experience and common sense as well as not contradicting other accepted theories. A lack of these consistencies would require the ideas to be modified or rejected. In this section, we do a reality check for the two-flux approximation in three special optical cases: an opaque, a transparent layer, and a purely scattering layer, corresponding to the conservative case. The reasons for these examples are made clear in the next section. These three special cases are applied to further approximations elsewhere in the book. An opaque material is by definition neither transparent nor translucent. It is characterized by zero transmission corresponding to Tμμ = 0. This means that an opaque material completely hides the substrate, which is mathematically equivalent to a layer of infinite thickness corresponding to d → ∞, and therefore likewise τ → ∞. Considering the exponential behavior of the hyperbolic functions in the denominator of Equation (5.2.10), the transmission vanishes: lim Tμμ = 0.

τ →∞

(5.2.13)

The same limit τ → ∞ in Equation (5.2.9) leads to the formula for the reflection of an opaque layer

320

5 Theories of Radiative Transfer

Rμμ =

ω0 . √ 2 − ω0 + 2 1 − ω0

(5.2.14)

This is a remarkable result for a variety of reasons: – the reflection of an opaque medium is only a function of albedo ω0 and independent of τ , – Rμμ does not depend on the angle of illumination μ, – it is identical to the corresponding formula for two diffuse fluxes (see Appendix A.3.2). As already shown, the range of albedo values is limited to 0 ≤ ω0 ≤ 1; see Equation (5.1.28). Expectedly, the reflection Rμμ is restricted to the same interval. This follows already from the general definition of the reflection in the last section. The second special case of interest is that of a transparent medium. For a transparent layer, part of the incident light is absorbed and the remainder is transmitted; that is, there is no reflection or scattering. The scattering coefficient S, together with the albedo ω0 , vanishes; in consequence, with Equation (5.2.9), the directed reflection Rμμ also vanishes: Rμμ = 0.

(5.2.15)

The absence of scattering reduces the characteristic root in Equation (5.2.3) to k = 1/μ; from Equation (5.2.10), the transmission then follows Tμμ = e−τ/μ .

(5.2.16)

In this case, the optical path takes the simple form τ = Kd. Expression (5.2.16) is, in fact, the general one-dimensional Lambert–Beer law for absent scattering written in Equation (5.1.7) with μ = 1 and Tμμ = I0 (z = d)/I0 . As a side note, the Lambert–Beer law also follows immediately by integration of Equation (5.2.1a) or one of the RTEs (5.1.30), (5.1.31), (5.1.35), and (5.1.37) when using ω0 = 0. The third special case to deal with is the conservative case. This is an absorption-free case and is specified by ω0 = 1. From the principle of the conservation of energy (2.1.14), in the absence of absorption, the relation Rμμ + Tμμ = 1 must hold. In order to verify this, we insert the expression for the characteristic root (5.2.3) into Equations (5.2.9) and (5.2.10) and determine the limits with ω0 → 1. We, thereby, obtain Rμμ =

τ , 2μ + τ

(5.2.17)

Tμμ =

2μ . 2μ + τ

(5.2.18)

5.2

Directional Two-Flux Approximation

321

It is clear, therefore, that in the two-flux approximation, the conservation of energy requirement is fulfilled. The bottom line here is that the results of the two-flux approximation above are consistent with experience and common sense. In the next section, these extreme cases are illustrated in particular with the help of the optical triangle.

5.2.3 Optical Triangle By definition in Sections 2.1.5 and 5.2.1, reflection R and transmission T are relative quantities; their values, therefore, are always in the range [0, 1]. Given this, it is possible to characterize the optical system with a representation of these values in Cartesian coordinates. In this case, T is interpreted as a function of R in the usual sense of an x, y coordinate system. The result of this is the optical triangle shown in Fig. 5.14. The legs and corner points of the triangle represent the special cases detailed in the previous section. To jump straight to the most important result here, each point within the area of the triangle represents the characteristics of an optical system and the area of the triangle contains the whole of possible combinations of R and T for all optical systems of linear materials. To elucidate these concepts further, we consider now the optical meanings of various cases. An opaque purely absorptive material without scattering is characterized by a zero albedo (ω0 = 0) and an infinite optical path length (τ →∞). The values Tμμ = 0 and Rμμ = 0 for this case, therefore, follow from Equations (5.2.13) and (5.2.14). These conditions correspond to an ideal black medium represented by the triangle corner labeled B in Fig. 5.14. Now, the case of absent absorption is called the conservative case. This case is represented by the line defined by Tμμ = 1 – Rμμ . This line extends from corner A for Rμμ = 0 to corner C for Rμμ = 1. Corner C is the special case of pure scattering, which, in addition to unity albedo, is characterized by an infinite optical path length (τ →∞). This corner, therefore, represents the case with Tμμ = 0 and Rμμ = 1, meaning all incident light is scattered back, synonymous with an ideal white material. Corner A represents ideal transparency characterized by a zero albedo and optical path length; this means zero reflection and unity transmission. It is a hypothetical material without any absorption or scattering (like vacuum). It is clear that the special cases just discussed should not depend on any particular approximation used to calculate transmission and reflection. We, therefore, leave out the subscripts of R and T in the following. Table 5.1 gives a survey of various values for the corner points. The legs of the optical triangle, not including the corner points, describe the following optical states:

322

5 Theories of Radiative Transfer

Transparent materials

Internal transmission T

A (0, 1)

C

on

se

rv at

ive

ca

se

Opaque materials

B (0, 0)

C (1, 0)

Internal reflection R

Fig. 5.14 The area of the optical triangle ABC is the geometric locus of all reflection– transmission combinations characterizing optical linear systems

Table 5.1 Three optical extreme cases, represented in the corner points A, B, C of the optical triangle Corner optical Optical special cases triangle Ideal transparency Pure absorption Pure scattering

A B C

Internal reflection and transmission

Optical constants

R

T

ω0

τ

K

S

0 0 1

1 0 0

0 0 1

0 →∞ →∞

0 →∞ 0

0 0 →∞

– AB: transparent materials; only absorption, no scattering; ω0 = 0, the optical path changes from τ = 0 at A toward infinity at B; – AC: conservative case; only scattering, no absorption, ω0 = 1; τ = 0 at A, τ → ∞ at C; – BC: opaque materials; absorption and scattering but no transparency, τ → ∞; the albedo changes nonlinearly between ω0 = 0 at B and ω0 = 1 at C. The optical triangle is somewhat analogous to the operating triangle of a machine or circuit where its operating state can be read from the coordinates

5.2

Directional Two-Flux Approximation

323

A, τ = 0 1.0

τ = 0.3

Internal transmission T

0.8 1.0 0.6

3.0

0.4

0.2

10 102 C, τ → ∞

B

0

ω0 = 0 0

0.5 0.2

0.8

ω0 = 1.0

0.95

0.4 0.6 Internal reflection R

0.8

1.0

R

Fig. 5.15 Optical triangle with families of curves calculated from the directional two-flux approximation; indicated are curves of constant optical path τ and constant albedo ω0 ; μ = 1, n = 1.5

of the working point. In a figurative sense, the “operating points of an optical system”, meaning the spectral (T, R) points, represent not only the values of reflection and transmission but also the accompanying values of albedo and optical path length. As previously mentioned, the coordinates and significance of the corner points are independent of any approximation being used. However, the shapes of any curves plotted with parameters of ω0 and τ within the boundaries of the optical triangle do, in fact, depend on the approximation. In Fig. 5.15, a family of curves for constant albedo and a family for constant optical path are plotted. These curves are calculated using Equations (5.2.9) and (5.2.10) of the twoflux approximation with value μ = 1 (representing perpendicular influx) and a refractive index of n = 1.5. Starting at point A, the triangle is ordered neatly by a set of curves with constant albedo ω0 ; they are intersected by curves of constant optical path τ . Note that at A, the albedo ω0 is mathematically not continuous, but this is unimportant for applications. As a result of the dependencies in Equations (5.2.9) and (5.2.10), the curve shapes in either family depend on ω0 and τ in a nonlinear fashion. It is remarkable that the recipe prediction (based on calculated values of reflection and transmission) gives quite acceptable results at least for absorption colorants.

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5 Theories of Radiative Transfer

It is worthwhile, however, to further improve the methods of recipe calculation, especially by taking into consideration the nonlinear spectral or colorimetric properties of colorants in dependence on concentration.

5.2.4 Determination of Optical Coefficients If accurate values of Rμμ and Tμμ are available from measurements and the quantities μ and d are also known from the measurement geometry and layer thickness, then it is possible to calculate the optical constants ω0 and τ , respectively. To do this, one may use the theory outlines in the preceding sections. It is important to remember that the calculated optical constants are specifically for the measuring wavelength λi and are normally different at other wavelengths. The knowledge of both coefficients, ω0 and τ , of a colored layer facilitates some important applications in colorant industry, such as: – determination of color strength, depth of color, or hiding power of absorption colorants; – comparative characterization of colored pigments; – numerical recipe prediction. With the albedo and optical path given, the absorption and scattering coefficients can also be calculated by using Equations (5.2.11) and (5.2.12). For the two-flux approximation, we now determine the optical constants ω0 and τ for the three special cases treated in Section 5.2.2, and for the general case of a translucent layer. In principle, for an opaque layer, only the albedo can be determined, for which one obtains ω0 =

4Rμμ (1 + Rμμ )2

(5.2.19)

using Equation (5.2.14). In the case of an opaque material, ω0 is only dependent on the measured reflection, whereas the optical path goes to infinity: τ → ∞ or equivalently d → ∞. If Equation (5.1.28) is considered, the following results: (1 − Rμμ )2 K = . S 4Rμμ

(5.2.20)

Note: It is a characteristic of an opaque medium that only the quotient of K and S can be ascertained; this is equivalent to the statement just above that only the albedo ω0 can be determined.

5.2

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325

According to definition, a transparent material acts as a partial absorber without any scattering. Using the value ω0 = 0 in Equation (5.2.16), the result gives the optical path length in dependence of the transmission τ = −μ ln (Tμμ ).

(5.2.21)

As a result of the fact that a transparent material has zero scattering, S = 0, expression (5.2.11) for τ reduces to τ = Kd; the absorption coefficient is therefore given by K=−

μ ln (Tμμ ). d

(5.2.22)

In the conservative case, using ω0 = 1 in Equations (5.2.15) and (5.2.16), one obtains τ = 2μ

Rμμ . Tμμ

(5.2.23)

In order to determine the optical coefficients of a translucent medium, one needs the values of reflection and transmission from two independent measurements. Often the direct measurement of transmission is quite uncertain due to the possibility of surface reflections or heterogeneities at the boundary layers. For this reason, the method of measuring the reflection over two different colored backgrounds is used quite often. In connection with Equations (4.2.13) and (4.2.14), we can write Rx = Rμμ +

2 R Tμμ bg,x

1 − Rμμ Rbg,x

,

(5.2.24)

where Rx is the measured reflection from the test layer optically contacted on top of background x and Rbg,x is the measured reflection directly from the uncovered background x. The insertion of Rμμ and Tμμ using Equations (5.2.9) and (5.2.10) into Equation (5.2.24) gives Rx =

ω0 − Rbg,x [2 − ω0 − 2μk coth (kτ )] . 2 − ω0 (1 + Rbg,x ) + 2μk coth (kτ )

(5.2.25)

Consider that reflection values for normally plain white and black backgrounds are indicated by Rbg,w , Rbg,b , respectively, and for a test layer covering these backgrounds by Rw , Rb , respectively. Four reflection measurements can determine these. By inserting the respective measured pairs into Equation (5.2.25), it is possible to eliminate the hyperbolic functions and obtain a term for the albedo

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5 Theories of Radiative Transfer

ω0 =

4D , (1 + Rbg,w Rw )(Rb − Rbg,b ) − (1 + Rbg,b Rb )(Rw − Rbg,w ) + 2D (5.2.26a)

where D = (Rbg,w Rb − Rbg,s Rw ).

(5.2.26b)

The right-hand side of Equation (5.2.26a) contains only measured parameters and, in particular, it is independent of μ and τ . Solving of expression (5.2.25) for τ results in ! ω0 (1 + Rbg,x Rx ) − (2 − ω0 )(Rbg,x + Rx ) 1 arccoth . τ= 2μk 2μk(Rx − Rbg,x ) This expression contains only known quantities, particularly since μ is known from the measuring geometry and k from Equation (5.2.3) with positive sign. The corresponding measured pairs Rbg,b , Rb and Rbg,w , Rw can be used in order to determine two values of τ . The reliability of results can be evaluated by error estimation of the measured values. If both quantities of τ differ by less than ±1%, experience shows that the mean is trustworthy. Finally it is important to once again stress that the resulting values of the optical constants depend on the approximation used. That is to say, the various approaches yield different results. Accordingly the question is legitimate, which approximation delivers the correct coefficients for a specified optical system? The question can only be answered empirically. However, experience shows that approximations of more than two fluxes lead to quite reliable results for recipe prediction of absorption and effect colorations. Nonetheless, the next section deals with the two-flux theory of Kubelka and Munk because it is widespread, but of restricted validity – a consequence of the simply structured evaluation formulas.

5.3 Theory of Kubelka and Munk The assumptions made of Kubelka and Munk [11,12] about light interactions in colored scattering materials are essentially identical with those of Schuster and Schwarzschild [9, 10]. This theory is, however, not based directly on the RTE (5.1.37) for plane parallel layers and is, therefore, not directly compatible with any of the further approximations of that equation. Kubelka and Munk work on an empirical premise of two diffuse fluxes – corresponding to a modified approximation which includes absorption and isotropic scattering. Due to the simple handling of this model, during the past few decades, some of the formulas have been implemented in algorithms for recipe prediction; generally, at the time, they were limited by the relatively low computing power for simulations.

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327

Simplicity, however, does not mean a completely reliable applicable description for the optical properties of real colorations. The limits of this theory become evident especially when corrections due to the different refractive indexes of a layer are necessary, which is clearly demonstrated with the aid of the optical triangle.

5.3.1 Empirical Approach Following Kubelka and Munk, we assume here that a homogenous, plane parallel layer of thickness z = d is illuminated of diffuse light at position z = 0. The layer is composed of an absorptive and isotropically scattering paint; see Fig. 5.16. The radiation field inside the layer is divided into two anti-parallel diffuse fluxes of intensities I+ (z) and I– (z). During a propagation through a distance dz, these intensities are modified by dI+ and dI– , respectively. The intensity I+ (z) is attenuated by absorption and scattering, but intensified by the scattering component of the oppositely oriented intensity I– (z). This attenuation and intensification is analogous for I– (z). The interplay between I+ (z) and I– (z) is described by Kubelka and Munk [11, 12] with the phenomenological system of two linear differential equations: z ≥ 0: +

dI+ = −(K + S)I+ + SI− , dz

(5.3.1a)

z ≤ 0: −

dI− = +SI+ − (K + S)I− . dz

(5.3.1b)

Wherein K and S denote the absorption and scattering coefficients of the paint layer. In comparison to Equations (5.1.22a) and (5.1.22b), this system only differs with regard to the weighting factor of the absorption constant.

R z=0

I– (z) K, S z

I+ (z) z=d T

Fig. 5.16 Diffuse radiation components in a plane parallel layer after Kubelka and Munk

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5 Theories of Radiative Transfer

In solving system (5.3.1a) and (5.3.1b), one first obtains the roots k of the corresponding characteristic equation k2 − (K + S)2 + S2 = 0.

(5.3.2)

The formalism is simplified introducing the abbreviations a=

K +1 S

(5.3.3)

and  b = + a2 − 1.

(5.3.4)

The characteristic roots are, therefore, given by k1/2 = ±k = ±bS.

(5.3.5)

Again, the integration constants follow from the boundary conditions which are, here, given by I+ (z = 0) = 1,

(5.3.6)

I− (z = d) = 0.

(5.3.7)

Using the definition of the reflection R in the irradiated surface R = I− (z = 0)/I+ (z = 0)

(5.3.8a)

and for the transmission T in the second surface T = I+ (z = d)/I+ (z = 0),

(5.3.8b)

cf. Fig. 5.16, the result follows 1 , a + b coth (kd)

(5.3.9)

b . a sinh (kd) + b cosh (kd)

(5.3.10)

R=

T=

Both expressions contain again hyperbolic functions. Equations (5.3.9) and (5.3.10) are, in fact, quite similar to those of the directional two-flux approximation in Section 5.2.1. They allow to calculate the diffuse reflection R and diffuse transmission T, requiring only the optical constants K and S and the layer thickness d. It must be pointed out that R and T, in fact, have different values

5.3

Theory of Kubelka and Munk

329

compared with Rμμ and Tμμ from Equations (5.2.9) and (5.2.10). These values and also K and S depend on the approximation used and are, therefore, not directly interchangeable between different approximations (cf. Section 5.4.2).

5.3.2 Exceptional Optical Cases It is useful now, in the context of the formalism of Kubelka and Munk, to consider the compliance for the three special cases previously considered for the directional two-flux approximation in Section 5.2.2. For this, we begin with Equations (5.3.9) and (5.3.10) and consider again the opaque, transparent, and absorption-free conservative cases. Already from ideas developed in the directional two-flux theory, we know that for an opaque layer the transmission vanishes. Again considering the exponential nature of the hyperbolic functions in Equation (5.3.10), the limiting case d → ∞ gives lim T = 0

d→∞

(5.3.11)

and therefore compliance to requirements for this case. The same condition in Equation (5.3.9) yields, for the reflection R of an opaque layer, the simple expression R=

1 , a+b

(5.3.12)

which, considering Equation (5.3.4), is equivalent to R = a − b.

(5.3.13)

Inserting the expressions for a and b from Equations (5.3.3) and (5.3.4) into the above results in an expression for the ratio of the absorption and scattering constants gives F=

K (1 − R)2 = S 2R

(5.3.14)

This relation is called Kubelka–Munk function. Although similarly structured to Equation (5.2.20), this result is larger by a factor of 2. This results from the empirical weighting of K and S in the system of equations (5.3.1a) and (5.3.1b). Expressions (5.3.14) and (5.2.20) each have a singularity for R = 0 and S = 0, respectively, and are, therefore, undefined in these cases. These special cases correspond to dark colors or colorants such as carbon black or dark violet. In order to avoid this difficulty in reality, a tiny amount of scattering white is added to the corresponding coloration.

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5 Theories of Radiative Transfer

A transparent medium absorbs light without any scattering; therefore, S = 0 holds, which means the reflection vanishes (since reflection means scattering, by definition). This connection follows additionally from Equation (5.3.9) for k=0 R=

S 1 = = 0, a K+S

(5.3.15)

provided K > 0. Here, the relations K = bS, a = 1, and b = 0 are valid; therefore, we obtain from Equation (5.3.10) in the limit b →0: T=

1 . sinh (Kd) + cosh (Kd)

(5.3.16)

This is equivalent to T = e−Kd ,

(5.3.17)

which again corresponds formally to the one-dimensional Lambert–Beer law of an absorptive layer, cf. Equation (5.1.7) using T = I0 (z = d)/I0 . Note that the absorption coefficient K here is twice as much as in Equation (5.2.16). Returning to the conservative case originally characterized by ω0 = 1 and K = 0, from Equation (5.3.3), it follows a = 1. The equation of energy (2.1.14) reduces to the simple relation R + T = 1. In order to reassess this expression, we take the limit of the expressions in Equations (5.3.9) and (5.3.10) for a → 1 and obtain R=

Sd , 1 + Sd

(5.3.18)

T=

1 , 1 + Sd

(5.3.19)

clearly complying with the conservation of energy requirement. In addition, for large values of S or d, the results from Equations (5.3.18) and (5.3.19) approach R = 1 and T = 0, respectively. This corresponds to the coordinates of corner point C of the optical triangle in Fig. 5.14. The coordinates of corner point A follow from expressions (5.3.18) and (5.3.19) for d = 0. On the other hand, the coordinates of the special points A and B, as well as the quantities K and S, follow directly from Equations (5.3.11), (5.3.12), (5.3.13), (5.3.14), (5.3.15), (5.3.16), and (5.3.17); all parameters mentioned agree with those listed in Table 5.1. Therefore, the special cases just discussed are again in agreement with experience and expectations.

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Theory of Kubelka and Munk

331

5.3.3 Determination of Optical Constants The methods outlined in Section 5.2.4 can be applied directly to the results of the Kubelka–Munk formalism. From Equations (5.3.9) and (5.3.10) and those that follow, it is possible to determine the coefficients K and S. In order to obtain the albedo ω0 or optical path τ , we refer to Equations (5.1.27) and (5.1.28). To begin, for the case of a transparent material, the absorption constant K follows immediately from Equation (5.3.17): 1 K = − · ln (T), d

(5.3.20)

with known layer thickness d and measured transmission T. The unit of K is an inverse length; this is in accordance with the considerations in Section 5.1.2. In the case of an opaque medium, only the ratio K/S can be calculated, that, using Equation (5.3.14). This ratio vanishes for R = 1; physically, this is because the incoming flux is completely backscattered and not absorbed at all (the conservative case represented by corner point C in Fig. 5.14). For translucent systems, it is possible to ascertain both coefficients K and S separately and, additionally, by way of known two different methods: either by single measurement of reflection and transmission or, most often applied, by measuring the reflection over two known but differently colored backgrounds. For the first of these methods, one can base calculations on Equations (5.3.9) and (5.3.10) suitably combined with (5.3.3) and (5.3.4) and solve for K and S. The second method refers to Equations (4.2.13) and (4.2.14), in which R1 stands for reflection R and T1 for transmission T. Together with Equations (5.3.9) and (5.3.10), the following formulas are obtained: Rw =

1 − Rbg,w [a − b coth (kd)] , a − Rbg,w + coth (kd)

(5.3.21)

Rb =

1 − Rbg,b [a − b coth (kd)] , a − Rbg,b + coth (kd)

(5.3.22)

where Rbg,w and Rbg,b are the measured reflection of the plain white and black background substrates, respectively, Rw and Rb are the reflection of the test layer over the white and black backgrounds; layer and background substrate have to be in optical contact (cf. Section 4.2.4). Combining Equations (5.3.21) and (5.3.22) results in a=

(1 + Rbg,w Rw )(Rb − Rbg,b ) − (1 + Rbg,b Rb )(Rw − Rbg,w ) . 2(Rbg,w Rb − Rbg,b Rw )

(5.3.23)

The parameter a, therefore, can be determined from four measured reflection quantities Rbg,w , Rbg,b , Rw , and Rb . If the black background substrate is really

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5 Theories of Radiative Transfer

black, i.e., Rbg,b ≈ 0, Rbg,b << Rb , and Rbg,b << Rbg,w , then Equation (5.3.23) can be simplified as

b − Rw + R R 1 bg,w Rw + . a= 2 Rbg,w Rb

(5.3.24)

In the case of an opaque material, the reflectivity of a white or black background can be assumed to be basically the same Rw ∼ = Rb ; this corresponds to a contrast ratio of about 1. In this case, Equation (5.3.24) simplifies further to   1 1 w a= R + w . 2 R

(5.3.25)

The quantity of b can be calculated from Equation (5.3.4) using one of the relations (5.3.23), (5.3.24), and (5.3.25). The transformation of Equation (5.3.21) with w = x (white) or alternatively of Equation (5.3.22) with b = x (black) gives ! 1 − aRbg,x + (Rbg,x − a)Rx 1 S= arccoth . bd b(Rx − Rbg,x )

(5.3.26)

The right-hand side of this relation, again, is comprised entirely of measured parameters. One can use measurement pairs Rbg,b , Rb and Rbg,w , Rw to obtain a pair of values for S; how close the values are to one another provides an evaluation of the accuracy. Finally, due to Equation (5.3.3), the absorption coefficient can be calculated from K = (a − 1)S.

(5.3.27)

Conversely, if the quantities K and S as well as the layer thickness d are known, then the corresponding reflection R and transmission T can be calculated using the appropriate equations of Section 5.3.1 or 5.3.2. The calculated theoretical quantities Rth and T th can be compared with the measured quantities of Rex and T ex . This is the basic idea of recipe prediction.

5.3.4 Boundary Layer Correction We have, thus far, not emphasized the fact that the derived relations for reflection or transmission are only strictly valid if the refractive indices inside and outside the layer are identical. This condition is nearly met for those colorants fixed directly to the surface of a chromophore such as to textile or paper fibers. In all other cases of colorants homogenized inside the volume of a binder, polymer, or ceramics, etc., the light intensity is altered at the surface boundary layer.

5.3

Theory of Kubelka and Munk

333

Consequently, due to the different refractive indices inside and outside the material (e.g., air touching binder), the calculated reflection and transmission may not agree well with the measured quantities. For these reasons, it is usually necessary to include the so-called boundary layer corrections. Because the quantities differ, we have to distinguish between the measured external reflection RE and external transmission TE and the previous given internal quantities R and T. The situation is somewhat complicated by the fact that the internal quantities are determined from the external and only afterward the accurate optical constants of K and S as well as ω0 and τ of the test layer are available. For the moment, we consider the case of two diffuse internal and external fluxes. We assume a layer with isotropic refractive index n2 surrounded on each side by media with isotropic refractive indices n1 and n3 ; see Fig. 5.17. These parameters are linked to the corresponding Fresnel coefficients for diffuse light rd , rd∗ , and od by an extension of formula (2.1.12) given by 1 − rd∗ 1 − od 1 − rd = = . 2 2 n1 n2 n23

(5.3.28)

The meaning and effect of reflection coefficients rd , rd∗ , and od follow from Fig. 5.17a. The quantities rd , od are the reflection coefficients for diffuse light at the inner boundary surfaces of a layer, rd∗ is the reflection coefficient for diffuse light at the illumination surface of the layer. Using values n1 = 1 (for normal air) and value of n2 = 1.5 (typical for binders and high polymers) as well as the precisely determined value of rd = 0.59635 in [13], we obtain rd∗ = 0.09178. From these values, three important facts come to light: first, about 9.2% of the incident diffuse flux is directly reflected from the surface; second, only about 90.8% is then available to interact with the inner layer material; third, 59.6% of the diffuse flux from the interior layer is reflected at the internal boundary and only the remainder exits the boundary. These flux components are visualized in Fig. 5.17b. At the interface surface, in addition to the diffuse components I+ and I– , there are two additional diffuse intensities jd and i–d ; see Fig. 5.18; here we introduce the definitions id = I+ and j–d = I– for formal reasons. Based on the nomenclature just discussed, equations for the relationships between the external quantities RE and TE and the internal quantities R and T are derived in the following. In Fig. 5.18, D represents the incident diffuse flux, id and i–d the left and right traveling fluxes internal to the layer and at the location of the left (input) interface, jd and j–d the right and left traveling fluxes internal to the layer and at the location of the right (output) interface. The symbols ˜i−d and ˜jd are the total reflected and transmitted fluxes,

334

5 Theories of Radiative Transfer

1

1 – rd

r *d

rd

1 1 – r *d

a) 1

od

1 – od 1 0.404

n1 = 1

0.092

0.596 Total reflection

n 2 = 1.5

b)

0.908 1 n3 = 1

Fig. 5.17 (a) Reflection coefficients of diffuse radiation at bounding surface of a layer and (b) reflected and transmitted diffuse intensities at bounding surfaces

respectively, from the layer. The internal diffuse reflection and transmission of the layer, R and T, are given by Equations (5.3.9) and (5.3.10). Based on the symbol definitions, it is possible to write out the relationships. The incident diffuse flux D at the boundary surface leads to the diffuse intensity id which is comprised of a transmitted (1−rd∗ )D and a reflected component rd i–d of intensity i–d in the medium refraction number n2 : id = (1 − rd∗ )D + rd i−d .

(5.3.29)

After transmission over the thickness of the layer, the flux id contributes Tid to jd . A second contribution to jd comes from internal reflection of j–d jd = T · id + R · j−d .

(5.3.30)

From the relationships shown in Figs. 5.17a and 5.18 additional expressions hold:

5.3

Theory of Kubelka and Munk

335

id

T

jd

~ jd

D

rd

od

R

~ i–d

j–d

i–d

n2

n1

n3

Fig. 5.18 Beams at the bounding surfaces as well as internal reflection R and transmission T corresponding to two-flux approximation with diffuse intensities of empirical Kubelka–Munk theory

j−d = od jd ,

(5.3.31)

i−d = T · j−d + R · id .

(5.3.32)

From the interrelated equations (5.3.29), (5.3.30), (5.3.31), and (5.3.32), the intensities i–d and jd are particularly of interest, because the resulting quantities ˜i−d and ˜jd , given by ˜i−d = (1 − rd )i−d ,

(5.3.33)

˜jd = (1 − od )jd ,

(5.3.34)

are responsible for the external reflection RE and external transmission TE in which we are interested. They are given by RE =

rd∗ D + ˜i−d , D

(5.3.35)

˜jd . D

(5.3.36)

TE =

Determining the relevant variables i–d and jd using Equations (5.3.29), (5.3.30), (5.3.31), and (5.3.32), we obtain a system of equations from which the results follow: RE = rd∗ + (1 − rd∗ )(1 − rd ) · TE =

(1 − od R)R + od T 2 , (1 − od R)(1 − rd R) − od rd T 2

(1 − rd∗ )(1 − od )T . (1 − od R)(1 − rd R) − od rd T 2

(5.3.37)

(5.3.38)

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5 Theories of Radiative Transfer

Based on these equations, we now consider both the three special optical cases discussed in Sections 5.2.2 and 5.3.2 and also the effects of the boundary surface corrections on the optical triangle. An opaque material is characterized by T = 0, from which TE = 0 also follows using Equation (5.3.38). The external reflection results from Equation (5.3.37), in this case giving RE = rd∗ +

(1 − rd∗ )(1 − rd )R . (1 − rd R)

(5.3.39)

This relation seems to resemble the formula of Saunderson [23]. Certainly, he manipulates Equation (5.3.39) by replacing the reflection coefficient of diffuse light rd∗ by the coefficient of directed light r. However, this model is physically inconsistent: the irradiated directionally flux D is “transformed” into two diffuse fluxes; moreover, the transmitted component is mysteriously again directed. The remarkable background of this manipulation is uncovered in the next section. Because the quantity RE is available from experiment, the internal reflection R of an opaque layer can be calculated from Equation (5.3.39) by R=

RE − rd∗ . (1 − rd∗ )(1 − rd ) + rd (RE − rd∗ )

(5.3.40)

Here, the conservative case has the simplicity that, if the measured quantities RE , TE , and d are available, the scattering coefficient S is directly calculated from Equations (5.3.18) and (5.3.19): S=

1 (1 − od )(RE − rd∗ ) − od (1 − rd )TE . · d (1 − od )(1 − rd )Te

(5.3.41)

A transparent layer is free of internal reflection; therefore, from Equations (5.3.37) and (5.3.38), the relations follow directly: RE = rd∗ + TE =

(1 − rd∗ )(1 − rd )od T 2 , 1 − od rd T 2

(1 − rd∗ )(1 − od )T . 1 − od rd T 2

(5.3.42) (5.3.43)

Combining both of these equations, the internal transmission of the layer results in the following expression: T=

(1 − od )(RE − rd∗ ) . od (1 − rd )TE

(5.3.44)

Using Equation (5.3.17), the absorption constant K is given, provided the layer thickness d is known.

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337

Finally, we emphasize the fact that the boundary surface corrections discussed above can be transferred without difficulty to the approximation with two directional fluxes discussed in Section 5.2. To do this, the quantities rd∗ , rd , and od have simply to be replaced consistently by the respective reflection coefficients r and o for directional flux (see Appendix A.3.1). The results concerning external reflection and transmission of the two-flux directional and diffuse approximations are applied in the next section, among other things.

5.3.5 Limits of Kubelka–Munk Theory As mentioned previously, strictly confining the Kubelka–Munk model to two diffuse fluxes results in a limited applicability, not only with respect to discontinuity at the surfaces of layers. In the example of a thin film with low scattering, the internal flux, upon closer examination, does not remain completely diffuse in the transmission area. Layers containing metallic or interference pigments produce diffuse and directed light components simultaneously; they cannot be described with this model. Diffuse contributions are caused by the edges of the flake-shaped particles. Directed light, on the other hand, is produced by reflection, for example, at the surfaces of metallic pigments, different boundary layers of interference pigments, and from blazed reflection grating pigments. Therefore, neither of the two-flux models already described is applicable to effect pigments; the three-flux approximation discussed in the next section alleviates some of these difficulties. It starts from two diffuse fluxes and a directed flux. The weakness and limited applicability of the Kubelka–Munk theory in the color industry are particularly evident from the consideration of the boundary surface corrections to the optical triangle. In order to better understand, we first outline the corrections for two directional fluxes; the resulting optical triangle is used later for reference. Now the corner points and legs of this optical triangle follow from Equations (5.3.37) and (5.3.38) by merely replacing the coefficients rd∗ , rd , od by the reflection constants r and o for directional flux. If we assume normal incidence of directed illumination, corresponding to μ = 1, and for the layer a refractive index of n = 1.5, r = o = 0.04 follows from Equation (2.1.8). In conjunction with the internal quantities R and T, we are now able to calculate the externals RE , TE and, in particular, the coordinates of the corner points of the optical triangle corresponding to the three extreme optical cases previously discussed. We will designate this new triangle as corrected optical triangle; it is shown in Fig. 5.19 containing corner points AG , BG , C. Point C maintains its original coordinates (1.0, 0), but corner points A and B move to new locations, now denoted as AG and BG in Fig. 5.19. The uncorrected straight segment AB is transformed into the curved element AG BG . The values of the optical coefficients corresponding to the three corner points are identical

338

5 Theories of Radiative Transfer 1.0

A G, τ = 0 0.3

External transmission TE

0.8

1.0 0.6

3.0

0.4

0.2

10 102 BG

0

ω0 = 0

0.5

0

0.2

0.8

C, τ → ∞

ω0 = 1.0

0.95

0.6 0.4 External reflection RE

0.8

1.0 RE

Fig. 5.19 Corrected optical triangle according to the directional two-flux approximation for μ = 1 and n = 1.5

in both the former and the new triangle; see Table 5.1. In addition, the optical interpretation of the new boundary lines is the same as given in the mentioned section: – AG BG : transparent materials; only absorption, absent scattering; – AG C : conservative case; pure scattering, without absorption, equation of energy conservation is reduced to RE + TE = 1; – BG C : opaque systems, no transparency; continuous change from pure absorption in BG to pure scattering in C. Compared to Fig. 5.15, the boundary surface correction results in a reduced triangle area in Fig. 5.19. This is as a consequence of external and internal reflections at the boundary surfaces. The decrease in area represents the extent of reflected flux. It is simply this amount responsible for the perceived color of a transparent medium observed in reflected light. The corner points AG and BG have new coordinates, for n = 1.5 and μ = 1, for example, AG = (0.0769, 0.9231) and BG = (0.0400, 0). With regard to the uncorrected triangle, the reflected component of 4.00% in BG is slightly greater, 7.69%, in AG , with a similarly further augmented transparency component. The reason for higher RE in AG compared to BG is that the internal boundary surface reflection due to

5.3

Theory of Kubelka and Munk

339

the increased transparency is missing in the second case. Families of constant albedo and optical path are drawn in the area of the corrected optical triangle in Fig. 5.19. In comparison to Fig. 5.15, the new sets are slightly displaced, compressed, and bent to a lesser extent. In switching to Kubelka–Munk formalism of two diffuse fluxes, a dramatic decrease in triangle area occurs. In order to understand this breakdown, we insert into Equations (5.3.37) and (5.3.38) – for the moment instead of the correct Fresnel coefficient rd∗ – the false quantity r for directional flux reflection according to Saunderson mix. It is sufficient to consider only the case of ideal transparency, because the coordinates of the corner points BG and C in Fig. 5.19 remain unchanged in this case. The dramatic consequences, regarding two diffuse fluxes, are shown graphically in Fig. 5.20. In comparison with reference triangle in Fig. 5.19, corner AG of coordinates (0.0769, 0.9231) is shifted into the new corner point ASM of coordinates (0.3986, 0.6014), and, in addition, curve section ASM BSM is far more bent than the corresponding section AG BG ; the coordinates of BG and BSM are identical. Furthermore, the corrected optical triangle resulting from Kubelka–Munk formalism is considerably compressed compared with the reference triangle AG BG C, whereas the vicinity of leg BSM C is nearly unchanged. 1.0 AG

External transmission TE

0.8 n = 1.5 ASM AD

0.6

0.4

0.2

0

BSM BD 0 0.2

0.4 0.6 External reflection RE

0.8

C 1.0

Fig. 5.20 Two corrected optical triangles according to Kubelka–Munk theory: excluded optical region AG BSM ASM by Saunderson mix; consistent correction reduces additionally the area of optical triangle AD BD C; n = 1.5

340

5 Theories of Radiative Transfer

The correction worsens further, if the right Fresnel coefficient rd∗ for diffuse flux is used as indicated in the exact equations (5.3.37) and (5.3.38). Inserting values rd∗ = 0.0918 and rd = od = 0.5964, points ASM and BSM are shifted to the accurate points AD and BD with coordinates AD = (0.4311, 0.5689) and BD = (0.0918, 0). The area of triangle AD BD C is further decreased by an amount of 8%. For these reasons, it is clear that with the inconsistent Saunderson mix, the reduced validity of the Kubelka and Munk approximation is not only questionably sugarcoated but even ineffective. Because the vicinity of triangle basis BSM C or BD C is nearly unaffected, both the boundary surface corrections corresponding to Kubelka–Munk theory are roughly applicable only to opaque materials. Nevertheless, for many transparent or translucent systems the corrections lead to wrong results. In particular, the further coordinates of measured pairs RE , TE are away from the basis BD C or BSM C, all the more incorrect are the resulting quantities K and S as well as internal values of R and T. These corrections fail completely, if the measured pair of RE , TE is represented by a point outside the corresponding triangles ASM BSM C or AD BD C in Fig. 5.20. The accompanying material cannot be assigned useful internal values, neither R and T nor optically accurate coefficients of K and S nor ω0 and τ . In this not-so-rare case of translucent layers, the only option is to switch to another approximation without this shortcoming. From there, it is better, in the future, to apply from the start neither the consistent nor the manipulated correction to Kubelka–Munk formalism. A fundamental reason for the limited applicability of this model is, namely, the restriction of two diffuse fluxes. A further aspect, but not explicitly taken into account, is the effect of total reflection. This phenomenon follows especially from diffuse radiation and occurs at all surface boundaries; in Kubelka–Munk theory, simply overall internal reflection is taken into consideration by internal reflection coefficient rd . In order to have a more precise description of radiation transfer, it is necessary, for the moment, to have a theory as simple as possible yet applicable to a corresponding boundary correction without any restrictions. A practical, reliable model satisfying these criteria is given by the three-flux approximation.

5.4 Three-Flux Approximation Considering the shortcomings of the two-flux approximation with diffuse radiation, an additional directional flux along with the original two diffuse fluxes improves the situation. The three-flux theory, outlined by Ryde [24], can be taken as synthesis of the two-flux approximation for diffuse radiation following from RTE (5.1.37) and one-dimensional Lambert–Beer law for directional

5.4

Three-Flux Approximation

341

flux including absorption and scattering. The outcome of this combination surmounts astonishingly well the most restrictive discrepancies of simpler approximations, in particular, that of Kubelka–Munk theory. The correction for the boundary surface influence, formulated by Pauli and Eitle [15], begins with directional and diffuse fluxes along with an additional partially directed flux arising from total reflection. For cases of isotropic scattering materials and additional inclusion of boundary surface corrections, this concept shows an accuracy at the level expected from a six-flux approximation using directional fluxes. Altogether, the three-flux formalism with and without boundary surface corrections is totally applicable to most of optical materials described thus far. This model, representing a logically consistent development of earlier approximations, has been applied successfully for decades in recipe prediction.

5.4.1 Conception of Three-Flux Theory The radiation field inside an isotropic scattering material is now divided into both the previous diffuse fluxes of intensities i and j and an additional directed flux of intensity J; see Fig. 5.21. Consequently, the integral of the source term in Equation (5.1.37) has to be approximated by a sum consisting of these three intensities i, j, and J. For simplicity, we assume axial symmetry as shown in Fig. 5.10. Now, both diffuse intensities i and j are only depending on optical path τ and direction μ with regard to coordinate axis z i(τ , μ ≥ 0) = i(τ ), j(τ , μ ≤ 0) = j(τ ).

(5.4.1)

With these considerations, a system of three linear differential equations follows describing the three-flux concept

τ=0 j ( τ)

τ

i (τ)

J (τ, μ)

ω0, τ τ = τd

Fig. 5.21 Intensities i(τ ), j(τ ), J(τ , μ) of the three-flux approximation

342

5 Theories of Radiative Transfer

μ ≥ 0:

μ

& ω0 # di(τ ) = −i(τ ) + i(τ ) + j(τ ) + J(τ , μ) , dτ 2

(5.4.2a)

μ ≤ 0:

μ

& dj(τ ) ω0 # = −j(τ ) + i(τ ) + j(τ ) + J(τ , μ) , dτ 2

(5.4.2b)

− 1 ≤ μ ≤ 1:

μ

dJ(τ , μ) = −J(τ , μ). dτ

(5.4.2c)

Both diffuse intensities i(τ ) and j(τ ) are absent in Equation (5.4.2c) because they do not contribute to the directed intensity J(τ , μ). In order to include all flux angles, we integrate Equations (5.4.2a) and (5.4.2b) over ranges 0 ≤ μ ≤ 1 and –1 ≤ μ ≤ 0, thus leaving intensities i(τ ) and j(τ ) as total average fluxes for all angles and thus independent of μ; this results in the equations +

di(τ ) = −(2 − ω0 )i(τ ) + ω0 j(τ ) + ω0 J(τ ) dτ

(5.4.3a)

dj(τ ) = ω0 i(τ ) − (2 − ω0 )j(τ ) + ω0 J(τ ). dτ

(5.4.3b)

−

The directional intensity J(τ ,μ) follows by integration of Equation (5.4.2c) with respect to τ leaving parameter μ as a constant resulting in the equation J(τ , μ) = J(0, μ) · e−τ /μ ,

(5.4.3c)

which corresponds to the one-dimensional Lambert–Beer law. From defining Equation (5.1.27) and consideration of layer thickness z = d we have again optical path length τ d given by Equation (5.2.6). The boundary conditions are i(τ = 0) = 0,

(5.4.4)

j(τ = τd ) = 0,

(5.4.5)

J(τ = 0, μ) = 1.

