ABSTRACT MODIFIED KUBELKA-MUNK ANALYSIS FOR THE APPROXIMATE THEORY OF MULTIPLE SCATTERING by Kaeo Songkhao The transmission and reflection of light in light-scattering layers is discussed on the basis of a set of simultaneous differential equations originally due to Schuster. Formulas previously developed by Kubelka and Munk are briefly recapitulated and then extended to describe the exponen- tial solutions, the semi-hyperbolic solutions, and hyperbolic solutions. The shortcomings of these formulas for practical applications is well known. To overcome these shortcomings, appropriate formulas have been developed. To test their validity, experiments were performed with pig- mented lacquer films as specimens. Experimental and calculated results for lacquers containing individual pigments, and for lacquer containing much of the pigment have been compared in terms of infinite reflectance and CIE chromaticity coordinates. Tables of various parameters necessary for application of the formulas developed in this study have been con- structed. Computer programs are given as well. Comparison of experimental and calculated results shows that the method developed in the present study seem to give reliable estimates of reflectance and color both in the individual pigment and mixed-pigment Kaeo Songkhao lacquer films. The method appears to be more accurate and convenient than the previous methods used. MODIFIED KUBELKA-MUNK ANALYSIS FOR THE APPROXIMATE THEORY OF MULTIPLE SCATTERING by KAEO SONGKHAO A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics, and Materials Science T975 ACKNOWLEDGEMENT I wish to extend my appreciation to Dr. Robert Summitt, my major professor for his presentation of the problem as a possible research topic and the helpful information and suggestion provided by him. I also wish to take this opportunity to thank Dr. D. J. Montgomery, Dr. J. 5. Frame, Dr. G. L. Cloud, Dr. G. A. Coulman and Dr. N. N. Sharpe, Jr., Dr. G. E. Mase for serving as my Guidance Committee. My graduate program was sponsored by the Royal Thai Air Force. I wish to thank Air Vice Marshall Bisudhi Ridhagni, Professor and Dean of the Faculty, the Royal Thai Air Force Academy, for his encouragement and his confidence in me. My grateful thanks to E. I. duPont-De Nemours & Company, American Cyanamid Company, Pfizer, Minerals, Pigments & Metals Division, Glidden- Durkee, Ceramic Group and FERRO Corporation Color Division for supplying the resins and pigments used in this study. I want to pay special tribute to my wife, Piyavat, for her wonderful patience, encouragement, and help and to our children, Mue, Mon and Mack, for giving up some time with daddy while he studied. Finally, I want to thank the excellent secretary, Ms. Beverly Anderson, for her superb aid in the preparation of this thesis. ii Chapter II III TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ...................... ii LIST OF TABLES ....................... v LIST OF FIGURES ...................... ix INTRODUCTION ........................ 1 l.l PRELIMINARY ...................... l 1.2 PURPOSE AND MOTIVATION ................ 3 1.3 ORGANIZATION OF DISSERTATION ............. 5 THEORY ........................... 7 2.1 REVIEW OF THEORIES OF TURBID MEDIA .......... 7 2.2 KUBELKA AND MUNK THEORY ............... 11 2.2.1 Derivation of the K.M Equations ........ 1] 2.2.2 Use of the K.M Equations ............ 23 2.2.3 Modification and Correction .......... 24 2.2.3.1 Absolute and Relative Measurements , , 24 2.2.3.2 Surface Correction .......... 25 2.3 SIMPLIFIED METHOD FOR COMPUTING K.M COEFFICIENTS 31 2.3.1 Developing of New Formulas . . ......... 32 2.3.2 Limitations and Scope of the New Technique . . . 43 2.4 THEORY OF COLOR MIXTURE ................ 50 2.4.1 The Single-constant Theory ........... 50 2.4.2 Two-constant Theory .............. 51 2.5 COLOR SPACE ..................... . 53 2.5.1 The CIE Colorimetric System .......... 53 2.5.2 Color Difference Formulas ........... 55 INSTRUMENTATION AND EXPERIMENTAL PROCEDURE ......... 58 3.1 SAMPLE PREPARATION ................. . 58 3.1.1 Vehicle .................... 58 3.1.2 Pigments .................... 61 3.1.2.1 White pigment ............. 62 3.1.2.2 Colored pigments ........... 64 iii Chapter IV TABLE OF CONTENTS (Continued) Page 3.1.3 Pigment Dispersion ............... 66 3.1.3.1 Pigment Volume Concentration ..... 69 3.1.3.2 Preparation of Paints . . ....... 70 3.1.3.3 Preparation of Films ......... 71 3.2 FILM MEASUREMENTS ................... 73 3.2.1 Measurement of Film Thickness ......... 74 3.2.2 Reflectance Measurements ............ 74 3.2.2.1 Spectrophotometric Components ..... 75 3.2.2.2 Reflectance Measurement of Films . . . 77 3.3 CALCULATION ...................... 79 3.3.1 Determination of R m, F(R0° ), K and S for Individual Pigment ............. . 80 3.3.2 Transforming Reflectance Spectra into CIE Coordinates .................. 81 3.3.3 Computing the Reflectance of Paint Films Containing More Than One Pigment ....... 85 3.3.4 Computing the Color of Paint Films Containing More Than One Pigment . . ..... 85 RESULTS AND DISCUSSION ................... 87 4.1 RESULTS OF CALCULATED VALUES OF R OF SINGLE PIGMENT LACQUERS ...................... 87 4.2 THE COEFFICIENTS K. S AND K. M FUNCTION K/S . ..... 107 4.3 RESULTS OF PREDICTED Rm AND COLOR OF MIXED PIGMENT LACQUERS .............. . ....... 12] 4.3.1 Application of the Theory in the Ultraviolet Region . . . . . . . . . . . . . . . . . . . J33 4.4 ERROR ANALYSIS ................... 435 4.4.1 Errors of the Measured Reflectance Spectra . . 136 4.4.2 Errors in Determinations of K/S, S and K . . . J38 4.4.3 Sources of Error in Predicting Reflectance and Color of the Mixed Pigment Lacquer Films . . 139 4.5 CONCLUSION ..................... 139 REFERENCES ........................ 142 APPENDIX I ........................ 150 APPENDIX II ....................... 182 iv Number 2.1 2.2 LIST OF TABLES Title Definition of symbols used in Table 2.2 .......... Formulas giving some of the relationships between Rm, R0], R02, X9 S, K, a, b, C and B ooooooooooooooo Lists of notations and abbreviations used in Tables 4.2 - 4.12 .......................... Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for TiO2 at 15 PVC Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for TiO2 at 10 PVC Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for TiO2 at 5 PVC . . . Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for TiO2 at 1 PVC . . . Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for Fe203 at 15 PVC . . Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for Fe203 at 10 PVC . . Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for Fe203 at 5 PVC Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for Fe203 at 1 PVC Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for Cr203 at 15 PVC . . V Page 46 94 95 96 97 98 99 100 lOl 102 Number 4.11 LIST OF TABLES (Continued) Title Page Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for Cr203 at 10 PVC . . 103 Comparison of calculated values in terms of reflectance and color for various cases with the average experimental values of infinite film thickness for Cr203 at 5 PVC . . . 104 Lists of notations and abbreviations used in Tables 4.14 - 4.24 . . . . ....................... 108 Absorption and scattering coefficients and K.M functions for various cases of TiO2 at 15 PVC ...... . ...... 109 Absorption and scattering coefficients and K.M functions for various cases of TiO2 at 10 PVC . . . . ......... 110 Absorption and scattering coefficients and K.M functions for various cases of TiO2 at 5 PVC .............. 111 Absorption and scattering coefficients and K.M functions for various cases of TiO2 at 1 PVC .............. 112 Absorption and scattering coefficients and K.M functions for various cases of Fe202 at 15 PVC ............. 113 Absorption and scattering coefficients and K.M functions for various cases of Fe203 at 10 PVC ............. 114 Absorption and scattering coefficients and K.M functions for various cases of Fe203 at 5 PVC ............. 115 Absorption and scattering coefficients and K.M functions for various cases of Fe203 at 1 PVC ............. 116 Absorption and scattering coefficients and K.M functions for various cases of Cr203 at 15 PVC ............. 117 Absorption and scattering coefficients and K.M functions for various cases of Cr203 at 10 PVC ............. 118 Absorption and scattering coefficients and K.M functions for various cases of Cr203 at 5 PVC ............. 119 Lists of notations and abbreviations used in Tables 4.26 - 4.32 ......................... . . 124 Number 4. 26 .27 .28 .29 .30 .31 .32 .33 LIST OF TABLES (Continued) Title Comparison of predicted values with the experimental values, the mixture of 25w + 756 ................. Comparison of predicted values with the experimental values, the mixture of 50w + 506 ................. Comparison of predicted values with the experimenta1 values, the mixture of 75w + 25R ........ . ........ Comparison of predicted values with the experimental values, the mixture of 50w + 50R ................. Comparison of predicted values with the experimental values, the mixture of 756 + 25R ................. Comparison of predicted values with the experimental values, the mixture of 25w + 20R + 606 .............. Comparison of predicted values with the experimental values, the mixture of 50R + 256 + 25w .............. Comparison of computed and experimental infinite reflectance values in ultraviolet region of PEMCO Virgo Blue stain and FERRO Low Infrared reflectance red lacquer pigment films . Table for inspecting R0], R02 ............... Values of infinite reflectance relative to R and R 01 02 ° ' ' Absolute reflectance values corresponding to measured reflectance values .................... Reflectance values with surface correction corresponding to measured reflectance values. These figures incorporate Saunderson correction using k1 = 0.04, k2 = 0.60 ..... Reflectance values with surface correction corresponding to measured reflectance values. These figures incorporate Duncan correction using k1 = 0.04, k2 = 0.555 ...... Table of Kubelka-Munk function, F(Rm) = K/S ......... Table of the K.M function F(Rm) corresponding to measured values R. These figures incorporate Saunderson correction using k1 = 0.04, k2 = 0.60. . .............. Table of the K.M function F(Rm) corresponding to measured values R. These figures incorporate Duncan correction using K] = 0.04, K2 = 0.555 ............... Page 126 127 128 129 130 131 132 135 152 153 161 165 166 167 170 173 LIST OF TABLES (Continued) Number Title Page A.9 Table of Kubelka-Munk parameter a vs. R infinity ...... 176 A.10 Table of Kubelka-Munk parameter b vs. R infinity ...... 179 viii Number 2.1 2.2 3.1 3.2 4.1 4.2 4.3 4.4 4.5 LIST OF FIGURES Title Page Schematic representation of a layer of absorbing and light scattering particles . . .......... 12 Reflection from front surfaces of the layer ....... 26 The optical diagram of the DB-GT spectrophotometer . . . 76 Flow Chart for Determining K/S and R in Various Cases by using Tables in APPENDIX 1 .......... 