(5.4.6)

The reflection resulting from this model consists of a directional and two diffuse components. It is called directed–diffuse reflection and is denoted by Rμd . This directed–diffuse quantity follows from the ratio of reflected and incident fluxes given by Rμd =

1 j(0). 2

(5.4.7)

The accompanying directed–diffuse transmission Tμd corresponds to the ratio of the transmitted flux and incident flux given by

5.4

Three-Flux Approximation

343

1 Tμd = J(τd ) + i(τd ). 2

(5.4.8)

The factor 1/2 in each of the last expressions stems from averaging of all diffuse parts over μ of the related half-space. In the following sections, we use the previously abbreviated notation τ rather than τ d .

5.4.2 Reflection and Transmission In consequence of the initial considerations in the last section, it is necessary presently to solve the homogeneous system (5.4.3a) and (5.4.3b). The corresponding characteristic roots now adopt the form  k1/2 = ±k = ±2 1 − ω0 . (5.4.9) If we confine ourselves merely to diffuse components, the so-called diffuse– diffuse reflection Rdd follows from Equation (5.4.7); it is given by Rdd =

ω0 . 2 − ω0 + k coth (kτ )

(5.4.10)

According to expression (5.4.8), the diffuse–diffuse transmission Tdd is given by Tdd =

k . (2 − ω0 ) sinh (kτ ) + k cosh (kτ )

(5.4.11)

Both formulas show technically the same structure as Equations (5.2.9) and (5.2.10) of the approximation with two directional fluxes, although Equations (5.4.10) and (5.4.11) are missing the factor of 2μ on k coefficients. This is a consequence of averaging over μ on the half-space. Regardless, it is quite remarkable that Equations (5.4.10), (5.4.11), (5.3.9), and (5.3.10) resemble one another. If we write out both systems (5.3.1a) and (5.3.1b) and (5.4.3a) and (5.4.3b) in dependence on absorption and scattering coefficients K and S for the current system and in notation KKM and SKM for the Kubelka–Munk formalism, then the two systems are mathematically identical for S = SKM , 2 K = KKM .

(5.4.12)

Strangely enough, it seems that the Kubelka–Munk theory turns out to be a modified version of the two-flux approximation of the RTE (5.1.37). We do not follow up on this realization here, but rather see Appendix A.3.2 for further details. The approach of Schuster and Schwarzschild, according to Equations (5.1.22a) and (5.1.22b), additionally results in Equations (5.4.10) and (5.4.11)

344

5 Theories of Radiative Transfer

as consequence of properties (5.4.12). Moreover, given conditions (5.4.12), it is again clear that the optical constants belonging to any certain approximation cannot be simply transferred to any other one. This warning remains even if both of the approximations are based on one of the RTEs in Section 5.1.5. In order to solve the inhomogeneous system (5.4.3a) and (5.4.3b), it is natural to use for i and j the formulation analogue to Chandrasekhar [1] with weight functions g1 , g2 i(τ ,μ) = g1 ω0 e−τ /μ , j(τ ,μ) = g2 ω0 e−τ /μ .

(5.4.13)

However, the particular solutions of i(τ ,μ) and j(τ ,μ) are not single valued if the albedo takes the critical value ω0 = 1 – 1/4μ2 , corresponding to ω0 = 3/4 for μ = 1. This imperfection is avoided by using the auxiliary function H(k1 ,k2 ,τ ) =

ek 1 τ − e k 2 τ k 1 − k2

(5.4.14)

containing the characteristic roots (5.4.9). Since k1 and k2 differ only in sign, the auxiliary function takes on the form H(k, − k,τ ) = ekτ

(1 − e−2 kτ ) 2k

(5.4.15)

with k ≥ 0. The conservative case is characterized by ω0 = 1 and therefore k = 0. The series expansion of H at point τ = 0 or application of l’Hospital rule gives H(0, 0, τ ) = τ . The H function is single valued in the entire optical range 0 ≤ ω0 ≤ 1 of interest. Using H, we carry out the variation of constants in the inhomogeneous system (5.4.3a) and (5.4.3b) using the boundary conditions (5.4.4) and (5.4.5). After some intermediate steps, the calculation for the reflection R results in the equation R=

1 A1 + D1 . ω0

(5.4.16)

The related transmission T is given by  T = A1 H(k, − k2 , τ ) − D2 H(k, − 1 μ, τ ).

(5.4.17)

The integration constant A1 and the coefficients D1 and D2 follow from relations  ω0 C1 (2μ + 1)H(k, − 1 μ, τ ) + μ(k + 2)e−τ /μ A1 = ω0 · , (kμ − 1)[ω0 C1 H(k, − k,τ ) + e−k τ ] ω0 C1 = 2 + k − ω0 ,

(5.4.18) (5.4.19)

5.4

Three-Flux Approximation

345

 2+k , D1 = −μ · kμ − 1   1 + 2μ D2 = ω0 · , kμ − 1 

(5.4.20) (5.4.21)

where k is the amount of characteristic root of Equation (5.4.9). According to Equation (5.4.7), the reflection contains no directional component; the entire directed–diffuse reflection Rμd is given by averaging of R over the entire range 0 ≤ μ ≤ 1: Rμd =

1 R. 2

(5.4.22)

In contrast, the transmission of three-flux approximation includes the directional intensity J(τ ); the entire directed–diffuse transmission Tμd follows from Equations (5.4.5b) to (5.4.3c) as Tμd =

1 T + e−τ /μ . 2

(5.4.23)

As in previous sections, we check over the results above using special optical cases. After this, the related boundary surface corrections are considered for the present approximation.

5.4.3 Optically Different Materials As for previous models, the results of the three-flux approximation must conform to what is expected from experience. We will check the validity of the relevant formulas for the three special cases known from the legs of the optical triangle. First, an opaque layer, mathematically described by the limit of τ → ∞, is considered. After some manipulation using Equations (5.4.17) and (5.4.23), we obtain the directed–diffuse transmission Tμd = 0 for this case. In addition, Equations (5.4.16) and (5.4.22) give the entire reflection of an opaque material Rμd =

μω0 . √ √ (1 + 2μ 1 − ω0 )(1 + 1 − ω0 )

(5.4.24)

Within the albedo range 0.1 ≤ ω0 ≤ 1, this formula leads to values which correspond quite accurately with those of a six-flux approximation [15]. It is, therefore, expected that for opaque layers the three-flux theory considered here results in an accuracy of the six-flux model. In fact, the reflection of an opaque medium as given by the three-flux approximation nearly agrees with that of a 32-flux theory of diffuse radiation field. Similar considerations are valid for

346

5 Theories of Radiative Transfer

opaque layers with surface boundary corrections; see Fig. 5.36 [15]. In an analogous procedure to that used for the two-flux approximations (Appendix A.3), the albedo ω0 can be directly determined by rearranging Equation (5.4.24): ω0 =

(1 + 2μ)[2μ + (2μ − 1)Rμd ]Rμd . μ2 (1 + 2Rμd )2

(5.4.25)

In the case of incident directional component normal to the surface, we have the value μ = 1, resulting in ω0 =

3 · (2 + Rμd )Rμd . (1 + 2Rμd )2

(5.4.26)

In all further approximations the resulting expressions for opaque materials can be solved in terms of the albedo ω0 or the ratio of K/S. A transparent material does not scatter; the albedo has the value ω0 = 0. The corresponding entire transmission follows from Equation (5.4.23). With the condition T = 0 in Equation (5.4.17), we obtain the expected one-dimensional Lambert–Beer law: Tgd = J(τ , μ) = e−τ /μ .

(5.4.27)

In the case of a purely transparent material, the radiative transfer described in the three-flux approximation is only assumed by directed intensity J(τ , μ). It is clear that this is in accordance with absent scattering, absent diffuse components, and zero reflection. The conservative case is characterized by pure scattering with unity albedo ω0 = 1. The reflection of an opaque layer can be calculated from Equation (5.4.25) resulting in a value Rμd = μ. In other words, a purely scattering medium reflects the influx without loss. Considering the general conservative case of a translucent material with finite optical path length τ < ∞, follows that the characteristic roots (5.4.9) vanish. According to the considerations in the previous section, the auxiliary function now gives H(0,0,τ ) = τ . If we use H(0, – 1/μ, τ ) from Equation (5.4.15) together with Equations (5.4.16), (5.4.17), (5.4.18), (5.4.19), (5.4.20), (5.4.21), (5.4.22), and (5.4.23), we obtain Rμd = Tμd =

2μτ + μ(1 − 2μ)(1 − e−τ/μ ) , 2(1 + τ )

2(1 + τ − μτ )e−τ/μ + μ(1 + 2μ)(1 − e−τ/μ ) . 2(1 + τ )

(5.4.28)

(5.4.29)

The sum of these can be simplified to Rμd +Tμd = μ+(1−μ)e−τ/μ ; this means, Rμd +Tμd = 1 for μ = 1, in complying with conservation of energy in Equation

5.4

Three-Flux Approximation

347

(2.1.14). On the basis of this result, it is useful always to illuminate a plane material perpendicular to the surface; this simplifies the numerical evaluation. Equations (5.4.10) and (5.4.11), valid in the general case of translucent layers, are difficult to directly solve in terms of ω0 or τ . These solutions can be more readily determined by suitable interpolation methods, for instance, using the features of the optical triangle where starting values could be provided by the two-flux approximation. If both reflection and transmission of a layer are determined from measurements over two different background surfaces, the analysis changes in comparison to Equations (4.2.13) and (4.2.14). The three-flux model requires the consideration of the directed–diffuse components Rμd , Tμd together with the diffuse components Rdd , Tdd ; see Fig. 5.22. In this illustration G and D are the directional and diffuse influx contributions; the diffuse intensities inside the layer are shown simplified for an improved overview. As in Section 5.2.4, the reflection of the background x is termed as Rbg, x and the reflection of the layer in optical contact with the background is Rx . With these terms the relation Rx = Rμd +

Rbg, x Tμd Tdd 1 − Rbg, x Rdd

(5.4.30)

holds. This equation follows from Equation (4.2.6) by inserting the appropriate quantities that are shown in Fig. 5.22 and follow from Fig. 4.13 with T3 = 1,

Rx G

D z=0 i

J

z

J–

j z=d

Tμd Rbg, x

Fig. 5.22 Determination of reflection over background surface x of diffuse reflectance Rbg, x according to the three-flux model

348

5 Theories of Radiative Transfer

R1 = Rμd , Rˆ 1 = Rdd , R2 = Rbg, x , etc. Usually in practice, two different opaque backgrounds of known diffuse reflections Rbg,w and Rbg,b are used. With the test layer in optical contact with the background, the external reflections Rw and Rb are then measured. The directed–diffuse reflection of the layer Rμd can then be determined by combination of the relations Rw = Rμd +

Rbg,w Tμd Tdd , 1 − Rbg,w Rdd

(5.4.31a)

Rb = Rμd +

Rbg,b Tμd Tdd . 1 − Rbg,b Rdd

(5.4.31b)

In order to get the quantity Rμd it is necessary to calculate Rdd using Equation (5.4.10). The directed–diffuse transmission Tμd follows from similar considerations applying twice relation (4.2.8).

5.4.4 Correction of Surface Effects In approximations with more than two diffuse fluxes, together with normal reflection, it is possible to take into account the total reflection from the external and internal boundary surfaces of the layer. Diffuse radiation is completely reflected for incident angles in the range between the critical angle γ cr and 90◦ ; such reflected diffuse radiation has been called partly directed. In the three-flux approximation, the surface correction has to be extended to three different types of fluxes: diffuse, directed, and partly directed [15]. Because each of these fluxes can be found at both internal surfaces of the layer, the correction for this three-flux model results in a modified six-flux approximation; as mentioned previously, the edge loss of scattered radiation is always left out of consideration. Each radiation type is associated with a specific reflection coefficient, which depends on the refractive indices of the corresponding materials in optical contact. The refractive indices for directed and diffuse light are shown schematically in Fig. 5.23, and they are organized in Table 5.2. The Fresnel quantities for directed radiation r1 , r2 , o1 , o2 are known from Equations (2.1.6), (2.1.7), and (2.1.8) and the coefficients for diffuse light rd∗ , rd , od follow from relation (5.3.28). The coefficients qd and pd correspond to the partly directed components from total reflection from the first and second surfaces, respectively. As mentioned above, total reflection for diffuse radiation occurs for incident angles in the interval γ cr ≤ γ ≤ 90◦ ; see also Fig. 5.17b. This radiation is, therefore, no longer completely diffuse, but rather partially directed and can be interpreted as a focused beam with direction cosine μcr = cos γcr .

(5.4.32)

5.4

Three-Flux Approximation n3

n2

n1

n3

n2

n1 1

349

1 – rd∗ 1

1 – r1, r2 rd∗

r1, r2 1

1 – od

1

od 1 – rd

1 – o1 , o2

1 o1 , o2 rd

Fig. 5.23 Fresnel coefficients for diffuse light (left) and directed light (right)

Table 5.2 Types of radiation in the context of the surface boundary correction according to three-flux approximation Different intensities at the surface boundary as in Fig. 5.24

Light

Fresnel reflection coefficients as in Fig. 5.23

Internal surfaces

External surfaces

Directed

r1 , r2 , o1 , o2

I1 , I–1 ; J1 , J–1

I˜−1 ; J˜ 1

Partly directed

qd ; pd

I2 , I–2 ; J2 , J–2

I˜−2 ; J˜ 2

Diffuse

rd∗ , rd ; od

id , i–d ; jd , j–d

˜i−d ; ˜jd

The corresponding reflection coefficients qd and pd are given by the ratio of the appropriate reflected flux to the incident flux at the boundary surface; therefore qd =

rd 2μcr

(5.4.33)

pd =

od . 2μcr

(5.4.34)

and

350

5 Theories of Radiative Transfer

G

J1

I1

~ I–1

I–1

J–1

I2

J2

~ J1

~ J2

~ I–2 I–2

D

~ i–d

J–2

id

jd

i–d

j–d

~ id

Fig. 5.24 Intensities at the boundary surfaces to correct the influence of different refractive indices. In the figure, indices denote ±1: directed, ±2: partly directed, ±d: diffuse intensities

In both relations the quantities rd and od represent the coefficients for purely diffuse light at the first and second boundaries; see Fig. 5.23. Diffuse light always induces partly directed components at different boundaries.4 In order to determine the corrected reflection and corrected transmission RGD and TGD outside the layer, the diffuse–diffuse and directed–diffuse reflections Rdd and Rμd as well as the corresponding transmission components Tdd and Tμd given in Section 5.4.2 need to be included. The various intensities referred to in the three-flux approximation, which appear at each of boundary surfaces of a layer, are shown in Fig. 5.24; the meanings of these quantities are summarized in Table 5.2. Note that the intensities indicated in Fig. 5.24 result from either the directional influx G component (above, indices ±1), the diffuse influx D component (below, index d), or the total reflection (middle, indices ±2). In addition, some of the various intensities of these three groups contribute to one another; this is in processes that will be roughly outlined here. The following examination leads to expressions for determining the external reflection RGD and external transmission TGD in analogy to Section 5.3.4. 4 They are not taken into consideration in the Kubelka–Munk theory because only diffuse radiation is allowed.

5.4

Three-Flux Approximation

351

The directed influx G is divided into two parts: one part, r1 G, is immediately reflected from the surface, the remaining, transmitted intensity I1 , which, after directional transmission Tμ = e−τ /μ

(5.4.35)

through the layer, ultimately becomes J1 . The intensity J1 is partially reflected into J–1 with reflection coefficient o1 for directional flux; J–1 becomes I–1 after the directional transmission Tμ back through the layer. Finally, the fraction r1 I–1 contributes back to the intensity I1 . Next we consider the partly directed intensity I2 resulting from total internal reflection of i–d ; the intensity I2 becomes J2 by after partly directed internal transmission through the layer Tpμcr = e−τ /μcr .

(5.4.36)

The intensity J2 contributes to J–2 by directional reflection with Fresnel coefficient o2 . From diffuse influx D, the part rd∗ D is initially reflected, the transmitted part (1 − rd∗ )D corresponds to the diffuse intensity id ; on the one hand it contributes to the diffuse intensity jd by diffuse–diffuse transmission Tdd , on the other hand to i–d by diffuse–diffuse reflection Rdd . Both the internal reflection Rdd and transmission Tdd are known from two-flux theory in Section 5.4.2. With these and further interrelated intensities at the boundary surfaces a system of linear equations results. From this system, the external reflection RGD and transmission TGD can be calculated [25]. Only the internal intensities I–1 , I–2 , i–d at the inner first boundary and J1 , J2 , jd at the second inner boundary are of interest. The external reflection is comprised of the intensities I˜−1 , I˜−2 , and ˜i−d at the influx surface, which are given by I˜−1 = (1 − r1 )I−1 + r1 G,

(5.4.37a)

I˜−2 = (1 − r2 )I−2 ,

(5.4.37b)

˜i−d = (1 − rd )i−d + rd∗ D.

(5.4.37c)

The external transmission can be determined by equations J˜ 1 = (1 − o1 )J1 ,

(5.4.38a)

J˜ 2 = (1 − o2 )J2 ,

(5.4.38b)

˜jd = (1 − od )jd .

(5.4.38c)

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5 Theories of Radiative Transfer

The diffuse intensity id follows from the relation id = (1 − rd∗ )D.

(5.4.39)

Equations (5.4.37a), (5.4.37c), and (5.4.39) contain the directional and diffuse components G and D of the total influx F simply given by F = G + D.

(5.4.40)

The mean of total influx F is comprised of directional flux μG and the mean of the diffuse part D resulting in D/2: 1 F = μG + D. 2

(5.4.41)

The external reflection RGD and transmission TGD are now given by   1 1 μI˜−1 + μcr I˜−2 + ˜i−d , RGD = 2 F   1 1 ˜ ˜ ˜ μJ1 + μcr J2 + jd . TGD = 2 F

(5.4.42)

(5.4.43)

The factor of 1/2 in the third term of each relation results from the averaging of the diffuse intensities ˜i−d and ˜jd over μ. Note: considering this and the explanations in Section 5.4.3, it is clear that each of the various approximations used to solve the radiative transfer equation needs a separate surface boundary correction.

5.4.5 Special Cases of External Reflection and Transmission As in the case of two fluxes, there needs to be a reality check for the external reflection and transmission expressions of the three-flux approximation in various special cases. The capabilities of the three-flux model particularly reveal themselves in conjunction with the surface boundary corrections. In order to minimize the effort, we now consider only materials of ideal transparency, ideal absorbance, and ideal opacity (caused by scattering); these special cases are represented in the corner points of the optical triangle. The corrected transmission TGD is given by TGD =

1 {(1 − r1 )[2μ(1 − o1 )Tμ + (1 − od )Tμd ] · G + (1 − od )(1 − rd∗ )Tdd · D}, 2F (5.4.44)

5.4

Three-Flux Approximation 1.0

353

AG A3

External transmission TGD

0.8

n = 1.5 μ=1

τ = 0.3

1.0

0.6

0.4 3.0

0.2 10 102 0 BG B3 0

0.95 0.99 ω0 = 0.5 0.8 0.2 0.4 0.6 0.8 External reflection RGD

C 1.0

Fig. 5.25 Corrected optical triangles: A3 B3 C corresponding to three-flux approximation and AG BG C to directional two-flux approximation of Fig. 5.19; μ = 1, n = 1.5

where the reflection coefficients r1 , o1 and od , rd∗ are known from Equations (2.1.6) and (5.3.28), and the transmissions Tμ , Tμd , and Tdd can be calculated from the expressions in previous sections. Using values μ = 1, n1 = n3 = 1, and n2 = 1.5, for the ideal transparent layer the transmission is TGD = 0.8658; the remaining 13.42% are completely reflected by the illuminated surfaces. It follows, therefore, that the corner point A3 in Fig. 5.25 has coordinates A3 = (0.1342, 0.8658). This result is instructive in so far that the external reflection of RGD = 0.1342 is only slightly larger than the sum of the coefficients r + rd∗ = 0.1318. An opaque material is characterized by zero internal and external transmission, corresponding to Tμ = Tpμcr = TGD = 0. This means that no intensity reaches the second surface and, in the end, I–1 = I–2 = 0. If Equations (5.4.10), (5.4.24), and (5.4.33) are taken into consideration, the external directed–diffuse reflection is given by

RGD

'  [(1 − r1 )Rμd G + (1 − rd∗ )Rdd D] 1 ∗ + rd D . = 2μr1 G + (1 − rd ) · (1 − qd Rpμcr ) 2F (5.4.45)

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5 Theories of Radiative Transfer

The denominator in this formula contains the product of reflection coefficient qd with partly directed reflection Rpμcr produced by total reflection. The quantity Rpμcr is given by Equations (5.4.16), (5.4.17), (5.4.18), (5.4.19), (5.4.20), (5.4.21), and (5.4.22) for μ = μcr . The efficiency of the three-flux model can hardly be better demonstrated than with Fig. 5.36: the boundary surface correction of an opaque layer shows an accuracy comparable to that of a 32-flux model with diffuse radiation. In the case of an absorbing opaque layer, there is additionally the fact that intensity i–d vanishes; from Equations (5.4.42) and (5.4.37c), the expression RGD

  1 1 ∗ = μr1 G + rd D 2 F

(5.4.46)

follows. The external reflection is only comprised of the directed and diffuse influxes reflected with Fresnel coefficients r1 and rd∗ at the boundary surface. With the previously used values r1 = 0.04 for perpendicular illumination, rd∗ = 0.09178, and presupposing G = D = 1, we obtain values for (RGD , TGD ) coordinates corresponding to corner point B3 = (0.0573, 0) shown in Fig. 5.25. A scattering opaque layer is represented by point C in Fig. 5.25 and belongs to the conservative case characterized by ω0 = 1. If we work from Equation (5.4.28) and use Equation (5.4.15), Rμd = 2μ and Rpμcr = 2μcr follow; the use of Equations (5.4.33) and (5.4.10) leads to Rdd = 1 and the simple result RGD =

  1 1 μG + D = 1; 2 F

(5.4.47)

this is in accordance with the law of energy conservation. With regard to the corrected optical triangle, the coordinates of corner points A3 and B3 from three-flux approximation are listed in Table 5.3 together with those of two-flux approximations previously described. The area of the triangle A3 B3 C in Fig. 5.25 is about 8% smaller than that of the ideal triangle AG BG C, which corresponds to the directional two-flux approximation (Fig. 5.19). The greater area of triangle A3 B3 C, compared with that of AD BD C in Fig. 5.20, is caused by the directional intensity of the three-flux model. Altogether, the approach of the three-flux model is the simplest consistent approximation which overcomes the disadvantages of Kubelka–Munk theory and Saunderson mix. There are two critical factors included in three-flux formalism: the directional flux and the partly directed component of total reflection. Both components cause the essential shift of corner point AD into A3 and the correlated expansion of triangle area. Consequently, the inconsistent Saunderson mix is unnecessary. Finally the external reflection with two different backgrounds has to be considered. Let us assume the standard refractive index of a test layer of n = 1.5 and background of reflections of Rbg,b = 0.04 and Rbg,w = 0.90. In the case of

5.4

Three-Flux Approximation

355

Table 5.3 Coordinates of corner points Ax , Bx of corrected optical triangles of four different approximations for μ = 1, n = 1.5; in all models the conservative case of an opaque layer, represented in corner point C, shows constant coordinates of C = (1, 0) Figure number

Approximation

Sort of flux

Two-flux

Directed–directed 5.19 Saunderson mix 5.20 Diffuse–diffuse 5.20

AG (0.0769, 0.9231) BG (0.0400, 0) ASM (0.3986, 0.6014) BSM (0.0400, 0) AD (0.4311, 0.5689) BD (0.0918, 0)

Three-flux

Directed–diffuse

A3

5.25

Corner point coordinates (RE , TE )

(0.1342, 0.8658) B3

1.0

ω0 = 1.0

0.8

Reflection over black R b

(0.0573, 0)

τ= 10 0.99

0.6

5.0

0.97 0.4 0.90 1.0

0.2 0.5

0.5

0 0

0

0 0.2

0.6 0.4 Reflection over white R w

0.8

1.0

Fig. 5.26 Reflection of a layer in optical contact with black or white undercoat according to the three-flux model; μ = 1, n = 1.5, Rbg,b = 0.04, Rbg,w = 0.90

optical contact between layer and background, we obtain results illustrated in Fig. 5.26: it is a modified optical triangle. The bisector of an angle represents the conservative case, the ends represent pure absorption (albedo equals zero) and pure scattering (albedo equals one); zero albedo corresponds to ideal transparency as previously. The accompanying coordinates (Rw , Rb ) of the corner points are (0.04, 0.04) and (0.96, 0.96). Interestingly, if layer and background are separated by an air gap, the above coordinates in fact remain the same, see Fig. 5.27; however, the curves for constant albedo are bent to a higher degree compared to those of Fig. 5.26.

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5 Theories of Radiative Transfer 1.0

ω0 = 1.0 τ

0.8 Reflection over black R b

10 5.0 0.99

0.6

0.97 0.4

1.0 0.90

0.5

0.2 0.5 0

0

0 0

0.2

0.4

0.6

0.8

1.0

Reflection over white R w

Fig. 5.27 Reflection of a layer in air without optical contact to black or white undercoat according to the three-flux model; μ = 1, n = 1.5, Rbg,b = 0.04, Rbg,w = 0.90

Additionally, both illustrations show families of curves of constant optical path or constant optical albedo which do not correspond exactly between the figures. On the basis of additional reflection loss in the second surface, the triangle area is smaller compared with that for the case of optical contact.

5.5 Approximation of Radiative Transfer by Multi-Flux Theory In this section, the ideas that have been encountered so far are generalized to more than three identical fluxes. The implementation follows the considerations of Chandrasekhar, who divides the radiation field into 2n fluxes: n in the forward direction and n in the backward direction. This method is called the multiflux approximation or nth approximation of radiative transfer [1]. In connection with this approximation, Rybicki [26] modified slightly the system in order to solve the RTE. The formalism became applicable in practice with the procedure of Mudget and Richards [27, 28]. This method was used by Billmeyer and Carter for the first time for the purpose of describing the anisotropic behavior of colorations with metallic pigments [29]. Comprehensive aspects concerning the radiative transfer of scattering materials are dealt in related works

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

357

[30, 31]. Further criteria regarding the numerical solution methods of the radiative transfer equation are given in the literature [32–36]. Experience shows that the procedure of Mudget and Richards describes accurately anisotropic scattering materials – especially those containing effect pigments – only if the number of fluxes is greater than 4. In particular, with the aid of the corner point coordinates of the corrected optical triangle, we illustrate that the power of approximation increases considerably with the number n of fluxes up to n = 256 fluxes. Industrial software for recipe prediction typically works with a maximum of n = 32 fluxes. In fact, the required computing effort increases as n3 . Fulfillment of this is usually justified by the accuracy achieved, especially considering that computing power has drastically increased in recent years. As of this writing, the largest number of workable arithmetic operations per second on a desktop PC is around 109 flops and for a supercomputer about 1015 flops. The unit flops is an acronym of “floating point operations per second” [37]. The strategy here is first to formulate exact solutions for the RTE in terms of integrals; after this, a set of appropriate approximations are possible which make final numerical calculations tractable. Therefore, first of all, we work out the accurate solutions for reflection and transmission, then both quantities are approximated. New to our representation is not only the extension to plane parallel layers but also the boundary surface correction in dependence on the number of fluxes n. In this procedure, different refractive indices and total reflection are taken into consideration.

5.5.1 Exact Solutions for Internal Reflection and Transmission To develop the formalism of the nth approximation, the exact solutions must first be determined. Therefore, in the following, we start out with the RTE (5.1.35), their angle-dependent intensities I(τ , μi ) can be determined with different methods. The simplest version is based on a system of 2n coupled linear differential equations. This system is obtained when the integral in Equation (5.1.35) is substituted by the quadrature formula of Gauss "+1 n  f (μ)dμ = wi f (μi ), −1

(5.5.1)

i = −n i = 0

where the integrand is a general function, not integrable, impossible to integrate, or only known at control points μi of the interval [–1, 1]. The values of μi correspond to the 2n zeros of the Legendre polynomial P2n (μi ) = 0, with i = ±1, ±2, . . . , ±n. The weight factors wi can be determined from relation

358

5 Theories of Radiative Transfer

1 wi = P2n (μi )

"+1 −1

P2n (μ) dμ, μ − μi

(5.5.2)

where P 2n (μi ) is the derivative of P2n (μ) with regard to μ evaluated at μ = μi . With P0 (μ) = 1 and P1 (μ) = μ the higher Legendre polynomials are given by the recursion formula (l + 1)Pl+1 (μ) = (2 l + 1)μPl (μ) − lPl−1 (μ) for l = 1, 2, . . . . The graphs of the first five Legendre polynomials are shown in Fig. 5.28. The continuous radiation field, in this representation, is divided into 2n fluxes with directions μi = cosϑ i . Note that here the antisymmetric properties μ–i = –μi and w–i = –wi are valid. The RTE (5.1.35) was received for rotational symmetric radiation field, and is, therefore independent of azimuth angle ϕ. This is why the intensities I(τ ,μi ) inside and outside a layer only depend on the optical path τ and the angle of declination μi . Figure 5.29 shows an example for magnitude and the direction of n = 16 light cones of intensities I(τ ,μi ) = Ii (τ ); there are 16 additional intensities for opposite direction which are not drawn [39]. In Fig. 5.30, the computed light cones at the illumination surface of an isotropic scattering layer (TiO2 ) and of a coating which contains only metallic pigments are represented on the same scale [18]. For fixed values of ϑ i or μi , the extent of the light cones depends on the sorts of colorants in the layer; their scattering behavior is represented by the corresponding phase function. Pn (μ) 1 P0 P1 P2

1μ

–1

P3

P4

–1

Fig. 5.28 Graphs of the first five Legendre polynomials of first kind

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

359

Illumination direction ϑi

Ii(τ)

Light cones

τ=0 Coating n = 1.5 τ = τd

Substrate

Fig. 5.29 Direction of μi = cosϑ i of intensity Ii (τ ) [22] Fig. 5.30 External light cones of constant declination angle; left: pure scattering medium (TiO2 ); right: strong reflective medium (metallic pigments) [18]

RTE (5.1.35), from which the intensities Ii (τ ) are to be determined, is transformed with quadrature formula (5.5.1) into μi

∞ n  ω0  dIi (τ ) = −Ii (τ ) + dl Pl (μi ) wj Pl (μj )Ij (τ ), dτ 2 l=0

j=−n

(5.5.3)

360

5 Theories of Radiative Transfer

where indices i and j take values of i, j = ±1, ±2, . . . , ±n, and j = 0. To solve system (5.5.3) of 2n linear differential equations, we assume, in agreement with Chandrasekhar [1], an exponential solution form. To each root kν of the characteristic equation with ν = ±1, ±2, . . . , ±n corresponds a special solution of the type Ii,ν (τ ) = g(μi ,kν ) · ekν τ ,

(5.5.4)

where g(μi , kν ) is a weight function. The general solution Ii (τ ) = I(τ , μi ) is given by the linear combination Ii (τ ) =

n 

Cν Ii,ν (τ ).

(5.5.5)

ν = −n ν = 0

The 2n integration constants Cν are given by the boundary conditions which we now formulate. For that, we consider the rays shown in Fig. 5.31. It is assumed that there are no particles in the space τ ≤ 0 outside of the layer – whether reflecting, scattering, or absorbing. The intensity I(0, μi ) = Ii (0) incident at the surface with angle μi is the influx intensity Ie (0, μi ), which, for simplicity, is assigned as 1; this boundary condition can be written as I(0, μi ) = Ie (0, μi ) = 1, 0 ≤ μi ≤ 1.

(5.5.6)

Fig. 5.31 Some representative beams at a layer to formulate the boundary conditions

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

361

In the region with τ ≥ τ d , it is also assumed that there are no scattering centers; therefore, no light can be scattered back from that half-space. This boundary condition can be written as Ie (0, μ) = I(0, μi )δ(μ − μi ),

− 1 ≤ μj ≤ 0.

(5.5.7)

The reflection R of a layer is given by the sum over all intensities I(0, –μj ) at the influx surface in the range 0 ≤ μj ≤ 1. The transmission T, on the other hand, follows from the sum over all intensities I(τ d , μj ) at the transmitted light boundary in the same angular range. To later simplify the relations for R and T, each of the intensities at the boundary surfaces is combined. Therefore the so-called reflection function S(μi ,μj ) is defined, which is given by the ratio of the weighted reflected flux I(0, –μj )μj to the incident intensity I(0, μi ) at position τ = 0 in the direction of μi I(0, −μj )μj . I(0, μi )

S(μi ,μj ) =

(5.5.8)

At position τ = τ d , the intensity I(τ d , μj ) leaves the layer in the direction μj . Analogous to the reflection, we define the transmission function T(μi ,μj ) given by the quotient of the weighted transmitted flux I(τ d , μj )μj in the direction of μj and the influx intensity of direction μi T(μi , μj ) =

I(τd , μj )μj . I(0, μi )

(5.5.9)

In both definition equations (5.5.8) and (5.5.9) μi and μj vary independently of one another in the ranges 0 ≤ μi ≤ 1 and 0 ≤ μj ≤ 1. The integration over the product of I(0,μi ) and the reflection function S(μi ,μj ) in the range 0 ≤ μi ≤ 1 leads to the reflected flux "1 I(0, − μj )μj =

S(μi ,μj )I(0,μi )dμi .

(5.5.10)

0

Accordingly, the transmitted flux at τ = τ d is given by "1 I(τd ,μj )μj =

T(μi ,μj )I(0,μi )dμi .

(5.5.11)

0

Whereas in the two- and three-flux approximations, the radiation field is divided into at most three different intensities; we now have to integrate over each of the n fluxes in the range of 0 ≤ μj ≤ 1.

362

5 Theories of Radiative Transfer

In the general case of continuously varying angles μ, we define the reflection of a layer by the ratio of the total reflected radiative flux at τ = 0 to the total incident radiative flux in the range of 0 ≤ μ ≤ 1: "1 I(0, −μ)μdμ R=

0

.

"1

(5.5.12)

I(0, μ)μdμ 0

Similarly, at layer position τ = τ d , the transmission T is defined by the ratio of the total transmitted radiative flux and the total incident flux "1 I(τd , μ)μdμ T=

0 "1

.

(5.5.13)

I(0, μ)μdμ 0

The quantities R and T denoted here are more precisely internal reflection and internal transmission in order to differentiate them from the external quantities which result from the boundary surface corrections (next section). The quantities R and T can be determined if the reflection and transmission functions (5.5.8) and (5.5.9) and the measuring geometry are known. The sort of illumination is directional or diffuse and is taken into account in the intensity I(0, μ). Generally the directional incident flux is scattered diffusely by the homogenized pigment particles of the layer; this state is called directed–diffuse and indicated by the arguments (μi , d). In the case of diffuse illumination, the resulting fluxes are also diffuse; this condition is called diffuse–diffuse and is indicated with (d, d). These illumination and measuring conditions correspond to different measuring geometries described in Section 4.1.2. According to Equation (5.5.6), the incident directional illumination of direction μi has a normalized intensity I(0, μi ) = 1. The corresponding directed– diffuse reflection R(μi ,d) follows from Equation (5.5.12) together with relation (5.5.10) and integration 1 R(μi ,d) = μi

"1 S(μi ,μj )dμj . 0

(5.5.14)

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

363

The corresponding directed–diffuse transmission yields with Equations (5.5.13) and (5.5.11) 1 T(μi ,d) = μi

"1 T(μi ,μj )dμj .

(5.5.15)

0

In the second case of diffuse illumination, the integral in the denominator of Equations (5.5.12) and (6.2.14) comes to a value of 1/2; this results from the averaging of the incident fluxes over μi in the semi-space above the reflection surface. Therefore, with Equations (5.5.12) and (5.5.14), respectively (5.5.15), the diffuse–diffuse reflection and transmission state is represented by the relations "1 R(d,d) = 2

μi R(μi ,d) dμi ,

(5.5.16)

μi T(μi ,d) dμi .

(5.5.17)

0

"1 T(d,d) = 2 0

The internal reflection and transmission, then, are given by integration of the reflection or transmission functions. The factor of 2 in front of the integral in Equations (5.5.16) and (5.5.17) is a consequence of the illumination conditions. At this stage, we refrain from checking the validity of the above results for special cases; such checks are performed in [26] with the three special cases of the optical triangle.

5.5.2 Surface Boundary Corrections In most practical cases, the refractive indices inside and outside of an optical layer are not identical; consequently, for accurate calculation, the reflection and transmission require a boundary surface correction. Therefore, we assume a plane parallel layer of refractive index ns ; outside the layer the refractive indices are no and nu for τ < 0 and τ > τ d , respectively; see Fig. 5.32. At the reflection surface, the quantities ns and no determine the angle-dependent Fresnel coefficient r(μ). With relations n = n1 = ns /no and (2.1.6), it follows 1 r(μ) = 2



μn1 − w μn1 + w

2

1 + 2



μ − n1 w μ + n1 w

2

 , w = + 1 − n21 (1 − μ2 ). (5.5.18)

364

5 Theories of Radiative Transfer

The reflection coefficient o(μ) is given, if n1 in this formula is substituted with n2 = ns /nu .

Fig. 5.32 Beams at the inner surface boundaries of a plane parallel layer

In the following, we divide the surface boundary corrections into three regions: inside the layer, at the front surface, and at the rear surface; these are now treated in detail. The intensities at the inner boundary surfaces of the layer result from the incoming beams Ie (0,μj ) with optical path τ d = 0, given by Ie (0,μj ) = I(0,μi )δ(μj − μi );

(5.5.19)

see Fig. 5.32. The intensity I(0, μj ) is comprised of the contributions Ie (0, μj ) and I(0, –μj ), the latter is reflected at the inner boundary surface with reflection coefficient r(μj ): I(0,μj ) = Ie (0,μj ) + r(μj )I(0, −μj ).

(5.5.20)

As in Fig. 5.32, the intensity I(τ d ,μj ) at the transmission surface is reflected by the fraction o(μj ) into I(τ d , –μj ) I(τd , −μj ) = o(μj )I(τd ,μj );

(5.5.21)

the remaining part 1 – I(τ d , –μj ) is transmitted through the surface boundary. Now, the intensity I(0, –μj ) is composed of all reflected components I(0,μi ) and all transmitted components I(τ d , –μi ), see Fig. 5.32. Together with the reflection and transmission functions S(μi ,μj ) and T(μi ,μj ) from Equations (5.5.8) and (5.5.9), the reflected flux is, therefore, given by "1 I(0, −μj )μj =

"1 S(μi ,μj )I(0,μi )dμi +

0

T(μi ,μj )I(τd , −μi )dμi . 0

(5.5.22)

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

365

Similar lines of reasoning result in the fact that the transmitted flux I(τ d ,μj )μj is comprised of all contributions of the internal intensities I(0,μi ) and I(τ d ,–μj ) "1 I(τd ,μj )μj =

"1 T(μi ,μj )I(0,μi )dμi +

0

S(μi ,μj )I(τd , −μi )dμi .

(5.5.23)

0

From these two equations, one can see immediately that the reflection and transmission functions have reversed roles in the integrals. In the following, the formalism is simplified if we introduce the inner reflection and transmission functions S(μi ,μj ) and T(μi ,μj ). These are analogs to Equations (5.5.8) and (5.5.9) defined by the expressions S(μi ,μj ) =

I(0, −μj )μj , I0 (μi )

(5.5.24)

T(μi ,μj ) =

I(τd ,μj )μj . I0 (μi )

(5.5.25)

In Equations (5.5.22) and (5.5.23), we substitute I(0, μj ) by Equation (5.5.20) and the intensity I(τ d , μj ) by Equation (5.5.21). Together with the definitions (5.5.8) and (5.5.9), we obtain two further integral equations with which the desired functions S(μi ,μj ) and T(μi ,μj ) can be approximately determined. 5.5.2.1 Corrected Reflection Function The rays arriving at the front surface determine the corrected reflection function. According to the designations in Fig. 5.33, the layer is illuminated from the upper semi-space at angle α by the intensity Je (0,ν), in which ν is defined as ν = cosα. The intensity Je (0,ν) is partly reflected by an amount Jr (0, –ν) and the remaining Ig (0,μ) enters the layer. The corresponding direction μ in the layer comes from the refraction angle β given by μ = cosβ. According to conservation of energy, the incident energy equals the sum of the contributions which are reflected and transmitted by the boundary surface Je (0,ν)νdν = Jr (0, −ν)νdν + Ig (0,μ)μdμ.

(5.5.26)

This assumes, of course, that the boundary surface itself does not absorb any light. In Equation (5.5.26), the variable ν can be expressed in terms of μ. To do this, the law of refraction sin α/sin β = ns /no = n1 is written in the form 1 − ν 2 = n21 (1 − μ2 ),

(5.5.27)

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5 Theories of Radiative Transfer

Fig. 5.33 Beam constellation at the reflection surface of a layer

which gives, by differentiation, νdν = n21 μdμ. The quantities ν and μ are related by the law of refraction (5.5.27); it is, therefore, possible to write all further expressions purely in dependence on μ. In analogy to Equation (5.5.24), we define the corrected reflection function J(0, −μj )μj  S(μi ,μj ) = . J0 (μi )

(5.5.28)

We obtain, therefore, the following expression:  S(μi ,μj ) = μi r(μi )δ(μi − μj ) + [1 − r(μi )][1 − r(μj )] · S(μi ,μj ).

(5.5.29)

The first term on the right-hand side of Equation (5.5.29) represents the reflected amount of the incident light at the reflection surface. The second term takes into account the scattering contribution from the inner reflection of directions μi and μj which pass through the boundary surface. 5.5.2.2 Corrected Transmission Function Finally, the fluxes at the transmission surface have to be considered; see Fig. 5.34. The incoming intensity I(τ d ,μ) at the inner boundary surface is split up into the transmitted component J(τ d ,μ) and the reflected component I(τ d , –μ). In the following, the intensities which leave the layer in the direction of μ are summed up by the corrected transmission function. Considerations similar to the corrected reflection function suggest that the intensity J(τ d ,μ) which leaves the layer can be described by the relation J(τd ,μ) = n22 J(τd ,ν ), where the primed variable is given by ν = cos β .