83 Chromaticity coordinates by experiment of pigment lacquer films Ti02, Fe203, and Cr203 . ........ 88 Chromaticity coordinates of TiO2 pigment lacquer films at 1 and 15 PVC. . . ... . ........... 89 Chromaticity coordinates of Fe203 pigment lacquer films at 5 and 15 PVC ......... . ....... 90 Chromaticity coordinates of Cr203 pigment lacquer films at 5 and 15 PVC ..... . . . . . . . . . . . . 91 Chromaticity coordinates of pigment lacquer films Tioz, Fe203, Cr203 and their mixtures ......... 122 ix CHAPTER I INTRODUCTION 1.1 Preliminary The optical properties of a material comprise all of the modes whereby it interacts with light, or, more generally, with electromagnetic radiation. Nhen radiant energy, in general, is transmitted through gases, liquids, or solids, radiation intensity, wavelength, direction of propa- gation, and plane of vibration may be affected by one or more of the following propagation phenomena [1]; reflection of radiation at the interface between two media which differ optically, refraction at the interface between the two media such that the direction of the beam under- goes a discontinuous change, absorption by which the energy of the beam is attenuated as it propagates through an absorbing medium, scattering of the radiation by particles of random distribution and polarization which changes or discriminates against the direction of the oscillating electro- magnetic field. By far the most widespread mechanisms of the interaction, of light with matter are scattering and absorption. These processes are responsible for a wide variety of phenomena, including translucency, opacity and color. 0f the many methods in use for the characterization of physical- chemical properties of materials, light scattering is potentially one of the most useful methods, yet much under utilized. A11 matter scatters l light, specular reflectance being merely a special case of backward scattering. This phenomenon is discussed in detail by Jenkins and White [2] in their chapter "Absorption and Scattering." If a scattering system is highly dense it may be called a turbid medium, where in the light scattered from one particle illuminate another particle at another point in the sample (secondary scattering), which particle in turn serves as the light source for still other particles (multiple scattering). Numerous authors have studied the propagation of light through such turbid media as a stellar atmosphere, translucent paper, opal glass, photographic emulsions, and so on. Although the optical phenomena of these are the same, the treat- ment may be different. The optics of turbid media have been widely studied, yet there is no satisfactory theory of their transmittance and reflectance properties. The exact theory of scattering from a turbid media is very complex. The mathematical treatment of multiple-scattering processes leads to a formidable mathematical situation, and solutions exist for only a limited number of situations. For this reason, in practice, it is of great importance to find simple analytical expressions which provide use- ful approximations to the exact solutions. The most widely accepted theory in the study of turbid media has been the Kubelka and Munk theory [3], which is, in fact, based on an earlier one put forward by the astronomer Schuster [4], who was studying the attenuation of light from stars by scattering and absorption before reaching an observer. The Kubelka and Munk theory contains quite a number of assumed simplifications. It is strictly applicable only to ideally diffuse light, i.e. light of unoriented radiation, both the incident light and that within the layer. In addition, the theory ignores optical phenomena at interfaces. Thus, other theories have been developed [5,6,7] which, in taking these deviations into account, are necessarily more complicated than the Kubelka and Munk theory. For several reasons, however, none of these theories have been accepted. A number of papers dealing with the analysis of the Kubelka and Munk theory have appeared from time to time in the literature of various fields. It is true, however, improvements in the Kubelka and Munk theory are still needed. 1.2 Purpose and Motivation Although the success of the Kubelka-Munk theory in solving a number of optical problems in turbid media is rather impressive, it should not be overlooked that it is based on a fairly simple model which some- times does not meet practical conditions. This theory assumes that the optical behavior of a layer which simultaneously scatters and absorbs can be described only by two terms, 315,, the scattering and absorption co- efficients. The theory results in solutions for the degrees of trans- mission and reflectance as a function of scattering coefficient, absorp- tion coefficient and layer thickness, and including an optional substrate as well as a special case. Often the question arises whether the Kubelka and Munk model is not too simple and should be improved. Instead of two fluxes, other authors [8,9,10] have studied models using four, six and more fluxes defined by different directions in space and associated with an increasing number of constants for every specimen or with assumptions about the spatial distri- bution of stray light from single scattering. The most complete theory is that of radiative transfer, leading to integral equations which can only be solved numerically with large computers. If the results of such calcu- lations are compared with those using the Kubelka-Munk solutions, the empirical experience is confirmed that the simple Kubelka-Munk solution is very good approximation for various applications[ll]. Recent literature on color in industry indicates a considerable need for an improvement in the Kubelka and Munk approach. Improvements of procedure and method in applying the Kubelka-Munk theory for the present- day practice is the purpose of this study. New formulas have been derived and we also have develOped a procedure that will lead to more accurate and precise use of spectrophotometers and colorimeters. Experimentation had been performed to test the validity of these formulas and procedure in a variety of cases of single and multiple pigment systems. The limitations of these formulas and methods have also been specified. Although the method and formulas developed in this study were tested by applying them to a pigmented film it is expected nevertheless that many of the relations obtained will be helpful in other applications as well. The pigmented film was selected as a model for this study not only because it is the simplest model but also because of the importance to the field of surface coatings. Since the solution of Kubelka-Munk analysis depends on the nature of the specimen including composition and preparation procedure, the specimen system including composition and preparation procedure, the specimen system used in this study is described fully. Although the Kubelka-Munk theory treats both transmission and reflect- ance, the present study is restricted to the reflectance problem only. All surfaces reflect a certain amount of the incident energy. The nature of reflected energy depends not only on the type of incident beam, but also on the physical properties of the material under examination. Hence measurement of reflectance be used to study the optical properties of materials. The study of reflectance spectra has not yet become wide- spread because of the difficulties of interpreting them. Reflectance measurements are used widely in many applications, however, for example, in the fields of textiles, paints, paper, ceramics, metals, mining, and printing. In this study, our primary objective is to determine for several pigments the reflectance of an infinite-thickness film, the scattering and absorption coefficients of individual pigmented films, and then use them to predict the reflectance for any mixture of these pigments in an opaque film. This is possible because the scattering and absorption coefficients are mathematically related to the reflectance, and their combined effect is proportional to concentration of the pigment in a mix- ture. 1.3 Organization of Dissertation The background of the problem and a review of turbid media theories are described in full detail in the beginning of Chapter II. The various derivations of the Kubelka-Munk theory and the method and formulas proposed in this study including their limitations are also described in that chapter. The related theories used in testing the validity of the method and formulas, color space, color mixture theories, and color differ- ence formulas are also described briefly in Chapter II. In testing the method and formulas proposed in this study, the pig- mented lacquer films are used as the experimental system. The background knowledge of pigmentation and the specimen preparation are described in full detail in the first part of Chapter III. The description of the instruments, precautions, and procedures in measuring the specimen are also in Chapter III. The methods of calculation by using the method and formulas proposed in this study have been shown in the last part of Chapter III. Results are given in Chapter IV. The appendix includes Tables and Computer programs. CHAPTER II THEORY 2.1 Review of Theories of Turbid Media The problem of calculating the optical properties of media which both scatter and absorb light is common to many disciplines [12] and it is related mathematically to the problem of describing the dif- fusion of neutrons [l3] and the phenomenon of radiative transfer [14]. Consequently, there is a large volume of widely dispersed and loosely connected literature. The major theoretical developments in turbid medium theory have resulted from the work of astrophysicists and geo- physicists interested in stars and planetary atmospheres. Important computation methods also have been developed by nuclear physicists, who face a similar problem when calculating the flux of neutrons in a reactor. Workers concerned with the optical properties of surface coatings, plastics, ceramics, photographic emulsions, papers, and textiles have developed a large literature which is poorly related to the broader material, but, nevertheless, has served the needs of these industries quite well [15]. Mathematical descriptions of the optics of turbid media may be divided into two categories [9]: (1) extremely general and broad treatments, often by astrophysicists, which, in most cases, lead to systems of integro-differential equations which cannot be solved 7 analytically; (2) restricted special cases which may not describe the physical situation precisely, but do so adequately with respect to observable phenomena, and are mathematically tractable. Excellent and unusually broad reviews of these literatures have been written by Billmeyer gt .21. [16] and by Orchard [15]. Exact solutions of the multiple-scattering problem in turbid media have been discussed by Chandrasekhar [l4], Grosjean [11], Case [18], Kourganoff [13], Sobolev [19], Van de Hulst [20], Chu and Churchill [21], Orchard [22] and Cheng [23]. Practical application of these integral solutions exist only for a limited number of situations, however, and the results so far are in a form too complex for ready application to cases of interest. Another, more recent, approach is the Monte Carlo method of random walk, using a digital computer [24]. This is not a good technique for thick, strongly scattering layers, where several hundred orders of scatter must be taken into account [25]. For this reason, in practice, it is convenient to use approximate theories since the latter can be used to obtain comparatively simple solutions, and correlations between the light scattering characteristics are discovered. The theoretical treatments applied to turbid media can be classi- fied roughly into those which deal with the problem in one dimension and those which consider it in two or three dimensions. The most widely adopted approach by far is the one-dimensional. In the one dimensional case, the medium is considered as bounded by two parallel plane and the propagation of the light is studied along a direction perpendicular to those planes. Thus the one-dimensional problem can be mathematically characterized by stating that everything depends on the thickness only, and on no other coordinate or angle. Hence planarity of the front and back surfaces is essential and any deviation from this condition would remove the one-dimensional nature of the problem. When the surfaces are irregular it is necessary to use the general three-dimensional theory of radiative transfer, which is well documented by Chandrasekhar [14]. Most theories for turbid media deal with plane parallel scattering layers and the two-flux method, partly because the theory for this geometry is simpler than for nearly all other cases, and also because most practi- cal problems involve this geometry. Besides the advantages which they present, however, such approximations also may suffer from drawbacks with various degrees of importance, so that several authors [5,16,26,27,28,29] in recent years have tried to construct increasingly better formalisms. For simplicity and uniformity, in this study we also adopt the one- dimensional, two-flux model. The earliest work related to (although not specifically concerned with) turbid media of one-dimensional theory appears to be a paper by Stokes in 1862 [30]. He was interested in evaluating the reflectance of a pile of glass plates each of which may or may not exhibit absorption. Beginning with the classic work of Stokes, a number of theories have been evolved during the past century to account for the observable optical properties of such light-scattering, light-absorbing media as piles of plates or films, translucent or partially opaque films, coatings and sheets. The first paper of importance on the theory of multiple scattering of light in turbid media was written in 1905 by the astrophysicist Schuster [4], who suggested that the complicated situation of light traveling in all directions and being scattered and absorbed at all places in a turbid medium could be simplified by focusing attention on 10 the rate of energy transfer, or flux, in the forward direction and in the reverse direction through the medium. Thus only two quantities are to be calculated at each point in the medium. It turns out that the differential equations which describe these two quantities are simple and easily solved. Several papers describing applications of these equations appeared before 1930, mostly published by astrophysicists and workers in the photographic industry [31,32,33,34]. In 1931, Kubelka and Munk [3] used essentially the same formalism in their treatment of light-scattering layers, suppressing the terms corresponding to the generation of light within the medium. Their work is very similar to that of Schuster, with the omission of the emission terms. Similar one-dimensional, two-quantities theories have been de— rived by several other authors [5,35,36,37,38]. The more inclusive historical review is summarized in the work of Beasley et_al_[9]. Ingle [39] has shown that the work by Smith, Amy and Stokes are merely special cases of the Kubelka and Munk theory. Kubelka [40] has shown that the work by Gurevic is also a special case of the Kubelka and Munk theory. The K.M hyperbolic equations for reflectance and trans- mittance of a turbid, isotropically scattering medium have been found formally identical with the solution for reflected and transmitted fluxes of Chandrasekhar's radiative transfer equation for isotrophically highly scattering media [41]. The general equations of Kubelka and Munk are re- garded as including all others of this type as special cases. This is why the most widely accepted theory in the study of multiple-scattering media is the Kubelka and Munk theory. Thus we will restrict our study to the Kubelka and Munk (hereafter K.M) theory. 11 2.2 Kubelka and Munk Theory Excellent reviews of the principal theories have been given by Judd and Wyszecki [42], Kortum [43], Kubelka [40] and Ingle [34], and the latter two have demonstrated the equivalence of some of the theories. Additionally, Kubelka has presented a number of very useful equations stemming from the K.M theory. The pioneering researches of Steele [44] and Judd [45], both of whom utilized the K.M theory, resulted in special solutions and various mathematical aids that have been responsible, at least in part, for the widespread use of this theory. Today, the K.M theory is employed for many scientific or industrial purposes, e.g., chromatographic systems [46], pharmaceutical products [47,48], paper and pulp material [44,49], text- iles [50,51], foodstuff [54], radiant energy transfer [56], color measurements and color comparison [45,57,58,59,60,61,62,63,64,65]. 2.2.1 Derivation of the K.M Equations The K.M theory, the approximate two-parameter theory for multiple light scattering in a turbid medium has become popular for studying scattering media. This popularity is attributed to relative simplicity of its end formulas. The theory examines the transmission of a light beam through an infinite plane-parallel sheet of an isotropic scattering material illuminated on one side by a diffuse beam. It de- scribes the light field by a system of two simple differential equations. The system of differential equations can be analysed under the assumption of an energy balance in the material from radiative transfer theory. The basic method is that of dividing the light flux into two parts: one flux, I, in the positive direction, and the other, J, in the negative direction 12 (Figure 2.1). The incident radiation is assuming for diffuse (i.e., the intensity is equal for all angles of incidence), and the radiation scattered sideways is assumed to be compensated for by an equal contri- bution from neighboring parts of the layer (i.e. the area investigated is either small in cross section compared with the total illuminated cross section of the sample or is large compared with the thickness of the sample). It also is assumed that the layer is illuminated with a monochromatic light source. For purposes of the calculus, the medium is taken homogeneous. The medium is assumed to be isotropic and not emit by radiation or fluoresce. Fig. 2.1 Schematic representation of a layer Of absorbing and light- scattering particles. 13 The following notation will be used: X is the thickness of the layer. The positive X direction is taken downward, with X = 0 as the illuminated surface. Ii is incident flux. K is an absorption coefficient, defined by requiring that (KIdX) be the amount of the radiation absorbed from the flux I on passing through an infinitesimal layer dX. S is a scattering coefficient, similarly defined by requiring that the flux scattered backward from I (and therefore added to J) in an infinitesimal layer dX is (SIdX). 0n passing this layer, I will then be diminished by the amount absorbed and the amount scattered, but will be increased by the flux lost by scattering from J or: dI - (K + S)IdX + SJdX (1) dJ + (K + S)JdX - SIdX (2) The signs have been chosen consistent with the fact that as X increas- es, I decreases, while J increases. Equations (1) and (2) are the two fundamental simultaneous differentail equations which describe the absorption and scattering process for K.M theory. It will be observed that only two directions of the incident and re- flected beam are considered here, namely, those perpendicular to the sur- face of the layer. This is an approximation, but is considered to intro- duce a small error compared with that resulting from the approximation of ideal diffuse illumination within the sample [34]. 14 These two fundamental simultaneous differential equations can be readily integrated, the general solutions being [56]: I = A(1 - e)eOx + 3(1 + e)e‘°X (3) and J = A(l + s)eOX + B(l - s)e'OX (4) where o = jk(k + 25$ (5) and a = o/(K + 2s) = J/K/(K + zs)‘ (6) A and B are constants of integration to be determined by boundary conditions, 0 is commonly referred to as the extinction coefficient, and B is an optical constant. If the integration is carried out over the entire thickness of the ideal diffusing sample layer, the appropriate boundary conditions are: p—c I1 H Solution for A and B under these conditions can be found, and it is now possible to obtain formulas of considerable practical importance from these solutions. For example, the transmittance of the layer is given by: 15 4B 0 (l + 8)2eOX - (l - B)2e -dX = 28 (1 + 82)sinhoX + ZBcoshoX and the diffuse reflectance is 2 o -o _ _ (1 - 8 )(e - e ) R — J /I - 0 0 (l + B)2eox- (l - (3)2e'OX (1 - 82)Sinho (l + 82)sinh0X + ZBcoshOX As case of considerable interest is that in which the layer thickness of the sample is allowed to increase without bound, commonly referred to as infinite layer thickness. Setting x = w in Equations (8) and (9), it is found that'T0° = 0, as to be expected, and R =1-e=1-JTl, so that Arcoth must be used and we have a - R %-(Arcoth a g R - Arcoth ——B——9) sx = (25) b2-( -R>( -R> =-l Arcoth a a b b(Rg - R) 1 - R (a - bcothbSX) from which R = a + bgothbSX _ R9 (26) Equation (26) is the hyperbolic solution obtained from K.M theory which corresponds to Eq. (23), the exponential solution of K.M theory. For an ideal black, non-reflecting background (R9 = 0, R = R0), Eqs. (25) and (26) may be simplified to: 1 1 ' 3R0 SX = '5 AY‘COth T'- (27) 0 _ sinhbSX Ro ‘ asinhbSX + bcoshbSX (28) 20 Using Eqs. (27) and (25), we obtain a, and hence Rco , as function of R, R0, and R9 such that R -R+R = 1/2(R + 0 R R g) (29) It may be noticed that in practical formulas, S always occurs together with the thickness X of the layer, that is as the product SX, called Scattering Power of the layer [45]. To find an analogous formula representing transmittance of such layer, = 30 10/1d ( ) we multiply Eq. (28) by I; then by means of Eq. (14), we obtain for a layer of thickness X with a black background I a + bcothbSX (31) J = ROI = Inserting this into Eq. (13) we obtain dI _ _ -T- adX dX a + bcothbSX Which now may be integrated. By replacing bSX = u, and dX = gg, we have bSX d1- 1"" aéjfdx + bdefa + bcothu d” wl—I 21 I . l d _ l l as1nhbSX + bcoshbSX and-S— ln—I—g-aX-E.-a-2—:t-)—2-(abSX-bln b ) Since according to Eq. (20), a2 - b2 = l, we obtain finally b T ‘ asinhbSX + bcoshbSX (32) The inverse function of Eq. (32) is SX = Arsinh b/T - Arsinh b (33) b The exponential functions for the transmittance can be found by expressing a and b by Eq. (19), and inserting them in Eq. (32) we obtain by replacing the hyperbolic functions with exponential functions for the transmittance of the layer the expression: (1 - Rm )2 exp(- bSX) 1 = 2 (34) 1 - Ra_ exp(- 2bSX) which corresponds to Eq. (24) for the reflectance of the same layer using a black background. In these two equations T and R0 are represented as functions of R0° and SX. From the relations so far derived, further equations can be obtained which relate the several variables. Thus, from Eqs. (28), (32) and (20), we arrive at 22 Many of the relations between the quantities R, R9, R0, R00 , SX and T of K.M theory have been given in a composite list in a form suitable for practical use by Kubelka [40] and have been repeated and summarized by Judd and Wyszecki [42], and Kortum [43]. The usual current forms of the equations of K.M theory are in a semi—hyperbolic form. It is possible to rewrite these equations in fully hyperbolic forms which are both elegant and very convenient for calcula- tion [60]. Define Y = 1n(l/Rn.) (36) Eq. (20) then becomes: a = coshY and b = sinhY (37) Eq. (26), Eq. (27) and Eq. (32) become: sinhW - Rgsinh(W - Y) R = sinh(W + Y) - RgsinhW (38) sinhW R0 = sifihTW + Y) (39) T = sinhY sinhTW + Y) (40) In these equations: W = bSX = SXsinhY (4]) 23 Two useful subsidiary relations are: (141-12 ) = 2R” = ScoshY - l (42) 2 ll coshY - sinhY (43) 2.2.2 Use of the K.M Equations The optical properties of turbid media can be calculated with fair success on the basis of K.M theory. This study is restricted to the consideration of turbid medium theory as applied to the problem of paint films. At the present time the K.M equations are almost universally used for multiple-scattering calculations in the paint industry. This wide acceptance is due to the fact that these calculations are easy and give rather good results in many situations. They have the drawback, however, that the scattering and absorption coefficients must be determined empirically, and then used only in similar systems under similar conditions of illumination and observation. Judd and Wyszecki [42] provide an explanation with examples how K.M equations should be used to solve problems involving reflectance and opacity of colarant layers. Billmeyer and Abrams [67] have summarized the K.M equations for its practical application to all cases of interest plus those for other closely-related operations together with step-by-step procedures for their use. There are many ways in which the solution to the basic K.M equations 24 can be expressed depending upon the particular application under study. Two important cases for paints may be considered. The first one relates S and K to the reflectance of a film so thick that further increase in thickness does not change its reflectance, that is, optically an infin- itely thick film. This is the use of Eq. (11). The second case applies to incompletely hiding films on a substrate. The equation of interest is Eq. (23) or Eq. (26). Note, however, that either case requires know- ledge of the parameter a and R00 . The application of these equations would be straightforward from the mathematical point of view except for the fact that, because of experimental errors it sometimes happens that a is slightly less than one, making b imaginary. This situation has no physical meaning, of course, but it is difficult to complete the compu- tation for this case. Ross [68] has suggested a method to overcome this difficulty, which involves a series transformation for the computation of the hyperbolic functions. In such a case, however, the absorption coefficient K turns out to be negative, which means a medium must amplify light or it must fluoresce. The other problem is the reproducibility of R00 . For dark paint, it usually is very simple to apply a sufficiently thick film to measure R0° directly, precisely, and accurately. Only as a paint becomes very white and low in hiding power does it become diffi- cult to apply a film thick enough to satisfy the definition for Rm . 2.2.3 Modification and Correction 2.2.3.1 Absolute and Relative Measurements All the quantities arising in the K.M theory for a simultaneously scattering and absorbing layer are defined as the ratios of reflected or transmitted radiation to the incident radiation. In 25 actual practice, however, only the transmittance of a layer may be determin- ed absolutely, and diffuse reflectance is measurable only relative to a suitable white standard, such as MgO, MgC03, or BaSO4. Under these con- ditions, one determines the ratio /R (44) Rmeasured = Rsample standard To determine the absolute R therefore, it is necessary to sample’ know the absolute reflectance R of the white standard. Many authors [69, 70,71,72] have measured and tabulated the absolute reflectance values of some white standards at various wave-lengths. 2.2.3.2 Surface Correction Most of the practical problems associated with the use of K.M theory are due to neglect of the original postulates laid down for its application. The original K.M equations for reflectance, R, and transmittance, T, apply only ixia planar scattering system of infinite extent, having no specularly reflecting boundaries and illuminated with diffuse light. In the derivation of the equations, Kubelka and Munk assumed that the refractive index of the layer is the same as that of the medium in which the measurements are made, which is not valid since usually the measurements are made in air. In practical use the K.M theory also must include the fact that light is reflected at the bound- aries of the medium where the index of refraction changes abruptly. It is convenient to consider first the energy balance just below the reflecting surface, and then to find the ratio of the incident and 26 emergent fluxes (see Figure 2.2) [22]. Assume here that the upper boundary of the layer is a surface with specified directional reflection properties and that the lower boundary is a perfect absorber. If the upper surface is a specular reflector, its directional reflecting proper- ties are described by Fresnel's equations. We now must take account of the interaction between the surface reflection and the true reflection of light by the diffuser. Fig. Let Rm = apparent reflectance, R = true reflectance, T = true transmittance, k1 = external reflectance of surface t1 = external transmittance of surface = l - k.l k2 = internal reflectance of surface t2 = internal transmittance of surface = l - k 2 Io = irradiance of surface. I o iokl A 2 k2 1 i I I J a. _—-—._“-hm‘—Q-. _. .---‘----~*O---------u‘-’ -o 2. 2 Reflection from front surfaces of the layer. 27 In Figure 2.2 I is the reduced irradiance and J is the net reflected radiance just below the reflecting surface. We have simply: J = 1R + szR . (45) On writing R* = J/I, we get R*'= R + R*k2R The apparent interior reflectance R* is thus R* = J/I = R/(l - k2R) . (46) Also we have I = Iot1 (47) Now the emergent radiance (excluding the surface component) is, from Eqs. (46) and (47) Jtz = Iot1t2R* (48) or tltzR RM = k1 + T—T R (49) 28 This fact has been recognized by many earlier authors [5,28,57,58, 64,73,74,75,76,77] , but the Saunderson [57] correction seems to be the most widely used. Saunderson used a General Electric recording spectrOphotometer for the Spectrophotometric measurements. In that instrument about half the specularly reflected light is lost through the entrance hole of the inte- grating sphere. His original surface correction was expressed as R=-k—‘—+(1-k)(1-k) R (50) m 2 l 2 I - K2R But the widely quoted equation that he used can be represented as Eq. (49) i.e. _ R Rm - k1 + (l - k]) (l - k2) 1‘771§;T (51) Both k1 and k2 depend on, and may be calculated from, the refractive index of the sample for which the refractive index of the binder may normally be substituted [11]. This quantity depends on the wavelength of the light, but this small dependence is often neglected. k1 is given by the well-known Fresnel relationship 2 k] = IELJLELLE (52) (n+1) where n is the ratio of the refractive index of the film to that of the surrounding medium. Corresponding values of the internal reflectance coefficient k2 can be obtained from the relation [22]: 29 2 t2 = tI/n (53) where t1 = external transmittance of surface = l - k1 t2 = internal transmittance of surface = l - k2 the refractive index of the film. 3 11 Eq. (53) follows directly from the definitions of t1 and t2 together with Snell's law of reflectance coefficients k1 and k2 with respect to the refractive index have been tabulated by Orchard [22]. For a glossy paint or other pigmented composition in a binder with a refractive index of about 1.5, k.l is about 0.04. Saunderson found, by trial and error, that the value of 0.4 for k2 produced the best color- matching prediction. A similar value has been obtained by Bridgeman [78]. Thus there is little doubt that k2 = 0.4 is the optimum value for use in the application of Eq. (51) to practical color-matching problems. How- ever, both theoretically and experimentally [74] k2 has a value of 0.5 - 0.6. Many authors [78,79] use k = 0.6. 2 Duncan [58] has developed from the correction equations of Saunderson [57] and Ryde [5], his formula for surface correction (1 - k1)(1 - k2)R Rm = (l - kZR) (54) He has used k1 = 0.04 and k2 = 0.555 for reflectance reading with the Beckman D-4 Spectrophotometer on glossy paint, and his formula then simplified to 30 _ 0.427R m ‘ ““1 - “0.5—“55R (55) R This value of Rm is then used in the K.M function and the corres- ponding values of Rm and F(RM) have been tabulated [59] for convenient use. Now if the lower boundary is not a perfect absorber, there will be a contribution from substrate reflectance. This modification has al- ready been introduced by Kubelka and Munk, but, the complexity of the substrate reflectance is not generally appreciated [74]. It should be remembered that two types of reflectance can occur from the substrate of a film. The diffuse light transmitted through the film first en- counters a boundary between the film and the bulk substrate. This boundary introduces a specular reflectance for the diffuse light, its magnitude depending on the relative refractive indexes of the film and the bulk substrate. Light that is transmitted across this boundary then can be reflected by scattering processes within the substrate. Generally, therefore, the reflectance of a substrate measured in air is not the value that applies when a film is laid on that substrate. This is due to the major changes in relative refractive index at the surface of the substrate and hence changes in the specular reflectance for diffuse light at the film-substrate boundary. An example of the magnitude of this effect has been given by Blevin [81]. To avoid this complicated phenomenon, it should be assumed that there is no incident flux on the substrate surface. This can be done in actual practice by using the ideal black substrate. In this study, therefore we will modify the technique of using K.M equa- tions by deriving the required equations only for the case of the ideal 31 black substrate. 