(5.5.30)

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

367

Fig. 5.34 Beams at the transmission surface of a layer without underground

With the designations given in Fig. 5.34, the indicated intensities lead now to the relation J(τd ,μ) = [1 − o(μ)]I(τd ,μ).

(5.5.31)

In this expression, o(μ) represents the Fresnel coefficient at the transmission surface. Following Equation (5.5.25), we define the corrected transmission function  T (μi ,μj )by the quotient  T (μi ,μj ) =

J(τd ,μj )μj , J0 (μi )

(5.5.32)

from which the expression  T (μi ,μj ) = [1 − r(μi )][1 − o(μj )] · T(μi ,μj )

(5.5.33)

is finally obtained if similar formulations are used as for derivation of the reflection function (5.5.29).

5.5.2.3 Reflection Over an Opaque Substrate If a transparent or translucent plane parallel layer is in optical contact with an opaque substrate, the corresponding conditions shown in Fig. 5.34 have to be taken into account. In this case, the incident intensities are altered at the latest at the upper boundary of the opaque substrate. From the incoming intensity J(τd , ν ) at this boundary, the part J(τd , −ν ) is reflected; it contributes to the fluxes at both boundary surfaces of the higher layer. The intensities at the upper surface of the substrate can be interpreted as a special case of the conditions

368

5 Theories of Radiative Transfer

given at the reflection surface of a layer with I(0, –μ) = Ig (0,μ) = 0 at τ = 0, cf. Fig. 5.33. In order to determine the reflection of the substrate for which we denote the Fresnel coefficient u(ν ), it is possible to follow after the above formalism for the corrected reflection function. Consequently, it follows J(τd , ν ) = Je (ν) ,

(5.5.34)

J(τd , −ν ) = Jr (0, −ν).

(5.5.35)

With both equations, if Equation (5.5.21) is taken into account and u(ν ) = r(μ) is used in Equation (5.5.29), we obtain the corrected reflection function which corresponds to the substrate used. In order to specify the corrected reflection and transmission functions (5.5.29) and (5.5.33), Equation (5.5.31) has to be modified by the addition of the reflected contribution of the substrate r(ν ) · J(τd , −ν ): J(τd , ν ) = r(ν ) · J(τd , −ν ) + [1 − o(μ )] · I(τd , μ ).

(5.5.36)

Additionally, the analogous equation I(τd , −μ ) = o(μ ) · I(τd , μ ) + [1 − o(ν )] · J(τd , −ν )

(5.5.37)

has to be taken into consideration. We refrain from going into further details at this stage. In connection with the corrected reflection and transmission functions (5.5.28) and (5.5.29), we are now able to give the exact solutions of the reflection and transmission of the interested transparent or translucent layer.

5.5.3 Total Reflection and Optical Extreme Cases In order to include total reflection, in all previous integrals the lower integration limit has to be replaced by the cosine of the appropriate critical angle. Optically, this means that light rays at angles beyond the critical angle are then excluded from the summations, meaning that they do not contribute to the total external quantities calculated. Light outside the layer can only be observed if the corresponding beams at the upper or lower inner boundary have angles smaller than the critical angles γ u or γl . With the relations μu = cosγ u and μl = cosγl coming from the refraction law (2.1.5), it follows μ = u



1 − n−2 1 ,

 μl = 1 − n−2 2 .

(5.5.38)

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

369

The functions S and T in relations (5.5.14), (5.5.15), (5.5.16), and (5.5.17) have to be substituted by the corrected functions  S and  T , respectively. The lower integration limits for the corrected external reflection  R and transmission  T now u have to be μ and μl . With these considerations follows for the directed–diffuse case  R(μi ,d) =

1 μi

1  T (μi ,d) = μi

"1

 S(μi ,μj ) dμj ,

(5.5.39)

 T (μi ,μj ) dμj .

(5.5.40)

μu

"1 μl

For diffuse–diffuse external reflection and transmission we obtain in a similar manner  R(d,d) =

"1 2n21

 T (d,d) = 2n21

"1 dμi

μu

μu

"1

"1 dμi

μu

 S(μi ,μj ) dμj ,

(5.5.41)

 T (μi ,μj ) dμj .

(5.5.42)

μl

This formally describes the surface boundary corrections. The solutions above, which correspond to the internal and external reflection and transmission, hold exactly. Among the special cases represented by the optical triangle, it is particularly critical to consider the conservative case. The validity of the balance equations  T (μi ,d) = 1, R(μi ,d) + 

(5.5.43)

 R(d,d) +  T (d,d) = 1

(5.5.44)

is proven in the literature [25]. In the second case of a transparent layer, the corrected reflection and transmission can be expressed in concrete terms. Because there arises no internal reflection in a transparent layer, it is clear that the internal reflection function (5.5.8) vanishes. From Equations (5.5.39) and (5.5.40) for the directed–diffuse reflection and transmission, it follows

370

5 Theories of Radiative Transfer

 R(μi ,d) =

r(μi ) + o(μi )[1 − 2r(μi )] · e−2τ /μi , 1 − o(μi )r(μi ) · e−2τ /μi

(5.5.45)

[1 − r(μi )][1 − o(μi )] · e−τ /μi . 1 − o(μi )r(μi ) · e−2τ /μi

(5.5.46)

 T (μi ,d) =

The diffuse–diffuse reflection and transmission also can be written in closed terms. From Equations (5.5.41) and (5.5.42), we have  R(d,d) =

"1 2n21

R(μi ,d) dμi , μi

(5.5.47)

T (μi ,d) dμi . μi

(5.5.48)

μu

 T (d,d) =

"1 2n21 μu

Both integrals are solvable by elementary methods. Because each Equation (5.5.45), (5.5.46), (5.5.47), and (5.5.48) is independent of the number of fluxes and generally applicable they can be used to check the exactness of an approximation chosen. For an opaque layer at the location τ = τ d , all internal intensities are zero and, therefore, the transmission function (5.5.53) is also zero. The reflection function S(μi ,μj ) reduces appropriately. If we use this expression in Equation (5.5.29) together with Equations (5.5.39), (5.5.40), and (5.5.41), it follows  R(μi ,d) = r(μi ) +

1 − r(μi ) μi

 R(d,d) =

"1 [1 − r(μi )] · S(μi ,μj ) dμj ,

(5.5.49)

μu

"1 2n21

R(μi ,d) dμi . μi

(5.5.50)

μu

With Equations (5.5.45), (5.5.46), (5.5.47), (5.5.48), (5.5.49), and (5.5.50), corresponding to transparent and opaque layers, the exact corner point coordinates of the optical triangle are given (Section 5.5.7).

5.5.4 Boundary Conditions, Matrices of Reflection and Transmission Before considering the evaluation relations of the nth approximation, we summarize the essential steps leading to the exact solutions given in the previous sections. By dividing the radiation field into 2n fluxes with directions μi and

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

371

together with the quadrature formula of Gauss, RTE (5.1.35) has been converted into a system of 2n linear differential equations (5.5.3). This system can be solved for 2n intensities from which the reflection and transmission of a layer are given, with or without boundary corrections including total reflection. Unfortunately, the exact integrals for the reflection and transmission usually are not solvable in closed form; therefore, they are approximated numerically by the Gauss quadrature formula "1 f (μ)dμ =

n 

wi f (μi ).

(5.5.51)

i=1

0

The Dirac delta function δ(μi – μj ), contained in the integrands, is replaced by the weighted Kronecker symbol δ(μi − μj ) ≈

1 δi, j , wi

(5.5.52)

where δ i, j is the original Kronecker symbol, which has the value 1 for i = j and 0 for i = j. The greater the number 2n of fluxes chosen, the better the approximation for the integrals as well as the internal and external reflection and transmission becomes. This is confirmed, for instance, in Section 5.5.7, with the corner point coordinates of the corresponding corrected optical triangles. In addition, we apply the 2n boundary conditions (5.5.6) and (5.5.7) to the general solution of the coupled differential equation system (5.5.5) Ii (0) = I(0,μi ) = Ii (τd ) = I(τd ,μi ) = 0,

1 δi,k , wi

i, k = 1, 2, ... , n

i = −n, − n + 1, ... , − 1.

(5.5.53) (5.5.54)

With these conditions, the 2n integration constants Cν contained in the general solution (5.5.5) can be determined from the following equations: n 

Cν Ii,ν (0) =

ν=−n ν =0 n 

1 δi,k , wi

Cν Ii,ν (τd ) = 0,

i, k = 1, 2, ... , n,

(5.5.55)

i = 1, 2, ... , n.

(5.5.56)

ν=−n ν =0

For the moment, we restrict ourselves to the uncorrected reflection and transmission. In order to approximate the integrals of the previous three sections, first the reflection function S(μi, μj ), from definition equation (5.5.8) with I0 (μi ) = 1,

372

5 Theories of Radiative Transfer

is transformed into the quadratic (n,n) matrix S = (sij ) given by S = (sij ) = S(μi ,μj ) = I−j (0)μj ,

i, j = 1, 2, ... , n,

(5.5.57)

where we use the abbreviation I−j (0) = I( −μj ,0). In an analogous manner, the transmission function (5.5.9) is replaced by the quadratic (n,n) matrix T with elements tij T = (tij ) = T(μi ,μj ) = Ij (τd )μj ,

i, j = 1, 2, ... , n.

(5.5.58)

In the next section, the internal reflection and transmission are given by corresponding summation formulas followed by the expressions which include the surface boundary corrections.

5.5.5 Directional and Diffuse Reflection and Transmission Following from the previous approximations (5.5.51) and (5.5.52), it is possible to give the internal reflection and transmission for directional as well as diffuse illumination in a simple manner. In order to keep track of integrals approximated by a sum, the evaluation formulas are mainly given in the form of sums rather than matrices. From the exact solutions (5.5.14) and (5.5.15) with quadrature equation (5.5.51) and Equations (5.5.57) and (5.5.58), for the case of directional illumination, the equations for reflection and transmission are given as n 1  wj sij , μi

(5.5.59)

n 1  T(μi ,d) = wj tij . μi

(5.5.60)

R(μi ,d) =

j=1

j=1

In the case of diffuse illumination, we obtain the corresponding reflection and transmission from Equations (5.5.16) and (5.5.17) as R(d,d) = 2

n 

wi μi R(μi ,d),

(5.5.61)

wi μi T(μi ,d).

(5.5.62)

i=1

T(d,d) = 2

n  i=1

In the conservative case, the relations R(μi ,d) + T(μi ,d) = 1 and R(d,d) + T(d,d) = 1 are exactly valid; this is proven in the literature [25].

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

373

Expressions (5.5.59), (5.5.60), (5.5.61), and (5.5.62) are applicable only to colored systems which are not embedded in a binder or if refractive indices inside and outside of a layer are identical. In the cases of paints, emulsions, colored plastics, ceramics etc., the boundary surface corrected relations have to be used. In order to obtain those functions, we apply the same approximation prescription to Equations (5.5.22) and (5.5.23), which are then cast into the following forms: I−j (0)μj =

n 

wi sij Ii (0) +

i=1

Ij (τd )μj =

n  i=1

wi sij I−i (τd ) +

n 

wj tij I−i (τd ),

(5.5.63)

i=1 n 

wj tij Ii (0), j = 1, 2, ... , n.

(5.5.64)

i=1

In the above equations, the matrix elements sij and tij follow from Equations (5.5.57) and (5.5.58), and the intensities Ii (0) = I(0,μi ) and I–i (τ d ) = I(τ d ,–μi ) from Equations (5.5.20) and (5.5.21). The reflection and transmission functions S(μi ,μj ) and T(μi ,μj ) are assigned to the quadratic (n,n)-matrices S = (¯sij ) and T = (¯tij ); these consist of the elements sij = S(μi ,μj ),

(5.5.65)

tij = T(μi ,μj ), i, j = 1, 2, ... , n.

(5.5.66)

Finally, the matrix elements s˜ij and ˜tij from  S and  T, which correspond to the reflection and transmission functions (5.5.29) and (5.5.33), are given by s˜ij = μi r(μi )Θi,j + [1 − r(μi )][1 − r(μj )] · s¯ij

(5.5.67)

˜tij = [1 − r(μi )][1 − o(μj )] · ¯tij

(5.5.68)

Consequently, all quantities are known in order to compute the corrected reflection and transmission. Below, the total reflection at the boundary surfaces is taken into consideration.

5.5.6 Inclusion of Total Reflection In Section 5.5.3, angles μl and μu have been used to characterize the total reflection at the internal boundaries; total internal reflection has then been taken into account by using these angles as upper and lower integration limits – effectively removing the non-contributing components from the integrals. Having non-zero lower integration limits in Equations (5.5.39), (5.5.40), (5.5.41), and (5.5.42)

374

5 Theories of Radiative Transfer

for the exact solutions of reflection and transmission requires the summation indices of the relevant quadrature formulas to be revised in a suitable manner. The numbering of the sampling points μi can be chosen in such a way that the first has a value μ1 = 1 and the nth μn = 0. Because of this, the approximations for the corresponding integrals have to be adjusted, at the front boundary, as "1

f (μ)dμ ∼ =

m 

m < n,

(5.5.69)

wi f (μi ), μm−1 > μl > μm , m < n.

(5.5.70)

wi f (μi ),

μm−1 > μu > μm ,

i=1

μu

and, at the rear boundary, as "1 μl

f (μ)dμ ∼ =

m  i=1

Due to Equations (5.5.39) and (5.5.40), in the case of directed–diffuse illumination, the corrected reflection and transmission now are given by  R(μi ,d) =

m 1  wj s˜ij , μi

(5.5.71)

m 1  wj˜tij . μi

(5.5.72)

j=1

 T (μi ,d) =

j=1

In diffuse–diffuse conditions, these equations, using Equations (5.5.41) and (5.5.42), become  R(d,d) = 2n21

m 

R(μi ,d), wi μi

(5.5.73)

T (μi ,d). wi μi

(5.5.74)

i=1

 T (d,d) = 2n21

m  i=1

In the conservative case, Equations (5.5.43) and (5.5.44) are exactly valid as well. This is proven simply by approximating the integrals in Section 5.5.3 using the Gauss quadrature formula. At this stage, it is necessary to go into two numerical aspects which arise during the nth approximation. First, we consider the results concerning the reflection of an opaque layer as in Section 5.5.3. From Equations (5.5.49) and (5.5.50), approximations for the corrected reflections for directional and diffuse illumination follow:

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

 R(μi ,d) = r(μi ) +

375

1 − r(μi )  wj [1 − r(μj )] · s¯ij , μi m

(5.5.75)

j=1

 R(d,d) = 2n21

m 

R(μi ,d); wi μi

(5.5.76)

i=1

in Equation (5.5.76), the quantity  R(μi ,d) corresponds to expression (5.5.75). Obviously, the fraction before the sum in Equation (5.5.75) is not defined for μi → 0. This case has to be considered separately: to determine the limit, for instance, Equation (5.5.18) can be taken into consideration. In addition, the question has to be clarified as to how accurate the integral in Equation (5.5.50) is approximated by the corresponding sum in Equation (5.5.76). Additionally, one must ask if there are additional methods with an enhanced accuracy. According to Equation (5.5.18) and illustrated in Fig. 5.35 for n = 1.5, the Fresnel coefficient r(μ) is extremely dependent on μ. The essential change takes place in the range 0.75 ≤ μ ≤ 0.9, the difference amounts about 95%. In order to estimate accuracy, it is advisable to use an opaque, a transparent, and a pure scattering layer; in this way, it is possible to check the method to find out which delivers the best correspondence to known external reflection values. Figure 5.36 shows the corrected reflection of an opaque layer in dependence of the albedo in the interval 0 ≤ ω0 ≤ 1 and three different approximations for Kubelka–Munk, three-flux, and 32-flux approximations. From the graph,

r (μ) 1.0

0.8

0.6 n = 1.5 0.4

0.2

Fig. 5.35 Fresnel reflection coefficient r of directional radiation in dependence on μ for n = 1.5

0 0

0.2

0.4

0.6

0.8

1.0

μ

376

5 Theories of Radiative Transfer 1.0

KM with Saunderson-Mix 0.8 Three-flux approximation 32-flux apparoximation

Reflection R

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1.0

Albedo ω0

Fig. 5.36 Corrected reflection of an opaque layer in dependence on ω0 in the cases of Kubelka–Munk approximation with Saunderson mix, three-flux, and 32-flux approximation with diffuse–diffuse fluxes; n = 1.5

one can see that only for the last two mentioned approximations, nearly equal corrected reflection values result in dependence on ω0 . Despite the inconsistent adjustment of Saunderson mix at ω0 = 0, the graph of the Kubelka–Munk formalism differs considerably from those of the three- and 32-flux approximations. This behavior is to put down to the real total reflection taken into consideration. Even for opaque materials, the superiority of the multiflux approximation over Kubelka–Munk theory is shown. In other words, the Kubelka–Munk theory is, these days, no longer widely applicable. The great limits in applicability of the Kubelka–Munk formalism outlined in Section 5.3.5 are, therefore, strongly supported here. The historical widespread use of this theory was a result of their simply structured formulas. These formulas can be solved directly with regard to the optical constants. In the threeand multi-flux approximations, the coefficients can be determined trouble-free

5.5

Approximation of Radiative Transfer by Multi-Flux Theory

377

by modern numerical methods. Altogether, the radiative transfer is only reliably described over a wide range of optical materials by the three- or multi-flux approximations. This becomes more clear when examining the corner point coordinates of the optical triangle in the next section.

5.5.7 Corrected Optical Triangle Finally, we take a look at the corrected optical triangle in the nth approximation in relation to both illumination conditions. In the case of directed–diffuse light, the corner points show constant coordinates which are independent of the number of fluxes; for refraction index of ns = 1.5 and influx angle of μi = 1 these are A = (0.0769, 0.9231), B = (0.0400, 0), C = (1, 0). These coordinates are already known from two-flux approximation with directed radiation; see Section 5.3.4 and Table 5.3. In comparison, the coordinates of the corner points A2n and B2n for diffuse– diffuse radiation depend on the number 2n of fluxes chosen; the coordinates can be taken from Table 5.4. Even the coordinates of A2n , which correspond to the four-flux approximation including total reflection corrections, differ insignificantly from those of the two-flux approximation. The area of the corrected triangle does not considerably increase until eight diffuse fluxes are used. From Table 5.4, it can also be seen that the coordinates, which correspond to both the lowest approximations, are strikingly different. Therefore, it is confirmed that both approximations are unsuited for reliable applications. With further subdivisions of the radiation field, the coordinates of both corner points A2n and B2n converge toward those for 2n = 256 fluxes. This large number of fluxes is intended to represent an infinite number for calculation purposes Table 5.4 Corrected optical triangle: coordinates of corner points A2n , B2n for 2n fluxes in the case of diffuse–diffuse radiation; ns = 1.5 Coordinates of corner points A2n , B2n of the corrected optical triangle for diffuse–diffuse fluxes Number 2n of fluxes

A2n = ( R,  T)

B2n = ( R,  T)

2 4 8 16 32 64 128 256 512

(0.43106, (0.36259, (0.14483, (0.15532, (0.15566, (0.15552, (0.15547, (0.15545, (0.15545,

(0.09178, (0.09178, (0.07864, (0.08897, (0.09135, (0.09172, (0.09177, (0.09178, (0.09178,

0.56894) 0.63741) 0.85517) 0.84468) 0.84434) 0.84448) 0.84453) 0.84455) 0.84455)

0) 0) 0) 0) 0) 0) 0) 0) 0)

378

5 Theories of Radiative Transfer

beyond which the results do not change practically; in this case, the radiation field nearly changes continuously between adjacent light cones. The first coordinate of point B256 corresponds to the value of reflection coefficient for diffuse flux of rd∗ = 0.09178, as expected. The first coordinate of point A256 represents the external reflection for ideal transparency; it is about 18% higher than the sum of reflection coefficients or directed and diffuse fluxes r + rd∗ = 0.1318 for index of refraction 1.5. Similar to the three-flux approximation, the amount 18% is a consequence of total reflection taken into consideration. In order to realize a reasonable accuracy with a suitable computing time, for practical applications a number of 2n = 64 fluxes is usually sufficient. Figure 5.37 shows the optical triangle for directed–diffuse light with 64 fluxes and Fig. 5.38 shows the corresponding optical triangle for diffuse–diffuse radiation. In the latter case, the area of the triangle is only slightly smaller compared with that of the ideal optical triangle. Obviously, from both illustrations, one can see that equal values of optical depth τ and albedo ω0 lead to different pairs of external reflection or transmission. Altogether, the effort to solve the numerical requirement of the multi-flux approximation is completely justified. For n ≥ 8, the method is applicable to all kinds of optical materials described. In recipe

1.0

A, τ = 0 0.3

External transmission

0.8

n = 1.5 μ=1 1.0

0.6

3.0 0.4

10 0.2 102 B

0 ω0 = 0 0

0.5

0.7 0.2

0.9 0.4

0.99 0.6

0.995 0.8

C 1.0 1.0

External reflection

Fig. 5.37 Optical triangle of the approximation with 2n = 64 directed–diffuse fluxes; refractive index n = 1.5

References

379 1.0 A64

τ=0

0.8

n = 1.5

External transmission

0.3

0.6

1.0

3.0

0.4

10

0.2

102 0

B64

ω0 = 0 0.7 0

0.2

0.9

0.97

0.4 0.6 External reflection

0.995 0.8

C 1.0 1.0

Fig. 5.38 Optical triangle of the approximation with 2n = 64 diffuse–diffuse fluxes; refractive index n = 1.5

prediction, the so-determined optical constants are more precise and trustworthy; in addition, the hit rate for colorations with absorption or effect colorants is considerably improved.

References 1. Chandrasekhar, S: “Radiative Transfer” (1950), reprint, Dover, New York (1960) 2. Born, M, Wolf, E: “Principles of Optics”, 7th ed, reprint, Cambridge Univ Press, Cambridge UK (2006) 3. Mie, G: “Beitraege zur Optik trueber Medien, speziell kolloidaler metalloesungen”, Annalen der Physik 25, 4th series (1908) 377 4. Kerker, M: “The Scattering of Light and Other Electromagnetic Radiation”, 9th ed, Academic Press, San Diego (1990) 5. Bohren, CF, Huffmann, DR: “Absorption and Scattering of Light by Small Particles”, Wiley-VCH, Weinheim (2004) 6. Greiner, W: “Classical Electrodynamics”, Springer, New York (1998) 7. Hedinger, H, Pauli, H: “Messung optischer Konstanten an aufgedampften Pigmentschichten und daraus berechnete Farbe pigmentierter PVC-Folien”, XIXth FATIPECCongress Aachen, Vol II (1988) 503 8. Joshi, JJ, Vaidya, DB, Shah, HS: “Application of multi-flux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films”, Col Res Appl 26 (2001) 234; Col Res Appl 28 (2003) 308 9. Schuster, A: “Radiation through a foggy atmosphere”, Astrophys J 21 (1905) 1

380

5 Theories of Radiative Transfer

10. Schwarzschild, K: “Ueber das Gleichgewicht der Sonnenstrahlung”, Goettinger Nachr (1906) p 41 11. Kubelka, P, Munk, F: “Ein Beitrag zur Optik der Farbanstriche”, Z Techn Physik 12 (1931) 593 12. Kubelka, P: “New contributions to the optics of intensely light-scattering materials part I”, J Opt Soc Am 38 (1947) 448 13. Kortuem, G: “Reflexionsspektroskopie”, Springer, Berlin (1969) 14. Voelz, HG: “Industrielle Farbpruefung”, 2nd ed, Wiley-VCH, Weinheim (2001) 15. Pauli, H, Eitle, D: “Comparison of Different Theoretical Models of Multiple Scattering of Pigmented Media”, Colour 73, Adam Hilger, London (1973) 423 16. Reichl, LE: “A modern Course in Statistical Physics”, Wiley-VCH, Weinheim (2009) 17. Peraiah, A: “An Introduction to Radiative Transfer”, Cambridge Univ Press, Cambridge UK (2002) 18. Gerber, WH: “Messung und Charakterisierung von Metallic-Lacken”, Proc XXII. FATIPEC Congress, Budapest, Vol I, (1994) 263 19. Quateroni, A, Sacco, R, Saleri, F: “Numerical Mathematics”, Springer, Berlin, Heidelberg (2007) 20. van de Hulst, HC: “Multiple Light Scattering: Tables, Formulas and Applications”, Vol I & II, Academic Press, New York (1980) 21. Henyey, LG, Greenstein, JL: “Diffuse radiation galaxy”, Astrophys J 93 (1941) 70 22. Gerber, WH: “Rezeptieren transluzenter Kunststoffe”, in: “Rationelles Verfahren zur Einfaerbung von Kunststoffen”, Deut Industrieforum Technologie, No 2140, Wuerzburg (1992) 1–24 23. Saunderson, JL: “Calculation of the color of pigmented plastics”, J Opt Soc Am 32 (1942) 727 24. Ryde, JW: “The scattering of light by turbid media. – Part I”, Proc Roy Soc Ser A, 131 (1931) 451 25. Klein, GA: “Farbenphysik fuer industrielle Anwendungen”, Springer, Berlin, Heidelberg (2004) 26. Rybicki, GB: “Radiative transfer”, J Astrophys Astr 17 (1996) 95 27. Richards, LW: “The calculation of the optical performance of paint films”, J Paint Techn 42 (1970) 276 28. Mudget, PS, Richards, LW: “Multiple scattering calculations for technology”, Appl Opt 10 (1971) 1485 29. Billmeyer, FW, Jr, Carter, EC: “Color and appearance of metallized paint films”, J Coat Techn, 48 (1976) 53 30. Ishimaru, A: “Wave Propagation and Scattering in Random Media”, IEEE Press, New York (2005) 31. Martin, PA: “Multiple Scattering”, Cambridge Univ Press, Cambridge (2006) 32. Lenoble, J, ed: “Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures”, Deepak Publishing, Hampton, VA (1985) 33. Dahlquist, G, Bjoerck, A: “Numerical Methods in Scientific Computing”, Soc Industrial and Applied Mathematics, Philadelphia, PA (2008) 34. Quateroni, A, Sacco, R, Saleri, F: “Numerical Mathematics”, Springer, Berlin, Heidelberg (2007) 35. Press, WH, Teukolsky, SA, Vetterling, WT, Flannery, BP: “Fortran Numerical Recipes”, Cambridge Univ Press, Cambridge (2008) 36. Bronstein, IN, Semendjajew, KA, Musiol, G, Muehlig, H: “Handbook of Mathematics”, 9th ed, Springer, Berlin (2007) 37. Wolf, W: “High Performance Embedded Computing”, Elsevier, Amsterdam (2007) 38. Gerber, WH, Pauli, HKA: “Accurate Model for Metallic Paints” (1989), without reference

Chapter 6

Recipe Prediction

The great progress in industrial color physics in the field of recipe prediction over the past few decades is due, in no small part, to the development of the sophisticated architecture and large memory of modern computers as well as user-friendly human interface design of software. A method of recipe prediction has clear utility. If one has such a method, this means that a given coloration of unknown colorant composition can be reproduced only from available colorants; this is accomplished by calculating, in advance, the necessary concentrations of the components. In literature this method is sometimes also called color-match prediction; here, we use the shorter term recipe prediction. The method finds its beginnings in the year 1944 and the fundamental concepts have been gradually developed since about 1960 with the emergence of digital computers [1–9]. Modern refined computation methods along with the introduction of new colorants have enabled the possibility of reproducing reference colors for demanding coloristic requirements including absorption and effect colorants. It is possible to divide applied computation procedures into classical and modern empirical methods. The classical procedures are based on the idea that the measured spectral values of the color sample can be calculated with the help of a suitable radiative transfer approximation. The design of the computation method delivers usable colorants, their concentrations, and even alternative recipes. In order to accomplish this, the optical constants and possibly the expansion coefficients of the phase functions have to be known for all available colorants. These values should, therefore, be determined in advance by suitable calibration methods. Occasionally, it is necessary to use an alternate algorithm for a numerical post-correction to a realized recipe. In addition to the classical recipe prediction procedure, empirical methods are also applied. These involve databases of recipe records, modified expert systems, or neural networks, thereby relying on catalogs empirical knowledge. With these approaches, recipes obtained are generally approximate and the

G.A. Klein, Industrial Color Physics, Springer Series in Optical Sciences 154, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1197-1_6, 

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6 Recipe Prediction

resulting component colorants and concentrations may need to be improved either by numerical interpolation or use of the classical correction method. In contrast to the classical method, the recipe prediction by databases is founded on an extensive number of previous realized and recorded recipes: at least 1,000 for absorption colorants and 3,000 for effect pigments. Among these numerous historical recipes, only those are searched and selected which possess desired properties, for example, those which do not exceed a specified color difference with regard to the reference color. To obtain an acceptable recipe for mixtures with absorption and effect colorants, modified expert systems have been utilized since about 1980. This kind of expert system consists of three things: first, a data bank with recorded recipes arranged based on various color physical criteria; second, an additional collection of reference pictures of the colorants in use, standard, or previous manufactured colors; third, a suitably equipped light microscope. Experience is also needed to work out recipes through this combined digital and visual approach. Another method for recipe prediction is the use of neural networks. A neural network consists of several programmed microprocessors, which are reticular interconnected similar to nerve cells. During an extensive training phase, the processors “learn”, by suitable algorithms, to transform the concentration values of a multitude of known colors at the output back to the known input spectral data of the corresponding color. This so-called back-propagation model is used particularly for textile dyes. Altogether modern color matching is performed by various prediction methods depending on the actual sort of color.

6.1 Classical Method Over the outdated methods of purely visual color tuning, recipe prediction using modern classical methods provides a series of advantages and considerable simplifications, even if it is required to perform unique preparations. The aggregate of procedures needed to carry out the classical method of recipe prediction can be divided simply into three steps: – calibration of available colorants; – measurement of the color sample to be reproduced and recipe computation; – possible correction of the manufactured recipe. Central to each of these steps is a specific computation procedure. Those procedures for the first and second steps are based on the same radiative transfer approximation. In the first step, the optical constants of the absorption or effect colorants used are determined during the so-called colorant calibration; see Fig. 6.1. For each

6.1

Classical Method

383

Measurement Calibration samp. R

R ex

λ

Radiative transfer approximation

R Optical constants, phase function λ Comparison R th, R ex

Data storage

Fig. 6.1 Calibration of available colorants: determination of the optical constants

available colorant, a calibration series is obtained with various criteria taken into account. Depending on the chosen radiative transfer approximation, the following optical quantities are generally determined: the specific absorption and scattering coefficients k and s, or alternatively, the specific optical path τ or specific albedo ω0 , and – in special cases of anisotropic scattering – the series coefficients dl of the phase function. The variables and process conditions used during the creation of calibration series should nearly agree with those of the future fabrication or processing terms. The more accurately are the optical constants determined, the more applicable are the computed recipes, and the fewer are the number of correction steps that may be required. This holds for both absorption and effect colorants. In the second step, the reflection and transmission of the sample color are measured; see Fig. 6.2. It must be stressed that the measurements in all steps of recipe prediction must be made solely with the same spectrophotometer. Afterward, with a suited prediction algorithm, the colorants and the corresponding concentrations which match the reference color are determined. The program is based on the same radiative transfer approximation already used in the calibration level.

384

6 Recipe Prediction Measurement reference color R

Colorants

R ex Optical constants, phase function

λ

Iteration

R

Prediction algorithm, approximation

Colorants, colorant concentrations

λ Comparison R th, R ex

Recipe output

Fig. 6.2 Measurement of the reference color for color reproduction: determination of the relevant colorants and concentrations

The spectral values of the manufactured color recipe are measured and the needed color values, color difference, and further coloristical characteristics are determined. The procedure is finished if the result of visual assessment is acceptable and if the preset numerical quantities of color values, color differences, or supplementary color physical criteria can be tolerated. If the most important of the needed color specifications are not sufficiently matched, a further step is then carried out; see Fig. 6.3. In this case, an improved color recipe using the spectral or colorimetric differences between reference and predicted colors is calculated using a correction algorithm. On account of the nonlinear colorimetric features of absorption and effect colorants, this procedure is limited to a few correction attempts. In the two sections that follow, we outline the various calibration series for dyes, absorption and effect pigments for determination of the optical constants.

6.1.1 Calibration Series with Absorbing Colorants The choice of colorants suitable for recipe prediction must take into account one or more of the criteria given in Sections 2.2, 2.3, 3.3, 3.4, and 3.5. Individual

6.1

Classical Method

385

Measurement recipe color R

Colorants

R ex Previous, additional or new

λ

Accurate correction algorithm

R Corrected colorant concentrations

λ Comparison R th, R ex

New correction

Fig. 6.3 Correction of the realized recipe on the basis of a suitable algorithm

requirements can generally be included provided that such inclusion results in no detrimental effects. For computation of optical constants, the following quantities have to be determined from each calibration series as accurately as possible: – concentrations of colorants and additives; – entire liquid or solid phase content; – density and mixture ratio; – layer or film thickness; for lacquers the dry film thickness; – depending on the colorant sort: either the spectral reflection over two different backgrounds or both reflection and transmission. The thickness of the layer or films is ascertained at several defect-free and homogeneous points outside of the plane measuring area intended for spectrophotometer measurement. This procedure avoids the rupturing of the inner measuring area permanently by the mechanical, pneumatic, or other detector used. At least three thickness measurements at different locations are generally made; the deviation from the mean should be only some micrometers. The massrelated or volume-related concentration of the colorants has to be determined

386

6 Recipe Prediction

with a maximum uncertainty of ±100 ppm. The results for the wavelength dependent reflection or transmission values are generally taken as the arithmetic mean of at least two separate measurements. It cannot be emphasized enough that in all three levels of recipe prediction – calibration, measurement of the color sample, and possible correction – the same spectrophotometer must always be used. For each available colorant, the calibration panels are manufactured using different concentrations and different components. Depending on the sort of the applied colorant, there are four distinct fundamental calibration series: – concentration series of dyes in solution; – absorption pigments in white and black blends; – metallic pigments or mixtures with colored pigments and carbon black; – series of interference pigments mixed with absorption and metallic pigments. The concentration-dependent calibration mixtures are necessary on account of the nonlinear behavior of the optical constants of the corresponding colorant; furthermore, this nonlinearity is clearly characterized by the coloration potential. It is possible to express, to a good approximation, this nonlinear relationship with a polynomial of order 3 or less [10]. Splines are unsuited for this due to the fact that the measured parameters are not balanced leading to undiscovered anomalies. The grading of the colorant concentrations can be tuned with future recipe formulations in mind. It is important to avoid critical manufacture conditions; this will be discussed further at the end of this section. The calibration colorations can also include standard colors or samples of known coloration composition; these additional calibration colors are only useful, if similar color shades are to be predicted in future. In the following, we focus on the four calibration series mentioned above. These series are primarily applicable to colors, lacquers, plastics, textiles, colored printing inks, or cosmetics. This section deals only with calibration series of absorption colorants. In the case of transparent dyes in solution, the transparent solvent should first be measured in a plane parallel glass cell of known interior width d; the same cells are used for the calibration series. The glass surfaces have to be clean and free from optical variation. It is then possible to test if the additives of a compound result in additional absorption or scattering light and, therefore, be included in the calibration. For every provided dye, a maximum of 10 samples should be mixed with nonlinear concentration changes considering the behavior of the Lambert–Beer law. For example, ratios 1, 2, 5, 10, 25, 50, 70, 80, 90, and 95 could be used, where the numbers stand for volume fractions in percent (of maximum dye concentration) in the same solvent. Although the scattering coefficients of the dyes or additives are expected to be small, both optical constants have to be determined for every measured spectral value of a calibration sample.

6.1

Classical Method

387

Afterward it can be determined if a single- or two-constant theory is preferable for recipe prediction. The calibration of dyes for transparent foils or layers of polymeric materials is realized in an analogous manner. First, the thickness d of each layer is determined. Afterward the natural material is measured over both known background surfaces; these measurements are repeated for all colored samples of the series. This procedure continues as described above. The unusual features of calibration series for colored textile fibers are described in the literature [10]. A similar careful approach is recommended for the manufacture of translucent or opaque colors containing absorption pigments. As before, the polymeric base material of known layer thickness is first calibrated again by measuring the reflection over two opaque backgrounds. After that, some manufactured white/black blends are measured over each background (cf. Table 6.1). To avoid confusion when using several white or black pigments, it is best to proceed systematically. For every colored pigment, in the range of about three to eight blends are to be manufactured with the same white pigment in different concentrations. The calibration series is then typically completed by the inclusion of at least one black mixture and the mass tone color (natural color); see Table 6.2.

Table 6.1 Examples of white/black blends for calibration with absorption pigments; see also Table 6.2

Table 6.2 Example of a calibration series for an absorption pigment in white blends and one black blend

Black/white mixtures; mass fraction % White

Black

99 98 95

1 2 5

Pigment/white mixtures, one pigment/black mixture; mass fraction % Pigment

2 5 10 20 50 75 90 99 100 (mass tone)

White

Black

100 98 95 90 80 50 25 10 1

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6 Recipe Prediction

In the tables, the numbers stand for mass fractions relative to 100. The calibration panels should show a similar translucency as expected of the final recipe colors. Special attention must be paid to organic absorption pigments from bright yellow to bright red; the scattering of these colorants decreases only minimally in dependence on concentration for wavelengths λ > 550 nm. This scattering is strongly reduced by the inclusion of even a slight fraction of black. As shown, for example, in Fig. 6.4, the addition of only one part black to the mass tone decreases the reflectance for wavelengths λ > 600 nm from about 0.7 to roughly 0.2. The scattering constant of such bright colorants, therefore, should be determined more carefully than for white mixtures. Although the black component has to be measured out precisely, a seemingly insignificantly higher content of black strongly distorts the hue of a color recipe (Section 6.3.2). Figure 6.4 shows, in addition, that the reflectance for λ < 500 nm increases proportionally to the white content. Consequently at lower wavelengths, the white pigment takes on – albeit in a reduced form – the function that the black performs at higher wavelengths. Because some organic absorption pigments are translucent, an opaque mass tone is obtained by either a sufficient layer thickness or a high colorant concentration. Both of these quantities are subject to technological and economical

1.00 R(λ)

a b c d e

0.75

P/W 5/95 10/90 20/80 50/50 80/20

d e Mass tone

0.50 a 0.25

b Pigment / black 99/1

c 0

400

500

600

700

λ nm

Fig. 6.4 Spectral reflection of the mass tone of a yellow absorption pigment, in five pigment/white mixtures P/W, and a pigment/black mixture 99/1

6.1

Classical Method

389

limits. Therefore, it is necessary to measure the translucent colors again in optical contact with both black and white backgrounds of known reflectance. As we will see later, the interactions of layer thickness, concentration, and given hiding criterion can be included directly into the recipe strategy (Section 6.2.4). In analogy to translucent cases, calibration colors should be checked again if the optical constants are possibly influenced by additives. The calibration measurements of colors containing absorption pigments start with both pure white and black pigmented samples. After this follow surfaces of blends with white and black, mixtures of the colored pigment with white, and possibly with the mass tone color. Different white or black pigments, which are used in addition to the first, may be interpreted as special gray colors; this trick avoids the need to produce new calibration series with all previous absorption pigments. The chromaticity of the calibration colorations should be nearly equally distributed on the boundary curve of the coloration potential. For improvement of the calibration results, it is possible to realize blends with various different colored pigments. Figure 6.5 shows, as an example, the spectral reflectance of a blue pigment in mixtures with the same white as well as the reflectance of the accompanying mass tone.

1.00 R (λ) a b c d e f

0.75

P/W 5/95 15/85 30/70 50/50 75/25 Mass tone

0.50

a 0.25 e

d

c

b

f 0

400

500

600

700

λ nm

Fig. 6.5 Spectral reflection factor of a blue absorption pigment: a–e: white blends, f: mass tone

390

6 Recipe Prediction

Independent of the chosen radiative transfer approximation, the absorption coefficient of an opaque coloration can only be determined as a relative quantity. In most cases the scattering coefficient of the calibrated white is used to serve as reference value; at each measuring wavelength, the scattering coefficient of the white is simply assigned to the quantity S = 1.0/l with dimension of reciprocal length l–1 (the same unit as for absorption coefficient). In many cases, it is advisable to manufacture translucent samples and to carry out measurements over two separate backgrounds, the resulting optical constants are then absolute quantities. Undoubtedly the careful, reproducible, and extensive determination of the optical coefficients is quite elaborate. Nonetheless, this effort is inevitably required for successful application of classical recipe prediction. The preliminary effort to apply this method is more than offset by a series of essential benefits, e.g.: – computation of alternative recipes; – qualitatively better and more exact recipes in a shorter time; – recipe choice after various criteria, for instance, coloristical, colorimetrical, applied technological, or economic points of view; – inclusion of individual color standards for calibration; – adaption of mixtures for recipes; – use of rest colors for new recipes; – consideration and management of historical recipes; – correction method for empirical procedures of recipe prediction. Not to be outdone, in comparison to the elaborate preliminary effort needed for the classical method, the modern empirical procedures require a substantially greater preliminary effort. This includes, for example, a greater number of historical recipes for databases compared with the number of calibration samples, or additional optical analysis for expert systems, or a time-consuming training phase of neural networks. These approaches differ greatly from the classical procedure and we deal with these procedures in Section 6.3.

6.1.2 Calibration and Reference Colorations for Effect Pigments The sort and the number of calibration colors for metallic, pearlescent, or interference pigments differ only slightly from those of translucent or opaque absorption pigments. The calibration series for effect pigments should also be well adapted to the specific optical properties of this sort of colorants. The calibration samples have to be measured with a directional geometry or rather the same angle-dependent spectrophotometer and under the same illumination and measuring angles used for recipe prediction procedure.