2.3 Simplified Method for Computing K.M Coefficients A knowledge of the absorption and scattering coefficients, K and S, respectively, of each optically significant component is re- quired in using K.M equations to predict other optical properties of material. Unfortunately, these two coefficients are not available for use directly. In paint technology these two coefficients also depend on the paint system, and specific measurements which have only limited application. For example, the K.M absorption and scattering coeffic- ients of a certain pigment from Manufacturer A might be measured and used for color formulation. If we wish to use instead another lot of pigment made by Manufacturer 8, we would have to redetermine the absorp- tion and scattering coefficients. It is, therefore, essential for any application of the K.M equations to determine these two coefficients. When considering opaque materials all the various methods are easy to apply, and with few exceptions, produce result of acceptable accuracy. With translucent materials, however,1nany difficulties arise. Even with the simple K.M analysis the absorption and scattering coefficients of the material are quite difficult to determine, although methods have been described by Billmeyer and Abrams [67], and Caldwell [82]. Also the results obtained are frequently too inaccurate. General improvements in the theory used for the calculations would have only a minor bearing on the whole process. Often the question arises whether the K.M model is not too simple and should be improved. Instead of the two fluxes, other authors studied models using four, six and more fluxes defined by 32 different directions in space and associated with an increasing number of constants. The most complete theory is that of radiative transfer, leading to integral equations that only can be solved numerically with large computers for limited situations. If the results of such calcu- lations are compared with those using the K.M equation, the empirical experience is confirmed that the simple K.M solution is a very good approximation [11]. Some of the difficulties in using K.M equations should be by improved computational or experimental technique. 2.3,] Developing of New Formulas Many difficulties arise in the practical use of the K.M equations. When the parameter a is less than unity the result is an imaginary value for the parameter b, which has no physical meaning. The problem of reproducibility of Ru leads to variability and uncertainly in the values of S and K. In the latter case, it has been shown by Hoffmann gt_al, [65] that the assumption of constancy of the absorption and scattering coefficients cannot be maintained, and that these seem to be constant only for paint films which are so thick that the contrast ratio is greater than 0.96. The complicated phenomena of the substrate reflectance are the third major difficulty. To overcome all these problems, we have developed the approach described in this section. In this technique the reflectance 0f two thickness 0f paint film should be measured over an ideal black substrate, and, by selecting the appropriate K.M equation for including the measured reflectance and thick- ness, two simultaneous equations will be obtained. These equations can be solved simultaneously for the scattering coefficient under the reasonable assumption that neither the S nor the R,o is a function of film thickness. 33 Caldwell [82] and Klien [53] have used a similar technique to find the K.M coefficients from transmittance measurements. In this study the technique will be used to find K.M coefficients from reflectance measurements. Eq. (9), Eq. (24) or Eq. (39) for the reflectance of the material must be solved for the constants. Let us consider first Eq. (39). Cross-multiplying in Eq. (39) gives: Rosinh(W + Y) sinhW 01‘ z + _< 11 R (56) sinhW] o Arsinh [ If two film thicknesses X1 and X2 on an ideal black substrate have reflectances R0] and R02 respectively, then from Eq. (41) we have: 2 ll 1 SX]s1nhY , and E II SXzsinhY , and then from Eq. (56) we have: sinhW1 W + Y = Arsinh (57) l R 01 sinhW2 W + Y = Arsinh (58) 2 R02 Subtracting Eq. (57) from Eq. (58) yields 34 . sinhW2 sinhW1 W2 - W1 = Ar51nh R - ArSinh R 02 01 Selecting a film thickness such that X2 = 2X1 = 2X, which implies that W2 = 2W1 = 2W, we then have that: w = Arsinh 5;"“2” - Arsinh S;”“”] 02 01 = Arsinh sinhZW‘jF]+ (sinhW)2 _ sinhW jflt (sinhZW)2 R02 R01 R01 202 sinh2W 2 sinh2WsinhW 2: (sinhW)2 + (sinthinhZW)2 or sinhW = (7f——-—)-+( R R ) R0] R01R02 02 01 02 . .2‘ = j/(ZsinthoshW)2 + (251nhgcoahW51nhW) R02 01 02 -_ J/Xéiflhfliz 25inthinthoshW)2 Ol R01202 Since W1 is not equal to W2, this implies that sinhW is not equal to zero. Then the above equation can be divided through by sinhW from which we get: j _ 2coshW 2 25inthoshW 2 l 2 2$inthoshW 2 , 1 - C—Tf—-9 + ( R R 1 - (fig?) ( 1 02 01 02 R01202 1 1/[( 1 )2 + (25inthoshW)2 _ J/ZZcoshW)2 + (25inthoshW)2 R R R ‘ R R R 01 01 02 02 01 02 35 Squaring both sides, and regrouping the terms, 2 jC—l—JZ + (25inthoshW)2 = (2coshW)2 _ _1_92 _ 1 R01 ROiRoz R02 Squaring again, we have: 2 2 4 2 2 2 2 4 )§l_) +(25inthoshW) = (ZCEShW) _2(2cgshW) (1 + )§l_) + 1+)fil‘) 01R01R02 02 02 01 01 or 2 2 4 2 2 2 2 1 4sinthoshW 2coshW 2coshW 2coshW l 4(-—) +( ) = (———) -2(——-—) -2( ) + 1+(——-) R01 R01 R02 R02 R02 ROiRoz ROi By using the trigonometric relation and regrouping the terms, we get: (4COShN)2 (COSHZW-1)- (ZCOShw)4+2(2C05hN)2+2(2COShw)i %( _)2+(1 1 )4 R01 R02 R02 R02 ROiRoz R01 R51 or 4 2 coshw coshW 16 4 8 2 8 2 _ 16W - 16(W) - r4(COShN) iF-24COShW) WCOShW) - 01 02 02 02 Ol 02 < < ‘ 12 )2 ———- - 1 R01 Multiplying through by RmaR02 4 and again regrouping the terms, 2 2 2 4 2 2 2 2 4 2 2 _ 36 This equation is the quadratic equation in costh in the form: Ax2 + BX + c = O , _ 2 where X - cosh W _ 2 2 2 A ' 16R01 (R02 ‘ R01 ) _ 2 2 2 B ' 8R01 R02 (1 ‘ ROi ) _ 4 2 2 C ’ ‘ R02 (1 ' ROi ) and the solution: X ’ 24 This will give us: 2 2 2 2 fig 4 4 2 2 2 4 2 2 2 cxbshzw = 8R01 R02 (1‘R01 ) 4R01 R02 (1‘ROJ ) +64ROi R02 (R02 'R01 )(1‘201 1 32R 2(R 22-R Z) 01 02 01 2 2 2 2 2 2 2 2* = 8R0] R02 (1-R0] ) + 8R0] R02 (l-RO] ) R0] + (R02 - R0] ) '7 2 2 2 32R01 (R02 ‘ ROi ) 2 2 g R02 (1‘R01 )(Roz T R01) 2 2? 4ROi(R02 'R01 ) 37 2 = R02 (1 ' ROi . (59) 4ROi(R02 ‘ ROi7 0r 2 ‘ R (1 - R ) 02 01 cash)! = --f (60) 2 RO1(R02 ' R01) From which we will have: w A h (1 2 R012) ( 1) = rccos -—-— 6 2 R01(R02 ‘ R01) Once W is known, R°° can be found by solving Eq. (57) for Y and then substitute the value of Y into Eq. (36) and solve for Rm or by using Eq. (43) directly. The other quantities in the K.M theory can then be found. Now if we consider Eq. (24), cross-multiplying, rearranging and solving for Sx we finally get: R R -1 00 5X = (Infi—7§—:T)/(1/Rm-Rm) 00 By substituting the values of R0], R02, x1 and x2 as before we will have: R Rm(R R0° - 1) ..Rm 01 01 (62) 38 x Rm Rm(R S = ————-——-1n 2 1 _ R 2 (I) ROD-l) 02 - R (63) Since we set X2 = 2X1 = 2X, then when we divide Eq. (63) by Eq. (62) we have: R (R R - l) Rm(R R - 1) Cross—multiplying and taking anti-log both sides yields: [Rm(RO]Rm - 1)]2 Rm(R02Rm - 1) R0] - R00 R02 - Rco 2 2 2 or R 2 - 2R R + R 2 2 R02 ‘RRw 01 01 m m ‘ Again cross-multiplying, 2Rm4_2R 3 2 _ = 2 2 3 4 2 3 01R” +R. )(R02 Rm) R0] ROZRm -2R01R02Rm +R02Rm -R01 Rm+2R01R (R 2 R n m 00 01 4-2R R R 3+R R 2 01 02 w 02 w - 2 5 Rm +2R 4 3 Rm -R 2 01 R R 2-2R R R 3 2 R R 02 m 01 02 m ROi 02 6 R01 01 6 =R 2 3 4 2 R- ‘Rs +R Rm -R0] Rm+2R 02 01 Rearranging and regrouping the terms we get: 39 2 3 Oi‘R01 2 R )R 2 _ 01 02 m - R0] - O 02-2R R02)Rw + (2R01-R02+R 2 01 2 ‘R 01 4 2 - and (l-Rm )R - (2R01 02+R R02)Rm(1‘Rm ) - 0 Since 1 - R»2 can never equal to zero, we divide through the above equation, to give 2 01 2 2 (1+Rm )R O] R - (2R01-R02+R 02)Rm = 0 2 2 2 or R0] R00 - (2R -R R 2 02)Rw+R = 0 +R 02 01 01 01 Again this equation is quadratic in Rco which can be solved to give j 2 7/ 2 2 4 - (ZROl'R02+R01 R02) ‘ (2R01‘R02+R01 R02) ' 4ROi Rm — 2 (64) 2R0] Once R0° is known, the scattering coefficient S can be found by Eq. (62). The other quantities can be found by using the appropriate equations. Now if we consider from Eq. (9) (1 - 82)sinhoX O ' 2) (1+8 sinhoX + 28coshox 4O Cross-multiplying in this equation and by regrouping the terms and solving for 8 gives: 2 2 2 . 2 2 =~ROCOShOX .1. /R0 COSh 0X + (1-R0 )51nh OX (65) 8 04+ R0)sinhoX' If two layers of thickness X1 and X2 and reflectances R0] and R02 respectively, are considered, such that X2 = 2X1 = 2X and replaced coshoX by C], cosh26X by C2, sinhoX by S1 and sinhZoX by S2 for convenience in writing, then -R c + /R 2c2+(1-R 2)s2 8 = 01 l 01 l Oi 1 (66) (l + R01)Sl 2 2 2 2'2 and e = 2R02C2 + J/ROZ C2 +(] 2 R02 )52 (67) T1+R02132 Dividing Eq. (66) by Eq. (67), and again cross-cultiplying and re- grouping the terms gives: _ 22 22 (1 T R01)R0251C2 ‘ (1 + R02)R015221 ‘ (1+R01)51 R02 C2 +(“Roz )52 22 22‘ _ (1+R02)SZ J/ROl C1 +(l-R0] )S1 Substitutin S = ZS C and C 2 = l + S 2 C 2 = l + S 2 in the above 92112 2:1 1 equation, arranging the terms and dividing through by S], we have: 41 2 _ 2 2 (1+R0])R02c2 - 2(1+R02)RO]C1 - (1+R01) R02 +S2 - 2(1+R02)C 0] +5 Squaring both sides and arranging the terms yields: 2 2 2 2 2 2 2 (1+R01) (R02 C -R02 -S2 ) + (1+R02) (4R0, C 2 —4R R C1 C2(1+R01)(1+R 01 02 02) _ 2 22 2 2“ ‘ ”421(‘TR01)“TR02) /R01 T S1 jRoz T S2 Substituting C22 = 1 + $22, C12-l = S12, arranging the terms and dividing through by -(1+R0])(1+R02) gives: 2 2 2 2 - (1+R0])(1-R02)S2 + 4(1-R0])(1+R02)C1 S1 +4R01R02C1 C2 - fl 2 2 2 2 4C1 jf (R01 + S1 )(R02 + $2 ) Again substituting $22 = 4C12512, multiplying out, arranging the terms and dividing through by 4C1 we have: ‘ 2 _ T 2 2 2 2 R0221 2 ‘ J/(R01 T S1 )(Roz T S2 ) 2(1-R R )C151 0102 TR 01 42 Substituting C2 = C]2 + 512, C12 - l = S12, multiplying out, and rearranging the terms, we have: 2 _ 2 2 2 22151 T R01R02Cl ‘ ‘jRR01 T S1 )(Roz T S2 Squaring both sides gives: 2 2 2 2 2 _ 2 2 2 C1 T 4R01R02Cl Si ‘ R01 R02 T R Substituting C]2 = l + $12, $22 = 4C12812, arranging the terms and dividing through by 512 we have: 2 2+4c 2s 2 2 4 2 2 2 2 C 01 1 1 l ’ 4S +4R R +4S +R +4R R S = R 1 01 02 1 01 R02 01 02 1 02 T4R 2 2 2 2 _ 2 1T51 )T4R01R02TROi R02 T4ROiR0251 ‘ R02 T4ROi 2 2 2 2 2 or 451 ( (1+51 )+4S1 (1+S1 ), Arranging and regrouping the terms we have: 2 2 2 2 +4R -4R0]R02-R0] R02 2 2 _ 4(R01R02"R01 )Sl ” R02 01 Again substituting $12 = C]2 - 1, and solving for C1 we have: 43 2 fl C 3.50—2. 'I-RO] 1 2 R01IRO2 ' R01) 2*. R 1 - R or coshoX = —gg-J/rR (R 91R 01 02 01 2a R 1 - R 1 02 01 and o = —-Arcosh ——-g/, X 2 R01(R02 ‘ R017 From Eq. (65) we have 8 in the form: 2 2 . ' - RmcoshoX + me + Sinh OX 8 = (1 + R01)sinhoX (68) (69) (70) One a and 8 are obtained, then the other parameters of K.M equations can be calculated. It is interesting to notice that oX is identical with W in Eq. (61). Some of the relationships between the quantities obtained from this work are given in Table 2.2. 2.3.2 Limitations and Scope Limitations of the technique described in this study depends on values of R01 and R02. This can be considered from Eqs. (60), (64) and (68). Let us consider Eq. (60) and Eq. (68). Since coshW and coshoX is greater than or equal to 1, from Eq. (60), or Eq. (68) we have the condition: 44 2 1‘R01) (R -R R02 4R0] 02 01) 2 2 R >, 4R 0] 'R 02 or R R 4R 02 01 01 02 - 2 2 2 Oi ' 4ROiR02 T R02 >/ R01 2 4R 02 R (2R -R)2>/ R2R2 01 02 Taking the square root of both sides gives: °" (2 ‘ R02)ROi >/ R02 R i.e. Rm >/ 7—92?— (77) This means that R01 and R02 must satisfy Eq.(77)otherwise Eq. (60) cannot be applied. From Eq. (64), the value of R0° will be real if and only if the value within the square root sign is positive, or at least equal to zero. This will give us the condition: \V or 2R R 01 ‘ R01 02 (2-R )R 2/ R 02 01 02 Again this gives us the condition: R 02 R > ———-—. which is the same as in Eq. (77). We therefore have established the condition for the limitation of our technique, which we have tabulated in Table A.l of Appendix A for use in inspecting R0] and R02 to determine whether they can be used. There are several advantages in this method. that always arise in using K.M equations by other methods can be over- come. One of the most advantageous is that one can decide immediately from a qualitative inspection of Spectrophotometric reflectances R01 45 and R02 of the sample whether they can be used. Many difficulties 46 Table 2.1 Definition of Symbols used in Table 2.2 Symbol Definition X Thickness of layer X], X2 Thickness of layer 1 and layer 2 such that 2X = X = 2X 1 2 R0], R02 Diffuse reflectance of layer 1 and of layer 2 with ideal black background Rm Diffuse reflectance of layer so thick that further increase in thickness fails to change the reflectance, called reflectivity T Internal transmittance of layer S = (dR /dx)x_>0 Coefficient of scatter, rate of increase of reflectance with thickness of a very thin layer for diffusely incident radiant energy K = (dT/dx)x+0 Coefficient of absorption, rate of de- crease of transmittance with thickness of a very thin layer for diffusely incident energy a = (S+K)/S, = coshY, Equal to (l/R + Rm)/2, whence: R0° = - (a2-1)T/2 = sinhY, .JK/(K+25) = o/(K+2$) IK(K+2$) sinhU m 0" II I 0 ll coshU ArsinhU a _ (a2 _1)172 Equal to (l/R0° - Rm)/2, whence: R0° II n) I 0' Optical constant Extinction coefficient U -U Abbreviation for §——é%51——, hyperbolic sine of U U -U Abbreviation for-E—{%11—- cosine of U , hyperbolic Inverse hyperbolic function of hyperbolic sine of U 47 Definition of Symbols used in Table 2.2 (continued) Table 2.1 Symbol Definition ArcoshU Inverse hyperbolic function of hyperbolic cosine of U Y = 1n(1/Rm) bSX = SXsinhY 2 ll 48 Table 2.2 Formulas giving some of the relationships between Rm, R0], Formula R X’s: KsAsbsOsB 02’ Eq. Number 2 fl R (1 - R ) N = AY‘COSh '-(2)—2— /; (R ‘0]R 7 O1 02 01 .< II Arsinh [fig-{NM} - W 01 24—1 31(-Arcosh[E-gg- /R (R 2 20;]; )3 01 02 01 Q ll 20X -R0]coshoX + (j R012 + sinh 8 = (1 + R0])sinHBX R = (1,- e2)sinhox (l + 62)sinhcx + 28coshox exp SX(l/Rm - Rm) - l ‘ (l/Ré) exp SX(1/Rm - Rm) - Rm sinhW sinh(WT4 Y) S/[S + K + ‘/K(K + 23) ] e"Y = l/exp {Arsinh [éiflhfl] - W} R01 Rm coshY - sinhY = 1 ‘ B l + B (61) (56) (69) (70) (24) (39) (18) (71) (43) (10) 49 Table 2.2 Formulas giving some of the relationships between Rm, R0], R02, X, S, K, a, b, o, 8 (continued) Formula Eq. Number 2 2 2 4 ‘ R = 2ROi‘R02TROi R02 ‘ (R2ROi'R02TROi R02) ‘ 4R01 (64) w 2 2R0] F(Rm) = (l-Rm)2/2Rw = coshY - l = a - l (72) S = W/XsinhY (4T) 2 R (R R - 1) = [Rm/X(l - Rm )] ln m 01 e (62) R0] - Rm = o - BK = 0(1 - 82) (73) 23 23 K = S(l - Rm)2/2Rw (74) = S(coshY - l) (75) = 08 (75) 50 2.4 Theory of Color Mixture The validity of K and S values obtained in this study, will be tested according to the theory of color mixture and the color-difference formulas. Lower values of color difference between experimental and calculated are assumed to indicate greater validity of the values obtained. The idea of being able to compute the color of a mixture of pigments in a paint extends back at least to 1940 [84]. Examples of calculating the color of pigmented plastics were given in 1942 by Saunderson [57]. By 1944 the whole theory of how to calculate color matches for the general case had been given by Park and Sterns [76]. Practical applications of the calculation of color matches for paints and textiles followed [58.59.83]. In K.M theory, K and S are additive, which implies that their values for mixtures can be calculated from the K and S of individual components in the mixture and the volume concentrations. An important step in this approach to the color of pigment dispersions is due to Duncan [84], who showed that if there is more than one kind of pigment dispersed in a film, the scattering and absorption coefficients of the film are additive func- tions of the scattering and absorption coefficients of the pigments. 2.4.1 The Single-Constant Theory The basic assumption for calculating color mixture is that the light scattering and the absorption of colorant are proportional to the concentration, c. Almost all instrumental color matching at present is based on K.M analysis, but the two constants K and S are usually re- duced to a single constant (K/S). Many people have pointed out the analogy between the equation in this form and Beer's Law. Davidson and 51 Hemmendinger [85] had proposed that if Fi(Rm) are functions of K.M parameters, K and S, of the pigment i, and c1 are the concentrations of Y! pigments i, such that .2101 = 1, then the K.M function of the mixture ('3 1 of these pigments should be: F(R ) = ch](Rm) + c2F2(Rm)+..... + CnFn(Rw) (78) For color prediction it is necessary, Of course, to predict reflect- ance at wavelengths in the visible region. 2.4.2 Two-Constant Theory For mixtures of pigments, Duncan [84] found that the effects of individual pigments are additive in proportion to their concentrations in the film: ll 0 —J 7< —J + O N 7? N + O I O O O + O 7< (79) (80) (I) II 0 —l U) —4 + 0 N (I) N + O O O O O + O (I) where K and S are the values for the pigments specified by subscripts l, 2, 3, . . . . ., n, c; is the proportion of the specified pigment in the mixture, and Km and Sm are the values for the film. The value of K/S for the mixture can be related in a simple way to values of K and S for the individual components: 52 (K/S)m = (CIK1 + CZKZ + ,,,,, + cnKn)/ (c151 + c282 + ..... + cnSn) (81) For a very wide gamut of colors in paints and plastics, the film is essentially opaque, and most of the scattering is supplied by the white pigment. In these cases, if we assume S1 = $2 = ..... = Sn and if we let w refer to the white pigment, the two-constant theory of Eq. (81) may be simplified as: K (K/S)m (c1K + c 1 2 2 + ..... + cnKn)/Sw c](K]/Sw) + c2(K2/Sw) + . . . . . + Cw(Kw/Sw)' (82) or more simply; (K/S)m c](K/S)] + c2(K/S)2 + ..... +cw(K/S)w (83) Thus a single-constant (K/S), rather than two constants K and S separately, characterizes each pigment at a specified wavelength. To predict the color of the known mixtures of pigments by the two- constant theory, the values of K and S for the pigments and the concen- trations are inserted into Eq. (81). Calculations are made at a suffic- ient number of wavelengths for each of the mixtures so that a spectro- photometric reflectance spectrum can be drawn. Tristimulus values for 53 the predicted spectrum then are computed and compared with the tristi- mulus values for the measured spectrum of the mixture to obtain a color difference. 2.5 Color Space Numerous color systems exist. Fairly detailed accounts of the important systems are given by Judd and Wyszecki [42], and Wyszecki and Stiles [86]. 2.5.1 The C.I.E. Colorimetric System 0f the various system of color measurement currently available, the C.I.E. method or trichromatic system is undoubtedly the most popular. In 1931 the Commission Internationale de l'Eclairage (C.I.E.) established a framework for color specification in which a given color may be expressed in terms of tristimulus values X, Y, 2 based on three synthetic primaries. These tristimulus values are deduced from the Spectrophotometric data. The tristimulus values of a colored material are expressed by three integrals of the form [42] k faR’E’ xadA , X = 0 00 Y = k fR’E’ yad?‘ , (84) 00 z = k (R’E, 2,” . . where k is a normalizing factor, which for object colors is conventiently 54 chosen as k=——-—-T-Q-9-—- . (85) 00 [Efiyfidfi d R, is a function of wavelength representing the reflection factor of the test sample, E,.is a function of wavelength representing the spectoral energy distribution of illuminant, and i, , y, , and 2) are the tristi- mulus values for the spectral response curves of the standard observer as specified by the International Commission on Illumination in 1931. Stand- ard Tables of these values are published in numerous books [42,86,87,88,89, 90]. These functions are too complex to formulate analytically. Hence, the indicated integrations are usually approximated by the finite sums 700 k £E:R)E’X.AUA. >< II 790 - k ZR,E,§,AA. (86) 400 .< I 700 kZR,E, 2,21). 400 and Z with AA usually taken uniformly as 10 or 20 millimicrons throughout the visible region 400-700 nm. The normalizing factor k is now defined as JEN—j.“ 55 If the relative distribution of EA is unchanged, but its intensity is increased, the values of X, Y and Z are of course all increased pro- portionally. To obtain a measure of chromaticity, that is, to make the coordinates independent of intensity, they may be normalized so that their sum is equal to unity. Such chromaticity coordinates are given by Y = W ' (88) _ Z and z — 7—;57—;77- , Since the sum of the coefficients x, y and z is unity, any color may be represented graphically on a plane diagram with only x and y coordinates. 