6.1

Classical Method

391

Calibrations of metallic pigments are carried out with the same metal pigment in two separate series, one over a black and the other over a white background (black/white contrast substrate). This approach amounts to different types of the same metallic pigment as well as the application in mixtures with absorption pigments. The first sequence consists of two to five mixtures of metallic pigments with carbon black, including the mass tone. The second sequence consists of three to five mixtures with metallic and absorption pigments, including a metallic/carbon black mixture. The absorption pigments can be of inorganic or organic nature. The optical constants of the absorption colorants are determined by a separate calibration using the same radiative transfer approximation as for effect colors. Optical constants of old calibrations need to be tested before being used; they can only be accepted if the previous and current manufacturing and measurement conditions are identical. Separate calibration series have to be performed for metallic pigments of different d50 specification or different particle size distributions. Table 6.3 lists a pair of series, both of which result from exchange of the white pigment with the metallic pigment following from the previous section for absorption pigments (cf. Table 6.2). In some sense, the metallic pigment takes on the role of the white pigment. Even fewer samples are, therefore, necessary, as indicated in Table 6.2. For computation of the optical constants, each carbon black mixture is interpreted as a special gray. The goniometric measurements of the calibration colors are carried out for five different observation angles. These angles are kept as a standard during further calibration and future recipe prediction. The calibration is somewhat simpler for pearlescent and transparent interference pigments over a black and white background. In the case of transparent pearl and liquid crystal pigments, a small number of colorations with carbon

Table 6.3 Examples of calibration series of a metallic pigment in mixtures with aluminum and a colored pigment Metallic pigment/aluminum pigment mixture; mass fraction %

Metallic pigment/colored pigment mixture; mass fraction %

Metallic pigment

Aluminum pigment

Metallic pigment

Absorption pigment

100 (mass tone) 98 90 75 60 40 25

2 (only carbon black) 10 25 40 60 75

98 90 60 50 25 10

2 (only carbon black) 10 40 50 75 90

392

6 Recipe Prediction

black is sufficient; see Table 6.4. The mass tone in a pigment-specific concentration is also included. Opaque multi-layer interference pigments should be blended in ratios with absorption pigments as shown in Table 6.4. Table 6.4 Example of a calibration series with a transparent interference pigment mixed with carbon black Content; mass fraction % Mixture

Interference pigment

Carbon black

Mass tone Black mixtures

100 99 98 97

0 1 2 3

In order to accurately compute the optical constants, the hiding layer thickness, the entire density of a formulation, or recipe price, the following specifications are necessary for each calibration color for absorption and effect colorants: – pigment identification including pigment codes; – pigment content of the mixture; – blending ratio; – dry film thickness; – total solid-state content; – pigment density; – price/kg. Experience accrued over the past few decades shows that the classical recipe prediction for effect pigments can be combined successfully with light microscopic methods, among other things (Section 4.2.5). Therefore, it is necessary for each effect pigment to manufacture a so-called reference coloration, which consists of the mass tone in a pigment-specific concentration. The effect pigment is homogenized together with carbon black of maximum of 1.0% and a transparent binder, applied over a black/white substrate, and finished with a clear coat. In cases of homologous series of metallic or interference pigments, reference colors are manufactured in dependence on the mean particle size or the particle size distributions. For each of the colorations with black background, light microscopic pictures are taken in bright-field illumination and dark-field illumination both with the same image area. The images registered with the two types of illumination are saved digitally in a reference database along with the quantities of spectral

6.1

Classical Method

393

reflectance and transmittance, optical constants, and further color specifications. These data are incorporated into a modified expert system for visual or optoelectronic analysis and identification of effect colorations (Section 6.3.5). Examples of images taken with both kinds of illumination are shown in Color plates 8, 9, and 10. Correspondingly, images of the original and reproduced colors can be taken and saved separately for comparison, analysis, or recipe prediction of similar colorations; see e.g., Color plates 11 and 12.1

6.1.3 Determination of Absorption and Scattering Coefficients As soon as all calibration samples have been manufactured, the optical coefficients of the colorants should be determined. Now, a suitable radiative transfer approximation needs to be determined. This is the approximation that is used in the following for computing the optical calibration parameters as well as the theoretical spectral reflection and transmission in recipe prediction. The optical constants are determined with appropriate formulas for each colorant of all single calibration samples. Naturally these constants belong only to the radiative transfer approximation being used and are by no means interchangeable with other approximations. In addition to the mostly different or comparable colorants – such as yellows, oranges, or reds of similar hue – the calibration samples contain one or more binders. If n colorants and one binder are given, then the optical constants of at least n + 1 components need to be determined. On the other hand, if the absorption and scattering constants are used for recipe prediction, a total of 2(n + 1) constants must be calculated, or for m constants m(n + 1). Because each calibration series of a colorant typically contains 3–10 different concentrations, the number of samples is generally greater than the number of constants to determine. The surplus of samples serves to improve the accuracy of the optical constants [11–15]. For calculating the optical parameters of each colorant, the minimum of the function 1 wi fi2 F= 2 N

(6.1.1)

i=1

needs to be determined. In this formula, the following notation is used: N the number of calibration samples of a calibration series, m < N, for m optical constants; wi the empirical weighting factor > 0; ex fi = Rth i − Ri 1 Color plates

are inserted between chapter 3 and 4.

(6.1.2)

394

6 Recipe Prediction

the difference of the experimentally measured reflection Rex i and the related reflection of the ith sample of a calibration series. theoretically calculated Rth i The minimum of F is obtained when the sum of the squared deviations fi2 is smallest.2 For the purpose of simplification for now, the term spectral value will be taken to mean either reflection or transmission. The simultaneous usage of both quantities is described at the end of the next section. In addition, we assume that ex the theoretical and experimental spectral values Rth i and Ri are both a function of the absorption and scattering constants Ki ≥ 0 and Si ≥ 0: th th th Rth i = Ri (Ki , Si ), ex ex ex Rex i = Ri (Ki , Si ),

(6.1.3)

i = 1, 2, ... , N.

(6.1.4)

Based on experience, the total absorption coefficient Kth of a colorant mixture is a linear combination of the specific absorption constants kj of each of the M mixed colorants. The total absorption coefficient Kith of the ith calibration sample is given by the expression Kith = di

M 

cij kj ,

i = 1, 2, ... , N, M + 1 ≤ N,

(6.1.5)

j=0

where the following notation is used: layer thickness of the ith sample of a calibration series, di : j = 0: index of the binder, measured concentration of the jth colorant in the ith sample of a cij : calibration series. Moreover, it is assumed that the total scattering coefficient Sith satisfies the relation Sith = di

M 

cij sj ,

i = 1, 2, ... , N,

M + 1 ≤ N,

(6.1.6)

j=0

in analogy with Equation (6.1.5), where sj denotes the specific scattering constant of the jth colorant.

2 The

summation index i is not related to wavelengths; as explained in Section 5.1, the procedure to obtain the optical coefficients has to be repeated separately for each of the l = 1, 2, ... , L experimental wavelengths.

6.1

Classical Method

395

There are further considerations for approximations with n > 2 fluxes. The evaluation equations for n = 2 fluxes are always solvable for the optical constants of interest (e.g., see Appendix A.3). In higher approximations, the optical constants merely follow from an indirect approach, one of the most often applied is shown in the following. In order to determine the specific absorption and scattering constants kj and sj from the system of equations as easily as possible, Equation (6.1.2) is substituted into two linear approximations. The first approximation concerns the theoretical quantityRth i . This quantity is expanded into a Taylor series in dependence on the experimental variableRex i . The series is terminated after the first derivative in each dimension: ex ∂Rex ex ∼ ∂Ri (Kith −Kiex )+ i (Sith −Siex ), fi = Rth i −Ri = ∂Ki ∂Si

i = 1, 2, ... , N. (6.1.7)

The two derivatives in this equation are to be replaced later by numerical difference quotients. Combining Equations (6.1.5), (6.1.6), and (6.1.7) yields ⎛ fi = αi di ⎝

M 

⎞

⎛

cij kj − Kiex ⎠ + βi di ⎝

M 

j=0

⎞ cij sj − Siex ⎠ ,

(6.1.8)

j=0

where abbreviations αi =

∂Rex i , ∂Ki

βi =

∂Rex i ∂Si

(6.1.9)

are used. The second approximation for Equation (6.1.2) is made under the requirement that the partial derivatives of fi2 with regard to km and sm approximately vanish ∂f 2 ∂fi2 ∼ = 0, i ∼ = 0, ∂km ∂sm

i = 1, 2, ... , N,

m = 0, 1, ... , M.

(6.1.10)

The resulting equations are reworked and summed over i = 1, 2, ... , N. This results in a system of 2(M +1) linear equations from which the 2(M+1) unknown coefficients kj and sj can be determined. These parameters are interpreted as elements of vectors k = (k0 , k1 , ... , kM )T and s = (s0 , s1 , ... , sM )T . The resulting system of equations can be written in matrix notation as AT Ck + GT Cs = AT kex + GT sex ,

(6.1.11a)

GT Ck + BT Cs = GT kex + BT sex .

(6.1.11b)

The definitions of the matrices and vectors are given in the following:

396

6 Recipe Prediction

– A = (αim ), B = (β im ), G = (γ im ) each are (N, M+1) matrices, the elements are given by αim = wi αi2 cim , βim = wi βi2 cim , γim = wi αi βi cim with weighting coefficient wi and concentration cim ; – C = (cij ) is a (N, M+1) matrix of elements cij , denoting the concentration of the jth colorant in the ith sample of a calibration series; – kex = (K1ex /d1 , K2ex /d2 , ... , KNex /dN )T and sex = (S1ex /d1 , S2ex /d2 , ... , SNex /dN )T are each vectors of N components, given by the ratio of the experimental absorption or scattering coefficient Kiex or Siex , and the layer thickness di of the ith sample. In conclusion, we focus on calibration colors which satisfy the standard three special cases of the optical triangle. For ideal transparent materials, all M + 1 scattering coefficients are zero and therefore s = sex =0. The remaining M + 1 unknown quantities kj of vector k can be determined by the relation k = {AT C}−1 AT kex .

(6.1.12)

The matrix in curly brackets has to be inverted as an entire product because, in general, neither A nor C can be inverted alone. The corresponding system of equations holds for a so-called one-constant theory. In this theory, the specific absorption constant is a relative quantity as mentioned previously. The case of opaque samples formally means di →∞ for all i = 1, 2, ... , N calibration patterns. Now, kex = sex = 0; therefore from Equations (6.1.11a) and (6.1.11b), a homogenous system of equations for the 2(M + 1) unknown elements of vectors k and s arises. This is why the scattering coefficient of a pigment or binder has to be determined by other methods. As mentioned previously, the remaining optical constants are often based on the scattering constant Sw of the white component used with highest numerical value. The rest of the 2M+1 optical constants are relative quantities. Figure 6.6 shows the reflection factor of a red absorption pigment measured over a spectral range and the relative absorption K/Sw and scattering S/Sw calculated with three-flux approximation in the same range. It can be seen that the distinct spectral scattering changes roughly in proportion to the reflection, whereas the relative absorption is high at those wavelengths with relatively small reflection. In the conservative case the system of equations (6.1.11) reduces to s = {BT C}−1 BT sex

(6.1.13)

on account of the values k = kex = 0. In the three special cases discussed, as well as in the case of translucent materials, it is important to pay attention in order to ensure sufficient numerical stability of the corresponding systems of equations. Here, we do not concentrate further on numerical questions, but rather refer to the relevant literature [11–14].

6.1

Classical Method

397

1.0

R

0.8

0.6

0.4 S/Sw 0.2 K/Sw 0

400

500

600

700

λ nm

Fig. 6.6 Reflectance R, relative absorption K/Sw , and scattering S/Sw of an inorganic red absorption pigment in dependence on wavelength λ, calculated with three-flux approximation

6.1.4 Optical Path and Albedo Rather than the absorption and scattering constants K and S, the optical path τ and albedo ω0 can alternatively be determined. In fact, τ and ω0 can be calculated directly from known K, S, and layer thickness d using Equations (5.1.27) and (5.1.28). However, it is often more advisable to determine the quantities τ and ω0 directly. In order to obtain the coefficients τ j and ωj of the jth colorant of a mixture, we start again from Equation (6.1.1). In analogy, the theoretical ex and experimental reflection values Rth i and Ri of the ith sample of a calibration series are now functions of the entire optical path Θ i and the entire albedo Ω i : th th th Rth i = Ri (Θi ,Ωi ), ex ex ex Rex i = Ri (Θi ,Ωi ),

i = 1, 2, ... , N.

(6.1.14) (6.1.15)

The quantities τ and ω0 are, therefore, replaced by Θ i and Ω i in the evaluation equations of the chosen radiative transfer approximation. Similar to the explanations in the previous section, we assume that the entire optical path Θith of M + 1 mixed components is described by the sum

398

6 Recipe Prediction

Θith

= di

M 

cij τj ,

M + 1 ≤ N.

(6.1.16)

j=0

The entire albedo Ωith of a coloration now corresponds to the weighted arithmetic mean Ωith =

M 1 

Θith

cij σj ,

M + 1 ≤ N,

(6.1.17)

j=0

using the following notation: di : τ j: σ j: cij : j = 0:

thickness of the ith layer of a calibration series; specific optical path of the jth pigment in the mixture; specific scattering constant of the jth pigment in the mixture, see Equation (6.1.18) below; concentration of the jth colorant in the ith sample; index of the binder.

The specific scattering constant σ j is defined as product of the scattering contribution ωj and specific optical path τ j , σj = ωj τj ,

j = 0, 1, ... , M.

(6.1.18)

The procedure continues as described in the previous section with regard to both approximations and a system of similar matrix equations. For these equations, a third approximation needs to be introduced. The denominator Θith in Equation (6.1.17) is limited to an empirical maximum of about 100; in the following, the system of equations has to be solved iteratively. To conclude, we mention the special case of translucent calibration samples. If we use in Equation (6.1.7) reflection Ri and transmission Ti separately, then both calculations should result in the same values of kj and sj or τ j and σ j , respectively. Otherwise, reflection and transmission are to use altogether in one equation: 1  r th 2 t th ex 2 F= [wi (Ri − Rex i ) + wi (Ti − Ti ) ], 2 N

(6.1.19)

i=1

where wri , wti > 0 denote weighting factors for reflection and transmission. The value of the difference Tith −Tiex in the theoretical and experimental transmission is developed in a terminated Taylor series in analogy to Equation (6.1.7). All further steps are the same as outlined above.

6.1

Classical Method

399

1.0

ω0 0.8 R 0.6

0.4

0.2

τ 0

400

500

600

700

λ nm

Fig. 6.7 Yellow inorganic absorption pigment: measured spectral reflection factor R, optical path τ , and albedo ω0 , calculated with three-flux approximation

Figure 6.7 shows, as an example, the optical path and albedo of a translucent yellow pigment in dependence on the wavelength. In this case, again the albedo changes in rough proportion to reflection and is nearly minimal at wavelengths where the optical path is highest. The increase of ω0 for wavelengths greater than 600 nm is caused by a small amount of absorption in this range and corresponds to a small value of τ . The fundamental principles have been outlined for determination of conventional optical coefficients. The following section deals with the calculation of the parameters for the phase function which enable, among other things, the approximate description of the anisotropic angle-dependent behavior of effect pigments.

6.1.5 Coefficients of the Phase Function The procedures for determination of two corresponding optical constants of a colorant can be extended to further characteristic quantities. Among these quantities are, for example, the anisotropy coefficient g of the Henyey–Greenstein function (5.1.38) and the expansion coefficient dl of the phase function. In this section, we restrict ourselves to the determination of parameters dl .

400

6 Recipe Prediction

In the following, we assume that the jth colorant of a mixture produces a phase function of lth Legendre coefficient dl,j , which, in accordance with Equation (5.1.39), is given by dl, j = (2 l + 1) · (gj )l ,

l = 0, 1, 2, ... , Λ,

j = 0, 1, ... , M.

(6.1.20)

According to Equations (6.1.17) and (6.1.24), the theoretical value of the lth Legendre coefficient of the ith sample Δth l,i follows from Δth l,i

=

1

M 

Θith Ωith

j=0

cij σj dl, j .

(6.1.21)

Accordingly, the relation equivalent to Equation (6.1.7) has now to be extended by the sum Λ  l=0

f˜i,l =

Λ  ∂Rex i (Δth − Δex i,l ). ∂Δi,l i,l

(6.1.22)

l=0

In the case of effect pigments, all optical coefficients are determined in a similar manner with dependence on each measuring angle. As an example, Fig. 6.8 shows the phase function p(μ,μ = −1) of an aluminum-cornflake pigment mixed with a dominant green pearlescent pigment. As an extension to Fig. 5.12, the phase function is represented simultaneously in dependence on wavelengths between 400 and 700 nm and measuring angle of declination over the interval 0◦ ≤ ϑ ≤ 40◦ . In the range where ϑ ≥ 30◦ , the backscattering is noticeably low and nearly independent of the wavelength. On the other hand, the scattering rises strongly for angles less than 30◦ . In the range of green wavelengths, a maximum is visible. This corresponds to the interference color of the pearlescent pigment. The intensities of the light cones of a metallic pigment exhibit a similar dependence on the scattering angle; this is shown in Fig. 5.30. With improved orientation of the effect pigment particles parallel to the undersurface, the curve of the phase function shows a narrower and higher behavior in the immediate vicinity of scattering angle ϑ = 0◦ . In the ideal case of entire parallel particles under perpendicular illumination, the phase function is a Dirac delta function.

6.1.6 Accuracy of Predicted Recipes In the end, it is important to consider how to assess the reliability and accuracy of the calculated constants and, therefore, the accuracy of the recipe prediction method used. In the following, we outline some aspects for appraisal of recipe accuracy. It goes without saying that all steps needed for the determination of

6.1

Classical Method

401

p

75

50

25

0 400

0 10 500

20 600

λ nm

ϑ/°

30

Fig. 6.8 Phase function p(μ, μ = −1) of a metallic pigment mixed with a green interference pigment in dependence on wavelength λ and angle of declination ϑ; μ = cosϑ

calibration parameters must be performed as carefully as those steps used for eventual recipe determination. If this is done, it is generally expected that less than 20% of the resulting bright and dark absorption colors need some sort of correction. To attain this level, gross and systematic errors should be eliminated from the prediction method. Such errors are already greatly reduced by the choice of suitable colorants and components. This is provided that these fulfill the necessary color strength, color fastness, compatibility of colors, or other color qualities. Likewise, it is critical to use compatible additives, binders, or plastics. It is always important to ensure that the components used originate from representative batches. In addition, the defined process parameters must be maintained over time. As mentioned previously at various stages of the text, two particularly serious systematic errors can be avoided by complying with the following three principles: 1. The measurement of all calibration samples, the reference color, the first reproduced color, and the corrected color(s) are all to be carried out with the same spectrophotometer.

402

6 Recipe Prediction

2. The calculations during the calibration procedure as well as the recipe prediction each have to be performed with the same radiative transfer approximation. 3. Colorimetric calculations must be performed with the same illuminant and standard observer, the same color space, and the same color difference formula. It must additionally be mentioned that if the spectrophotometers are readjusted, it becomes necessary to reassess the applicability of the previous calibration constants of all colorants in use. Furthermore, if previously used colorants are replaced or if new colorants are added then the related calibration colorations must be realized under the same conditions as those of the entire first calibration series. Using new software generally assumes that the same optical approximation model is used as previously. In all other cases, the calibration procedure has to be repeated. At this stage, it must be decided carefully whether or not to use the original calibration samples or to prepare new ones. With the choice of aptly programmed software, an essential source of systematic errors is eliminated in recipe prediction. In order to reduce random errors, quantifying the various aspects of the components used has to be carried out as precise and particularly reproducible as possible. The components also have to be well homogenized. In addition, all calibration colors are to be applied, processed, or cured under the same conditions so that surface and volume properties between samples are as consistent as possible. Above and beyond this, the sample thickness and the spectral values must each be measured with the highest possible precision. In cases of doubt, several calibration sets should be produced. Afterward, a reasonable set of data are determined with statistical error computation. It must be emphasized that there is a commonly held – yet incorrect – opinion that calibration errors can be completely compensated by recipe correction procedures. In some sense, the reality is that recipe correction merely reduces the influence of relatively insignificant systematic errors: calibration errors are by no means comprehensively eliminated. Even random errors are not addressed at all by the various correction methods (Section 6.3.3). If sensible colors need repeated corrections, then all random error influences have to be kept as low as possible in advance. The mean error of the optical constants can be determined using meaningful quantities of error computation [13]. The overall error can be estimated using the Gaussian law of error propagation or other suitable assessment. This approach offers, in fact, the chance to directly determine the resulting influence of faulty input quantities. This procedure does not, however, result in an assessment of the total error expected for the later recipe prediction calculation. A reasonable method for assessing the total error of recipe prediction consists of calculating back the spectral reflection and transmission of the calibration colorations by using the previously determined optical parameters. For that,

6.2

Strategies for Recipe Prediction

403

typical standards or critical test colorations of known composition can be used which are not necessarily included in the calibration procedure. Not all commercial programs for recipe prediction have routines implemented which enable such backcalculation. From the calculated and measured color values a color difference results; this corresponds to the underlying illuminant. The corresponding deviation ellipses are, dependent of chromaticity, presumably oriented in different directions of a color plane. The magnitude and direction of all color differences is a realistic measure for the quality of the calibrated optical coefficients. The mean color difference from all calibration colors can be considered to be a measure for the deviations of colors subsequently produced. For formulations with absorption colorants and missing effect pigments, the color differences are distributed, for the most part, non-uniformly in color space. With a sufficient number of random tests are to observe accumulation regions of nearly equal color differences. It is clear that this conclusion is accurate only if the color matching is performed under the same conditions as the calibration procedure, particularly in view of binders, colorants, production and application parameters, or spectrophotometers. Experience shows that it is quite painstaking to maintain the parameters and process steps during calibration, adjustment of calculated color recipes, and further production. The numerical color difference of reproduced first recipes with ∗ ≤ 0.5; with effect pigments a absorption colorants should be in the range ΔEab ∗ value of ΔEab ≤ 2.5 for a positive aspecular angle of μas = 15◦ is normally reached. The median color difference of the measured and calculated color differences obtained from all calibration samples is, on the whole, a realistic quantification of the expected accuracy of the first recipe. However, the numerical color difference can be taken only as an informative quantity; it is by no means an absolute measure for the color prediction with absorption colorants, let alone for effect colors. Quite a few more aspects, such as coloristic, technical, or economic criteria, must be taken into account. This conclusion is correct to a greater extent in the assessment of colorations containing effect pigments. With alternative recipe prediction strategies – the focus of the following section – an adjustment as good as possible for the spectral values of the reference color and the prediction color is the aim where the smallest possible value of ΔE is the criterion. The numerical color difference conveys no information with regard to the needed effect characteristics such as brightness, flop, sparkle, or distinctiveness of image. This topic is continued in Section 6.3.5.

6.2 Strategies for Recipe Prediction The first level of the classical recipe prediction procedure is finished after the determination of optical constants or coefficients of the phase function

404

6 Recipe Prediction

of available colorants. Now, we turn toward the second level and delve into three different methods of classical recipe prediction which, in the past, have proven to be quite successful: the spectrometric, the plain colorimetric, and the weighted or balanced colorimetric procedure. Each of these methods uses different criteria for reproduction of the reference color; this is why the individual processes are suitable for different applications. Above and beyond this, additional coloristical requirements can be included in the corresponding algorithms of each of the three systems mentioned.

6.2.1 Numerical Methods Among the classical recipe methods developed over the past tens of years, it is possible to distinguish two principal categories: spectrometric and colorimetric. The spectrometric procedure is based on the idea of calculating the recipe parameters in such a way that the measured spectral values of the reference color match as well as possible to the calculated spectral values. The colorimetric algorithms can further be classified into two groups: in the simplest method, the color difference is minimized for a single illuminant; in the second, the minimum of the color difference is determined with regard to various illuminants. In order to reach the relative minimum, adjustments must be computed; from these the necessary recipe parameters follow. In comparison to the methods for determining optical constants, the determination of the colorant concentrations needs some modifications. The straightforward spectrometric strategy is based on the minimization of the function 1  2 th 2 wk (Rk − Rex k ) , 2 L

FS =

(6.2.1)

k=1

where the summation index extends over k = 1, 2, ... , L measured wavelengths. In words, this equation means that the weighted sum of squared deviations of ex theoretical and experimental reflection values Rth k , Rk is to be minimized. The weighting factor wk in Equation (6.2.1) is proportional to the sum of the three standard color-matching functions (SCMFs) x¯ (λk ), y¯ (λk ), and z¯(λk ) at wavelength λk , with which the different visual dependence of the three cones is taken into consideration. In fact, compared with the colorimetric strategies, this approach delivers larger color differences for the selected illuminant [3]. However, for practical applications, this procedure has a significant advantage: in the weighting factor, the energy distribution of several illuminants – such as D65, A, FL 11 – can simultaneously be taken into account. Using this advantage, for example, metamerism between the reference and reproduced color is reduced directly for the applied illuminants.

6.2

Strategies for Recipe Prediction

405

The simplest colorimetric method uses only one illuminant, which, for example, consists of the strategy of making the differences vanish, ( L, a, b) → (0, 0, 0),

(6.2.2)

for the chosen illuminant. Experience shows, however, that for other illuminants the color differences tend to be higher compared with those resulting from the use of the spectrometric minimization (6.2.1). In other words, a change of illuminant causes new recipe concentrations and possibly colorants, and changes of metamerism indexes for further illuminants. It is not surprising that, in this case, the resulting smallest color difference for a given single illuminant is achieved at the expense of higher differences for other illuminants. Because only three independent color values belong to an illuminant, for the moment only mixtures with a maximum of four colorants can be handled with the colorimetric prediction method. For obvious reasons, it is useful to minimize the color difference not just for one, but for several illuminants [16]. Considering altogether Λ illuminants denoted as l = 1, 2, ... , Λ, the expression Λ

1 2 FB = wl (ΔEl )2 2

(6.2.3)

l=1

should be made to be as small as possible. In this formula, wl ≥ 0 stands for the weighting factor of illuminant l. Each illuminant is assigned to an index l, for example, l = 1 for D65, l = 2 for A, l = 3 for FL 11. The weighting factors are chosen individually and appropriate to the subjective estimation suitable for further applications. The quantity ΔEl indicates the color difference related to illuminant l. In Equation (6.2.3), the factors ΔEl should consistently use the same color difference formula, where this formula can be chosen from those in Sections 3.2 and 3.3. Strictly speaking, the color difference value ΔEl is controlled with the weighting factors wl . Generally, only similar color differences result if all wl are of equal value [16]. The color differences shown in the last row of Table 6.5 are for w1 = w2 = w3 = 1.0. According to the results given in this table, the spectrometric method (1) shows higher color differences than for the weighted colorimetric strategy (3). On the other hand, the simple colorimetric algorithm (2) delivers a zero color difference for the single chosen illuminant D65; for both other illuminants, the differences are the highest of these sets. More straightforwardly: method (2) is a special case of the more comprehensive method (3); in method (2) only w1 = 1.0 is specified, whereas all other weighting factors are zero allowing for higher color differences for the corresponding illuminants.

406

6 Recipe Prediction

Table 6.5 Calculated mean color differences from three different recipe strategies [16] ∗ , each of three recipes Mean color difference ΔEab

Strategy

Illuminant Number

Identification

D65

A

FL 11

1 2 3

Spectrometric weighted Colorimetric with D65 Colorimetric weighted w1 = w2 = w3 = 1.0

0.68 0.00 0.60

1.06 1.21 0.74

0.49 0.91 0.44

Of course, if the same reference color is used, all three strategies produce unequal recipes. An essential reason for that is the nonlinear dependency on colorant concentration of the spectrometric or colorimetric quantities. For clarification, Fig. 6.9 represents schematically the build loci in CIELAB or DIN99 coordinates of two similar absorption colorants and some positions of equal concentrations are indicated. In addition, the color point Fx of an unknown original color is plotted. In order to match Fx , mostly linear approximations in three directions of color space are applied. These approximations are weighted differently in the three algorithms above and deliver, therefore, unequal results with regard to the calculated concentrations and coupled colorants. L

Lx

bx

b

Fx

ax a

Fig. 6.9 Build loci of two similar colorants and color point Fx of a given color

All three classical methods possess a remarkable property with regard to coloristic applications: several alternative recipes are simultaneously calculated and selected to reproduce the reference color. The number of possible recipes is restricted by the total number of calibrated colorants and the fixed number of

6.2

Strategies for Recipe Prediction

407

colorants per recipe. If from n available colorants m < n are designated for each recipe, then the total number of different recipes Cn (m) is given on the basis of combinatorics by  Cn (m) =

n m

 =

n(n − 1) · · · ( n − m + 1) . 1·2 ··· m

(6.2.4)

In the case of n = 25 and m = 3, for example, the number of recipes already amounts to C25 (3) = 2,300. Not all of these, however, are realistic considering some would require negative concentrations. Either such components are set to zero and a new calculation is performed with the remaining colorants of the recipe or such recipes are sorted out and the combinatorics is continued. The number of such recipes can be reduced in advance if some of the possible colorants are excluded initially for prediction. For instance, it is not useful to predict a bright orange color with the inclusion of blue or violet colorants. In this context, it is often desirable to have f defined colorants – absorption and especially effect colorants – specified which remain fixed for all possible recipes; clearly the secondary condition 0 < f ≤ m holds. Now, only n – f colorants from the n initial stock are available for the combination; the number of free miscible colorants per recipe is, therefore, reduced to m – f. The total number of possible recipes obeys the inequality 

n−f m−f



 <

 n . m

(6.2.5)

If f in the example above is set to unity: f = 1, the number of recipes reduces by 88% to C24 (2) = 276. The preliminary selection of colorants is necessary if only a few colorants contribute to the desired color qualities or different kinds of colorants are to be mixed. A rigorous preselection is essential for formulations with effect pigments alone or together with absorption colorants (cf. Section 6.3.5). If, for example, an effect coloration needs three of five possible colored pigments, two of four suitable metallic pigments, and one of three similar interference pigments, then the result is still 180 different recipes.3 This selection is reduced in practice to a manageable number of about 5. They are further worked out in view of typical effect qualities such as brightness, flop, or others. In each individual case, the preselection procedure requires a great deal of coloristic experience. Beforehand, the original effect color has to be analyzed thoroughly with a suitable light microscope in bright-field and dark-field illumination with regard to the sorts of colorants contained and the individual amounts. 3 The

total number of combinations is, in this case, the product of the three single combina(3) (2) (1) tions C5 , C4 , C3 .

408

6 Recipe Prediction Table 6.6 Order of magnitude of pigment quantities for automotive paint coatings

Pigment sort

Identification

Monochromatic coating

Effect coating

Absorption pigment

Inorganic Organic

30 100

100

Metallic pigment

Cornflakes, silver dollar, leafing, non-leafing flakes

–

20

Interference pigment

Pearlescent, multilayer, optical variable interference pigments

–

90

Without this preparation, pigment systems for automotive colors, for example, are controllable only with higher computing effort; this follows already from Table 6.6, which shows the realistic extent of a pigment palette. An effect formulation seldom contains solely one pigment, demanding color shades are realized often only with a variety of colorants. The pigmentation of modern effect lacquers, for example, can be composed of any or all of the following constituents: – metallic pigments; – pearlescent, interference, and diffraction pigments; – inorganic and organic absorption pigments; – carbon black; – opacity pigments; – nano-titanium dioxide; – graphite; – flop-control pigments. Naturally the final four pigments given in the list are not included in combinatorial analysis because they behave mostly as “additives” for the fine adjustment of effect qualities. Opacity pigments often consist of dark organic blue or green pigments. They are added in small quantities to enhance the covering capacity of colorations based on metallic, pearlescent, or interference pigments. In fact, opacity pigments change the hue; however, in most cases they are favored over carbon black or black pigments which deteriorate chroma and brightness to a large degree. Inorganic pigments with high scattering power used to improve covering capacity are often unsuited due to poor dispersion capability. Flop control pigments such as bismuth vanadate or titanium oxide usually have sufficient covering capacity even for observation angles close to the horizontal. With these pigments an effect coloration of given flop behavior can be closely matched.

6.2

Strategies for Recipe Prediction

409

6.2.2 Spectrometric Strategy A recipe for a paint color normally contains about three to five absorption colorants, a recipe with effect pigments could contain as many as five additional colorants. From each color sample, a set of L = 16, 31, or 40 reflection or transmission values are measured – depending on the spectrophotometer used. Accordingly, more spectrally measured parameters are then obtained than numbers of unknown colorants. For determination of the colorants and concentrations cm , it is appropriate to apply the method of computation of adjustment or gradient descent, taking into consideration all measured spectral values [3]. If the number of measured variables and unknowns are equal – which is generally unreasonable from color physical or economic points of view – then the experimental and theoretical spectral values should be identical. On the other hand, the smaller the number of colorants that exist in a recipe, generally, the higher the resulting metamerism between reference and reproduced color. The example in Fig. 6.10 shows that the curve of the measured spectral values Rex winds around the curve of theoretical values Rth . The more often the curves intersect, the smaller the difference between two adjacent intersection points and, altogether, the smaller the index of metamerism. Moreover, the nonidentical kurtosis of the curves at about 500 and 600 nm indicates that reference and reproduction contain different yellow and orange pigments. R(λ) R th 0.7

R ex

0.6

0.5

0.4

0.3

0.2

400

500

600

700

λ nm

Fig. 6.10 Experimental reflection Rex of a reference color and theoretical reflection Rth calculated by computation of adjustment

410

6 Recipe Prediction

In order to develop the spectrometric strategy, we start from Equation (6.2.1) and assume, in addition, that a recipe contains altogether M + 1 colored components of unknown concentrations cm ; these are combined in the vector c = ex (c0 , c1 , ... , cM )T . The spectral values Rth k and Rk are interpreted as reflection or transmission (k = 1, 2, ... , L), with M+1 < L. For weighting factors of the chosen illuminant wk = w(λk ), modified sums of the color-matching functions x¯ k , y¯ k , z¯k of the 10◦ standard observer are used: −a wk = [(¯xk + y¯ k + z¯k )Sˆ k ]1/4 · (Rex k ) , k = 1, 2, ... , L.

(6.2.6)

In this equation, Sˆ k denotes the spectral energy density at wavelength λk of the illuminant and a > 0 an empirical exponent. For Λ different illuminants, l = 1, 2, ... , Λ, Schmid and Strocka suggested the summation wk =

Λ 

−a [(¯xk + y¯ k + z¯k )Sˆ l,k ]1/4 · (Rex k ) ,

k = 1, 2, ... , L,

(6.2.7)

l=1

where Sˆ l,k stands for the energy density of the lth illuminant at wavelength λk [6]. The sole variable in Equation (6.2.1) is the theoretical spectral value Rth k, which is a function of the sought-after concentrations c0 , c1 , ... , cM : th Rth k = Rk (c0 , c1 , ... , cM ).

(6.2.8)

In order to determine the components cm , we linearize Equation (6.2.1) using an approximation. In view of the M + 1 quantities, cm is used in an extension to Equation (6.1.19) resulting in the approximation ex ΔRk = Rth k − Rk =

M  ∂Rth k · Δcm , ∂cm

k = 1, 2, ... , L.

(6.2.9)

m=0

This relation contains the concentration differences Δcm . The partial derivative of Rth k with regard to cm follows, in view of the absorption and scattering coefficients Kk and Sk , from th th ∂Rth ∂Rth ∂Sm ∂Rth k k ∂Km = Rkm = + k , ∂cm ∂Km ∂cm ∂Sm ∂cm

k = 1, 2, ... , L,

m = 0, 1, ... , M. (6.2.10)

The left-hand side of this expression is abbreviated by Rkm . The partial derivatives of the function in Equation (6.2.1) with regard to cn along with Equations (6.2.9) and (6.2.10) lead to the identity

6.2

Strategies for Recipe Prediction M L  

w2k Rkn Rkm Δcm

k=1 m=0

=

411

L 

w2k Rkn ΔRk ,

n = 0, 1, ... , M.

(6.2.11)

k=1

This relation can be used to determine the M + 1 concentration differences Δcm . This is because the quantities Δcm in Equation (6.2.11) are the only unknowns; th Rex k is a measured quantity, the theoretical spectral quantity Rk follows from the chosen radiative transfer approximation, and the partial derivatives of the reflection values are given either by an analytical or numerical procedure. The formal solution to the system of equations (6.2.11) is outlined again by using matrix notation [2]; this formalism is also applied to the colorimetric strategy. The coefficients on both sides of Equation (6.2.11) suggest the introduction of the following matrices and vectors. The partial derivatives in Equation (6.2.10) are interpreted as elements of the (L, M + 1) matrix R = (Rkm ),

k = 1, 2, ... , L;

m = 0, 1, ... , M,

(6.2.12)

the L weighting factors wk , given by Equation (6.2.7), are assigned to the vector w = (w1 , w2 , ... , wL )T ;

(6.2.13)

with the unit matrix E, therefore, the diagonal matrix W = Ew

(6.2.14)

results. Furthermore, we define a vector for the L spectral differences ΔRk of Equation (6.2.9) as ex th ex th ex T ΔR = (Rth 1 − R1 , R2 − R2 , ... , RL − RL ) .

(6.2.15)

With the vector of the M + 1 concentration differences Δc = (Δc0 , Δc1 , ... , ΔcM )T

(6.2.16)

A = RW

(6.2.17)

and the abbreviation

the system of equations (6.2.11) reads {AT A}Δc = AT WΔR.

(6.2.18)

412

6 Recipe Prediction

From this, the formal solution is given by Δc = {AT A}−1 AT WΔR.

(6.2.19)

The originally sought-after concentration vector c = (c0 , c1 , ... , cM )T follows by iteration using suitable starting values. For this, we use empirical values; to begin with, we set, for example, all M + 1 components to cm = 0.2 thereby establishing a starting vector denoted as cold . With that, either the absorption and scattering constants or optical path and albedo of the mixture are calculated using Equations (6.1.5) and (6.1.6) or (6.1.16) and (6.1.17). The spectral values Rth k are determined with the chosen radiative transfer approximation. The partial derivatives in Equation (6.2.9) are given analytically or approximately by difference quotients from calibration data obtained by methods discussed in Sections 6.1.1 and 6.1.2. The solution Δc of Equation (6.2.19) is added to cold resulting in an improved approximate solution cnew = cold + Δc.

(6.2.20)

The iteration procedure is continued as long as the value of all L spectral differences is less or equal to the measuring uncertainty ε of the used spectrophotometer, as in ex |Rth k − Rk | ≤ ε,

k = 1, 2, ... , L.

(6.2.21)

Alternatively, the iteration is stopped if some or all calculated concentration differences are smaller than the technical dosing for the colorants. Generally, no more than five iteration cycles are necessary for sufficient precision. The flowchart in Fig. 6.11 shows the most important steps of this strategy. In rare combinations of absorption colorants, the system of equations (6.2.19) delivers a non-satisfactory coloristic solution. The reasons for this are often the partial derivatives of first order, calculated anew in each iteration step, which do not lead to the expected minimum. In those cases, derivatives of second order should also be taken into account in the terminated Taylor series (6.2.9); this corresponds, for example, to the method of Levenberg and Marquardt [14, 17]. Similar considerations also hold in both of the following sections. The spectrometric recipe prediction method described can be formally applied to transparent or opaque colorations. In the case of translucent materials, further modifications are necessary if reflection and transmission are measured separately. This is also an issue if the separately used reflection and transmission values result in different concentrations. In this case, an approach with an analog to Equation (6.1.19) with other weighting factors is necessary. For that, in Equation (6.2.9) it is necessary to add the difference between the theoretical and experimental transmission values as in

6.2

Strategies for Recipe Prediction

413

Begin

Estimate values for M + 1 concentrations cm, m = 0, 1, ... , M

Optical constants kj, sj or τj, ωj, dj of M + 1 colorants Iteration

th Calculation of R th j , ∂Rk / ∂cm

with chosen radiative transfer approximation

⏐Rkth – Rkex ⏐≤ εk ?

cnew = cold + Δc No

Yes

Recipe output

Fig. 6.11 Simplified flow diagram for iterative determination of colorant concentrations by spectrometric strategy

ΔTk =

Tkth

− Tkex

M  ∂Tkth = · Δcm , k = 1, 2, ... , L, ∂cm

(6.2.22)

m=0

and to continue as previously. The layer thickness is determined separately based on the procedure given in Section 3.4.3 or with a direct method. The spectrometric procedure described is, of course, applicable to colorations containing effect pigments. In this case, the additional dependence on the spectral values of the measuring angle μas has to be considered. Now, it is advisable to seek the minimum of the following function:

414

6 Recipe Prediction

1  2 ex 2 w μ k (Rth μ k − Rμ k ) ≡ min . 2 L

Fe =

A

(6.2.23)

k=1 μ=1

In this expression, wμk is a weighting factor, which depends on the measuring angle μas and the wavelength λk . Furthermore, the coefficients of the phase function have to be taken into account. In each iteration step, A · L spectral values Rth μ k and A · L · (M + 1) partial derivatives of the first order with regard to the M + 1 concentrations cm are established so that the condition of Equation (6.2.23) is fulfilled as well as possible.

6.2.3 Colorimetric Method The colorimetric procedure differs from the spectrometric strategy by the fact that now the color difference assumes the role of the spectral value differences. Accordingly, the colorimetric system, the relevant illuminant, and the standard observer are chosen in advance. The advantages of this method, as compared to the spectrometric strategy, are the direct specification of illuminant and observer as well as the improvement in matching of the predicted recipe to the visual impression on the basis of an included color difference criterion. Primary disadvantages are the restriction to a single illuminant – which limits a recipe, for the moment, to at least four colorants – and the consequence that metamerism is minimal only for this illuminant. The colorimetric method based on the CIE standard color values X, Y, Z generally leads to constrained coloristic results. For this, the CIELAB or DIN99o color values are better suited because the accompanying color values are subject to a further weighting by the corresponding color model. Without loss of generality, we denote in the following the color values again by L, a, b. The simplest colorimetric procedure for prediction comes from the function FC =

1 2 th [w (L − Lex )2 + w2a (ath − aex )2 + w2b (bth − bex )2 + w2c (Δc)2 ]. (6.2.24) 2 L

In this expression, Lth , ath , bth are the theoretical color values which follow from the chosen radiative transfer approximation. The experimental values Lex , aex , bex are known from the measured spectral values of the reference color. The control variable Δc corresponds to the summed concentration differences of at most four colorants Δc =

3  m=0

Δcm .