2.5.2 Color Difference Formulas A number of color difference formulas are in use today, and a considerably larger number have been proposed over the years and then superseded. Detailed information and discussions of variOus color- difference formulas have been discussed by numerous authors [91,92,93,94]. The existence of so many color-difference equations has produced consider- able confusion. No single formula can be considered as the best because no basis of judgment has been universally accepted. Most published dis- cussion is concerned with the use of total or 'single-number' color- difference units. A total color-difference unit is not very revealing 56 in many instances, for it gives no indication of the character of the difference. It does not show the relative size and direction of hue, saturation, and lightness differences. The present study adopts the Hunter color-difference formula based on projective transformations of the C.I.E. chromaticity diagram in com- bination with a lightness scale. This formula was chosen in view of the wide use of the Hunter color-difference meter in industry [95]. In the Hunter color coordinate-space system [96] of L, a, and b, "L" measures lightness/darkness (100 = pure white, 0 = pure black), 'a' measures redness if positive and greeness if negative, and 'b' measures yellowness if positive and blueness if negative. These coordinate values can be transformed from the C.I.E. ones by: L = lO(Y)T/2 17.5(1.02x - Y)/(v)‘/2 (89) D.) II 7.0(Y - O.847Z)/(Y)T/2 and b where X, Y, and Z are tristimulus for the C.I.E. standard observer and standard source C. The total color difference, AE, is derived from the following equation: ( 2 2 2)l/2 (90) AE(Hunter) = AL + Aa + Ab where AL, 8a, Ab are the difference between the Hunter coordinates of the two colors, and AL Aa Ab i. e. L1 - L2 a1 - a2 b - b 57 (91) CHAPTER III INSTRUMENTATION AND EXPERIMENTAL PROCEDURE 3.1 Sample Preparation To apply the formulas developed in the previous chapter, a number of conditions must be satisfied in accordance with the assumptions in the development. The model takes the samples as a set of plane parallel layers. This choice has been made because the geometry is simple, and because most practical problems involve it. The model for the layers is a fairly uniform dispersion of particles in a turbid medium. A paint film is the most appropriate specimen for this study. 3.1.1 Vehicle In the manufacture of a paint, pigments are dispersed in a fluid mixture consisting of binder, solvents, and other components. This mixture is known as the MEDIUM or VEHICLE. The binder is uSually a resinous material solid at room temperature and serving as the medium to hold pigments in a dispersed state in the finished paint film. Sometimes the term "film former" is used instead of "binder." The solvent is a volatile liquid whose function is to dissolve the binder so that it will be a liquid state when the paint is formulated and when it is applied. The mode of drying, and the adhesive and mechanical properties of the films, are determined by the type of binders. 58 59 A vehicle must be selected carefully in order to satisfy the re- quirements for the intended application. For the present study a High Gloss Overprint Lacquer based on ELVACITE 2045/2046 (duPont ELVACITE Acrylic resins formulas 23) [97] was selected as the vehicle because of its clarity, gloss, adhesion, and non-yellowing characteristics. This specific vehicle, moreover, is colorless and almost completely trans- parent, thus minimizing optical interference, and it is quick-drying. Finally, it is poorly adherent to Mylar film and hence nonsupported lacquer films are easily prepared. Compositions The composition of the duPont ELVACITE Acrylic resins form- ula 23, High Gloss Overprint Lacquer based on ELVACITE 2045/2046 is as follows: ELVACITE 2045 15% by weight; 30%. 11.79% by volume} 23.6% ELVACITE 2046 15% by weight 5°T‘d5 11.85% by volume 5°‘Td5 Mineral Spirits 35% by weight 38.44% by volume Acetone 35% by weight 37.92% by volume 100% 100% Properties ELVACITE 2045 is a bead polymer of isobutyl methacrylate with density 9.08 lb/gal, bulking value 0.1101, and specific gravity 1.09. It is the hardest of the weak-solvent-soluble butyl grades. ELVACITE 2046 is a bead polymer of n-butyl/isobutyl methacrylate 50/50 copolymer with density 9.04 lb/gal, bulking value 0.1106, and specific gravity 1.09. It is a medium-hardness butyl grade. Both ELVACITE 2045 and ELVACITE 2046 60 are soluble in mineral spirits, VM&P naphtha, and some alcohols. In lacquer technology two or more solvents are mixed so as (a) to control the flash point, price, etc., of the product, and (b) to ensure that the film deposited is free from defects such as the occurrence of undis- solved resin in the film [98]. In this lacquer-type vehicle the solvents were acetone and mineral spirits. Acetone, industrially the most important member of the acetone group, is dimethyl ketone, CH3.CO.CH3. It has a density of 0.791 at 20°C, and a boiling point of 56.2°C. It is extremely volatile and in- flammable, and flashes at ordinary atmospheric temperature, with a flash point well below 0°C. The solvent power is extremely great, and extends to a very wide range of substances. It is completely miscible with water, and with many hydrocarbons and other organic liquids. It can therefore be used to convert many two-layer mixtures into one layer, i.e., single-phase, mixtures. As a rule, solutions in acetone have low visco- sity. Mineral spirits are petroleum fractions boiling in the 150°-190°C range, although a small percentage may boil up to 210°C. They are the most widely used solvents in the industry. They are also sometimes re— ferred to as petroleum spirits, or mineral turpentine, or 'turps sub- stitute', or VM & P naphtha and are known in British usage as White Spirits. Preparation ELVACITE bead resins dissolve at room temperature, but re- quire constant agitation to prevent solvent-swollen granules of polymer from forming agglomerates and sticking to the walls of the container. Acetone 350 gm and Mineral Spirits 350 gm were mixed in a 2000 ml 61 beaker and ELVACITE 2045 and ELVACITE 2046 (150 gm each) were slowly added. Solvents and resins were mixed thoroughly with a glass rod. The beaker was put on a magnetic hot-plate stirrer, covered with a watch glass, and maintained with constant stirring until a clear solution formed. Time required for this was about 1-3 hours, depending on the conditions of the resins. It is important that the polymer beads should be sifted directly into the vertex of the stirred solvent to speed wetting-out and dispersion. Solution time can be reduced by heating; but care is required to avoid over-heating and excessive solvent loss. 3.1.2 Pigments Within the many basic types of pigment available to the modern paint formulator there are numerous subtypes. Pigments of the same nominal type and subtype as supplies by the various manufactures may be significantly different in characteristics [99]. In general, pigments may be classified into two main groups: white and colored. The pigment influences the working properties and the durability of the paint, as well as imparting color and hiding power to the paint. The reflection, refraction, and absorption of light by a pigment are the fundamental properties which determine hiding power; selective absorption is assoc- iated with the production of color. Knowledge of the optical pigment properties in physical terms is a basis for initial selection of a pig- ment. It must be stated, regrettably, that few useful data are known, even though they would be helpful in all calculations of optical effic- iency [100]. In the present study the pigments were selected primarily for 62 for their optical characteristics, with the other physical, mechanical, and chemical properties being of secondary importance. Pigments for studies in the visible study were titanium dioxide (white), ferric oxide (red), and chromic oxide (green). Pigments for studies in the ultra- violet were V-8295 Low-infrared-reflectance and L6-3503-Virgo Blue Stain. 3.1.2.1 White Pigment The characteristic "whiteness" of white pigment which the eye sees is the result of the diffuse scattering and reflectance of light by a myriad of fine particles which do not selectively absorb any of the wavelengths of visible light. Naturally, then, under red light they appear red; under blue light, blue; under yellow light, yellow. Only when the light by which they are illuminated is itself perceived as white can the pigments appear white. The opacity of white pigments when dis- persed in a vehicle results from two factors: particle size, and the disparity between the refractive index and that of the paint vehicle, the greater the mismatch, the greater the opacity. If a colorless pig- ment has a refractive index the same as that of a paint vehicle, it be- comes transparent when dispersed. When the difference in refractive index is relatively small, the pigment is referred to as extender rather than a white pigment. .flflilfi: 'Titanium Dioxide (Ti02) Titanium dioxide has three inherent properties which make it suitable as a pigment: high refractive index, "whiteness", chemical inertness, and non-toxicity. The titanium dioxide in this study, Cyanamid Unitane 0R-560 Titanium Dioxide (American Cyanamid Comapny), is an 63 alumina-silica-treated rutile titanium dioxide. It is a chalk-resistant rutile grade designed to give greater hiding power and tinting strength in emulsion paints at higher levels of pigment volume concentration. Note: Special Properties of Cyanamid Unitane OR-560 Titanium Dioxide: -High brightness -Greater hiding power and tinting strength at elevated PVC levels -High oil absorption -Low sheen -GOod dispersion in all types of equipment -Good chalk resistance and color retention. Typical Physical and Chemical Data [101]: TiO2 content 90% minimum Additives: A1203, SiO2 Specific Gravity 4.0 Solid Bulking (lb/gal) 33.3 Bulking value (gal/1b) 0.0300 Approximate pH 7.8 Oil absorption 26-28 Average Particle size 0.23 micron Refractive index (Rutile Ti02) 2.76 Titanium dioxide pigments are manufactured products and are of two distinct types, anatase and rutile (a-titania and r-titania). Both are composed of very fine, colorless crystals of the same compound, Ti02, but the structure of the crystals may assume either of two forms, which leads to two types of pigment with different properties. A third form, "brookite," also exists but 64 as yet has no technical importance [99]. 3.1.2.2 Colored Pigments The field of colored pigments is vast with respect to both types of materials and shades of color. Hiding power of color pig- ments does not lend itself to a numerical system.. Several colored pig- ments from Pfizer Minerals Pigments & Metals Division have been selected for this study. Their brand names and their properties are as follows [102]. RED: Pure Red Iron Oxide R-2900 Pure REd Iron Oxides, Fe203, are produced by several methods which impart individual properties for specific applications. The R Type Reds, made by the thermal decomposition and oxidation of iron salts, are avail- able in a range of colors from a light salmon red to a deep bluish maroon. Thesepigments are high in chemical purity, and contain a minimum of silica, alumina, calcium, and soluble salts. Properties of Pure Red Iron Oxide R-2900 Iron Oxide 99.5% Specific Gravity 5.15-5.20 Hiding 700-1100 sq.ft/lb 325 Mesh Retention less than 0.10% Oil Absorption 13-23 Particle Shape ‘ Spheroidal GREEN: Pure Chromium Oxide Green 6-4099 Pure Chromium Oxide Green, Cr203, the most stable of all green pig- ments is made in a range of shades for use in paints, ceramics, rubber, 65 plastics, concrete products, and other applications requiring superior durability. Properties of Pure Chromium Oxide Green G-4099 Chromium Oxide 99.0% Specific Gravity 5.20 Hiding 475-575 sq.ft/lb. 325 Mesh Rentention less than 0.10% Oil Absorption 13-19 Particle Shape Rhombohedral. V-8295 LOW-INFRARED-REFLECTANCE PIGMENT V-8295 is a reddish-blue inorganic pigment manufactured without arsenic an important consideration in worker and consumer protection. It is the production of FERRO Corporation Color Division. Typical Physical Properties Composition CoLiPO4 Specific gravity 3.80 Average particle size (u) 2. Bulk density (gm/cu. in) 11.7 Weight/solid ga1(1b) 31.7 Residue (+ 325 mesh screen) 0 Oil absorption (lb oil/100 lb pigment) 28.7 PEMCO LG-3503 Virgo Blue Stain This stain is a dry blend of two of SCM Glidden-Durkee standard glaze stains [127]. Turq. GS-ll9 95.0% B/Green GS-ll7 5.0% The Turq. 68-119 is a calcined inorganic stain that is made from 66 Zr0 V O and SiO 2’ 2 5 2' The B/Green 65-117 is also a calcined, inorganic stain, but is manu- factured using a combination of COD and Cr203. Both V-8295 and LG-3503 pigments were used to test our method in the ultraviolet region. 