(6.2.25)

6.2

Strategies for Recipe Prediction

415

The requirement that the function FC of Equation (6.2.24) should be as small as possible yields the values of the four independent variables, Δc0 , Δc1 , ... , Δc3 . The weighting factors for the color difference values are typically set to wL = wa = wb = 1.0,

wc = 100.0

(6.2.26)

[18]. The linear system of equations for determining the concentration differences of the colorants Δc0 , Δc1 , ... , Δc3 is established in a similar way to the approach in the previous section. The colorant concentrations are arranged in the four-vector c = (c0 , c1 , c2 , c3 )T ,

(6.2.27)

and the theoretical color values fith of FC are interpreted as functions of this vector fith = fith (c),

i = 1, 2, 3 ≡ L, a, b.

(6.2.28)

Similar to the explanations in the previous section, relation (6.2.24) is minimized most simply by linear approximations. Correspondingly, the differences between the theoretical and experimental color values in Equation (6.2.24) are approximated by the equation

Δfi =

fith

− fiex

3  ∂fith = Δcm . ∂cm

(6.2.29)

m=0

Furthermore, the partial derivatives ∂FC /∂ck , k = 0, 1, 2, 3, follow from statement (6.2.24). We simplify the notation for the partial derivative as fim =

∂fith , ∂cm

i = 1, 2, 3, m = 0, 1, 2, 3.

(6.2.30)

Consequently, together with Equations (6.2.25) and (6.2.29), a linear system of equations for determining the concentration differences Δcm follows. This system is formally identical with that of Equation (6.2.11) provided that the quantity Rkl is replaced by fkl and the upper summation limits are set to L = M = 3. In the following, we formalize this system. The partial derivatives of the three theoretical color values with regard to the concentration cm are given by

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6 Recipe Prediction

∂Lth ∂Lth ∂Y th = , ∂cm ∂Y ∂cm

(6.2.31)

∂ath ∂X th ∂ath ∂Y th ∂ath = + , ∂cm ∂X ∂cm ∂Y ∂cm

(6.2.32)

∂bth ∂Y th ∂bth ∂Z th ∂bth = + . ∂cm ∂Y ∂cm ∂Z ∂cm

(6.2.33)

The four sought-after concentration differences Δcm are interpreted as components of the solution vector Δc = (Δc0 , Δ c1 , Δ c2 , Δc3 )T .

(6.2.34)

In order to determine the components of Δc, the weighting factors in Equation (6.2.24) are arranged into the further four-vector w = (wL , wa , wb , wc )T ,

(6.2.35)

with the unit matrix E the diagonal matrix W = Ew

(6.2.36)

is generated. The partial derivatives of the color values and concentrations are gathered in the quadrature matrix ⎛

∂Lth ⎜ ⎜ ∂c0 ⎜ th ⎜ ∂a ⎜ F=⎜ ⎜ ∂c0 ⎜ th ⎜ ∂b ⎜ ⎝ ∂c0 1

⎞ ∂Lth ∂Lth ··· ⎟ ∂c1 ∂c3 ⎟ ⎟ ∂ath ⎟ ∂ath ⎟ ··· ∂c1 ∂c3 ⎟ ⎟. ⎟ th th ∂b ⎟ ∂b ⎟ ··· ∂c1 ∂c3 ⎠ 1 ··· 1

(6.2.37)

Finally, we introduce the vector Δf given by Δf = (Lth − Lex , ath − aex , bth − bex , Δc)T .

(6.2.38)

This vector is composed of the three color difference distributions and the component Δc from Equation (6.2.25). Together with the product of both matrices (6.2.36) and (6.2.37), B = WF, (6.2.39) the linear system of normal equations {BT B}Δc = BT WΔf

(6.2.40)

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417

follows with the formal solution Δc = {BT B−1 }BT WΔf ;

(6.2.41)

this is provided that the inverse {BT B}–1 exists. Equation (6.2.41) is similar to (6.2.19) of the spectrometric strategy. The determination of vector (6.2.27) is carried out by iteration in analogy to that of Section 6.2.2. Again, we start with equal values of cm = 0.2, m = 0, 1, 2, 3, calculate Lth , ath , bth , determine the partial derivatives contained in matrix (6.2.37), etc. The iteration process is terminated if the numeric color difference with regard to the chosen illuminant and observer is in the range of |ΔE| < 10−2 , for example (cf. Fig. 6.11). The formalism above can be extended to more than four colorants. Certainly, then a secondary illuminant has to be chosen in addition. This secondary illuminant should differ considerably from the primary illuminant. For example, the combinations of D65 with D55 or D75 are to be avoided. This is because the corresponding coefficient matrix {BT B} of the normal equations (6.2.40) will be unstable. The added number of secondary color values in Equation (6.2.24) is identical to the number of additional colorants. The secondary color values are completed in sequence of L, a, b and belong naturally to the same illuminant and observer as in the primary system. In conclusion, it has to be mentioned that some of the modifications given at the end of the previous section can also be applied to the present colorimetric method. A further variant is given if directly the squared color difference (ΔE)2 is introduced in Equation (6.2.24) rather than the three weighted color difference contributions. Disadvantages of this approach are certainly the omission of different possible weighting factors and the reduced number of colorants. As it turns out, this is a special case of the recipe prediction method with balanced color differences and is discussed in the next section.

6.2.4 Balanced Color Differences To conclude the discussion of classical recipe prediction method, we bring together the previous calculation procedures in order to tap into each of their prevailing advantages. The corresponding strategy is called method of balanced color differences. The basis of our considerations is Equation (6.2.3). The squared color differences (ΔEl )2 summed over l = 1, 2, . . . , Λ different illuminants have to be as small as possible. Generally, however, the resulting color differences are nearly of equal amount if all weighting factors wl ≥ 0 in Equation (6.2.3) have identical values [16]. Any preferred illuminant will have the smallest color difference as compared with some other illuminants. Because of the fact that color constancy as well as metamerism of the recipes can be

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controlled by the weighting factors, this method exceeds the efficiency of both prediction procedures discussed up until now. We designate the color values of illuminant l by Ll , al , bl and write the differences between the corresponding theoretical and experimental values as ΔLl , Δal , Δbl . From Equation (6.2.3), it follows Λ

1 2 FB = wl [(ΔLl )2 + (Δal )2 + (Δbl )2 ]. 2

(6.2.42)

l=1

It is, perhaps, self-evident that the next step should be to linearize the color differences in this equation in a similar way to that of Equation (6.2.29) for each illuminant l and to apply the iteration method as described in the previous section. This would then possibly result in more equations available than colorants provided in a recipe. Therefore, we follow another solution in which the number of sought-after concentrations is preset as in the spectrometric strategy. In order to deduce the corresponding algorithm, we form the partial derivatives of Equation (6.2.42) with regard to the M + 1 colorant concentrations ck and assume by approximation vanishing values: ∂FB ∼ = 0, ∂ck

k = 0, 1, ... , M.

(6.2.43)

The analytical description again needs the partial derivatives of Ll , al , bl , Xl , Yl , Zl , and Rth j with regard to concentration ck . Instead of the absorption and scattering coefficients Kjth and Sjth – as used in Section 6.2.2 – we now develop the formalism in dependence of the entire optical path Θjth and the entire albedo Ωjth at wavelength λj . Similar to Equations (6.1.16) and (6.1.17), we assume that both quantities can be written as linear combinations: Θjth

=d

M 

cm τjm ,

(6.2.44)

m=0

Ωjth

=

M d 

Θjth

cm τjm ωjm ,

j = 1, 2, ... , L,

M + 1 < L;

(6.2.45)

m=0

therefore, M + 1 quantities cm are to be determined for each recipe. In Equations (6.2.44) and (6.2.45), the following notation is used: d: layer thickness of the reproduced color; concentration of the mth colorant; cm : τ mj , ωmj : specific optical path and specific albedo of the mth colorant at wavelength λj ; m = 0: index of the binder.

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419

For the partial derivatives of Equations (6.2.44) and (6.2.45) with regard to ck , it follows ∂Θjth ∂ck ∂Ωjth ∂ck

=

d2

M 

(Θjth )2

m=0

= dτjk ,

(6.2.46)

cm τjk τjm (ωjk − ωjm ), j = 1, 2, ... , L, k = 0, 1, ... , M. (6.2.47)

After some intermediate steps, we obtain two expressions termed as α km and β k for k, m = 0, 1, ..., M. From each expression α km and β k , a system of equations for the determination of the unknown concentrations follows. The quantity α km consists, essentially, of the radiation energy density Sˆ jl of the lth illuminant at wavelength λj multiplied by Equation (6.2.47). The quantity β k contains the optical path τ jk as well as a factor resulting from the partial derivatives of the color values. For formulation of the system of equations for the M + 1 unknown concentrations cm , we establish the vector c = (c0 , c1 , ... , cM )T

(6.2.48)

and arrange the quantities α km in the quadratic (M + 1, M + 1) matrix A = (αkm ),

k, m = 0, 1, ... , M.

(6.2.49)

In this matrix, all elements of the main diagonal are zero on account of Equation (6.2.47), therefore resulting in α mm = 0. From Equation (6.2.47), it follows that the additional condition α km = – α mk holds. In other words, the matrix A is skew-symmetric – meaning that instead of (M + 1)2 , matrix A contains only M(M + 1)/2 different elements. Finally, we combine the quantities β k into the vector b = (β0 , β1 , ... , βM )T .

(6.2.50)

The system of equations for direct determination of the solution vector c of Equation (6.2.48) takes the form d2 Ac = b.

(6.2.51)

Once again, this system lends itself only to solution by iteration. The entire optical path Θjth is contained in the denominator of Equation (6.2.47) and the theoretical optical constants are determined using Equations (6.2.44) and (6.2.45). In analogy to the explanations in previous two sections, we start with a suitable estimate for values of the concentrations cm . These are inserted into

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6 Recipe Prediction

Equations (6.2.44), (6.2.45), and (6.2.47). We limit the numerical value of the denominator in Equation (6.2.47) again to a maximum of 100. Afterward, the respective solution is used to determine the coefficients of the system in Equation (6.2.51) while maintaining the cm values within a preset range of variation, and respectively, the function FB of Equation (6.2.42) is minimized. Above and beyond the properties mentioned at the beginning of this section, the present method has further advantages such as: – the maximum number of colorants is limited only by M + 1 ≤ L; L denotes the measuring frequencies of the used spectrophotometer; – the color difference formula is optional; therefore, the corresponding weighting factor is changed by the partial derivatives of the color values; – instead of the usual color distributions ΔL, Δa, Δb, differences such as ΔL, ΔC, ΔH, or others, for example, from CIE color appearance models can also be used. An essential point should be clear if translucent colorants are to be predicted [7, 15]. In this case, the system of equations in Equation (6.2.51) delivers, for the moment, concentration values that are associated with a specified layer thickness d > 0. On the other hand, the covering thickness of the layer can be determined separately by procedures described in Section 3.4.3. But the hiding criterion, which depends on the color difference over black and white backgrounds, can be included directly into formula (6.2.42). We denote the color difference over both backgrounds and corresponding to illuminant l by ΔElb,w . Using the mentioned criterion, all three color contributions can be changed independent of one another in order to achieve hiding with the possible consequence of undesired color changes. To avoid this, it is advisable to take into consideration only the lightness difference ΔLlb,w and to minimize the expression Λ

FL =

1 2 [wl (ΔEl )2 + g2l (ΔLlb,w − sl )2 ]. 2

(6.2.52)

l=1

In this equation, ΔLlb,w represents the tolerable lightness difference over both backgrounds and the corresponding illuminant l, sl is a load factor, and gl a suitable weighting factor for illuminant l. Further details of this comprehensive method are beyond the scope of this text.

6.3 Realization of Recipes Independent of the chosen strategy, classical recipe prediction has the particular property that the method potentially delivers numerous alternative recipes on

6.3

Realization of Recipes

421

the basis of combinatorial analysis. Among those recipes, the most promising should be selected. In order to make such a selection, many reliable coloristical and colorimetrical criteria are available with which the features of the reference color can be sufficiently approximated. Recently, two further colorimetrical selection criteria can be applied: the so-called sensitivity and correctability of a recipe. Both quantities enable the estimation of the extent that small concentration variations have an effect on the color of a recipe, and, whether a recipe is corrigible at all. In spite of the high dependability of the classical recipe prediction for absorption colorants normally attained, in a few cases, however, it is still necessary to have additional correction for the components of the first recipe. For such additional correction, we describe three correcting procedures. These procedures have been successfully applied to all kinds of colorant mixtures of the color industry. To an ever-increasing degree, additional methods of information technology are applied for color reproduction containing absorption and effect colorants. Among these, we emphasize databases with recorded historical recipes, expert systems combined with optical equipment, or neural networks working with a great number of pre-trained microprocessors. The underlying software can be partially supported by elements of fuzzy logic. The reproduction of effect colors is generally successful either by using classical or empirical methods alone or rather by using combination of the two. In this combination, reliable correcting methods are indispensable for adjustment of suitable recipes for absorption colorants or the optimization of effect colors. Consequently, the classical recipe prediction procedure, alongside modern methods, remains a universally applicable instrument of industrial color physics.

6.3.1 Selection of Suitable Recipes The widespread usage of the classical recipe prediction methods for absorption and effect colorants essentially comes down to three important advantages: versatile applicability, dependability of suitable recipes, and simultaneous computation of alternative recipes. In fact, the wealth of obtained recipe proposals increases with the extent of available colorants and with the permitted number of colorants in a recipe. Because of this, the number of allowable colorants in a recipe has to be restricted reasonably. From the combinatorially obtained recipes, three to five suitable recipes are generally selected on the basis of various criteria outlined in Sections 3.2, 3.3, 3.4, and 4.1.6, among other things. The determination of the “best recipes” is oriented, in practice, not only toward color physical and coloristical criteria but also toward technical processes and economical points of view. The relationships of these primary categories are shown in Fig. 6.12. It can be taken from this illustration that recipe prediction has precedence over technical and economic considerations.

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Fig. 6.12 The three primary categories for selection of color recipes and their mutual interrelation

Color physical criterions

Technical aspects

Economic necessities

First is the fixing of three to five most suitable coloristical recipes; all further criteria are subordinate, but not to be neglected. One might get the impression that recipe prediction confines the working space of colorists; however, it is quite the opposite! The high-quality standards, which color users combine with this efficient method, require especially wide coloristical, color technical, and color physical experience, a sensible evaluation of the efficiency and limits of the used colorants, additives, as well as all employed computation methods. In short, anyone needing to determine the most suitable color recipe require: enhanced flexibility, excellent knowledge, and broad experience with the coloristic and numerical procedures used. In the case of absorption colors, for example, some of the color properties explained in Sections 3.2, 3.3, and 3.4 are often fulfilled simultaneously. Consequently, it is often necessary to balance against one another: – minimal color difference; – low metamerism; – high color constancy; – desired color fastness; – high covering capacity; – low turbidity; – low tinting power; – special tolerance agreements. In spite of the combined control of some above-mentioned properties is important to point out: every recipe prediction fails if the previous selection and calibration of the available colorants were not performed with the utmost care.

6.3

Realization of Recipes

423

The detection of the suitable recipes containing effect pigments alone or in mixtures with absorption pigments sometimes requires a great deal of experience. On the basis of the special angle and particle size dependent properties of effect colorants, at least some or all of the following qualities have to be matched: – brilliance; – reflection lightness; – lightness flop or color flop; – hue extinction; – sparkle or glittering; – distinctiveness of image, DOI; – covering capacity. Except of covering capacity, all other criteria listed above cannot be optimized straightaway by methods of numerical color recipe prediction. This is because they cannot, in a simple manner, be included in recipe algorithms. These properties are to be carried out separately after the first coloration is manufactured. Certainly, this task is much like a balancing act for recipes containing effect pigments: some of the above-mentioned effect properties change contrary in dependence on the particle size, as already shown in Tables 2.9 and 2.18. Furthermore, there are additional technical facts which should not be underestimated and are to be taken into account by the transition from laboratory to production scale and by coating or processing of the produced color. These process conditions are united in the literature by the vague term “process stability.” In the context of recipe realization, it is important to take into account the following quantities, among other things: – recipe sensitivity and correctability; – dosing exactness of color components and additives; – homogenization conditions like rpm, agitator, fullness, time, temperature; – compatibility of all components, especially with regard to additives; – fluctuations of processing or application parameters. Details of these aspects should be known from preliminary examinations or from experience. Finally, the selection of computed recipes depends on the given economic frame. Similar to the categories mentioned above, it is also necessary here to find an acceptable compromise by taking into consideration criteria such as: – costs for colorants, technical equipment, processing; – availability of colorants; – substitution of components;

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6 Recipe Prediction

– quality consistency of added constituents; – using of remaining colors; – environmental aspects. Experience shows that – with all listed points in this section – the potential of the recipe prediction procedure is widely exploited for nearly all color applications. On the other hand, the power of these tools is sometimes unknowingly overestimated. With regard to spectrometric and colorimetric emphasis, the sections that follow are intended to avoid this.

6.3.2 Sensitivity and Correctability of Color Recipes Colorimetric criteria for recipe selection – such as color difference, degree of metamerism, color constancy, or covering capacity – are largely included in recipe prediction procedures as outlined in several previous sections. The combinatorially obtained recipes can be further sorted out with regard to processing parameters of colorimetric influence. Those color components which react sensitively to variations in dosing as well as other production or processing quantities turn out to be critical. Among them are especially: dark colorants such as black, dark violet, or dark blue. In addition, the question arises as to what extent a recipe is still corrigible if unforeseen process fluctuations occur. The assessment of such influences can be performed with two different quantities which are called sensitivity and correctability [19, 20]. For the determination of the numerical sensitivity of a recipe we start, for simplicity, with the colorimetric strategy. We assume, in addition, that the concentration ck of the kth color component has an uncertainty of Δck . This uncertainty ultimately results in a change of some or all color values of the recipe. In order to determine this dependence, we linearize the color difference amounts Δfl with regard to the illuminant l (l = 1, 2, ... , Λ), similar to Equation (6.2.29), by Δfl = flth − flex =

∂flth Δck , ∂ck

f = 1, 2, 3 ≡ L, a, b, k = 0, 1, ... , M. (6.3.1)

The corresponding color difference ΔEl (Δck ) is assumed to be directly proportional to Δck : ΔEl (Δck ) = sl,k · Δck .

(6.3.2)

The proportional factor sl,k in Equation (6.3.2) is called sensitivity of the kth recipe component with regard to illuminant l and corresponding observer. The value of sl,k is a measure of the change of color difference with the variation

6.3

Realization of Recipes

425

of the kth colorant component. With Equations (3.1.18), (6.3.1), and (6.3.2) follows 

2

2   ∂Lth 2 ∂ath ∂bth  l l l + + . (6.3.3) sl,k = ∂ck ∂ck ∂ck Please recall that L, a, b stand for color values of different color spaces and especially also for the color value triple L, C, H. From Equation (6.3.3), it follows that the sensitivity of the kth colorant is directly proportional to the change of the three color values with regard to the colorant concentration. From the values of each summand it is possible to infer which color value is influenced to the greatest (smallest) amount by concentration variation and, therefore, produces the highest (lowest) color shift. The dependence of illuminant and standard observer is of secondary importance. This is the reason why we leave out the index l in the following. With Equation (6.3.1), on the other hand, neither the color values nor the color difference formula ΔE(Δck ) in Equation (6.3.2) is restricted. Each of the M + 1 concentrations c = (c0 , c1 , ... , cM )T

(6.3.4)

of a recipe is able to vary by ck , combined in the vector Δc = (Δc0 , Δc1 , ... , ΔcM )T .

(6.3.5)

The resultant color difference ΔE(Δc) is given clearly, if the variation of |Δck | is not greater than the technical dosage uncertainty Δcu , i.e., |Δck | ≤ Δcu ,

k = 0, 1, ... , M.

(6.3.6)

The use of Equations (6.3.2) and (6.3.6) with regard to M + 1 colorants results in an estimate for the highest possible resulting color difference  M  √ ΔE(Δc) =  s2k (Δck )2 ≤ |s| · M + 1 · Δcu .

(6.3.7)

k=0

The quantity s in this expression is called the entire sensitivity of a recipe, where the squared value is given by using Equations (6.3.6) and (6.3.7): s2 =

M 

s2k .

(6.3.8)

k=0

This definition equation is certainly unsuited for numerical assessment of the entire sensitivity. In order to obtain a reliable approximate value for s, we use the

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6 Recipe Prediction

matrix notation of the colorimetric strategy (Section 6.2.3). Each summand s2k of Equation (6.3.8) consists of the partial derivative of a color value with regard to ck . Similar to Equation (6.2.36), the partial derivatives with regard to the M + 1 colorants are organized in the (3, M + 1) matrix ⎛

∂Lth ⎜ ∂c ⎜ 0 ⎜ th ⎜ ∂a D=⎜ ⎜ ∂c ⎜ 0 ⎜ ⎝ ∂bth ∂c0

⎞ ∂Lth ∂Lth ··· ∂c1 ∂cM ⎟ ⎟ ⎟ th th ∂a ∂a ⎟ ⎟; ··· ∂c1 ∂cM ⎟ ⎟ ⎟ th th ∂b ⎠ ∂b ··· ∂c1 ∂cM

(6.3.9)

now the quadratic (3, 3) matrix follows H = DT D which possesses the dominant eigenvalue |λ|max . With expression  sLab = |λ|max ,

(6.3.10)

(6.3.11)

an approximate value for the entire sensitivity of a calculated recipe with regard to the color values L, a, b is given. To complete the picture, the reciprocal value r = 1/s, called robustness, is sometimes also used. The concentration variation of a sensitive colorant can definitely cause great color deviation. Therefore, the question arises as to what extent such a colorant mixture is correctable within an acceptable color tolerance. Clearly, all numerical correction procedures only work efficiently if the corresponding recipe is really corrigible (Section 6.3.3). A dimensional value for this characteristic is given by the so-called correctability. This quantity corresponds to the summed concentration variations Δck of all M + 1 colorants used which produce a maximum color difference of ΔE(Δc) = 1.0. The correctability of a recipe κ is defined, therefore, by the expression κ=

max

ΔE(Δc)=1.0

 M   (Δck )2 .

(6.3.12)

k=0

Similar to sensitivity s, the quantity of κ can also be assessed. For such an assessment, the value of the smallest eigenvalue |λ|min of matrix H in Equation (6.3.10) is used. The quantity  κLab = |λ|min (6.3.13) is called entire correctability of a recipe [19].

6.3

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427

In accordance to these considerations, each predicted recipe for absorption colorants should possess a sensitivity as low as possible and, in contrast, a correctability as high as possible. But in most cases, both requirements are not easily fulfilled simultaneously. This is shown, for example, in Table 6.7 for three recipes which produce nearly the same color. Only the first of these recipes complies with both desired properties. Table 6.7 Entire sensitivity, robustness, and correctability of three recipes producing virtually equal color [19]

Recipe

Entire sensitivity sL∗ a∗ b∗ /(1/conc.)

Entire robustness rL∗ a∗ b∗ /conc.

Entire correctability κL∗ a∗ b∗ /(1/conc.)

1 2 3

13.9 15.1 21.7

0.0719 0.0662 0.0461

1.91 1.10 1.14

Finally, a critical remark is necessary. The real correctability cannot be inferred alone from the numerical value of κ. For example, it may be possible to have a coloration combination which produces a color difference of ΔE(Δc) = 1.0 only for one or more negative concentration variations. Furthermore, a large entire correctability can be accompanied with a high sensitivity. This is the case if the matrix elements of Equation (6.3.9) are of nearly equal value. In addition, if the necessary concentration changes are smaller than the dosing uncertainty Δcu , i.e., |Δck |<Δcu according to Equation (6.3.6), the recipe fails on the basis of this unrealistic parameter. These considerations also come into play, if reduction of metamerism can only be realized by the addition of extra colorants in tiny amounts. In spite of this, sensitivity and correctability are indispensable further characteristics for easier selection and technical handling of color recipes.

6.3.3 Numerical Procedures for Recipe Correction The described methods of operation for color measurements, calibration of colorants, radiative transfer approximation, as well as recipe strategy are effective apparatus to specifically reproduce given colors according to different criteria as outlined in Section 6.3.1. Nevertheless, the first coloration of a recipe sometimes shows an unacceptable difference compared to the reference color or other color specifications are unsatisfactory. Accordingly, this so-called first recipe of off-shade formulation is to be corrected one or several times. At this stage, we have to stress that the following methods are applicable to absorption and effect colorants. They can be considered to be indispensable constituents of recipe prediction also by databases or modified expert systems (next two sections).

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Generally for absorption colorants, the reasons for mismatch between prediction and reference (sample and standard, color sample and target) are not immediately recognizable. In general, they are caused by systematic or random errors. Both types of errors need to be evaluated and kept as small as possible. Some possible systematic errors in recipe prediction are as follows: – wrongly chosen radiative transfer approximation, illuminant, observer, or prediction strategy; – deviation of optical behavior of colorant with regard to the assumed nonlinearity; – interactions between colorants; – deviation of coloristic colorant properties from calibration; – altered production parameters; – incorrect dosing; – confusion between similar colorants, additives, or binders. Systematic errors should be eliminated from the onset although they are partially averaged out by numerical recipe corrections. Certainly, the consequences of accumulated errors of this kind are not entirely correctable. An inappropriate radiative transfer approximation already leads, systematically, to off-shade formulations. Further deviations result from measuring, production, or handling conditions having been altered between the times of coloration and colorant calibration. In contrast to these errors, random errors cannot be corrected so easily. Because random errors cannot be avoided, the influences of these are to be kept as small as possible. The following random errors might occur in recipe prediction: – measuring error caused by fluctuations within the components of the spectrophotometer or other measuring instruments; – random variations of manufacturing parameters of colorants; – changes of the surface of the interior of the reference or substrate by physical or chemical processes; – change of surrounding conditions; – contaminated colorants. It is, therefore, essential to perform the entire set of process steps during numerical recipe prediction and manufacturing exceedingly careful and reproducibly, as in the calibration procedure. By keeping these requirements, first recipes for absorption colors generally obtain an average color differ∗ ≤ 0.5 and first recipes for mixtures of effect pigments ence of about ΔEab ∗ of about ΔEab ≤ 2.5. Certainly, a small color difference – especially in cases of effect colors – is not a sufficient criterion alone of an acceptable color

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429

reproduction. For effect colors, colorimetric methods have limited meaning. It is very rare that the characteristic coloristic distinctions of effect colorations with regard to absorption colors can be described by a color difference alone. Among the several developed correction operations which work effectively for absorption colors as well as effect colors, we restrict ourselves to the description of the factor method, colorimetric procedure, and spectrometric operation. The factor method is largely applied to dyes of textile fibers. This method requires the dyes and their concentrations of the reference as well as the prediction to be known [6]. The varying dye intake of sample and target fibers may be caused by the molecular structure, conformation, and as consequences to separate fabrication and conditioning, among other reasons [10]. The correction factor α for a dye in the reproduced color is given by the ratio of the reference concentration cR to the predicted concentration cP . For k = 0, 1, ... , M mixed dyes, the kth correction factor α k then follows from the relation  αk = cRk cPk . Both reference and predicted concentrations are multiplied by α k . This results in the concentrations βkP = αk · cPk and βkR = αk · cRk . Now the concentration adjustments Δck , specific to the dye intake process, follow from Δck = βkR − βkP = αk (cRk − 1). Unreasonably high or low correction factors usually indicate serious systematic errors. A negative adjustment Δck indicates that the dye lot of the prediction contains too much of the kth dye and has to be reduced possibly before further correction steps. In contrast, the colorimetric and spectrometric correction procedures are usable for all other sorts of colorants. The colorimetric correction method turns out to be effective provided that the color difference of the off-shade formula∗ = 3.0. This procedure starts from tion and the reference is not higher than ΔEab the known color value differences Δf = f P − f R ,

f = L, a, b or f = L, C, H, etc.

(6.3.14)

between prediction and reference; the differences are interpreted as components of the three-vector Δf = (LP − LR , aP − aR , bP − bR )T . In analogy with matrix F in Equation (6.2.37) or A in Equation (6.2.49), the partial derivatives of the color values are calculated with regard to the concentrations of the first recipe. These coefficients constitute new matrices F1 , A1 with which the corresponding system of equations (6.2.41) or (6.2.51) is once again solved. The concentration adjustments Δc from Equation (6.2.41) are taken as a correction to the offshade formulation according to Equation (6.2.20); the concentrations resulting from Equation (6.2.51) are taken directly for correction. Generally, no more than three correction steps should be performed. This is because in most cases the algorithm iteration will not result in convergence of the final recipes. We come back to this aspect further down.

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The final procedure mentioned above is the spectrometric correction method. It is likewise generally applicable and moreover – like the factor method – independent of the recipe strategy used. Within this procedure, the differences between the l = 1, 2, ... , L spectral values of reference RRl and prediction RPl are used immediately for recipe improvement. The essential correction step consists in determining corrected spectral values Rcor l according to the relation Rcor = RRl + ΔRl l

(6.3.15)

ΔRl = fe · (RRl − RPl ), l = 1, 2, ... , L.

(6.3.16)

with

The corrected values Rcor l are immediately used for correction computation; see Fig. 6.13. The quantity fe in Equation (6.3.16) denotes an empirical adaption parameter in the range 0.6 ≤ fe ≤ 0.8. Now, an improved recipe is calculated together with the fixed colorants from with the corrected spectral values Rcor l the last prediction. After each correction step it is necessary to decide if the new color is acceptable or not. It has also to be mentioned that in both of the previously described correction methods, generally no more than three correction steps are to perform. Experience shows that further correction computations likely lead again to higher color differences! The reasons for this divergent behavior are not only R(λ)

RR RP R cor

λ Fig. 6.13 The spectrometric method of recipe correction: RR : reflectance of reference color, RP : predicted color, Rcor : corrected recipe

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in use of linear approximations but also in the flat topology of the color space in the vicinity of the color locus corresponding to the reference color. After three unsuccessful corrections, a remedy is provided by an increased colorant number – generally in a gradual way. Each increase should be followed by a new prediction with one or more fixed colorants. The possibly very low concentration of a further colorant certainly may turn out disadvantageous.4 This marginal amount is restricted by correctability and is confined by technical dosing and dispersing process. On the other hand, it is necessary to assess if the improvement of coloristical properties is justified also economically. The sole use of colorimetric or spectrometric correction methods normally fails for colorations with effect colorants. For successful adjustment of such colorations to the reference color, additional visual methods are required. We concentrate on these methods in connection with modified expert systems in Section 6.3.5.

6.3.4 Databases In addition to classical methods of recipe prediction, further procedures have been developed after about 1980. Since then, these modern methods have been increasingly introduced into the color industry. This is because color recipes have accumulated over the decades and progress in these respects has been additionally facilitated by the advancement of digital information technology. These methods are essentially founded on the concept of the database. A database represents a systematic and long-term available collection of structured information. This includes the necessary software to edit and recall the information [21, 22]. A database is always an integral element of general information systems. The corresponding software should be able to search and sort, especially with regard to different colorimetric, spectrometric, or economic criteria. Databases for recipe prediction can be assembled from different records such as: – calibration samples or standards of mixtures containing absorption and effect colorants; – extensive collections of historical recipes, well ordered in groups of different colorant mixtures; – basic mixtures of so-called master batches; – remaining colors for utilization in new recipes.

4 Similar

to minimizing of metamerism or color improvement of color constancy.

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A complete data record of a historical recipe includes, among other things: – measured spectral values of the accepted color; – employed colorants and their concentrations; – optical constants of the colorants at each measuring wavelength; – measurement geometry used; – tristimulus values of the recipe and color difference with regard to the original reference color; – illuminant and observer; – layer thickness; – economic and further parameters. Historical recipes are only applicable for recipe prediction if at least two further prerequisites are fulfilled: first, a sufficiently large number of recorded color mixtures is available; second, all of these recipes were manufactured exclusively in the same experimental conditions. As expected, the procedures for finding a suitable recipe are differently structured for absorption and effect colorants. For absorption colorants at least 1,000 historical recipes are necessary. This depends on the number of available colorants as well as their color locus density or rather distribution in color space. The available colorants and binder for the relevant recipe should possibly be the same as in the previously manufactured recipes. Furthermore, the historical and relevant data should at least meet the following requirements: the color measurements should have been carried out with the same spectrophotometer, all color values of stored recipes should belong to the same illuminant, the same standard observer, and the same color space. The historical data may be usable with reservation if some of the mentioned requirements are not fulfilled. The “rough recipe” found in such a way is to be taken as the first recipe and then improvement can be attempted using one of the correcting procedures in the previous section. Generally, with a suitable search program, such a recipe can be determined, which shows, for example, a color difference of |ΔEx,i | < 5 with regard to the reference color. The color locus of the reference and the resultant recipe are denoted by Fx and Fi , respectively. Figure 6.14 illustrates, for instance, the narrowing of Fx by three color points Fi , Fi+1 , Fi+2 of known recipes. The color locus Fx , in this case, forms the top of a skew-symmetric tetrahedron. We restrict ourselves to an outline of a search routine. This is based on weighted color differences between adjacent color points of historical recipes with regard to the reference point Fx . In general, there are i = 1, 2, ... , N resultant recipes each of color point Fi adjacent to Fx and color difference condition |ΔEx,i | < 5. For simplicity, we denote the color difference of adjacent color points Fi , Fi+1 by |ΔEi,i+1 | and set up the relative difference δi =

|ΔEx,i | − |ΔEx,i+1 | , |ΔEi,i+1 |

i = 1, 2, ... , N,

(6.3.17)

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433 L Fx ΔEx,i Fi + 2

Fi ΔEi,i +1

Fi +1 b

a

Fig. 6.14 Narrowing of color locus Fx for a reference by three neighboring color loci Fi , Fi+1 , Fi+2 of known recipes

which is used as a weighting factor. For clear assignment in this equation, |ΔEx,N+1 | = |ΔEx,1 | and |ΔEN,N+1 | = |ΔEN,1 | are used. The k = 0, 1, ... , M + 1 known colorant concentrations cik of the recipe represented by Fi are interpreted as components of the vector ci = (ci0 , ci1 , ... , ciM )T , i = 1, 2, ... , N.

(6.3.18)

Finally, the unknown concentrations ck of the reference color are arranged in the vector c = (c0 , c1 , ... , cM )T .

(6.3.19)

Considerations of analytical geometry led to an approximate solution for this vector using formula c=

! N 1 δi ci − (ci − ci+1 ) . N 2

(6.3.20)

i=1

It should be taken into account that in Equation (6.3.20), the quantity cN+1 = c1 is set. The formation of the colorant components cik is the same in each vector ci . In the case of different colorant numbers, the components of each vector ci are arranged in the same order as before and filled up with zeros up to M + 1 components. This prediction method may fail for nearly achromatic or brilliant absorption colors if the color locus Fx of the reference is located in the vicinity of the lightness axis or represents a color of high chroma. Moreover, empirical correction

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factors, which were used to achieve historical recipes, are clearly not to transfer to an actual searched coloration. If the rough recipe seems to be appropriate, a recipe correction with the determined colorants is meaningful. However, if the color difference is not yet sufficiently small or another color quality with regard to the reference is unacceptable, then further steps must be undertaken, such as: – immediate correction computation for rough recipes of adjacent color points with regard to reference color; – combinatorics with colorants of all rough recipes, perhaps by specification of one or more of these colorants; – combinatorics with one or more additional available colorants. The utilization of remaining colors of absorption colorants always needs a new recipe formulation. In the first step, the known concentrations of the remaining colorants are fixed input quantities for the spectrometric or colorimetric prediction procedure. Afterward, correction factors are determined as in the previous section. In case of one negative concentration adjustment, the ratio of the actual concentration ca of the corresponding colorant to the predicted lower concentration gives the correction factor for all other components. If more than one negative adjustment is necessary, then the lowest predicted concentration is taken for the calculation of the overall correction factor. Remaining colors are, on the other hand, only of limited usability, because coloristic and additional quality criteria need to be met. For example, the existing and newly added components should have the necessary consistency during production, processing, and storage. Just to mention a few qualities, at least the stability and fastness of the predicted color should be ensured. A similar successive approach is necessary using conditioned coloration mixtures such as the so-called pigment preparations or master batches. They consist mainly of a colorant in a predefined concentration in a binder and mostly suited additives. In each case, the pigment concentration as well as the compatibility of all components with further colorants should be examined in advance. If such a mixture is intended to match a given recipe, initially only 80% of the final quantity is generally produced. After checking this mixture by color measurement, the remaining 20% can be mixed using an overall correction to result in the final 100% of the intended quantity with the correct color. For recipe determination of colors with several effect pigments and possibly additional absorption colorants, at least 3,000 historical recipes should be recorded in a database. This effort is justified in light of the complex color physical properties of this type of colorant. It is quite useful to include known recipes of color standards, repeatedly realizable similar effect shades or expected effect colorations. The corresponding recipes are to be characterized by spectrometric and colorimetric procedures. Colored light microscopic reference pictures

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435

of the samples are taken. Together with the recipe data mentioned above, the light microscopic pictures are recorded in the database. On the basis of the high density of color loci with minimum number of recipes mentioned above, the corresponding color space is filled, so to speak, with “calibration colorations.” Like absorption colors, also all stored recipes for effect colorations have to share the same illumination and measuring geometry, illuminant, standard observer, and of course the same color space. During formulation, the interim results are checked with a light microscope until an acceptable result is achieved. In the majority of cases, the adjustment of special color effects such as brilliance, flop, or distinctiveness of image, is carried out in the end of the procedure. A modern method of recipe determination for effect colors is based on the idea of only correcting the color difference of the obtained “best recipes” with regard to the reference color. This method is quite similar to the approach outlined above. The corresponding software requires two essential components: an optimal search strategy and a convergent interpolation algorithm. Further concentration on details about databases is beyond the scope of this text.

6.3.5 Modified Expert Systems As we have seen in the last section, a successful recipe search for colorations containing effect pigments can be achieved by a database search and accompanied by light microscopic analysis. The light microscope can also be used for a preliminary selection of a limited number of colorants for database search as well as following classical recipe prediction. The combination of recorded recipe data together with suitable search and correction programs as well as additional light microscope equipment represents a specially modified expert system. Beyond this, an expert system, in a general sense, contains the simulated know-how of an expert; complex problems of his/her special area are handled by dialog with this system.5 In the following, we outline some essential aspects of a modified expert system to perform recipe prediction of effect colors. The light microscopic analysis of a mixture with effect pigments in both the bright- and dark-field illumination delivers details for: – constituent effect pigments, consulting a reference library; – morphology of the particles; – geometry and possibly size distribution of the pigments; – transparency of the particles; – constituent absorption pigments; – concentration ratios of the different particle types. 5 Expert systems were developed, such as database, already in the year 1960 at the University of California, Berkeley.

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The combination of both types of field illumination allows for the identification of existing effect and absorption pigments. Fully automatic optoelectronic particle identification has, at this time, not been implemented in color industry. The photos in Color plate 11 show the pigment mixture of a reference effect color taken in bright- and dark-field illumination. There are different particles which one can distinguish: two metallic pigments (cornflake and silver dollar aluminum, d50 ≈ 25 μm and 15 μm), a gold yellow pearlescent pigment, two absorption pigments (yellow, reddish brown), white, and carbon black. Color plate 12 shows another reference color. In the bright-field picture, already recognizable are three pearlescent pigments (red, golden, blue); in dark-field illumination, three absorption pigments (two blue, one violet), and hiding carbon black. Again, these pictures illustrate that the size of effect pigments is about 10–100 times greater than that of absorption pigments (cf. Fig. 2.36). In practice, proven recipe prediction procedures for effect colorations extend from classical methods followed by reduced combinatorics to extravagant database search accompanied by light microscopic control. Modern first recipes ∗ ≤ for demanding automotive coatings show a mean color difference of ΔEab ∗ ≤ 2.5, whereas the more empirical procedure produces a value of about ΔEab 1.0, both for aspecular measurement angle of μas = 15◦ . As previously emphasized, characteristic color effects such as gloss, brilliance, sparkle, flop, hue extinction cannot be described by common color difference formulas. The necessary corrections and adjustments of the first recipe come down to partial experimental information or inadequate knowledge of optical behavior of the mixed particles. Only in rare cases it is possible to use the light microscope alone to determine several specific effect properties in such a manner as to allow the first recipe prediction to be accepted. In fact, the identified effect pigments, colored pigments, and mixture ratios may actually reproduce some of the required color effects, but further effect properties such as gloss or pearl luster, brilliance or color flop often are not tolerable. In addition, an identified effect colorant sometimes does not meet expectations. This pigment is then partially or entirely substituted by one of a similar sort. For example, a cornflake type might be exchanged for a pale silver dollar pigment. In cases of low brilliance or bright flop, for example, generally an “overdoing” effect type is added. If brilliance is too bright, a scattering pigment or an additive is used to reduce the orientation of the flakes. For sole flop correction, bismuth vanadate or possibly nano-titanium dioxide is suitable. Certainly these actions are of limited applicability because other effect properties can be unintentionally affected. All determined special case information should be retained in the expert system for future use in similar recipes. An instructive example with regard to the angle-dependent reflection of an effect color is shown in Fig. 6.15. The curves belong to the reference and predicted color with pigmentation reproduced in Color plate 11. The measurements

6.3

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R (%) μas = 15° 120

Reference color Reproduced color

100

80

25° 60

40 45°

75° 20 110°

0 400

500

600

λ nm

700

Fig. 6.15 Angle-dependent reflection of a yellow-gold effect reference color represented in Color plate 11 and corresponding predicted color of non-corrected lightness

are carried out with an illumination angle of β = 45◦ and five standardized aspecular measuring angles μas . With increasing measuring angle, the systematic deviations from the predicted color decrease. This is caused by a lower brightness and brilliance relative to the reference. On the other hand, the hue

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of the predicted color, determined by the family of curves with similar curvatures and shoulders, matches well. These conclusions are in accord with the corresponding differences of color values. For μas = 15◦ and D65/10◦ , a color ∗ = 3.70 and a lightness difference of ΔL∗ = 3.54 result. In difference of ΔEab other words, the hue is precisely reproduced. The remaining lightness adjustment is simply achieved by adding a single light pigment or mixtures of white, silvery pearlescent, or brilliant metallic pigments. The rising reflection difference of the set of curves with increasing wavelength in Fig. 6.15 is of little importance to the perceived shade. It is known that deviations at wavelengths in the range from yellow to bright red are visually more tolerated than for violet, blue, or green wavelengths. The flanks, plateaus, or maximums in Fig. 6.15 cannot be assigned to specific pigments without a supplementary light microscopy analysis. From Color plate 11 can be assigned, however, cornflake and silver dollar aluminum pigments, a yellow brass pigment and at least a reddish brown pearlescent pigment. Because the curve pairs do not intersect, the reference and predicted colors are even free of metamerism. This objective is achieved by exactly reproducing the maximum height, slope, and curvatures at both ends of a flank of the curves. The reference curve to the smallest aspecular measuring angle serves for orientation. This is because, here, the deviations are easiest to recognize. These considerations are underlined by Fig. 6.16 for reference and reproduced color shown in Color plate 12. The reflection maximum at λ = 460 nm is caused by the blue pearlescent pigment, the shoulder at 560 nm by a yellow pearlescent pigment; the further shoulders at about 605 and 680 nm represent a red and brown absorption pigments, respectively. Notice the 10 times lower reflectance values in comparison with Fig. 6.15 – caused by the brown absorption pigment; it is impressive that the lightness and total color difference are ∗ = 1.35and ΔL∗ = 1.00, respectively, for μ = 15◦ and D65/10◦ . merely ΔEab as ∗ = 0.59 is scarce to perceive. The corresponding chroma difference of ΔCab Typical effect properties are controllable primarily by morphology, size, size distribution, and orientation of the particles. It is important to pay careful attention to these properties especially for high concentrations of metallic or silvery pearlescent pigments. These conclusions are relevant to water-based and refinishing paints as well as high-solid systems or high-viscosity plastics. Unlike colored pigments, effect pigments are almost totally dispersible in most binders or plastic materials. Some effect properties are, therefore, often uncontrollably altered especially transparency, hue, or even color strength; sometimes also color flop, metamerism, and color constancy are affected. With regard to these effects, all applied absorption and effect pigments, binders, or plastic materials as well as dispersion processes need to be appropriately coordinated for the desired effect. In this context, changes in process variables are also significant. The alignment and mechanical degradation of effect particles are mainly caused by

6.3

Realization of Recipes

439

R (%)

12.0

μas = 15°

Reference color Reproduced color

10.0

8.0

6.0

25°

4.0

45° 2.0 75° 110° 0 400

500

600

λ nm

700

Fig. 6.16 Angle-dependent reflection of a brown effect reference color represented in Color plate 12 and corresponding predicted color of non-corrected lightness

flow conditions. Inhomogeneous colorations can be caused by altered die geometries, dissimilar shear velocities, as well as temperature or pressure gradients. Changes of effect colors are also created by modified application

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methods such as pneumatic or electrostatic paint coating, different printing processes, or altered forming of press, injection molding, or extrusion installations. Information about all progression during stepwise recipe determination, whether positive or negative progress, should be retained – together with further recipe parameters – in the database for prediction of similar effect colorations in the future by the color expert. The range of influence of other technical parameters, in terms of a positive or negative effect, on the coloristic properties can be assessed additionally by using methods of fuzzy logic. This strategy also allows for the approximate determination of effect pigments contained in a coloration of unknown composition. The basic ideas of fuzzy logic go back to Zadeh, who expanded the familiar sentential logic in a simple way [23]. In the normal sentential logic, only two sharp sentential values are allowed, namely false and true (so-called binary logic). Both sentential values are arbitrarily assigned by digits: 0 for false and 1 for true. A relevant algebra is available for these sentential values. This logic is applied to design logic circuits, among other things. Compared to the normal binary logic, the fuzzy logic is based on diffuse or vague sentential values like rough, perhaps, virtually. Accordingly, the assessment of a sentential is characterized in fuzzy logic by an arbitrary numerical value – the so-called premise which lies inside the interval [0, 1]. The numerical values in fuzzy logic, which are not to be interpreted as probable values, have their own algebra. With this formalism, it is possible to obtain obvious decisions from imprecisely characterized premises like 0.53 or 0.85. The task of an expert now, among other things, is to judge correctly the fuzziness of the premises so that the resulting outcome corresponds, at least satisfactorily, with expectations. Furthermore, on the basis of microscopically found pigments, he/she has to solve the corresponding algebraic equations and to interpret the resulting premises. With regard to mixtures of effect pigments it is ultimately needed to determine if the given premises will result in a recipe for the required coloristical properties. This method has been applied with increasing success not only for coloristical questions with effect colorants but also for the solutions of economical and technical problems. Further details are given in [24, 25]. To summarize, in comparison to absorption colors, the prediction of effect colors at present requires substantially greater effort in using the classical method or database. The reasons for this are that typical effect properties partly change with particle size and also due to optical interactions of absorption and effect pigments. The precise optical adjustment of effect shades can be worked out after a first recipe by an expert using light microscopic and additional information technology. This is done mainly with modified expert systems, but also with classical correction methods.

6.3

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441

6.3.6 Neural Networks Normally, only one or just a few microprocessors are used to carry out classical algorithms for prediction and correction of color recipes, as well as for recipe search using digital databases. In comparison, a neural network consists of a greater number of series and parallel connected processors or computers of varying hierarchy and such a network can be used in special cases for recipe prediction. A neural network essentially serves as a model of connected nerve cells and processes information similar to the brain, although the neural processes of the brain are as yet not fully understood. It is clear, however, that the signals arriving a neuron (nerve cell of the brain) are transformed by localized synapses (places of stimulus sharing), by a manner of learn strategy, and forwarded to adjacent neurons. Instead of networked neurons, artificial neural networks consist of interconnected processors; the functions of the synapses are simulated by a suitably designed “learning software”. This procedure is used in particular for recipe determination of textile fibers [26]. For these reasons, we give a brief discussion. Among realized artificial neural networks one of the most frequently used structure is based on the learning algorithm called back-propagation model [27]. For this, microprocessors are connected in a series of layers. This structure consists of an input layer, several hidden layers, and followed by an output layer. Figure 6.17 shows an example of a simplified neural network of three layers. Each layer is directly connected with the next higher layer. In addition, each neuron of a layer is directly connected with all processors of the next higher layer, but not with neurons of its own layer. Each new layer following the input layer transforms the activity pattern of the preceding layer in a suitable manner. Therefore with back-propagation algorithm, each neuron is assigned to a continuous initial activity σ (x) with range 0 ≤ σ (x) ≤ 1. For this, in most cases the Fermi function σ (x) =

1 1 + e−x

(6.3.21)

with any real number x is used. A plot of the function is shown in Fig. 6.18. Equivalently, suitable fuzzy algebra can be implemented. The jth neuron of the hidden layer in Fig. 6.17 has an activity sj which, in dependence of σ (x), is given by the expression

sj = σ

N  i=1

Wij ei , j = 1, 2, ... , M,

(6.3.22)

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6 Recipe Prediction

Wij

Wjk

Input layer

Hidden layer

Output layer

Fig. 6.17 Simplified neural network with three layers with synapse strengths Wij and Wjk

σ (x) 1.0

0.5

–5

0

5

x

Fig. 6.18 Graph of the Fermi function σ (x), the approximated response function of a neuron

where ei denotes the activity of the ith input neuron, Wij stands for the coupling of the ith input neuron to the jth hidden neuron of the next layer. Wij is called synapse strength. The output neurons show an activity of

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⎛ ⎞ M  ak = σ ⎝ Wjk sj ⎠ ,

k = 1, 2, ... , L.

(6.3.23)

j=1

In this relation, Wjk denotes the synapse strength of the jth neuron of the last hidden layer and the kth neuron of the output layer. Each input pattern e, characterized by the vector e = (e1 , e2 , ... , eN )T ,

(6.3.24)

is assigned, therefore, to an output pattern a a = (a1 , a2 , ... , aL )T

(6.3.25)

by combining Equations (6.3.21), (6.3.22), and (6.3.23). Before the neural network can be used for the actual task, a great number P of input patterns et = (et1 , et2 , ... , etN )T ,

t = 1, 2, ... , P,

(6.3.26)

has to be trained onto the required output patterns at = (at1 , at2 , ... , atL )T .

(6.3.27)

For the above neural network of three layers, now the synapse strengths Wij and Wjk are sought under the condition that the training leads to the minimization of the function 1  [atk − ak (et )]2 . FN = 2 P

L

(6.3.28)

t=1 k=1

With regard to training for recipe prediction, the following details need to be taken into consideration. From each of the t = 1, 2, ... , P reference colors, i = 1, 2, ... , N spectral values Rex t (λi ) at wavelength λi are measured. These quantities correspond to the P input patterns eti : eti = Rex t (λi ),

i = 1, 2, ... , N,

t = 1, 2, ... , P.

(6.3.29)

Each input pattern (6.3.25) is propagated to the output pattern atk . These are assigned to the known concentrations ctk of the corresponding reference color: atk = ctk .

(6.3.30)

The propagation is repeated until the function FN in Equation (6.3.28) has reached a minimum. To achieve this, the synapse strengths Wij and Wjk are

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6 Recipe Prediction

altered stepwise by ΔWij = −γ

∂F ∂Wij

(6.3.31)

and new values Wijnew = Wijold + ΔWij are determined. With a suitably slow learning rate γ > 0, the change is carried out along the steepest slope of FN . In other words, the training of the widespread back-propagation algorithm comes back to the known minimization procedure outlined in Sections 6.2.2 and 6.2.3. Instead of optical constants and the phase function, synapse strengths Wij are determined. These have unknown physical significance but can be assumed variables of the function FN . Consequently, FN can have numerous local minima of unknown position and extent. Because of this, the gradient method can lead directly to some local minimum and never actually obtain the real absolute minimum. The determined synapse strengths are not always reliable quantities. Therefore the entire training period can be prematurely finished, before the neural network is prepared carefully and effectively for its original task. Concerning recipe prediction, this method seems to manage without any calibration series; in reality, systematically manufactured colorations are indispensable for the training period. Additionally, the requirements for a successful training concerning the number of colorations and concentrations of the color components are not known. Furthermore, the optimal number of layers and neurons per layer is not entirely clear. This method has the further disadvantage of not delivering any alternative recipes because combinatorics is not possible. Accordingly, this procedure is applicable only with reservation for recipe prediction. To conclude, we summarize that each kind of reference color is quite precisely predictable with one or more of the discussed numerical procedures. Table 6.8 gives a survey over the various prediction methods recommended for reference colors of different colorant types. Experience and success are decisive factors for the application of one of the given alternatives. Problems of modern industrial color recipe prediction can be successfully addressed by applying, Table 6.8 Colorants of a reference color and recommended recipe prediction methods Colorants of the reference

Recipe prediction method

Dyes

One-constant theory, two- or three-flux approximation; databases, neural networks

Absorption pigments

Two-, three-, and multi-flux approximations, databases

Effect pigments or mixed with absorption pigments

Multi-flux approximation, databases, modified expert systems

References

445

for absorption and effect colors, the appropriate radiative transfer approximations, several prediction and correction methods, as well as databases or expert systems with additional optical equipment. Looking back, from the rock and cave paintings with natural colors to the modern absorption and flake-shaped effect colorants, without doubt, there has been an impressive historical development. For the color industry, this was accompanied by sophisticated research and development as well as continually improved measurement and manufacturing techniques, process engineering, and application methods. The variety of non-self-luminous colors in modern civilization is based on fundamental properties of color physics which will – doubtlessly – be of ever greater importance in the color industry in future.

References 1. Park, RH, Stearns, EI: “Spectrometric formulation”, J Opt Soc Am 34 (1944) 112 2. Allen, E: “Basic equations used in computer color matching”, J Opt Soc Am 56 (1966) 1256 3. McGinnis, PH, Jr: “Spectrophotometric color matching with the least squares technique”, Color Engg 5 (1967) 22 4. Gall, L: “Computer Colour Matching”, Colour 73, Adam Hilger, London (1973) 153 5. Allen, E: “Basic equations used in computer color matching, II, tristimulus match, two constant theory”, J Opt Soc Am 64 (1974) 991 6. Kuehni, RG: “Computer Colorant Formulation”, Lexington Books, Lexington, MA (1975) 7. Eitle, D, Pauli, H: “Rezeptvorausberechnung unter Beruecksichtigung des Deckvermoegens”, XIV FATIPEC-Congress Budapest (1978) 209 8. Allen, E: “Colorant Formulation and Shading”, in: Grum, F, Bartelson, CJ, eds: “Optical Radiation Measurement”, Vol 2, “Color Measurement”, Academic Press, New York (1980) 290 9. McLaren, K: “The Colour Science of Dyes and Pigments”, 2nd ed, Hilger, Bristol (1986) 10. McDonald, R: “Recipe prediction for textiles”, in: McDonald, R, ed: “Colour Physics for Industry”, 2nd ed, Soc of Dyers and Colourists, Bradford (1997) 11. Lawson, CL, Hanson, RJ: “Solving least squares problems”, SIAM, Philadelphia (1995) 12. Wolberg, J: “Data Analysis Using the Method of Least Squares”, Springer, Berlin (2006) 13. Grossmann, W: “Grundzuege der Ausgleichsrechnung”, 3rd ed, Springer, Berlin (1969) 14. Press, WH, Teukolsky, SA, Vetterling, WT, Flannery, BP: “Fortran numerical Recipes”, Cambridge Univ Press, Cambridge UK (2008) 15. Klein, GA: “Farbenphysik fuer industrielle Anwendungen”, Springer, Berlin, Heidelberg (2004) 16. Sluban, B: “Comparison of Colorimetric and Spectrometric Algorithms for Computer Match Prediction”, Col Res Appl 18 (1993) 74 17. Marquardt, DW: “An algorithm for least squares estimation of nonlinear parameters”, J Soc Ind Appl Math 11 (1963) 431 18. Nobbs, JH: “Colour-match prediction for pigmented materials”, in: McDonald, R, ed: “Colour Physics for Industry”, 2nd ed, Soc of Dyers and Colourists, Bradford (1997) 19. Sluban, B, Nobbs, JH: “Colour Correctibility of a Colour Matching Recipe”, Col Res Appl 22 (1997) 88 20. Sluban, B, Sauperl, O: “A sensitivity model and repeatability of the recipe colour”, Croatic Chem Acta 74 (2001) 315

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21. Won, K: “Introduction to object-oriented databases”, MIT Press, Cambridge, MA (1991) 22. Lausen, G: “Datenbanken”, Elsevier, Spektrum Akademischer Verlag, Muenchen, Heidelberg (2005) 23. Zadeh, LA, ed: “Fuzzy logic for the management of uncertainty”, Wiley, New York (1992) 24. Nedjah, N, Mourelle, L: “Fuzzy systems engineering: theory and practice”, Springer, Berlin, Heidelberg (2005) 25. Spitzer, D, Gottenbos, R, van Hensbergen, P, Lucassen, M: “A novel Approach to Color Matching of Automotive Coatings”, Prog Org Coat 29 (1996) 235 26. Westland, S: “Advances in artificial intelligence for the colour Industry”, J Soc Dyers Col 110 (1994) 370 27. Priddy, KL, Leller, PE: “Artificial neural networks”, SPIE Press, Bellingham, WA (2005)

Appendix A.1 Non-colored Applications of Effect Pigments A.1.1 Metallic Pigments

Physical effect

Metallic pigment

Application

Mechanical

Coated aluminum flakes

Influence of impact strength and tensile strength of organic materials Inhibition of diffusion of gases, vapors, and corrosive mediums

Leafing pigments Thermal

Carbon black with Al flakes Elongated Al flakes, Ni powder, steel flakes Leafing Al flakes of high reflectivity TiO2 on mica Aluminum flakes <15 μm

Microwave heating Increase of thermal conductivity of organic materials Reflection of UV and IR radiation of translucent plastics Reflection of light and thermal energy Flame-retardant finishing of textiles

Optical

Aluminum flakes; TiO2 , SnO2 /SbO2 on mica Fine aluminum flakes

Laser marking of plastics and organic materials UV absorption in degradable high polymers X-ray absorption

Lead flakes Electric/ magnetic

Physicochemical

Fibers consisting of steel, silver, gold bronze, Cu, Al: Φ = 10–150 μm, l = 0.5–10 mm; Carbon black, Ni, Cu, Ag, steel, Zn; Al flakes coated with Ag; graphite, carbon fiber or mica coated with nickel Iron-III-oxide, chromium-III-oxide, Mica coated with iron-III-oxide Zinc powder, leafing zinc flakes, Al, Ni, steel Non-stabilized aluminum flakes, aluminum powder

Reduction of electrostatic charging as well as electrically conducting organic materials Avoidance of electromagnetic interference (EMI) of electric circuits

Magnetic domains in organic materials Magnetic domains in organic foils Lacquers or coatings: corrosion prevention of metals or metallic alloys In connection with H2 O: release of H2 as propellant for pore former in concrete: 2Al + 6H2 O = 2Al(OH)3 + 3H2 ↑

G.A. Klein, Industrial Color Physics, Springer Series in Optical Sciences 154, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1197-1, 

447

448

Appendix

A.1.2 Pearlescent Pigments Physical effect

Coating of mica substrate

Mechanical

TiO2

Increase of tensile strength of synthetic high polymers

Thermal

TiO2

Reflection of IR radiation of transparent plastics

Optical

TiO2 , (Sn, Sb)O2

Marking and welding of coatings or plastics with laser (gas laser with CO2 , excimer laser, solid-state laser Nd:YAG laser)

Electric

(Sn, Sb)O2 , Sn(O,F)2 , SiO2

Antistatic finish of foils and floor covering, electric conductive foils for clean rooms for dust reduction; electrostatic lacquer application of moldings

Magnetic

Co in γ-Fe2 O3 ; Fe3 O4 , Cr2 O3

Magnetic surfaces

Application

A.2 Chromatic Adaption Transform CAT02 The transform CAT02 needs to perform the following calculation steps in forward and backward modes:

A.2.1 Forward Mode Input data: Sample in test illuminant: X, Y, Z; Sample in reference illuminant: XR , YR , ZR ; Adopted white in test illuminant: XW , YW , ZW ; Reference white in reference illuminant: XWR , YWR , ZWR ; Luminance of test and reference adapted surfaces: LA (cd/m2 ). Output data: Standard color values in the corresponding illuminant Xc , Yc , Zc , Color-inconstancy index CCI (ΔE) in dependence on a color-difference formula ΔE. 1. Step: Determination of cone responses R, G, B in test illumination of absolute white in test and reference illuminants RW , GW , BW and RWR , GWR , BWR , respectively:

A.2

Chromatic Adaption Transform CAT02

449

⎛

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎞ R X XW RW ⎝ G ⎠ = MCAT02 ⎝ Y ⎠ , ⎝ GW ⎠ = MCAT02 ⎝ YW ⎠ , B Z BW ZW ⎞ ⎛ ⎞ ⎛ XWR RWR ⎝ GWR ⎠ = MCAT02 ⎝ YWR ⎠ ; BWR ZWR

(A.2.1)

the empirical coefficient matrix is given by ⎛

MCAT02

⎞ 0.7328 0.4296 −0.1624 = ⎝ −0.7036 1.6975 0.0061 ⎠ . 0.0030 0.0136 0.9834

(A.2.2)

2. Step: Calculation of the degree of adaption D: D = F · {1 − 0.01584 · e−[(LA +42)/92] };

(A.2.3)

if D is outside of 0 ≤ D ≤ 1, then  D=

1 for D > 1 . 0 for D < 0

(A.2.4a)

Furthermore, F is given as ⎧ ⎨ 1.0 for middle illumination F = 0.9 for dim illumination . ⎩ 0.8 for dark illumination

(A.2.4b)

The luminance LA of adapted field of sample and reference colors follows from  LA = (YH YWR ) · LWR ,

(A.2.4c)

where YH is the lightness of surround color and LWR for luminance of reference white of units cd/m2 .

450

Appendix

3. Step: Determination of the corresponding response in test illuminant: Rc = R · [1 + α · (RWR /RW ) − D],

(A.2.5a)

Gc = G · [1 + α · (GWR /GW ) − D],

(A.2.5b)

Bc = B · [1 + α · (BWR /BW ) − D],

(A.2.5c)

 α = D · (YW YWR ).

(A.2.5d)

where

4. Step: Calculation of the corresponding standard color values ⎛

⎛ ⎞ ⎞ Xc Rc ⎝ Yc ⎠ = M−1 · ⎝ Gc ⎠ CAT02 Zc Bc

(A.2.6a)

using the inverse matrix of Equation (A.2.2): ⎛

M−1 CAT02

⎞ 1.096124 −0.278869 0.182745 = ⎝ 0.454369 0.473533 0.072098 ⎠ . −0.009628 −0.005698 1.015326

(A.2.6b)

Determination of the color values, for example, Lc , Cc , hc and LR , CR , hR ; calculation of color difference ΔE by a suited color difference formula from color contributions ΔL = Lc – LR , ΔC = Cc – CR , and ΔH.

A.2.2 Reverse Mode In the reverse case, the standard color values in test illuminant X, Y, Z are found according to the following procedure: Input data: Standard color values of corresponding color Xc , Yc , Zc and remaining input quantities listed of forward mode. Output data: Standard color values in test illumination X, Y, Z. 1. Step: Calculation of

A.3

Two-Flux Approximations

451

⎛ ⎞ ⎛ ⎛ ⎞ ⎞ ⎞ Xc XW Rc RW ⎝ Gc ⎠ = MCAT02 · ⎝ Yc ⎠ , ⎝ GW ⎠ = MCAT02 · ⎝ YW ⎠ , Bc Zc BW ZW ⎛ ⎞ ⎞ ⎛ XWR RWR ⎝ GWR ⎠ = MCAT02 · ⎝ YWR ⎠ BWR ZWR ⎛

(A.2.7)

2. Step: The degree of adaption D is determined using Equations (A.2.3), (A.2.4a), (A.2.4b), and (A.2.4c). 3. Step: R, G, B are calculated according to Equations (A.2.5a), (A.2.5b), and (A.2.5c). 4. Step: The standard color values in test illumination X, Y, Z follow from ⎛ ⎞ ⎛ ⎞ R X ⎝ Y ⎠ = M−1 · ⎝ G ⎠ , CAT02 B Z etc.

A.3 Two-Flux Approximations A.3.1 Directional Fluxes

Quantity

From radiative transfer equation (5.1.37)

Optical constants

Characteristic roots

Optical path: τ = (K + S)d S Albedo: ω0 = K+S  Absorption coefficient: K = (1  − ω0 )τ d Scattering coefficient: S = ω0 τ d 1 √1 − ω k = ±μ 0

Translucent layer

Reflection Rμμ and transmission Tμμ : Rμμ =

ω0 2 − ω0 + 2μk coth (kτ )

Tμμ =

2μk (2 − ω0 ) sinh (kτ ) + 2μk cosh (kτ )

(A.2.8)

452

Appendix

Quantity

From radiative transfer equation (5.1.37)

Opaque layer

Rμμ =

ω0 , √ 2 − ω0 + 2 1 − ω0

(1 + Rμμ )2 1 = , ω0 4Rμμ

Tμμ = 0

(1 − Rμμ )2 1 − ω0 K = = S ω0 4Rμμ

Transparent layer

Rμμ = 0, Tμμ = e−τ/μ , ω0 = 0,

Conservative case

Rμμ =

Determination of optical constants over two different backgrounds Rbg,b , Rbg,w

τ = −μ ln (Tμμ )

 τ 2μ , Tμμ = , ω0 = 1, τ = 2μRμμ Tμμ 2μ + τ 2μ + τ

1 = ω0 (1 + Rbg,w )(1 + Rw )(Rb − Rbg,b ) − (1 + Rbg,b )(1 + Rb )(Rw − Rbg,w ) 4(Rbg,w Rb − Rgb,b Rw ) Rbg,b ≈ 0, Rgb,b << Rb : (1 + Rbg,w )(1 + Rw )Rb − (1 + Rb )(Rw − Rbg,w ) 1 = ω0 4Rbg,w Rb Rb ∼ = Rw : (1 + Rw ) 2 1 = ω0 4Rw   ω0 (1 + Rbg,x Rx ) − (2 − ω0 )(Rbg,x + Rx ) 1 τ= arcoth , 2μk 2μk(Rx − Rbg,x ) with x = b or x = w

Optical layer

Surface boundary corrected relations

Transparent

Ra = r +

Translucent

Ra = r + (1 − r)2 · Ta =

Opaque

(1 − r)2 rTμμ (1 − r)2 Tμμ Ra − r , T = , Tμμ = a 2 2 rTa 1 − r2 Tμμ 1 − r2 Tμμ 2 (1 − rRμμ )Rμμ + rTμμ 2 (1 − rRμμ )2 − r2 Tμμ

(1 − r)2 Tμμ 2 (1 − rRμμ )2 − r2 Tμμ

Ra = r +

(1 − r)2 Rμμ , 1 − rRμμ

Rμμ =

Ra − r , (1 − r)2 + r(Ra − r)

Ta = Tμμ = 0 Conservative case

S=

,

1 (1 − r)(Ra − r) − r(1 − r)Ta d (1 − r)2 Ta

A.3

Two-Flux Approximations

453

A.3.2 Diffuse Fluxes Quantity

From radiative transfer equation (5.1.37)

Optical path: τ = (K + S)d S Albedo: ω0 = K+S  Absorption coefficient: K = (1  − ω0 )τ d Scattering coefficient: S = ω0 τ d √ Characteristic roots k = ±2 1 − ω0 Optical constants

Transparent layer

Reflection Rdd and transmission Tdd :

1 Rdd = 0, Tdd = e−2τ , ω0 = 0, τ = − ln (Tdd ), 2 1 S = 0, K = − ln (Tdd ) 2d Translucent layer

Rdd =

ω0 H(k, − k,τ ) ω0 = 2 − ω0 + k coth (kτ ) (2 − ω0 + k)H(k, − k,τ ) + e−kτ

k (2 − ω0 ) sinh (kτ ) + k cosh (kτ ) 1 = (2 − ω0 + k)H(k, − k,τ ) + e−kτ

Tdd =

Opaque layer

Rdd =

ω0 , Tdd = 0 √ 2 − ω0 + 2 1 − ω0

1 (1 + Rdd )2 = , ω0 4Rdd Conservative case

Rdd =

τ 1 , Tdd = , ω0 = 1, 1+τ 1+τ

K = 0, S = Determination of optical constants over two different backgrounds Rbg,b , Rbg,w

1 − ω0 K (1 − Rdd )2 = = S ω0 4Rdd  τ = Rdd Tdd ,

1 Rdd d Tdd

1 = ω0 (1 + Rbg,w )(1 + Rw )(Rb − Rbg,b ) − (1 + Rgb,b )(1 + Rb )(Rw − Rbg,w ) 4(Rbg,w Rb − Rbg,b Rw ) Rbg,b ≈ 0, Rbg,b << Rb : (1 + Rbg,w )(1 + Rw )Rb − (1 + Rb )(Rw − Rbg,w ) 1 = ω0 4Rbg,w Rb Rb ∼ = Rw : 1 (1 + Rw ) 2 = ω0 4Rw   ω0 (1 + Rbg,x Rx ) − (2 − ω0 )(Rbg,x + Rx ) 1 τ = arcoth , k k(Rx − Rbg,x ) with x = b or x = w

454

Appendix

Quantity

Following Kubelka–Munk theory, Section 5.3

Optical constants

Absorption coefficient: KKM Scattering coefficient: SKM Combining with two diffuse fluxes from radiative transfer equation: KKM = 2K,

SKM = S

Characteristic roots

k = ±bS

Transparent layer

R = 0,

Translucent layer

R=

b 1 , T= a + b coth (kd) a sinh (kd) + b cosh (kd)

Opaque layer

R=

1 = a − b, T = 0, a+b

Conservative case

a = 1, b = 0, R = S=

Determination of optical constants over two different backgrounds Rbg,b, Rbg,w

a=

T = e−KKM d ,

1R dT

S = 0,

Sd , 1 + Sd

1 KKM = − ln (T) d

KKM (1 − R)2 = SKM 2R T=

1 , KKM = 0, 1 + Sd

(1 + Rbg,w Rw )(Rb − Rbg,b ) − (1 + Rbg,b Rb )(Rw − Rbg,w ) 2(Rbg,w Rb − Rbg,b Rw )

Rbg,b << Rb :

Rb − Rw + Rbg,w 1 w a= R + 2 Rbg,w Rb Rb ∼ = Rw :   1 1 a= Rw + w 2 R   1 − aRbg,x + (Rbg,x − a)Rx 1 S= arcoth , with x = b or x = w bd b(Rx − Rbg,x ) KKM = (a − 1)S

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Name Index

Names mentioned exclusively in the text:

B Beer, A, 299–300, 302, 306–307, 309, 320, 330, 340, 342, 346, 386 Bethe, H.A., 16 Billmeyer, F.W., 356 Boltzmann, L., 15, 306 Born, M., 302 Bouguer, P., 299 Brewster, D., 29–30 Brillouin, L., 32, 302 Burch, J.M., 126 C Carter, E.C., 356 Chandrasekhar, S., 7, 295, 344, 356, 360 D Dirac, P., 298, 314, 371, 400 E Eitle, D., 341 Euler, L., 286 F Fabry, Ch., 36–37, 86, 89–90 Fechner, G.T., 142 Fermi, E., 208, 441–442 Fraunhofer, J., 38 Fresnel, A.J., 27, 29–31, 333, 339–340, 348–349, 351, 354, 363, 367–368, 375 G Gauss, C.F., 282–286, 290, 357, 371, 374, 402 Goethe, J.W. von, 2 Grassmann, H.G., 2, 109, 116, 118–121 Greenstein, J.L., 303, 313–314, 399 Guild, J., 120–121, 125

H Helmholtz, H. von, 2, 114 Henyey, L.G., 303, 313–314, 399 Hering, E., 3, 49, 57, 114, 141 Huygens, Ch., 39 I Ishihara, S., 238 J Jacquin, F., 6 Jeans, J.H., 16 Joule, J.P., 305, 316 Judd, D.B., 114 K Kronecker, L., 371 Kubelka, P., 7, 182, 305, 326–327, 329, 331, 335, 337, 339–341, 343, 350, 354, 375–376 L Lambert, J.H., 235, 299–300, 302, 306–307, 309, 320, 330, 340, 342, 346, 386 Land, E.H., 3, 114, 156 Legendre, A.-M., 311, 313, 357–358, 400 Levenberg, K., 412 Littrow, K.L., 42 M Marquardt, D.W., 412 Maxwell, J.C., 2, 27–28, 302 McAdam, D.L., 139, 141–143 Michelson, A.A., 104 Mie, G., 7, 32, 302–304, 313 Mudget, P.S., 356–357 M¨uller, G.E., 114 467

468

Munk, F., 7, 182, 305, 326–327, 329, 331, 335, 337, 339–341, 343, 350, 354, 375–376 Munsell, A.H., 57–58, 141 N Nagel, W.A., 238 Newton, I., 2, 109 O Ostwald, W., 169, 171 P Pauli, H.K.A., 142, 341 P´erot, J.B., 36–37, 86, 89–90 Planck, M., 15–16, 23, 139 Plinius C. Secundus the E., 5 Poynting, J.H., 25, 296 Purkinje, J.E., 127 R Raman, Ch., 32, 302 Rayleigh, J.W., 16–17, 32, 81, 302 Richards, L.W., 356–357 Ryde, J.W., 340 S Saunderson, J.L., 336, 339–340, 354–355, 376

Name Index

Schroedinger, E., 3 Schuster, A., 305, 326, 343 Schwarzschild, K.S., 305, 326, 343 Snellius, W., Snel (Snell) van Royen, 26 Speranskaja, N.I., 126 Stiles, W.S., 126 Stokes, G.G., 274–275 T Taylor, B., 395, 398, 412 Theophilus, , 5 U Ulbricht, R., 243, 248, 250, 260, 262 W Weber, E.H., 142 Weiss, P.E., 108 Weizs¨acker, C.F. von, 16 Wien, W.K.W., 16 Wright, W.D., 121, 125 Y Young, Th., 2, 114 Z Zadeh, L.A., 440

Index

A Abrasion, 69, 193 Abridged spectrophotometer, 260 Absorption, 299–302, 393, 418 coefficient, 28, 49, 51–52, 299–300, 305, 325, 330, 332, 390, 394, 451, 453–454 pigments, 4, 5–6, 14, 26, 43–49, 53, 58, 64, 67–68, 73, 76–77, 80–81, 85, 89, 92, 98–99, 101–103, 128–129, 179, 185, 197–199, 201, 207, 214–216, 218–221, 238, 243, 248–249, 313, 315, 386–392, 396–399, 408, 423, 435–436, 438, 444 selective, 4, 13, 46–47, 50, 85, 102–103, 257 Acceptability, 255–256, 278 accuracy of, 278 certainty of, 283 of colorant, 161–162, 192, 234, 255–257, 278, 426–430, 435 of colors, 161–162, 234, 255–256, 278, 384, 427–430 of color strength, 185 error of, 162, 255–256, 278 relative, 95, 404 value, 287, 297, 324 Accuracy absolute, 259, 263, 266, 280 of optical constant, 332, 393 of recipe-prediction, 400–403 of spectrophotometers, 162, 263, 265 Achromatic axis, 142–143, 211, 214 color, 3, 53–54, 57, 59, 111, 116, 142, 186, 207, 215, 222 point, 138, 142, 144, 194, 210–211, 222, 227–228

Achromatopsy, 113 Acid fatty, 68 oleic, 68–69 palmitinic, 68–69 stearic, 68–69 Action time, 248 Activity continuous of neurons, 441–442 photo, 200 Adaptation chromatic, 164–167 degree of, 167–168 kinds of, 164–165 mesopic, 111, 123 parameter, 430 phase, 164–165 photopic, 111, 120, 123, 165 scotopic, 115, 128, 133, 176 See also Adaption Adaption chromatic, 116, 165, 238, 448–451 dark, 116, 165 light, 165 mesopic, 123 phase, 238 photopic, 128 transform, chromatic, 167, 448–451 See also Adaptation Adaptive shift, 165 Additive color mixing, 2, 11, 92, 103, 116–120 laws of, 2, 11, 92, 119–121 Additives, 44, 136, 138, 169, 171, 186, 192–193, 199, 215, 233, 249, 277, 279, 302, 385–386, 389, 401, 408, 422–423, 428, 434, 436 suitable, 69, 249 Adhesion, 199–200 Adjacent color, 58, 432, 434

469

470 Aerosol, 17 After-image, 114–115 Aging artificial, 193 effects, 18, 192, 198–199 process, 189, 192–193 See also Weather resistance Albedo, 309, 315–316, 318, 320–325, 331, 339, 344–346, 356, 375–376, 378, 383, 397–399, 412, 418, 451, 453 definition of, 309 entire, 397–398, 418 specific, 383, 418 Alternative recipes, 381, 390, 403, 406, 420–421, 444 Aluminum bronze, 66–67 cornflakes, 69, 74, 77, 92–93, 221, 400 flakes, 67, 73–74, 199, 219–220, 447 pigment, 5, 66, 78, 92–93, 199, 219–221, 391, 438 silver dollar flakes, 75, 80, 198 Amblyopy, 112, 238 Anastas, 47, 83–84, 90, 93, 97, 98 Anchor curve, 218–221 Angle azimuth, 144, 296, 302–303, 308, 312, 315, 358 blaze, 40–42 counting, 201, 242 of declination, 303, 317, 358, 400–401 specular, 26, 78, 83, 91, 95, 101, 201–202, 204, 222–223, 241–242, 252, 403 Anisotropic force, 108 Anisotropic scattering, 313, 357, 383 Anisotropy coefficient, 313, 399 degree of, 309 extreme, 313 Anti-reflection coating, 36 Anti-Stokes emission, 275 Aperture measuring, 193, 199, 249, 251, 277–278 modus, 156 Application, 1–3, 6–8, 11, 22, 36, 43–44, 50, 57, 63–64, 66, 69, 85, 103, 109, 114, 123, 156, 159, 167, 244, 260, 288, 306, 316, 324, 344, 377–378, 390, 406, 439 Approximation linear, 395, 406, 415, 431 multi-flux, 51, 295, 312, 376, 378, 444 See also Approximation, nth

Index nonlinear, 178 nth, 356–357, 370, 374, 377 See also Approximation, multi-flux of radiative transfer, 49, 356–379 three-flux, 312, 340–356, 361, 376, 378, 396–397, 399, 444 two-flux, 316–326, 328–329, 335, 338, 340, 343, 346–347, 353–354, 377, 444, 451–454 Arched surface, 249 Arithmetic mean, 61, 147, 154, 158, 193, 207, 281–282, 291, 386, 398 Artificial aging, 193 neural networks, 441 source, 18, 128 weathering, 192 Aspecular angle, 78, 95, 100, 201–202, 204, 222, 242, 252, 403 Assessment of color differences, 115–116 color inconstancy, 253 colors, 4, 11–12, 19, 23–24, 32, 63, 115, 126, 136, 155–156, 159–160, 164–165, 207, 238, 253, 403 effect pigments, 26, 103, 403 fluorescence, 20, 253 ASTM standards, 455–456 Attenuation coefficient, 28, 299, 309, 327 Autocollimation configuration, 42 Auxiliary function, 148, 344, 346 Auxochromes, 45 Average flux, 342 Axial symmetry, 303, 310, 341 Axiom, 33 Azimuth angle, 144, 296, 302–303, 308, 312, 315, 358 B Background black, 91, 93–102, 186, 189, 211, 215, 218, 222, 267, 272, 325, 331–332, 392 color/colored, 50–51, 81, 89, 91, 100–101, 185, 252, 325, 331 differently colored, 331 light, 252 white, 91–92, 100, 186, 189, 191, 268–269, 389, 391, 420 Back propagation algorithm, 441, 444 Back propagation model, 382, 441 Backscattering, 53, 235, 248, 400 Baking finishes, 198 Ball mill, 69, 72

Index Band spectrum, 19–20 BCRA II ceramic, 237, 260, 264–265 Beer-Lambert law of absorption, 299–302 generalized, 300, 306, 309 multi-dimensional, 307 one-dimensional, 320, 330, 340, 342, 346 Best recipes, 421, 435 Binary logic, 440 Binder, 5, 26, 27, 44–45, 50, 60, 64–65, 67, 69, 109, 136, 159, 162, 171, 176, 186, 189, 192, 198–200, 225, 238, 248–250, 252, 256, 267, 277, 279, 302–304, 332–333, 373, 392–394, 396, 398, 401, 403, 418, 428, 432, 434, 438 Blackbody radiator, 14–16, 18, 139–140, 236 Black mixture, 77, 176–178, 387–389, 391–392 Black pigments, 45, 47, 86, 177–178, 181, 387, 389, 408 Black standard, 236, 264–265 Black trap, 236 Blaze angle, 40–42 Blaze grating, 43 Blaze technique, 40, 43, 105, 223 Blending ratio, 177, 392 Blends, 62, 66, 177, 386–387, 389, 392 Blinding glare, 156 Blindness, 113 Blistering, 193, 199, 249 Blooming, 61, 254 Blow molding, 70 Boltzmann constant, 15 equation, 306 Bouguer-Lambert law, 299 Boundary condition, 310, 316–317, 328, 342, 344, 360–361, 370–371 Boundary-layer correction, 332–333 Boundary surface, 25, 27–29, 31–32, 50–51, 59, 60, 64, 82, 333–334, 336–338, 340–341, 345, 348–351, 354, 357, 361–367, 373 corrections, 336–338, 340–341, 345, 354, 357, 362–363 Brass, 5, 67, 198 Brass pigment, 438 Brewster angle, 29–30 Bright-field illumination, 87, 93, 215, 392, 407, 436

471 Brightness, 15, 52, 72, 77–78, 81, 107, 201, 403, 407–408, 437 Brilliance, 57, 65, 67–70, 75, 77, 81, 83, 89, 96–98, 100–101, 103–104, 153, 197, 199–201, 222, 226, 228, 238, 248, 271, 273, 423, 435–437 high, 69–70, 97, 226 low, 436 metallic, 65, 68, 75, 197, 201 Brilliant colors, 6, 82–83, 95, 97, 99, 156, 159 Bronze, 5, 66–67, 90, 95, 447 Bronzing, 61 Brookit, 93 Build loci, 406 Build-up, 176–179, 200, 210 C Calibration of colorants, 427 colors, 386, 389–392, 396, 402–403, 435 errors, 402 of interference pigments, 392 metallic pigments, 391 methods, 381 pearlescent, 410 procedure, 263, 402–403, 428 samples, 56, 386, 390, 393–394, 398, 402–403, 431 series, 45, 383–387, 390–394, 396–398, 402, 444 temporary, 262, 264 wavelengths, 263 CAM, see Color appearance model (CAM) Candela (cd), 24 definition of, 18 Carbon-arc lamp, 18, 192 Carbon black, 47, 101, 102, 179, 329, 386, 391–392, 408, 436, 447 CAT, see Chromatic adaptation transform (CAT) CAT02, 167–168, 448–451 Cavitation, 271 CCD camera, 204–205, 272 Chalking, 193, 248 Chemical vapor deposition (CVD), 68, 73, 85 Cholesteric phase, 84, 88 texture, 88–89, 277 Chroma azimuthal anisotropy, 224 curves, 200–201, 210–212, 214, 217, 221–228 difference, 146, 151, 159–160, 438 maximum, 222–223

472 Chroma (cont.) natural, 248 potential, 211 Chromatic adaptation, 164–167 colorant, 159, 178 colors, 24, 33, 57, 115–116, 142–144, 193, 200, 207 pigments, 54, 176–178, 219, 273 shift, 165–166 Chromatic adaptation transform (CAT), 167 Chromaticity, 58, 121, 125, 137–141, 156, 159, 162, 184, 195, 290, 389, 403 coordinates, 136–140, 194–195 diagram, 58, 137–141, 159, 162, 184, 234 Chromophores, 45, 332 CIE, see International Commission on Illumination (CIE) CIE94 formula, 158 CIE 1931 color space, 143 CIE 1931 standard observer, 125–129 CIE 1931 system, see CIE 1931 color space CIE 1964 standard observer, 125–129 CIE 1964 system, 125–129 CIE 1976 color spaces, 141–148 CIE 1976 system, see CIE 1976 color spaces CIEDE2000 formula, 152, 157–160 CIELAB color differences, 159 color values, 157, 174, 206, 222 differences, 159, 193, 198 space, 143–144, 152 system, 3–4, 49, 57–59, 141, 143–144, 147–149, 152, 159 values, 161, 174 CIELUV system, 141 CIE reference conditions, 61, 149, 155–156, 158, 238, 250, 253 CIE standards, 458 Classical recipe prediction, 381, 390, 392, 403–404, 417, 420–421, 435 advantages, 421 benefits, 390 Clear coat, 201, 254, 392 Coarseness, 77, 79, 201 Coat blooming, 61, 254 Coating, 36, 39, 44, 59, 62, 64, 67–70, 73, 76, 79, 82, 85, 90–91, 94, 97–98, 101–102, 106–107, 109, 162–163, 185–186, 191, 195, 198–200, 202, 205–206, 208–209, 225, 249, 262, 271–273, 308, 358, 408, 423, 436, 440, 447–448

Index anti-reflection, 36 covering, 91, 101, 186, 199 CVD method, 85 effect, 102, 199, 205, 209, 408, 448 multi-layered, 36 paint, 62, 76, 162–163, 208, 408, 440 of plastics, 79 PVD method, 70, 80, 85, 110 reference, 209, 272 thickness, 90, 94, 98, 185 Coefficient of absorption, see Absorption, coefficient of reflection, 29–32, 60, 253–254, 333–334, 336–337, 340, 348–349, 351, 353–354, 364, 375, 378 See also Fresnel coefficient of scattering, see Scattering, coefficient Co-extrusion, 89 Coherent light, 14, 104 Color achromatic, 3, 53–54, 57, 59, 111, 116, 142, 186, 207, 215, 222 adjacent, 58, 432, 434 aging, 192 amblyopy, 113, 238 anomaly, 113 assessment, 11, 20, 23, 115, 126, 156, 165 asthenopy, 113 attributes, 43, 51–57, 59, 67, 76, 141, 146, 161 blindness, 113 brilliant, 6, 82–83, 95, 97, 99, 156, 159 calibration series, 45, 383–387, 389–394, 396–398, 402, 444 chromatic, 3, 24, 33, 57, 115–116, 142, 144, 159, 165, 178, 193, 200, 207 constancy, see Color inconstancy corresponding, 35, 43, 57, 59, 119–120, 124–125, 138, 142, 153, 161, 166–168, 174, 176, 187–188, 193–194, 227, 266, 329, 382, 386, 414, 424, 434–435, 450 dark, 53, 56–57, 60, 140, 144, 156, 159, 162, 178, 182, 261, 329, 424 fluorescent, 247, 274–277 impression, 1, 3, 11–13, 18, 23–24, 35, 43, 52–53, 59, 61, 63, 67, 81, 89, 91–95, 99, 103, 107, 115, 116–117, 119, 120, 133, 136, 155–156, 159, 161–162, 164–166, 170, 179, 227, 233–234, 249–250, 253–254, 258 inconstancy, 155, 161, 164, 166–169, 172, 253, 260, 448

Index inconstancy index, 161, 166–168, 172, 260, 448 laws of mixing, 11, 92, 102–103, 110, 119–121 matching, 3, 23, 120–127, 134, 182, 194, 234, 252–255, 257, 278–279, 382, 403–404, 410 natural, 2, 43, 53, 57, 65, 81, 91, 103, 110, 177, 191, 214–215, 218, 387, 445 See also Mass tone nonlinear, 155, 292, 384 non-self-luminous, 1, 11–12, 14, 18, 25, 32, 43, 51–53, 57, 86, 119, 123–124, 134, 136–137, 141, 145, 162, 237, 445 non-uniform, 93, 145, 156 pale, dirty, dull, 53 pastel, 207 perception, 1, 3, 11, 13, 58–59, 110–118, 126, 141, 156, 164, 238, 250, 266 photochromic, 274–278 production, 11, 25, 34, 43, 45–47, 63–64, 82, 85, 110, 274 properties, 2, 7, 44, 58, 104, 136, 175, 185, 197, 209, 257, 422 pure, 53 reproduction of, 171–172, 178, 184, 266, 384, 404, 421, 428–429 sensation, 2–4, 8, 11–12, 32, 35, 51, 63–64, 110, 113, 115–116, 121, 165, 170–171, 235 shade, 49, 91, 116, 125, 129, 162, 189, 214, 386, 408 synthesized, 43, 110 thermochromic, 13, 249, 257, 265, 274–278 uncolored, achromatic, 3, 53–54, 57, 59, 111, 115–116, 121, 134, 138, 142–144, 159, 162, 177–178, 184, 186, 193–194, 207, 210–211, 214–215, 222–223, 226–227, 265–266, 319, 433 See also Colors Colorant matrix, 49 Colorants absorbing, see Color tolerance, absorption colorants of batches, 171, 401, 431, 434 calibration of, 382–390, 392–393, 402, 427–428 concentration, 52, 170, 179, 185, 200, 250, 276, 384–386, 388, 404, 406, 413, 415, 418, 425, 433 mass tone, 92, 178, 388, 392

473 mixtures, 118, 219, 244, 273, 315, 394, 421, 426, 431 modified, 110 particle size, 66–67, 70, 74–82, 101–104, 106, 181, 215, 271, 392, 423, 440 selection of, 172, 407, 422–423, 435 substitution of, 4, 169, 184, 266, 269–270, 273 synthesized, 110 Color appearance, 12, 25–26, 83, 133, 141, 146, 152, 156, 159–161, 166–167, 209, 258, 420 Color appearance model (CAM), 133, 141, 146, 152, 156, 159–161, 167, 258, 420 CIECAM02, 152, 160–161, 167 Color assessment, 11, 20, 23, 115, 126, 156, 165 Coloration, 26, 33, 43–45, 49–51, 57, 62–64, 79–81, 92, 101, 102, 119, 125, 127–128, 133, 137, 152–153, 159, 162, 164, 166–167, 169–170, 179, 185–189, 192–194, 197–200, 204–207, 209, 218, 228, 236, 238, 241, 247–250, 252–255, 257, 259–261, 265–267, 272, 278, 292, 315, 326–327, 329, 356, 381, 386, 389–393, 398, 402–403, 407–408, 412–413, 423, 427–429, 431, 434–436, 440, 444 effect, 26, 62, 80, 152, 162, 200, 204–206, 209, 241, 252–254, 261, 272, 326, 393, 407–408, 429, 434–436, 440 opaque, 252, 257, 267, 390, 412 potential, 179, 386, 389 reference, 198, 206, 252, 265, 390–393 Color attributes, 43, 51, 59, 67, 76, 141, 146, 161 Color blindness blue-yellow, 113 complete, 113 green, 113 red, 113 Color build-up, 179, 200, 210 Color capability of colorists, 253 Color change(s), 6, 13, 15, 24, 34, 62, 81, 155, 156, 164, 166, 192–195, 201, 211, 213, 274, 277–278, 420 nonlinear, 156 phototropic, 13, 257 thermochromic, 13, 274, 277

474 Color collection, 59, 153, 159–161, 167, 207, 238, 250, 255, 266 Color component, 36, 60, 120, 136, 423–424, 444 Color constancy, 49, 114, 133, 156, 164–166, 184, 199, 417, 422, 424, 431, 438 Color contrast, 115 simultaneous, 114–115 Color depth, 52, 179–181, 183–184 equal, 180 standard, 181, 183–184 Color development, 43, 176, 201, 216, 222, 228, 292 diffraction pigment, 222–228 effect pigment, 43, 201, 216, 292 nonlinear, 292 Color difference formulas, 133, 152, 156–157, 159–161, 168, 175, 206, 436 angle-dependent, 200, 209 CIE94, 152, 154–156, 158–160 CIEDE2000, 152, 157–160 CIELAB, 142–143, 145, 149, 152–153, 159 CMC, 153–154, 159–160 DIN99o, 160–161 metallic pigments, 206 Color differences acceptable, 255–256 amounts of, 424 assessment of, 249 balanced, 417–420 CIE94, 152, 154–156 CIEDE2000, 152, 157–160 CIELAB, 149–150, 158–159 CMC, 153, 159–160 contributions, 144, 147, 152–153, 159–160, 193, 207, 256, 265–266, 288–290, 292, 417 DIN99o, 159 discernable, 209, 255–256 equations, see Color difference formulas mean, 289, 291, 403, 406, 436 minimal, 162, 422 numerical, 121, 145, 153, 158, 167, 403 perceived, 57–58, 114, 120, 145, 152–153, 160 smallest, 162, 183, 255–256, 403, 405, 417 smallest visually discernable, 162, 255–256 tolerated, 149, 156, 207 Colored layer, 7, 29, 32–33, 60, 189, 250–251, 267, 295, 308, 316, 324

Index Color effects, 6, 34, 38, 64–65, 68, 76, 99, 102–103, 116, 192, 435–436 Color equation, 119–121 Color fastness, 20, 49, 172, 175, 189, 192–197, 254, 401, 422 Color flop, 62, 63, 80–81, 85–87, 89, 92, 97, 99, 102, 207, 214, 238, 249, 423, 436, 438 distinct, 85–87, 89, 97 Colorfulness, 44, 52–53, 58, 161 Colorimetric correction method, 429 measurement, 215 method, 3–4, 164, 175, 405, 414–417, 429 procedure, 133, 185, 404, 414, 429, 434 strategy, 405, 411, 424, 426 Colorimetrical-matching criteria, 183 Colorimetry, 2–3, 8, 11, 23–24, 103, 114, 124, 133, 152, 161, 185, 285 Color impression, 1, 3, 11–13, 18, 23–24, 35, 43, 52–53, 59, 61, 63, 67, 81, 89, 91–93, 95, 99, 103, 107, 114–116, 119–120, 136, 155–156, 159, 161–162, 164–166, 170, 179, 227, 233–234, 249–250, 253–254, 258 angle-dependent, 162, 179 different, 3, 18, 170 entire, 52, 59, 119 quantification of, 136, 233 three factors of, 61, 155, 234 trichromatic, 159 visual, 11, 23, 254, 258 Color inconstancy, 155, 161, 164, 166–169, 172, 253, 260, 448 degree of, 167 index of, 161, 166–168, 172, 260, 448 Color index (C.I.), 58 Coloring body, 138 Coloring potential, 176–179 Coloring power value, 191 Color intensity, 52 Colorist, 4, 43, 45, 49–50, 57–58, 61, 76–82, 102, 125, 144, 159, 163, 172, 178–179, 184–185, 198, 201–206, 209, 228, 381, 384, 390, 403–404, 406–407, 412, 414, 422, 428–429, 434, 440 Coloristic assessment, 61 Coloristic properties, 4, 43, 45, 76–82, 440 Color locus, 140–143, 145–147, 149–151, 159, 162, 177, 184, 186, 215–216, 219, 225, 256, 265, 431–433

Index Color matching, 3, 23, 120–123, 125–128, 134, 182, 194, 234, 252–255, 257, 278–279, 382, 403–404, 410 visual, 234, 253, 257 Color-matching functions (CMFs), 121–122, 126–127, 134, 257, 404, 410 Color-matching values, 121, 124, 127 Color measurement, 60, 62, 92, 118, 120, 206, 234, 239, 247–249, 251–253, 256–257, 265, 275, 277–292, 427, 432, 434 double meaning, 234, 292 uncertainty of, 233–234, 252–253, 256, 263–265, 271–272, 278–281, 292, 412 Color mixing, 2, 11, 92, 102–103, 117–120, 214 Color opponent theory, 49, 141 Color order systems, 43, 56–59, 238 colorimetric, 57–58 coloristic, 57–58 Color pattern, 11, 24, 43, 50, 57, 58–59, 61–63, 109, 136–137, 141, 149, 156–157, 160, 164, 175, 195, 233, 234, 236, 239, 244, 246–248, 250, 252–255, 257, 261–263, 266, 277–280, 288–289, 292 Color perception, 1, 3, 11, 13, 58–59, 110–118, 126, 141, 156, 164, 238, 250, 266 normal, 112, 266 subjective, 59 Color plane, 49, 141–144, 147, 149–151, 154, 158, 160, 176–177, 207, 210–211, 222, 236 CIELAB, 49, 143 DIN99o, 148–152 Color plate, 49, 58, 85, 87, 89, 93, 95, 99, 118, 121, 137–138, 140–144, 154, 184, 214–215, 273, 393, 436–439 Color production, 11, 25, 34, 43, 45–47, 63–64, 82, 85, 114, 274 interference pigments, 64 pearlescent, 34, 82 Color receptors, 112–114, 118, 122, 167 Color recipe correctability of, 178, 421, 423–427, 431 correction of, 243, 427–431, 441 prediction, 170, 184, 187–188, 244, 246, 250, 259, 316, 423, 444 sensitivity of, 178, 421, 423, 424–427 technical handling of, 427 Color rendering index, 21, 24, 161 Color reproduction, 184, 266, 384, 421

475 Colors absorption, 34, 43, 45, 53, 56, 61, 86, 91–92, 95, 97, 99, 133, 136–137, 142–144, 148, 152–153, 155, 161–162, 166–168, 172, 175, 197, 207, 266, 275, 401, 422, 428–429, 433, 435, 440 achromatic, 3, 53–54, 57, 59, 111, 115–116, 142, 144, 159, 186, 193–194, 207, 215, 222 additive mixing of, 110, 117–120 adjacent, 58, 434–435 amblyopy, 113, 238 assessment of, 4, 11, 12, 14, 19, 23–24, 115, 126, 136, 155–156, 159, 164–165, 178, 207, 238, 253 attributes of, 24, 43, 51–57, 59, 67, 76, 141, 146, 161 brilliant, 6, 82–83, 95, 97, 99, 156, 159 chromatic, 3, 24, 33, 53–54, 57, 59, 111, 114–116, 123, 133, 142, 144, 159, 165, 178, 186, 193, 200, 207, 215, 222 clear, 53, 55, 56 corresponding, see Corresponding colors dark, 53, 56, 57, 60, 140, 144, 156, 159, 162, 178, 182, 261, 329, 424 effect, 4, 59, 63, 153, 162, 200, 206, 391, 403, 407, 421, 428–429, 435–436, 440 filter, 257 fluorescent, 274–276 impression, 1, 3, 11–13, 18, 23–24, 35, 43, 51–53, 59, 61, 63, 81, 89, 91, 99, 103, 107, 115–117, 119–120, 133, 136, 155, 156–159, 161–162, 164–166, 170, 179, 227, 233–234, 249–250, 254, 258 interference, 34, 36, 45, 64, 82–87, 89, 90–91, 93–96, 102, 104, 179, 209–211, 213, 252, 400 light, 19–20, 48, 53, 56, 117, 119, 140, 166, 194, 237 metameric, 169–170, 174, 259 natural, 2, 43, 53, 57, 65, 81, 91, 103, 109, 177, 191, 214, 218, 445 non-self-luminous, 1, 11–12, 14, 18, 25, 32, 43, 51–53, 57, 86, 118, 123–124, 134, 136–137, 141, 145, 162, 237, 445 pale, dirty, dull, 53 pastel, 207

476 Colors (cont.) perception of, 1, 3, 11, 13, 58–59, 110–120, 126, 141, 156, 164, 237–238, 250, 266 photochromic, 274, 278 production of, 11, 25, 34, 43, 45–47, 63–64, 82, 85, 109, 274 pure, 53 saturated, 121 sensation, 2–4, 8, 11–12, 32, 35, 51, 110, 112, 114–115, 121, 165, 170–171, 235 synthesized, 43, 109 thermochromic, 13, 249, 257, 265, 274–278 See also Color Color saturation, 102, 115, 278 Color sensation, 2–4, 8, 11–12, 32, 35, 51, 110, 112, 114–115, 121, 165, 170–171, 235 Color sense simulation of, 134 subjective, 171, 235 see also Human color sense Color shade, 49, 91, 115, 125, 129, 162, 189, 214, 386, 408 Color shift, 45, 60, 81, 98, 156, 165, 167–168, 189, 195, 198, 425 Color space CAM, 161 CIELAB, 141–148 CIELUV, 141 CIE x, y, Y, 133–136, 141–142, 184 DIN99o, 148–152 Euclidian, 152, 158–159 UCS uniform color scales, 58, 141, 144 visual non-uniform, 292 Color stimulus, 2, 115, 117–126, 128, 133–134, 142, 166, 170, 233 function, 125, 170 specification, 119–122, 123, 142 Color streaking, 248 Color strength, 44, 52, 133, 172, 175, 179–181, 183–185, 324, 401, 438 definition of, 179 relative, 52, 180–181, 183–185 Color systematic, 92 Color temperature, 16–18, 20–21, 23–24, 139, 261 correlated, 16, 21, 24, 139 daylight, 17 definition of, 16 evening light, 21, 23

Index similar, 16 true, 24 Color tolerance, 141, 152–153, 155, 157, 159, 161–163, 171, 233, 255–257, 266, 278, 426 absorption colorants, 162 acceptable, 161–162, 426 agreement, 141, 161–162, 171, 233, 255, 257 effect colorations, 162 requirements, 163 Color travel, 102–103, 214–215, 252 Color triangle, 58, 137–138 Color value, 1–2, 24, 109, 118–121, 123–129, 133–148, 150, 152, 156–157, 161, 166–168, 170–171, 173–176, 183–184, 186, 193, 200, 206, 210, 222, 233, 239, 243–244, 246, 248–249, 253, 257–259, 261–265, 281, 284, 287–290, 292, 295, 384, 403, 405, 414–420, 424–426, 448, 450–451 of CIE 1931 system, 120, 125–127, 134–137, 174, 184 of CIE 1964 system, 125–127 of CIELAB system, 157, 174, 206, 222 of DIN99o system, 150, 414 of systems, 133–134, 141, 233 Color vision, 2, 110–113, 118, 156, 165 defective, 112 normal, 113 Combination pigments, 85, 86, 91, 93, 97–99, 200, 212, 273 Combinatorics, 407, 434, 436, 444 Community Reference Bureau (CRB), 236 Comparability of results, 175 Comparison of colors, 52, 162 Compatibility, 3, 44, 49, 86, 102–103, 172, 401, 423, 434 of components, 423 Complementary color, 45, 48–49, 51, 53, 83, 91–92, 94, 98, 102, 103, 115, 179, 267, 316 Computation of adjustment, 190–191, 409 Concentration, 7, 45, 58, 75, 102, 170, 172, 176–181, 185–186, 188, 200, 219–221, 250, 276, 292, 299, 304, 316, 324, 381–389, 392–394, 396, 398, 404–407, 409–416, 418–421, 424–427, 429, 431–435, 438, 443–444

Index adjustments, 429–430, 434 colorant, 52, 58, 170, 179, 185, 200, 250, 276, 384–386, 388, 404, 406, 413, 415, 418, 425, 433 dependence, 176, 178–179, 180, 200, 219–220, 292, 324, 388 differences, 410–412, 414–416 mass related, 385 pigment, 220, 434 predicted, 429, 434 ratio, 181, 186, 188, 435 variation, 421, 425–427 volume related, 385 Condensation water resistance, 199 Cones, 7, 16, 88, 110–112, 122, 156, 165, 312, 358–359, 378, 400, 404 of light, 7, 312, 358–359, 378, 400 S, M, L types, 111 types in retina, 111, 167 Confidence, level of, 286–287, 290 Conformation, molecular, 87, 88, 429 Conservation of energy, 78, 309, 320–321, 330, 346, 365 Conservative case, 309, 316, 319–322, 325, 329–331, 336, 338, 344, 346, 354–355, 369, 372, 374, 396, 452 definition of, 309 Constructive interference, 6, 35–37, 82–83 Contrast ratio, 185–186, 189, 250, 252, 332 Contrast substrate, 186, 191, 391 Cornflakes, 69–70, 72, 74–75, 77–78, 80, 92–93, 198, 221, 273, 400, 408, 436, 438 leafing of, 80 light scattering of, 69 pigment, 70, 74, 93, 400 Correctability absolute, 164–165 definition of, 424 entire, 426–427 of measuring, 263 of recipe, 178, 421, 423–424, 426–427, 431 Corrected optical triangle, 337–339, 353–355, 357, 371, 377–379 Corrected reflection, 350, 365–366, 368–369, 371, 373–376 Corrected reflection function, 365–366, 368 Corrected transmission, 350, 352, 366–367 Corrected transmission function, 366–367 Correction algorithm, 384–385 classical, 382, 440 colorimetric method, 414–417

477 computation, 430, 434 method, 382, 390, 402, 429, 430–431, 440, 445 operations, 429 of recipe, 402, 427–431, 434 spectrometric method, 430 steps, 383, 429–430 Correlated color temperature, 16, 21, 24, 139 Corresponding colors, 35, 57, 59, 118, 123–124, 138, 142, 153, 161, 166–168, 174, 187, 193–194, 227, 266, 382, 414, 424, 435, 450 Corresponding illuminant, 135, 167, 405, 420, 448 Covariance, 284–287, 289 Cover, 81, 185, 227, 237, 252 layer, 227 Covering capacity, 22, 44, 50–51, 77, 81, 96, 104, 175, 185–189, 201, 204, 238, 260, 267, 272–273, 278, 295, 408 film thickness, 185–186, 188 layer, 102, 186, 252, 267 potential, 184 power, 133, 191 spreading rate, 175, 185 value, 185–188, 267 Cracking, 193, 248 Crazing, 193 Criterion, 126, 155–156, 162, 166, 181–183, 197, 209, 389, 414, 420, 422, 428 of adjustment, 127, 181 hiding, 428 quality, 166 Critical angle, see Total reflection Crosslinking, 65, 109, 225 Cross section at fracture, 105–106 of grooves, 39–40, 105, 222–223 scattering, 301 Curve fitting, 190 Cuvette, 270–271 CVD, see Chemical vapor deposition (CVD) D Damage of colors, 198, 274 Dark adaptation, 165 See also Scotopic vision Dark-field illumination, 87, 215, 272–273, 392, 407, 435–436 Databases, 8, 381–382, 390, 392, 421, 427, 431–436, 440–441, 445–445 for recipe prediction, 381–382

478 Daylight bluish white, 21, 24 cold white, 20–21 middle, 21, 155, 192, 261 natural, 11–12, 165 simulator, 24, 155, 192 Decay constant, 278 Declination angle, 302, 308, 315, 359 Degradation, 193, 197–200, 224, 438 of aluminum flakes, 198 artificial, 200 natural, 200 Dehydration of polymeric materials, 199 Delta function, 298, 314–315, 371, 400 Density function, 283–284 Dependent scattering, 302, 304–305 Depth of color, 179, 183–184, 324 equal, 180 standard, 181, 183–184 Destructive Interference, 35–36 Detergents, 192 Deviation mean, 281 squared, 182, 281, 285, 394, 404 standard (SD), 281–282, 288, 290 Deviation ellipses, 403 Dichromasy, 113 Dielectric mirror coating, 36 Difference acceptable, 427 of chroma, 146, 151, 157–159, 193, 438 of hue, 146, 153, 157–159, 193 lightness, 80, 141, 146, 157, 198, 200, 202, 420, 438 unacceptable, 427 Diffracted light, 40, 273 Diffraction, 38–43 color, 34, 209, 228 effect, 6, 226 grating, 38, 260 intensity of, 108 laws of, 39–41 maxima, 39–41 of optimized pigments, 39 orders, 39–40, 222–224, 226–228, 241 particles, 39, 106, 223 pigments, 4–6, 14, 34, 38–39, 43, 62, 64–65, 80–85, 95, 102, 104–110, 201, 209, 222–228, 241, 243, 252, 260, 314–315, 408 spectrum, 39, 43, 223, 260

Index Diffuse flux, 196–197, 298, 305, 320, 326–327, 333–334, 336–337, 339–341, 348, 376–379, 454 geometry, 215, 244, 247 illumination, 30–32, 79, 93–94, 98, 107, 109, 205, 210, 226–227, 239, 243–244, 248, 252, 254, 272, 362–363, 372, 374 measuring geometries, 243–246, 252–253 radiation, 7, 31, 298, 327, 334, 340, 345, 348, 350, 354, 377–378 reflection, 60, 62, 64, 76–77, 235, 328, 334, 342–343, 345, 348, 350–351, 353, 362–363, 369–370, 372 transmission, 60, 62, 64, 77–78, 247, 328, 345, 348, 351, 363 Diffuse-diffuse geometry, 247 radiation, 377–378 reflection, 343, 351, 363, 370 transmission, 343, 351, 363, 369–370 DIN99 color space, 148 DIN99o, 148–152 color difference formula, 160 color space, 52, 133, 148–152, 170 color values, 150, 414 system, 148, 264 DIN color chart, 58 DIN standards, 163, 241–242, 458–459 DIN EN ISO standards, 459 Diode, 14, 20, 258 Dirac delta function, 298, 314, 371, 400 Directed-diffuse reflection, 342, 345, 348, 350, 353, 362, 369 transmission, 342, 345, 348, 363 Directed flux, 337, 341 Directed radiation, 297, 348, 377 Directional flux, 337, 339–341, 343, 351–352, 354, 451 geometry, 240–242, 261, 390 influx, 350 measurement, 210, 243, 252, 254 Discharge lamp, 19–20, 49, 222 helium, 237 mercury vapor, 19–20, 192 sodium vapor, 23–24, 237 xenon, 20, 49, 222 Discrimination of color difference, 186

Index ellipse, 158 threshold, 165 Dispersing process, 431 Dispersion of light, 26, 65–67, 84, 251, 408, 438 Distinctiveness of image (DOI), 77, 82, 199–201, 204, 271–272, 403, 423, 435 measuring of, 271 measuring uncertainty of, 272 value of, 82, 200, 204 Distraction effect, 156 Distribution function, 283–284, 286, 315 DOI, see Distinctiveness of image (DOI) Domains, magnetic, 108, 447 Dosage uncertainty, 425 Dosing, 178, 412, 423–424, 427–428, 431 Double bond structure, 48 Dry film thickness, 185, 385, 392 Dual-beam spectrophotometer, 259, 261 Dull colors, 44, 53, 56, 92, 156 pigments, 56 surface, 244 Dullness, 52 Dust, 17, 68, 197, 249, 448 Dyes, 1, 5, 14, 22, 43–45, 48–49, 58, 154, 176–179, 181, 183–184, 247, 261, 264, 382, 384, 386–387, 429, 444 calibration series, 384, 386 fluorescence, 48–49 natural, 22, 48 E Echelette grating, 40–43 Edge loss, 249, 280, 348 peeling, 193 phenomenon, 249 reflection, 249 scattering, 75–76, 103, 215 Effect, characteristics, 403 Effect coating, 205, 209, 408, 448 Effect colorants, 6, 8, 13, 22, 52, 63–64, 197, 209, 214, 228, 243–244, 271, 273, 292, 312, 314, 379, 381–384, 392, 407, 421, 423, 427, 431–432, 436, 440, 445 Effect colors, 4, 59, 63, 152–153, 162, 200, 206, 391, 403, 407, 421, 428–429, 435–436, 440, 444–445 Effect lacquers, 408 Effect line, 211–214, 216–223

479 Effect pigments, 4–7, 26, 34, 43–44, 49, 51, 53, 57, 62–103, 133–228, 234, 239, 244, 247–249, 252, 257, 259–260, 271–273, 292, 302, 312, 337, 357, 382, 384, 390, 392, 399–400, 403, 407, 409, 413, 423, 428, 434–436, 438, 440, 447 climate resistance of, 200 color physical properties, 62, 197, 200, 206, 434 definition of, 63 morphology, 197, 271–273 recipe determination of, 434–435 size, 74, 271–273, 436 See also Effect colorants Efflorescence, 193 Efflux, 318 Electromagnetic wave, 12, 15, 25, 43, 109, 116, 296 Electron gas, 65 spin, 107–108 wavelength, 274 Electrostatic charging, 249, 447 Ellipsoid, 141–142, 153–154, 157, 159–160, 285–286, 288–292 three-dimensional, 153, 285 Emission of light, 16, 19, 22 stimulated, 22 Emulsion paint, 50, 59, 102, 254 Energy conservation law, 33, 235, 338, 354 Energy flux density, 25 Energy release, 277 Entire correctability of recipe, 426 Entire sensitivity of recipe, 425–426 Environmental aspects, 424 Equality method, 238, 257 Equienergy spectrum, 121–122, 125, 138 chromaticity of, 121 Equipment metamerism, 171 Error absolute, 282 first kind, 288 gross, 279, 401 mean, 281, 402 probable, 282 propagation, 264, 402 random, 279–280, 282, 402, 428 recipe prediction, 428 relative, 190, 264–265, 282 root-mean-square, 282 of second kind, 288, 292

480 Error (cont.) systematic, 249, 256, 279–280, 401–402, 428–429 See also Errors Error-free value, 280, 282–283 Error propagation Gaussian law of, 402 Errors gross, 279, 401 random, 279–280, 282, 402, 428 stochastic, 280 systematic, 249, 256, 279–280, 401–402, 428–429 Estimate of concentration, 406, 413 of error, 190–191, 270 non-covering film thickness, 187 Estimation of errors, 326 of uncertainty, 280 Etalon, 37, 86, 89 Etching, 38, 60 Evening light, 21, 23 Exchange force, 108 Excitation energy, 19, 277 Excluded Optical region, 339 Expert system, 8, 221, 274, 381–382, 390, 393, 421, 427, 431, 435–436, 440, 444–445 modified, 381–382, 393, 427, 431, 435–440, 444 Exposure test, 16, 200 See also Aging Extended interference films, 84, 89 Extinction, 49, 77, 81–82, 181, 261, 277, 423, 436 coefficient, 181 of fluorescence radiation, 49, 277 of hue, 57, 77, 81–82, 423, 436 Extrusion, 70, 89, 440 F Fabry–P´erot etalon, 86, 89, 90 Fabry-P´erot interferometer, 36–37 Factor method, 429–430 Fastness tests absorption colorants, 48, 197 after effects, 197 of color, 20, 254 of effect colorants, 197 Fatty acids, 68 Fermi function, 208, 441–442 Ferromagnetic

Index diffraction pigments, 109, 225–228 domains, 108 material, 107–108, 225 state, 107–108 substrate, 107, 109, 201 Fibers of carbon, 467 of metals, 467 of paper, 287, 349 of textile, 49, 159, 247–248, 254–255, 278, 387, 429, 441 Filter, 37, 83, 116–117, 227, 257–259 Fingerprint of colored pigments, 178 of effect pigments, 95, 249 of pigments, 95 Fire colors, 67–68, 74 First paint coating, 162–163, 208 First recipe, 403, 421, 427–429, 432, 436, 440 absorption colorants, 409 absorption colors, 435 accuracy of, 409 correction of, 427 effect pigments, 409, 436 off-shade formulation, 427–429 Flake degradation, 198, 200, 224, 438 glitter, 70, 72–74, 77 pigment, 63, 74, 76–77, 93, 400 shear stable, 70, 198 Flake-shaped pigment, 4, 62–65, 63, 68, 76, 79, 84–90, 85, 197, 206, 228, 271, 337, 445 Flip, see Flop Flocculation, 76 Flooding, 61 Flop, 62–63, 77–82, 85–90, 92, 97, 99, 102, 104, 153, 179, 198, 200–204, 207, 214–215, 238, 249, 252, 271–273, 357, 403, 407–408, 423, 435–436, 438 behavior, 271, 273, 408 bright, 436 character, 80 colors, 204, 215 See also Color flop -control pigments, 408 correction, 436 dark, 79, 202 index, 81, 198, 200–204, 272 light, 80 See also Lightness, flop

Index Florida test, 200 Flow conditions, 76, 439 front, 76 line, 76 structures, 247 temperature, 66 Fluorescence colorant, 20, 236, 276 dye, 48–49 emission, 45, 100, 238, 275–277 extinction, 49 light, 23, 165, 277 radiation, 45, 277 spectrum, 276 stimulation, 49, 275–278 Fluorescent brightener, 194 Fluorescent lamp, 14, 20–21, 23 Flux, 25, 297 density, 25 diffuse, 196, 298, 305, 320, 326–327, 333–334, 336–337, 339–341, 348, 376–379, 453–454 directed, 337, 341 inward, 316 outward, 240, 316 partly-directed, 348 Fogging, 193 Foreign absorption, 277 Form factor, 65 Forward mode, 448, 450 Forward scattering, 303–304 Fraunhofer diffraction, 38 Fresnel coefficient, 333, 339, 349, 351, 354, 363, 367–368, 375 Fresnel equations, 27, 29 Frost effect, 81 Fuzzy algebra, 441 Fuzzy logic, 421, 440 G Gas discharge, 237 Gas discharge lamp, 19 Gas discharge tube, 14 Gaussian distribution, 283–284, 286 Gauss quadrature formula, 357, 359, 371, 374 General radiative transfer equation (GRTE), 306–307 Geometrical optics, 12, 25, 27, 64, 239 Geometric mean, 205, 207 Geometry bidirectional, 241, 244, 258 circular, 241, 244, 258

481 diffuse, 215, 244, 247 diffuse-diffuse, 247 directional, 240–242, 261, 390 metamerism, 175 normal-diffuse, 247 spherical, 246, 280 variable directional, 241–242, 261 See also Measuring geometry Glass of borosilicate, 89, 90 inorganic, 50, 83, 104, 247 organic, 50, 195, 247 transition temperature, 67, 92, 198 Glint, 72, 201 Glitter effect, 70 See also Sparkle Glitter flakes, 70, 72–74, 77 Glittering, 22, 70, 72, 77, 82, 423 Glitter particles, 198 Gloss, 4, 22, 59, 61–62, 64, 67, 76–77, 81, 86, 89, 92, 103, 189, 193, 206, 244, 247, 249, 253–255, 436 metallic, 4, 64, 67, 76–77 specular, 59, 244 subtraction, 247 of surface, 22, 61, 206 Gloss meter, 62, 81, 253 Gold bronze, 66–67, 447 Gold flitter, 5 Gonio spectrophotometer, 222, 237, 241–244, 247, 253, 263, 271–272 Gradient, 300 of curve, 189 descent, 409 method, 444 shear, 225 of velocity, 68 Graininess, 77, 79, 200–201, 204–206, 271–272, 287 Grain quality, 206 Granulate, 248 Graphite, 408, 447 Grating constant, 38, 105–109, 201, 222–227 crossed, 225 geometry, 38, 221, 222 linear, 223–224, 225 normal of, 41–42 patterns, 105 period, 38–39 structure, 6, 14, 39, 65, 221

482 Gray, 52–53, 55–56, 61, 65, 67, 69, 75–78, 86, 92–93, 115–116, 145, 155–156, 193–194, 197–198, 205–206, 215, 262, 274, 389, 391 dark, 55–56, 78, 206 light, 55, 65, 92–93 middle, 55, 61, 115, 145, 155 Gray level, 205, 274 Gray scale, 52, 193–194, 198, 205, 237, 262 linearity, 237 rating, 193–194, 198 Grinding, 68, 70, 75 Groove cross section, 222–223 distance, 42, 224 geometry, 105 Gross errors, 279, 401 GRTE, see General radiative transfer equation (GRTE) H Hall milling method of, 68–69, 74 Hametag milling method of, 68 Haze, 193, 195–197 index, 195–197 Hematite, 83, 86, 95–96, 210 layer, 210 Hiding criterion, 389, 420 Hiding layer, 101, 392 Hiding power, 44, 81, 103, 185, 324 See also Covering, capacity High polymers, 44, 50, 70, 76, 83, 243, 247, 333, 447–448 crystalline, 50 degradable, 447 synthetic, 44, 83, 448 See also Plastic materials High solid lacquers, 243 paints, 304 systems, 438 High-viscosity, 438 Historical recipes, 382, 390, 421, 431–432, 434 absorption colorants, 432 complete data record, 432 for recipe prediction, 432 Homogenization, 423 Homogenizing, 248 Hue angle, 143, 144, 146–147, 149, 154, 158, 161, 177, 184, 209, 289, 291

Index contribution, 147, 152, 157, 289, 291 dark, 77, 249 difference, 146, 153, 157–159, 193 extinction, 57, 77, 81–82, 423, 436 Human color perception, 1, 3, 11, 13, 18, 58, 109–114, 117, 141, 164, 238–239, 266 Human color sense, 11, 64, 103, 109, 114–116, 134, 139, 164, 169, 235, 238 Human eye, 17, 56, 110, 116, 126, 128, 135, 140 optical components of, 110 Humidity, 192, 199–200, 237–238, 250, 265, 278 Hygroscopic polymers, 198 Hypothesis, statistical, 287–289 I Illuminant, 414, 417–420 A, 18–19, 21, 23–24, 135 C, 138, 140 change, 115, 164, 167, 171, 174 corresponding, 135, 167–168, 405, 420, 448 D65, 19–21, 23–24, 155, 210–211, 222, 253, 258, 261, 405 energy distribution of, 115, 404 fluorescence, 20–21, 23–25, 61, 100, 165, 253, 276 reference, 166–167, 170, 172–174, 261, 448 spectral power distribution, 12, 23, 123–124, 136, 155, 170, 173, 253 standard, 18–21, 23–24, 140, 155, 171, 186, 210–211, 222, 253, 258, 261 test, 161, 166–168, 172–174, 448, 450 Illuminant/observer combination, 168, 186, 193 Illumination bright-field, 87, 93, 392, 407, 436–437 changes, 24, 165–166, 173, 200 dark-field, 87, 93, 99, 205, 273, 392, 407, 435–436 diffuse, 30–32, 79, 93–95, 98, 105, 107, 109, 205, 210, 225–227, 239, 243–244, 248, 252, 254, 272, 362–363, 372, 374 directional, 32, 60, 95, 210, 215, 225–227, 252, 272, 362, 372 metamerism, 171–172, 175 modus, 156 Incident energy, 13, 33, 365 Incident flux, 318, 342, 349, 362–363 Incident intensity, 17, 40, 196, 299, 301, 361

Index Incident light, 25, 27, 32, 39, 116, 227, 236, 250, 258, 260–261, 302, 314, 320–321, 366 Index of color inconstancy, 161, 166–168, 172, 260, 448 of color rendering, 24, 161 of metamerism, 172–173, 175, 260, 405 Indicatrix, 59, 69, 244 Influx, 43, 103, 192, 226, 239, 318, 323, 346, 347, 350–352, 360–361, 377 diffuse, 347, 350–351, 354 directional, 350 energy, 192 total, 352 Injection molding, 70, 247–248, 440 Inorganic pigments, 44–47, 73, 178, 408 colorants, 44, 178 dyes, 22 glasses, 83, 178, 408 Intensity, 296 definition of, 296 reflected, 62 scattered, 301–302 transmitted, 196, 351 Interface reflection, 237, 254–255, 270 Interface surface, 333 Interference color, 45, 83, 86, 90, 93, 95–96, 98, 102–103, 179, 209–211, 400 constructive, 6, 35–37, 82–83 destructive, 35–36 filter, 37, 83, 227 laws of, 82 of LCP, 80, 84, 87 of light, 34–38 line, 211–215, 217–221 lithography, 105 order, 35–36, 82, 89 of OVIP, 86 pigment, 4, 34–36, 38, 45, 64, 68, 80, 82–87, 89–93, 95–96, 98–99, 101–104, 179, 197–199, 201, 204, 209–217, 386, 390–392, 407–408 pigment substrates, 90 of plane parallel layer, 34–35 of transparent pigments, 91–92, 396 wavelength, 36, 89, 93–94, 100 Interferometer, 36–37 Inter-instrument correspondence, 264, 266 Internal energy, 305 International Commission on Illumination (CIE), 3, 18–21, 23–24, 43,

483 58, 61, 109, 120, 125–128, 134–139, 141–149, 154–161, 163, 173–175, 184 chromaticity diagram, 58, 137–139, 184 color space, 43 illuminant, 21, 23–24 reference conditions, 61, 149, 155–156, 158, 238, 250, 253 simulator, 21, 23 source, 23 standard illuminant, 19, 21, 23–24 standards, 414 x, y, Y system, 137, 141 X, Y, Z values, 128, 133–134 See also CIE Interpolation, 261, 347, 382, 435 Intrinsic absorption, 276 IR radiation, 13–14, 69, 192, 447–448 range, 13, 15, 192, 253, 274 Ishihara color panels, 238 ISO standards, 461 Iteration, 184, 187, 384, 412–414, 417–419, 429 K Kinetic theory of gases, 305 Kries coefficient law, 167 Kronecker symbol, 371 Kubelka–Munk function, 182, 329 Kubelka–Munk theory, 335, 341, 343, 350, 354, 376, 454 limited applicability, 337, 340, 376 L Lacquers, 44, 49–51, 61, 65, 76, 102, 152–153, 159, 162, 243, 247–249, 274, 385–386, 408, 447 Lambert–Beer law, 299–300, 302, 306–307, 309, 320, 330, 340, 342, 346, 386 of absorption, 307, 309, 330, 340 generalized, 300, 306 multi-dimensional, 307 one-dimensional, 320, 330, 340, 342, 346 Lambert law, 235 Lamp fluorescent, 14, 20–21, 23 gas discharge, 19 incandescent, 14, 117, 118 metal halide, 23 sodium vapor, 23–24, 237 three-band, 20–21 tungsten arc, 18 tungsten filament, 18–21, 261 xenon, 19–21, 23, 258–262

484 Laser, 13–14, 22, 34, 74, 104, 447–448 advantages, 22 classification, 22 granulometry, 74 light of, 34 pump sources, 22 Lateral inhibition, 115 Law of additive color mixing, 2, 11, 92, 118–120 of energy conservation, 33, 235, 354 of error propagation, 402 of large numbers, 282 of radiation, 15–16, 23, 139 of refraction, 26–29, 35, 365–366, 368 Layer chromatic, 49, 255 colored, 7, 29, 32–33, 60, 189, 250–251, 267, 295, 308, 316, 324 covering, 102, 186, 252, 267 opaque, 316, 319, 324, 329, 336, 345–346, 354–355, 370, 374–376, 452–454 plane parallel, 34–35, 76, 251, 255, 296, 299, 308–313, 316–317, 326–327, 357, 363–364, 367 translucent, 83, 91, 251–252, 267–268, 324, 340, 347, 368, 451, 453–454 transparent, 50, 197, 316, 319–320, 336, 353, 369, 452–454 LCP, see Liquid crystal pigment (LCP) Leaches, 192 Leafing, 68–69, 77, 79–80, 199, 271–273, 408, 447 flakes, 69, 79 stability, 199, 271 Learning software, 441 Legendre coefficients, 311, 313, 400 Legendre polynomials, 311, 357–358 Light adaptation, 165 booth, 20, 61–62, 238, 253 circularly polarized, 88 coherent, 14, 104 collimated, 26 colors, 19–20, 48, 53, 56, 116, 118, 140, 166, 194, 237 cones, 7, 312, 358–359, 378, 400 diffraction of, 104 directed, 299, 316, 336–337, 349 directional, 26, 28–30, 32, 59, 72, 195, 204, 250 dispersion of, 26, 65 efficacy, 18, 21, 24 emission, 12, 20, 22–23, 258

Index flop, see Lightness, flop flux, 121 interactions, 7, 11, 28, 32, 43–44, 51, 59–60, 82–83, 235, 253–254, 306, 308, 326 polarization, 25–26, 29–31 polarized, 29, 88–89 sources, 2, 11–129 visible, 12–13, 117, 273–274 wave nature of, 14, 65 Light emitting diode (LED), 14, 20, 201 Light microscope, 87, 271–274, 382, 407, 435–436 Light microscopic analysis, 273, 435 Lightness, 1–3, 45, 52–53, 57–58, 62–63, 77–82, 110, 113, 115–116, 121, 125, 127, 129, 135, 137–142, 144, 146, 148, 151, 153, 156–158, 161, 165–166, 169, 176–179, 183–184, 186, 195, 198–205, 207, 209, 222, 236, 238 axes, 138, 158, 178 constancy, 166 contrast, 115 difference, 80, 141, 146, 157, 198, 200, 202, 420, 438 flop, 62–63, 77–82, 179, 200–202, 272, 423 scale, 45 Light path, 79, 239 reversibility of, 27 Light scattering, 16, 51, 56, 69, 91, 224 See also Scattering Light source, 2, 11–129 monochromatic, 116, 118, 121, 166 Light transmission, 184, 250, 255, 257 Light trap, 244–246, 254, 259 Line spectrum, 19–20 Liquid crystalline phase, 87 polymer, 277 texture, 87 Liquid crystal pigment (LCP), 26, 29, 50, 84, 87, 89, 93, 99–101, 211, 391 Liquid crystal structure, 6, 38 Load factor, 189, 420 Long-term drift, 262 repeatability, 265 reproducibility, 263–265, 280 Low-bake paints, 206, 209 Lumen (lm), 21, 24 definition of, 24

Index Luminance, 18, 110, 118, 121, 165, 168, 235, 448–449 Luminescence colorant, 13, 275 lamp, 19 pigments, 45 radiator, 14, 16, 19, 34, 139, 155 source, 24 Luminous efficiency, 126, 128 flux, 18, 24 Luster decrease, 198 of pearls, 6 M Macro-cracks, 248 Magnetic elementary dipoles, 108 Magnetic field, 25, 65, 108–109, 225 Marbleizing effect, 243 Mass coloration, 44 Mass tone, 91–92, 177–178, 387–389, 391–392 See also Natural color Master batch, 431, 434 Matching colorimetrical, 183 of colors, 3, 23, 120–123, 125–129, 134, 182, 194, 234, 252–255, 257, 278–279, 382, 403–404, 410 criteria, 181 experiment, 120–123 spectrophotometric, 183 stimuli, 121 trichromatic, 123 See also Color matching Material opaque, 51, 239, 244, 255, 296, 319, 322, 324, 332, 336, 340, 345–346, 353, 376 purely absorptive, 321 science, 184, 192 translucent, 51, 189, 193, 246, 251–252, 267, 346, 396, 412 transparent, 134, 189, 191, 197, 237, 240, 246, 250–251, 267, 313, 322, 325, 331, 338, 346, 396 Mathematical statistics, 279, 292 Maxwell equations, 27, 302 Mean color difference, 289, 291, 403, 406, 436 deviation, 281

485 lateral particle dimension, 65, 82, 87 particle size, 72, 74, 106, 181, 302, 392 Measurement, 19, 60, 69, 95–98, 120, 124, 136, 153–154, 186, 191, 193, 196, 204–206, 210, 218, 233–292 anomalies, 279 aperture, 193, 199, 237, 249, 251, 277–278 deviations, 403, 437 errors, 136, 279–280, 288 geometries, 234, 239–248 of opaque material, 239 of reflectance, 257 of reflecting material, 234–257 result, 233, 239, 246, 248, 251, 253–254, 274, 278–280, 282 series, 253, 279, 289 spectrometric, 49, 249 of transmittance, 237, 240 of transmitting material, 234–257 of transparent material, 240, 246, 251 uncertainty of, 234, 252, 256, 263–264, 271, 278–279 Measuring angle, 51, 78, 81, 95, 100, 198, 200–201, 204–208, 210–211, 219, 222–223, 227, 241–243, 261, 272, 390, 400, 413–414, 437 aspecular, 78, 96, 100–101, 198, 202–203, 206, 217, 242, 437–438 Measuring geometry, 94, 97–99, 210–211, 239, 241–242, 244, 246–247, 249–250, 252–254, 260, 277, 310, 326, 362, 435 diffuse, 243–246, 252–253 directional, 240–243, 259 Measuring instrument, 13, 20, 33, 171, 193, 199, 233–237, 246, 249–250, 253, 257–258, 272, 279–280, 288, 428 Measuring plane, 226–227 Measuring sensitivity, 262 Measuring uncertainty, 233, 253, 265, 272, 280, 292, 412 Measuring wavelengths, 260, 278 Mechanical degradation, 224, 438 Mercury spectrum, 20 vapor lamp, 19–20, 192, 237, 275 Mesomeric phase, 87 Mesopic vision, 110, 123 Metal effect value (MV), 200–202 Metal flakes, 5, 65, 67–68, 70, 72–73, 90 See also Metallic flakes Metal gloss, 67 Metallic brilliance, 65, 68, 75, 197, 201

486 Metallic character, 69, 74, 76–77, 198 Metallic colors, 67, 209 Metallic flakes, 6, 67, 76–78, 81–82, 100–101, 202, 219, 315 Metallic gloss, 4, 64, 76–77 Metallic paints, 209 Metallic pigments, 4–6, 28, 34, 57, 62, 64–73, 75–82, 101–103, 197–198, 201–204, 206, 209, 236, 241, 252, 272, 314–316, 337, 356, 358–359, 386, 391, 407–408, 436, 438, 447 brilliance, 77, 201, 438 classification of, 67 color difference formula, 206 colored, 67, 69, 73 light resistance, 198 non-colored applications, 69, 447 Metallics, 70, 202, 206–209 Metallic value, 81, 201–202 Metal-oxide, 210 Metal powder, 75 Metameric colors, 169, 259 pairs, 169, 171 Metamerism, 49, 133, 161, 164–175, 184, 238, 253, 260, 404–405, 409, 414, 417, 422, 424, 427, 431, 438 appearance of, 170 definition of, 169 degree of, 169, 172–173, 424 index, 172–173, 175, 260, 405 kinds of, 169–172 minimal, 172 reduction of, 427 Michelson configuration, 104 Micro-brightness, 72 Micro-crack, 248 Microhardness, 189, 193 Microprocessor, 233, 258, 261, 382, 421, 441 Microscope bright-field illumination, 87, 215, 273, 392, 407, 435 dark-field illumination, 87, 215, 273, 392, 407, 435 optical magnification, 273–274 resolution capability, 273–274 scanning electron (SEM), 69, 74, 271, 274 Middle daylight, 21, 155, 192, 261 Mixer machine, 198 Mixing of colors, 109, 116–120 Mixing ratio, 20, 178, 218

Index Mixtures of absorption pigments, 77 black, 177, 387–389, 391–392 of effect colorants, 214 of metallic and absorption pigments, 220–221, 391 white, 177, 387, 389 white/black, 77, 176, 178, 387 Modified expert system, 381–382, 393, 427, 431, 435–440, 444 Molecular configuration, 88 nematic, 88 smectic, 88 See also Texture, cholesteric Monochromatic illumination, 275–276, 278 light, 20, 115–116, 118, 121, 138, 166 wavelength, 22, 138 Morphology, 62, 64, 66, 69–76, 79, 83, 103, 197, 215, 271–273, 302, 435, 438 diffraction pigments, 64 interference pigments, 64, 103 metallic pigments, 64, 76–82, 438 pearlescent pigments, 64, 438 Mottling, 76, 199 Multi-flux approximation, 51, 295, 312, 376–378, 444 Multi-flux theory, see Multi-flux approximation Multi-layer coatings, 5, 36 films, 84, 90 particles, 214 pigments, 6, 86, 91 Multiple scattering, 7, 295, 302–306, 318–319 N Nagel anomaloscope, 238 Nano-engineering, 104 Nanoparticle, 81, 200 Nano-titanium dioxide, 80, 408, 436 National Institute of Standards and Technology (NIST), 236 Natural color, 2, 43, 53, 65, 81, 91, 103, 109, 177, 191, 214–215, 218, 387, 445 See also Mass tone Natural color system (NCS), 57–58 Nematic phase, 88 Neural network, 381–382, 390, 421, 441–444 Neuron, 441–444 Night light, 166

Index Non-leafing, 68–69, 77, 80, 408 Non-uniform color space, 145 Normal distribution, 283–287 See also Gaussian distribution Normal equations, 416–417 Normal vision, 110, 126 North skylight, 17, 238 Notch effect, 224 Null hypothesis, 287, 289, 292 Numerical recipe prediction, 7–8, 295, 324, 428 O Object modus, 156 Observation angle, 34–35, 62, 78–80, 82, 85, 95, 98–100, 122–123, 126, 205, 209, 215, 228, 254, 316, 391, 408 condition, 149, 155, 161, 168 Observer 2◦ observer, 122, 125–128, 135, 137–138, 194 10◦ observer, 126–127, 129, 135, 137–138, 156, 194 geometry metamerism, 171, 175 metamerism, 171–172, 238 Off shade, 162 Off-shade formulation, 427–429 One-constant theory, 396, 444 Opacity, 77, 352 pigments, 408 Opaque, 6, 49–51, 82, 91, 101–104, 134, 186, 189, 191, 212, 218, 239, 244, 247, 250–252, 255, 257, 267, 296, 307, 316, 319–322, 324, 329, 331–332, 336, 338, 340, 345–346, 348, 353–355, 367–368, 370, 374–376, 387–388, 390, 392, 396, 412, 452 color, 134, 186, 239, 244, 250–252, 267, 387 coloration, 252, 257, 267, 390, 412 films, 101–104 layer, 316, 319, 324, 329, 336, 345–346, 354–355, 370, 374–376, 452–454 pigments, 82 Operating point, 323 Opponent color theory, 3, 49, 57, 114, 141 Optical anisotropy, 225 coating, 36 coefficients, see Optical, constants constants, 33, 186, 252, 316, 319, 322, 324, 326, 328, 331–333, 344, 376,

487 379, 381–386, 389–393, 395–396, 399, 402–404, 413, 419, 432, 444, 451–454 contact, 222, 252, 255, 267–269, 331, 347–348, 355–356, 367, 389 Optically anisotropic, 87, 241, 252 Optical path, 14, 17, 34, 37, 308–310, 312, 315–316, 318, 320–325, 331, 339, 341–342, 346, 356, 358, 364, 383, 397–399, 412, 418–419, 451, 453 difference, 37 entire, 397, 418–419 length, 14, 34, 308–310, 321, 323, 325, 342, 346 specific, 383, 398, 418 Optical region, 339 excluded, 339 Optical roughness, 72, 77, 79 See also Graininess Optical shielding, 254 Optical triangle, 295, 316, 321–324, 327, 330, 336–339, 345, 347, 352, 354–355, 357, 363, 369–370, 377–379 corrected, 337–339, 353–355, 357, 371, 377–379 modified, 355 nth approximation, 377 Optical variable interference pigment (OVIP), 84, 86–87, 90, 408 Optic chiasma, 113 Optic nerves, 111, 136 crossing over, 114 Optic tract, 113 Organic colorants, 44, 178 dyes, 22 glasses, 50, 193, 195 pigments, 44, 50, 67, 179 solvents, 192 Orientation distribution, 205, 315 of flakes, 76, 79, 82, 243, 436 of particles, 75, 80, 108, 243, 438 OSA-UCS system, 58 See also Color space, UCS uniform color scales Outdoor exposure, 16, 200 weathering, 192 weather stability, 199 Outer mixture, 121–122 OVIP, see Optical variable interference pigment (OVIP)

488 P Paint coating of automotives, 408 electrostatic, 440 first, 162–163, 208 metallic, 62, 163 refinishing, 162–163, 208 Paint refinishing, 162–163, 209 Parent population, 256, 281–282, 284, 292 Partial absorption, 13 Partial derivatives, 395, 410–412, 414–420, 426, 429 Partially absorbed, 299 Partially directed flux, 341 Partially oxidized, 67, 73, 90, 199 Partially reflected, 351 Partial waves, 37 Particle alignment, 76, 227 morphology, 271 packing, 304 size distribution, 66, 70, 74–78, 80–81, 92, 100, 215, 248, 271, 391–392 Pastel colors, 207 Path difference, 37, 39 Path length, optical, 14, 34, 308–310, 321, 323, 325, 342, 346 Pauli extension, 142 Pearlescent pigments, 6, 64, 80–85, 95, 107, 200, 209, 215, 436, 438, 448 Pearl essence, 6, 83–84 Pearl luster, 64, 82–85, 89, 91, 93, 95, 101–103, 197, 199–201, 209–210, 212, 214–215, 218–219, 221, 436 effect of, 98, 103 Pellets of plastics, 248 Penetration depth, 65 Perception of color, 1, 3, 11, 13, 58–59, 110–117, 126, 141, 156, 164, 238, 250, 266 visual, 114, 214, 292 Phase function, 17, 50–51, 296–297, 301, 307, 309–316, 358, 383–384, 399–401, 403, 414, 444 see also Scattering, function Phosphene, 114–116 Phosphorescence, 13, 45 Photo activity, 200 Photochromic, 274–278 See also Phototropic Photochromism, 278 Photoluminescence, 19–20 Photometer scale, 236

Index Photometrical resolution, 264 Photon, 22, 114 absorption, 114 energy, 22 Photopic adaption, 128 Photopic vision, 110, 120, 123, 165 Photo pigments, 113, 117 Photosensitive pigments, 13, 111–113 receptors, 110 Phototropic, 13, 249, 253, 257 See also Photochromic Physical vapor deposition (PVD), 36, 60, 67, 70–71, 80, 84–85, 105, 109 coating, 70, 109 films, 67 method, 67 techniques, 105 Physikalisch-Technische Bundesanstalt (PTB), 236 Pigment absorption, 4–6, 14, 26, 43–46, 49, 53, 58, 64, 67–68, 73, 76–77, 80–81, 85–86, 89, 92, 99, 101–103, 129, 176–180, 185, 197–199, 201, 207, 214–215, 217–219, 221, 238, 243, 248–249, 313, 315, 386–392, 396–397, 399, 408, 423, 435–436, 438, 444 calibration series of, 45, 391, 393 corrosion prevention, 69 diffraction, 4–6, 14, 34, 38–39, 43, 62, 64–65, 80–82, 84–85, 95, 101, 104–110, 201, 209, 222–228, 241, 242, 252, 260, 314–315, 408 flake-shaped, 63, 76, 79, 85, 206 interference, 5–6, 34–36, 38, 45, 64, 68, 80, 82–93, 95–96, 98, 100, 103–104, 179, 197–199, 201, 204, 209–217, 222, 225, 242, 252, 267, 316, 337, 386, 390–392, 401, 407–408 loading, 176–178, 185–186, 188–189, 218 metallic, 4–6, 28, 34, 57, 62, 64–82, 102–103, 197–198, 200–204, 206, 209, 220, 225, 236, 242, 252, 271–272, 314–316, 337, 356, 358–359, 386, 391, 400–401, 407–408, 436, 438, 447 pearlescent, 6, 64, 80–85, 92–93, 95, 98, 107, 200, 209, 211, 215, 218, 221, 400, 436, 438, 448 scattering, 436

Index Pigmentation, effect pigments, 102 Pigment preparation, 434 Pigment volume concentration (PVC), 61, 198 Pigment volume content, 102 Pitch, 88–89, 99 Planck constant, 15 Planckian locus, 139 Planck law of radiation, 15, 23, 139 Plastic materials, 44, 49, 51, 59, 61, 65, 69, 153, 159, 192, 195, 198, 247–249, 254, 267, 274, 438 See also Polymers Plastic melts, 197, 252 Plastics, 5, 27, 50, 102, 152, 159, 249, 373, 386, 401, 438, 447–448 Polyacrylate, 73, 84 Polyamide, 199 Polycarbonate, 84, 199 Polychromatic illumination, 275–276 light, 104, 117 measurement, 275 Polyethylene terephthalate, 67, 84, 199 Polymeric materials, 154, 199, 247, 387 dehydration of, 199 Polymers, 26, 44–45, 50, 60, 66–67, 69–72, 76–77, 83, 87–90, 198–199, 243, 247, 277, 332–333, 387, 447–448 Polynomial, 190, 311, 357–358, 386 Polypropylene, 67, 84 Polystyrene, 67, 84 Polyvinylchloride, 61, 198 Poynting vector, 25, 296 Prediction procedures, 418, 424, 436 See also Recipe prediction Pressure gradient, 197, 439 Primaries, 118–123, 125–126, 137 real, 125 virtual, 125 2◦ observer, 126 10◦ observer, 126 Primary standard, 236 Printing process, 440 Prism spectrum, 39 Probability, 280, 282–288, 311 of light scattering, 296, 301, 309, 313 Process control, 73, 233 Processing, 3, 49, 64, 67, 76, 79, 85, 109, 141, 161–163, 192, 197, 199, 205, 224, 252, 383, 423–424, 434 techniques, 70, 76 Process variables, 248, 438 Producer risk, 288

489 PTB, see Physikalisch-Technische Bundesanstalt (PTB) Purkinje effect, 127 Purple boundary, 137–138, 140 PVC, see Pigment volume concentration (PVC), Polyvinylchloride PVD, see Physical vapor deposition (PVD) Q Quadrature formula of Gauss, 357, 371 Quality consistency, 424 control, 233–234, 248, 252, 259–260, 279, 292 Quantum emission, 49 energy, 13, 16, 45 hypothesis, 16 optics, 12 R Radiance factor, 239–240, 243 Radiant energy, 121, 136, 192 Radiant flux, 192, 297, 306 Radiation constant, 15 cosmic, 4, 12 diffuse, 7, 31, 298, 327, 334, 340, 345, 348, 350, 354, 377–378 diffuse–diffuse, 377–378 directed, 297, 348, 377 electromagnetic, 13 energy, 13, 15, 24, 32, 192, 274, 296, 419 energy density, 419 field, 16, 296, 305, 312, 316–317, 327, 341, 345, 356, 358, 361, 370–371, 377–378 IR, 13–14, 69, 192, 447–448 optical source, 11–43 Planck law of, 15, 23, 139 radioactive, 192 subdivision of field, 377 thermal, 278 UV, 13, 19, 69, 88, 274, 447 Radiative transfer, 7, 33, 49–50, 187, 295–379, 381–383, 390–391, 393, 397, 402, 411–414, 427–428, 445, 451–454 approximation, 187, 381–383, 390–391, 393, 397, 402, 411–414, 427–428, 445 theory of, 7

490 Radiative transfer equation (RTE)295–296, 302, 306–309, 311–312, 316, 326, 340, 343–344, 352, 356–359, 371, 451–454 general (GRTE), 307–309 plane parallel layer, 308–313, 326 special, 311 Radiator blackbody, 14–16, 18, 139–140, 236 luminescence, 14, 16, 19, 34, 139, 155 temperature, 14, 16, 18–19, 23–24, 140 xenon, 192 Radioactive radiation, 192 RAL, 58–59 Random errors, 279–280, 282, 402, 428 measurement error, 280 spot check, 287 test, 292, 403 variable, 283–284, 288 variations, 428 See also Stochastic Rayleigh–Jeans radiation formula, 16 Rayleigh law, 16–17 Rayleigh scattering, 32, 302 Receptors, photosensitive, 3, 110 response time, 116 Recipe accuracy of, 400–403 alternative, 381, 390, 403, 406, 420–421, 444 best, 421, 435 correctability of, 178, 421, 424–427 different, 406–407, 421 with effect pigments, 409 selection, 424 strategy, 389, 406, 427, 430 Recipe prediction, 4, 7–8, 56, 118–119, 161, 169–170, 172, 178, 184, 187–188, 243–244, 246, 250, 259–261, 263, 265, 267, 277–278, 295, 316, 323–324, 326, 332, 341, 357, 381–445 basic idea of, 332 classical, 381, 390, 392, 403–404, 417, 420–421, 435 different methods of, 404 heart of, 170 modern classical methods, 382 modern methods, 421, 431 three levels of, 386 total error of, 402 Red to green contribution, 141

Index Reference color, 49, 122–123, 144–147, 151, 155, 157, 169–171, 179, 207–208, 235, 255–256, 259, 266, 288–290, 382–384, 401, 403–404, 406, 409, 414, 421, 427, 430–437, 439, 443–444, 449 Reference coloration, 198, 200, 206, 252, 265, 390–393 Reference database, 392 Reference illuminant, 166–167, 170, 172–174, 261, 448 Reference pictures, 382, 434 Reference spectrophotometer characteristics of, 260 specifications of, 260 Refinishing paint, 208, 438 Reflectance, 28–29, 33, 49–56, 77–78, 81, 91, 93–101, 105–107, 109, 134–136, 169–171, 174, 182, 186–189, 201–202, 204, 206, 211, 233, 236, 239, 241, 246, 248, 251–252, 257, 261, 266–271, 275–276, 295, 347, 388–389, 393, 397, 430, 438 angle dependent, 81, 95 apparent values, 236 curve, 54, 95, 98–100, 107, 109 diffuse, 50, 94, 347 of ferromagnetic diffraction pigments, 109 measurement, 81, 95, 189, 206, 211, 236, 241, 248 standardized measurement, 241 wavelength-dependent, 236 Reflected light, 26, 29, 36, 49, 69, 77, 79, 156, 250, 261, 338 Reflection boundary surface, 32, 59, 338 coefficient, 29–32, 60, 253–254, 333–334, 336–337, 340, 348–349, 351, 353–354, 364, 375, 378 corrected, 350, 365–366, 368–369, 371, 373–376 corrected function, 369 definition of, 320, 328 diffuse, 60, 62, 64, 76–77, 235, 328, 334, 372–373 diffuse–diffuse, 343, 350–351, 363, 369–370 directed, 26, 59–60, 309, 314, 318, 320, 354 directed–diffuse, 342, 345, 348, 350, 353, 362, 369 directional, 26, 59, 77, 253–254, 351 experimental, 397, 404, 409

Index external, 267, 333, 335–339, 348, 350–356, 369, 371, 375, 378–379 factor, 30–31, 33, 239, 277, 389, 396, 399 function, 361, 365–371 grating, 38–43, 104–105, 233, 259–260, 337 indicatrix, 59, 244 inner, 30–31, 365–366 at interfaces, 83 internal, 267, 322–323, 334–336, 338, 340, 351, 357–363, 369, 372–373 law of, 26, 59 metallic, 4–5, 65, 74–76, 82, 202 multiple, 25, 34, 36, 63–64, 68, 82–83, 86, 249, 251, 267–268 outer, 30 single, 83, 268 spectral, 53, 56, 77, 92, 233, 252, 258, 267, 269, 274, 316, 385, 388–389, 393, 399, 402 specular, 26, 59–62, 236, 244, 246–247, 249, 253–254, 277 theoretical, 409 Reflectivity, 36, 53, 65–66, 89, 332, 447 of metals, 65, 107 Refraction index, 377 law of, 26–29, 365–366 Refractive index, 6, 17, 26–32, 34, 36–37, 60–61, 66, 82–83, 85–86, 93, 95–96, 189, 197, 209, 250, 303–304, 323, 333, 337, 354, 363, 378–379 complex, 28 of metals, 66 See also Fresnel coefficient Relative color strength, 52, 180–181, 183–185 Relative tinting strength, 52 Remaining colors, 424, 431, 434 Repeatability, short-term, 162, 264–265 Repetition accuracy, 162 Representatives of color, 1, 119, 124, 133, 136, 167, 234, 255 Reproducibility, 4, 18, 178, 259, 278 long-term, 263–265, 280 Residual stresses, 192 Resistance, 4, 44, 48, 69, 102, 189, 192–193, 197–200 abrasion, 69, 193 climate, 200 condensation water, 199

491 light, 198 weather, 44, 48 See also Color fastness Retina, 2–3, 8, 13–14, 109–118, 122, 126, 136, 156, 165, 167, 234 Retinex theory, 3, 114 Ring pumps, 197 Robustness of recipe, 426–427 Rods, 110–112, 126, 156, 165 Root-mean-square error (r.m.s), 281–282 Rough recipe, 432, 434 See also First recipe Rutil, 47, 81, 83–84, 90, 93, 97, 98 S Safety colors, 166, 274 Salt spray testing, 199 Sample color, 8, 11, 14, 18, 20, 24, 28, 31, 43, 49–50, 57, 61–62, 87, 125–126, 129, 136, 144–146, 154, 160, 162–163, 167–169, 179–180, 192, 206–207, 234, 238–240, 244, 246, 249–250, 258, 260–261, 279, 289, 381–383, 386, 409, 428 requirements, 248–250 semi-glossy, 254 structured, 254 surface, 26, 239, 242–244, 246, 249, 261, 272 Sanding, 60 Saturation, 2, 52–53, 58, 81, 102, 115, 161, 179, 184, 276, 278 Scanning electron microscope (SEM), 69–71, 73, 86–87, 211, 271, 274 Scanning electron microscopy, 74 Scatter ellipsoid, 285, 290 Scattering, 4–5, 7, 11, 13–14, 16–17, 25, 32–34, 50–51, 53, 56–57, 60, 90–92, 96, 102–103, 107, 178, 182, 185, 189, 195–197, 299–306, 313–316, 352, 354–361, 400, 408, 410, 412, 418 anisotropic, 313–316, 357, 383 coefficient, 50–51, 182, 300, 312, 316, 319–320, 324, 327, 336, 343, 383, 386, 390, 393–397, 410, 418, 451, 453–454 constant, see Scattering, coefficient cross section, 301 dependent, 304 diffuse, 195, 248 edge, 75–76, 103, 215

492 Scattering (cont.) of edges, 50, 83, 224 elastic, 32 function, 296, 301, 303 See also Phase function inelastic, 32 isotropic, 60, 247, 303, 312–316, 326, 341, 357–358, 383 multiple, 7, 295, 302–306, 318–319 pure, 309, 321–322, 338, 346, 355, 359, 375 selective, 46, 81 simple, 295 single, 7, 302–304, 313 Scatter term, 307 See also Source, function Scotopic vision, 110, 165 Scratches, 197, 249 Search program, 432 Search strategy, 435 Secondary color values, 417 Secondary illuminant, 417 Secondary standard, 236–237, 243, 255, 262 Selection of colorants, 407 of recipes, 407 Semiconductor diode, 14 Semiconductor laser, 22 Sensitivity absolute, 164–165 of color receptors, 112, 122 definition of, 424 entire, 425–427 of human eye, 17, 56, 135, 140 of measuring, 247, 262 of recipe, 425–426, 427 Sequential analysis, 292 Serial production, 256 Shade, see Color shade Shear, 3, 69–70, 85, 192, 197–198, 200, 225, 248, 252, 439 gradient, 225 rates, 192, 197, 252 stability, 198 stable, 70, 198 stress, 197 velocities, 248, 439 Significance, statistical, 288 level, 289 point, 287 testing, 288

Index Silver, 5, 66–67, 70–72, 75, 77, 80, 84–86, 89–90, 92–94, 106–107, 109, 198, 207, 215–216, 218–220, 408, 436, 438, 447 Silver bronze, 5, 66–67 Silver colored, 92, 107, 109 Silver dollar aluminum pigment, 218–220, 438 flakes, 75, 80, 198 pigments, 70–71, 75 Simultaneous contrast, 114–115, 156, 250 Single-constant theory, see One-constant theory Single scattering, 7, 302–304, 313 Size distribution, see Particle, size distribution Skylight, 17, 238 north, 17, 238 overcast, 17 south, 238 zenith, 17 Smectic, 88, 225 Sodium lines, 27 vapor lamp, 23–24, 237 Software, 8, 234, 261, 272, 357, 381, 402, 421, 431, 435, 441 Solar constant, 16, 192 Solar energy, 192 Source of artificial light, 18 function, 302, 306–307, 312 technical, 18, 23 term, 307, 311–313, 317, 341 working life of, 18 Sparkle, 70, 72, 77, 79, 82, 89–90, 99, 103, 153, 200–201, 204–206, 271–273, 287, 403, 423, 436 area, 272, 287 definition of, 72 degree, 200, 205, 272 effect, 89–90, 201, 204–205 grade, 205–206 intensity, 272 particle, 72, 99, 205, 272 pigments, 90 uniformity of, 273 See also Glitter effect Sparkling, see Sparkle Special metamerism indices, 172–175 Special optical cases, 295, 307, 309, 319, 336, 345 See also Optical triangle

Index Spectral colors, 13, 20, 93, 121, 137–138, 278–292 Spectral energy distribution, 14, 17–21, 115, 118, 125 Spectral lines, 237 Spectral-matching criteria, 182 Spectral power distribution, 11–12, 14–15, 17–18, 23–24, 33, 49, 123–124, 128, 134, 136, 138, 155, 165–166, 170, 173, 192, 253, 261 Spectral reflectance, 53–56, 78, 93, 95–100, 105–107, 109, 134, 136, 169–170, 186–188, 233, 236, 239, 251–252, 257, 276, 389 Spectral reflection, 53, 56, 77, 92, 233, 252, 258, 267, 269, 274, 316, 385, 388–389, 393, 399, 402 Spectral transmission, 189 Spectral transmittance, 134, 257 Spectral value, 52, 170, 172, 210, 236, 244, 248–249, 252–255, 258–259, 262, 264–266, 275–276, 288, 295, 297, 310, 381, 384, 386, 394, 402–404, 409–410, 412–414, 430, 432, 443 calculated, 404 corrected, 430 experimental, 394 measured, 252, 266, 275, 288, 381, 386, 404, 409, 414, 432 theoretical, 409–410 Spectrometric correction, 429–431 Spectrometric procedure, 264, 404, 413 Spectrometric recipe prediction, 412 Spectrometric strategy, 404, 409–414, 417–418 Spectrophotometer, 39, 124, 162, 183, 222, 237, 241–244, 247, 251, 253, 257, 259–267, 271–272, 288, 383, 385–386, 390, 401–403, 409, 412, 420, 428, 432 accuracy of, 263–266 characteristics of, 260 error limits, 264 photometrical resolution, 264 short-term repeatability, 162, 264–265 wavelength resolution, 260–261 Spectrum colors, 121 continuous, 18–19 diffraction, 39, 43, 223, 260 electromagnetic, 12 locus, 137–138 visible, 6, 12–14, 44–45, 52, 89, 111, 127, 138, 316

493 Specular angle, 26, 78, 83, 91, 95, 222–223, 241–242, 252 Specular color, 226, 228 Specular component, 244, 246–247, 254, 262, 277 excluded, 244, 254 Specular reflection, 26, 59–62, 236, 244, 246–247, 249, 253–254, 277 Stable colors, 116 Standard black, 236, 264–265 color, 237, 390, 434 metallic, 78 primary, 236 reference, 238, 266 secondary, 236–237, 243, 255, 262 transfer, 236 white, 202, 236, 262–265 Standard climate, 237, 250, 265, 278 Standard color-matching functions (SCMFs), 125–126, 134–136, 138–139, 257–258, 288, 404 Standard color values, see CIE 1931 color space Standard deviation (SD), 281–282, 287–288, 290 Standard illuminant A, 18–19, 21, 23–24, 140, 253 D65, 19–21, 23–24, 140, 171, 186, 155, 210–211, 222, 253, 258, 261 Standard observer, 128, 134–136, 138, 142, 168, 170–171, 173–175, 183–184, 194–195, 238, 253, 258, 261, 402, 414, 425, 432, 435 2◦ standard observer, 122, 124–125, 127, 135, 138, 170–171, 182, 194, 257 10◦ standard observer, 127, 135, 137–138, 156, 163, 168, 171, 182, 194, 210–211, 222, 257, 410 Standards, ordered ASTM, 185, 241–242, 455–456 CIE, 3, 18–21, 23–24, 109, 125–129, 134–135, 137, 163, 174, 184, 238, 253, 261, 414, 458 DIN, 163, 241–242, 458–459 DIN EN ISO, 459 ISO, 461 JIS, 461 Starting values, 168, 347, 412 Statistical error, 280, 402 Statistical methods, 287 Statistical significance testing, 288

494 Statistical test, 256, 279, 287–292 color differences, 279, 287–292 Statistical testing color differences, 279, 287–292 Steradian (sr) definition of, 24, 297 Stimulation, 13, 49, 238, 260, 275 of fluorescence, 49, 238, 275 selective, 13 Stochastic errors, 280 variable, 283–284, 287–288 variations, 256, 280 See also Random Stokes emission, 275 Storage, 4, 233, 237, 248, 250, 261, 266, 278, 383, 434 color collections, 250 color patterns, 250 data, 261, 383 standards, 278 Storing, 192, 262 Strength of color, relative, 52, 180–181, 183–185 Striations, 193 Substitution of colorants, 4 Substrate aluminum, 106–107, 215 curved, 62 ferromagnetic, 107, 109, 201 glass, 89 metal, 68, 104 mica, 6, 64, 85–86, 91, 93–99, 200, 215, 239–240, 448 muscovite, 86, 209 nickel, 80 Subtractive color mixing, 102–103, 116–120 Subtractive color mixture, 215 Superposition additive, 103 of colored lights, 117 of diffraction and interference, 228 of interference and absorption, 95 of waves, 39, 81 Superstructure, 189 Surface boundary corrections, 346, 352, 363–369, 372 boundary effect, 60 boundary layer, 332 correction, 336–338, 340–341, 345, 348, 354, 357, 362–363 forms, 60

Index gloss, 22, 61, 206 glossy, 26, 58, 246, 255 high-glossy, 59–60, 244, 255 irregular, 50 matt, 60, 235, 249 phenomenon, 59–63 reflection, 32, 59, 62, 72, 253–254, 325, 338 roughness, 60, 107 slightly glossy, 247 structure, 59, 69, 73, 162, 238–239, 246, 248–249, 253, 255–256, 271, 274, 279 structured, 59, 156, 197, 240 Surrounding color, 115, 156 Surrounding polymer, 69, 77 Swelling, 199 Synapse strength, 442–444 Systematic errors, 256, 279–280, 401–402, 428–429 recipe prediction, 402, 428 T Target color, 172 See also Reference color Taylor series, 395, 398, 412 Temperature drift, 264–265 Temperature radiator, 14, 16, 18–19, 23–24, 140 Test colorations, critical, 403 Test quantity, 287–288, 290–292 statistical, 291 Tetrachromacy, 112 Textile fibers, 49, 159, 247–248, 254–255, 278, 387, 429, 441 Texture, 77, 79, 87–89, 201, 225, 246, 277 cholesteric, 88–89, 277 nematic, 88 smectic, 88, 225 Theory of errors, 279–280, 282 of radiative transfer, 7, 295–379 Thermochromic color, 249, 277–278 Three-flux approximation, 312, 337, 340–356, 361, 376, 378, 396–397, 399, 444 Three-flux theory, 340–343, 345 Threshold of acceptability, 256 of discrimination, 164–165 Tinting power, 44, 189, 422 Tinting strength, 52 Titanium dioxide anastas, 47, 83–84, 90, 93, 97–98

Index brookit, 93 rutil, 47, 81, 83–84, 90, 93, 97–98, 210, 239–240 Tolerance agreement, 141, 161–163, 171, 233–234, 255–257, 266, 422 ellipse, 140, 205, 238 parameter, 153, 256 ranges, 163, 256 specification, 205 value, 162, 256 Topcoat, 200, 249 See also Clear coat Total error of recipe prediction, 402 Total influx, 352 Total reflection, 28, 30, 334, 340–341, 348, 350, 354, 357, 368–371, 373–378 critical angle of, 28, 30 partly directed component of, 354 Training colorimetrical, 163 coloristical, 163 color perception, 238 of neural networks, 382, 390 Transfer standard, 236 Translucent, 49–51, 83, 91, 134, 184, 189, 191, 193, 197, 246–247, 250–252, 255, 257, 266–268, 296, 319, 324–325, 331, 340, 346–347, 367–368, 387–390, 396, 398–399, 412, 420, 447, 451–454 color, 50, 134, 255, 389 colorants, 420 diaphanous, 51 gleaming, 51 ideal, 51 layer, 83, 91, 251–252, 267–268, 324, 340, 347, 368, 451, 453–454 materials, 51, 189, 193, 246, 251–252, 267, 346, 396, 412 system, 50, 252, 331, 340 Transmission, 11, 32–33, 38–43, 53, 59, 116, 118, 124, 128, 182, 184, 189, 196, 233, 235, 237, 246–248, 250–251, 255, 257, 261, 267–271, 277, 295, 297, 310, 316–323, 325, 328–239, 331–339, 343–345, 346–348, 352–374, 383, 385–386, 393–394, 398, 402, 409–410, 412, 451, 453 corrected, 350, 352, 366–367 definition of, 32, 328 determination of, 50, 237, 251, 261, 270

495 diffuse, 50, 247, 328 diffuse–diffuse, 343, 351 directed, 318 directed–diffuse, 342, 345, 348, 363 directional, 247, 351 experimental, 398, 412 external, 333, 335, 338–339, 350–351, 353, 378–379 function, 361–368, 370, 372–373 grating, 38–40 internal, 322–323, 336, 351, 362 spectral, 189 Transmissivity, 195–196 Transmittance, see Transmission factor, 33, 240, 247 Transmitted light, 49, 88, 239, 250, 252, 361 Transparency ideal, 135, 138, 189, 321–322, 339, 352, 355, 378 index, 189–191, 197, 250 Transparent colors, 49, 86, 134, 250 Transparent layer, 50, 197, 316, 319–320, 336, 353, 369, 452–454 ideal, 353 Transparent liquid, 89, 99, 211, 270–271 Transparent material, 134, 189, 191, 197, 237, 250, 267, 313, 322, 325, 331, 338, 346, 396 measurement, 240, 246, 251 Transparent medium, 189, 191, 250, 320, 330, 338 Travel, see Flop Trichromasy, 112 anomalous, 113 Trichromatic color sense, 133 Trichromatic color theory, 114 Trichromatic matching, 123 Trichromatic stimulus, 123, 234 Tristimulus colorimeter, 257–259, 265 Tristimulus color matching, 120–123 Tristimulus values, 123–125, 133–228, 432 Tungsten filament lamp, 18–21, 261 Turbidity, 50, 192–197, 422 Twilight, 165–166 Two-color pigments, 85 See also Color flop Two-constant theory, 387 Two-flux approximation diffuse–diffuse, 355

496 Two-flux approximation (cont.) directional, 316–326, 328–329, 338, 353–354 See also Two-flux theory Two-flux theory, 7, 326, 329, 351 See also Two-flux approximation Two-mode spectroscopy, 275–276 Two-tone, 77, 79 See also Lightness, flop U UCS Uniform color scales, 58, 141, 144, 160–161 Ulbricht sphere, 243, 248, 250, 260, 262 Uncertainty, 12, 186, 190, 233–234, 252–253, 256, 263–265, 271–272, 278–281, 292, 386, 412, 424–425, 427 of measurement, 234, 252, 256, 263–264, 271, 278–279 relative, 190 Uniform chromaticity scale, 141 UV absorption, 53, 200, 447 UV catastrophe, 16 UV radiation, 19, 88 UV range, 15–16, 20 V Variable directional geometry, 241–242, 261 Variance, 281, 284–287, 289, 291 Varnish power value, 191 Viscosity, 69, 76, 438 Newtonian, 69 structural, 69, 252 Visible range, see Visible spectrum Visible spectrum, 6, 12–14, 44–45, 52, 89, 111, 127, 138, 316 Visible wavelengths, 6, 11, 13, 36, 56, 69, 82, 252, 257, 297 Vision of colors, 2, 110, 112–113, 118, 156, 165, 176 mesopic, 110, 123 photopic, 110, 120, 123, 165 scotopic, 110, 165 Visual acuity, 164–165 Visual angle, 110–111, 122, 126, 156, 171 Visual assessment, 19–20, 23–24, 26, 62, 115, 126, 128, 145, 164–165, 181, 238, 253, 255, 384 Visual cells, 110–111 Visual cortex, 3, 8, 109, 113–114, 116, 159, 234 Visual perception, 114, 214, 292 Visual sensation, 8

Index Volume, 17, 28, 32, 34, 43, 59–61, 74–76, 102, 108, 138, 142, 160, 185–186, 189, 195, 197, 235, 240–241, 246–248, 250, 253–254, 267, 299, 309, 332, 385, 402 fraction, 386 Von Kries coefficient law, 167 W Water-based paint, 199 Water-borne system, 198 Water resistance, 199 Water vapor, 56, 199 Wavelength band, 36, 265 discrete, 19, 24, 123, 136, 257, 275 long, 16–17, 111 monochromatic, 22, 138 reflected, 228 scale, 20, 237 short, 16, 24 visible, 6, 11, 13, 36, 56, 69, 82, 252, 257, 297 Wave optics, 64 Weather durability, 45, 200 resistance, 44, 48 stability, 199 Weathering artificial, 192 natural, 200 outdoor, 192 resistance, 102, 189, 200 Weather resistance, 44, 48 Weighted colorimetric strategy, 405 Weighting factor, 153, 170, 182–183, 305, 327, 398, 404–405, 410–412, 414–418, 420, 433 empirical, 195, 207–208, 393 Weiss areas, 108 White balance, 121 White mixtures, 176–177, 387–388 Whiteness index, 193–195 Whitening index, 194–195 White pigment, 45, 47, 181, 193, 387–388, 391 White standard, 202, 236, 262–265 Wien’s law of radiation, 16 X Xenon lamp, 19–21, 260–262 of high pressure, 23, 258–259 Xenon radiator, 192

Index x, y, Y system, see CIE 1931 color space X, Y, Z standard color values, see CIE 1931 color space Y Yellow to blue contribution, 141 Yellowing, 193, 195, 199

497 determination of, 199 index, 195 Yellowness, 193, 195 Young–Helmholtz theory, 2, 114 Z Zone theory, 114