3.1.3 Pigment Dispersion We have already seen that paint consists essentially of two components, the dry pigment and the liquid vehicle, which must be mech- anically mixed. The basic operation is the dispersion of a pigment into the vehicle. For a stable dispersion the liquid must wet the solid particles. In this study the pigments are powders. In the dry condition the particles are largely aggregated into clusters. Making paint is breaking-down the aggregates, and dispersing the individual particles into the vehicle. The dispersion process is to be distinguished from the grinding process, a reduction of the primary particle size. Modern treatments of pigment dispersion are given in references [103,104,105, 106,107,108]. In earlier times, and even well into the present century, the professional painter often mixed his own paints by hand, stirring the pigments into oil, varnish, or aqueous medium with a stick in a bucket. In such hand-mixed paints, the degree of dispersion of the pigment is [far from complete. Larger or smaller amounts of the pigment are present as large, badly wetted agglomerates, and the resultant film usually has a sandy finish. To obtain a smooth, glossy surface the pigment must be properly dispersed, or ground, a process for which a wide variety of machines exists [109,110,111]. A machine-ground paint has the advantage 67 over a hand-mixed paint that the pigment content may be very high and yet the paint will be easily brushable. Starting with a dry pigment and a liquid vehicle, several separate stages in the dispersion process may be distinguished [105,112]. First is wetting, defined as the replacement of the pigment/air interface by the pigment/vehicle interface. Second is mechanical disaggregation. Third is stabilization of the dispersion, defined as the uniform distri- bution of dispersed particles throughout the entire workable mass. These three processes, though distinct, may occur almost simultaneously. Dispersion is a generic term which does not fully describe the state of the material. A dispersion may be deflocculated, flocculated, or agglomerated, or a combination of the three. Breakdown of the aggregates takes place before dispersion of the individual particles. It is not always necessary, however, to break down the aggregates completely, and for each type of paint there is an acceptable degree of dispersion. This quantity, commonly known as fineness of grind, is measured usually by a Fineness of Grind Gage as described in ASTM D 1210-64 [113]. The gage consists of a hardened steel block (178 x 64 x 12.7 mm) whose top surface is ground flat and smooth. Centered in the top of this block is a shallow channel 12.7 mm wide by 128 mm long which is tapered uniformly in depth (lengthwise) from 100 u at one end to zero depth at the other. The scraper is a wedge-shaped steel blade (89 x 38 x 6.4 mm) with the straight edges on the 89-mm side rounded to a radius of 0.25p. The test procedure is to place a puddle of the test paint in the deep end of the channel (gage resting on a flat horizontal surface). With the rounded edge of the scraper, the paint material is drawn down the length of the channel with uniform motion and with sufficient pressure to U 68 wipe clean the flat face of the gage. The thin wedge of paint remaining in the tapered groove is then immediately inspected. At some point along the channel, coarse particles or agglomerates will become visible. The finer the dispersion, the greater the distance from the deep end. The degree of pigment dispersion as measured by the fineness-of- grind gage, is directly related to surface appearance; gloss, smoothness, texture, opacity, tinting strength, floating, flooding settling, sagging, film durability, gloss retentation, and other special properties. It should be remembered, however, that the fineness-of-grind reading is only indicative of the largest pigment agglomerates present in the pigment dis- persion. It does not furnish any real index as to the particle distribution within the pigment dispersion. The performance characteristics of paints in a given formulation are significantly controlled by the relationship of pigment and vehicle, specifically their respective volumes. The pigment-vehicle volume re- lationship will determine gloss, drying, stain removal, brushing, flow, holdout, viscosity, and other properties. This relationship is called Pigment Volume Concentration (PVC), and is expressed as the percentage of pigment volume to total volume. Thus PVC = Pigment Volume x 100 Pigment Volume + Vehicle-Solid Volume When successive small increments of binder are added to a paint with extremely high pigment volume, a point is reached at which the paste or semi-paste product undergoes a sudden change in such properties as consis- tency, leveling, gloss, and permeability of the dry film [114]. This point, known as the Critical Pigment Volume Concentration (CPVC), is the specific 69 PVC at which the vehicle demand of the pigment is precisely satisfied; i.e., there are no voids among the pigment particles and there is no excess vehicle. The concept of CPVC has been most intimately associated with its effects on the performance properties of cured coatings. Such paint characteristics as gloss, blistering, rusting, hiding, scrubbability, stain removal, tensile strength, and many others are known to change radically with a transition through this point. Many excellent reviews of the subject of pigment concentration have been published [104,105, 114,115,116]. 3.1.3.1 Pigment Volume Concentration Practicable application and flow properties usually require that the pigment volume be reduced slightly below the critical value. In this study the highest PVC is 15% PVC, and the other PVC's are 1%, 5%, 10% PVC. The proportion of pigments to 1 gm of vehicle dePont ELVACITE formula 23 for 1%, 5%, 10%, and 15% PVC calculated from the relation, PVC = Pigment Volume x 100 Pigment VOTUme + vehiETe-Solid VOTume ' is as follows: PVC TiO2 Fe203 Cr203 white red green 1% 0.00955 gm 0.01236 gm 0.01242 gm 5% 0.04977 gm 0.06439 gm 0.06470 gm 10% 0.10507 gm 0.13593 gm 0.13659 gm 15% 0.16687 gm 0.21589 gm 0.21693 gm 70 It is convenient to obtain the lower-PVC paint by adding the vehicle to the higher one. Consequently, we prepared paints of 15% PVC, and then obtained the lower PVC's by adding the vehicle (duPont ELVACITE formula 23) in the ratios calculated according to the above information. Starting from 1 gm of paint at 15% PVC, we added the following amounts of vehicle to obtain the required PVC of various paints. For white TiO 2 , add 14.11751 gm to obtain 1% PVC add 2.01635 gm to obtain 5% PVC add 0.50407 gm to obtain 10% PVC . For red Fe203 , add 13.54656 gm to obtain 1% PVC , add 1.93516 gm to obtain 5% PVC , add 0.48379 gm to obtain 10% PVC For green Cr203 , add 13.53449 gm to obtain 1% PVC , add 1.93351 gm to obtain 5% PVC , add 0.48338 gm to obtain 10% PVC 3.1.3.2 Preparation of Paints The required quantities of pigment and vehicle were mixed and dispersed in a Waring Commercial Blender - model 5011. This equipment is designed for high-speed blending, pulping, mixing, or reducing of materials. The container and jar lid were made of stainless steel. There are two switch settings for speed, L0 and HI. Sharp stainless-steel blades whirling as fast as 23,000 rpm. almost instantaneously reduce ingredients 71 to a homogeneous mass. The capacity of this blender is one quart mixing capacity. The procedure was the following: (1) Measure the required quantity of vehicle into the blender con- tainer. (2) Weigh the required quantity of pigment and add to the vehicle in the container. (3) Mix the pigment-vehicle combination with a glass rod and ob- serve for fluidity. If fluid, proceed to Step 4, otherwise continue stirring until fluid, then proceed to Step 4. (4) Mix for 5 minutes in disperser. Start at L0 speed for two min- utes or so, follow at HI speed for about three minutes. A free-flowing paste composition is obtained, which is left to stand for a while to allow entrapped air to escape. (5) Measure fineness of grind with Fineness-Of-Grind Gage. If the degree of dispersion is not in the required range, go back to Step 4. Repeat until the required degree of dispersion is obtained. Fill the paint in the tin can and cap tightly. 3.1.3.3 Preparation of Films Many methods and devices for preparation of films have been developed. A fairly complete description of these methods and devices will be found in Gardner and Sward [117]. In the present study the films were cast with Microm Film Applicator Adjustable-Clearance Model (Paul N. Gardner Company) with 23 x 30 cm vacuum plate. In previous studies most workers prepared films on a Morest chart, black and white glass, ceramic tiles, or transparent plastic film, and then measured film reflectance without removing the substrate. Because the reflectance of the substrate 72 is often changed by the applied paint there are unavoidable optical effects of such supported films -- even though there is a method for correcting for these effects [118]. Therefore in this study, the films were free, i.e., they are removed from the substrate after drying. Many methods have been suggested for obtaining free films [119]. Since the vehicle in this study adheres poorly to certain plastic films, a duPont Mylar Polyester Film Type S Gauge 100 with nominal thickness 0.001 in. was chosen as substrate for casting the paint films to obtain free films. Mylar films were cut to the same size as the vacuum plate, i.e., 23 cm x 30 cm. The mylar film was placed on the vacuum plate lying on a level bench top. Drawdowns of each paint were made on the Mylar sheet with the Film Applicator. The cast films were allowed to air-dry at room tempera- ture for about a day or more, depending on the thickness of the film. Without removing the substrate the dry films were then cut to obtain three 2 by 2 in. pieces for each film thickness. To obtain the double-thickness film, two pieces were patched together, paint side toward paint side, with mineral oil of refractive index 1.5 as a glue, and placed down on the base plate and carefully rolled over with the glass rod to ensure adhesion be- tween those two films. The substrates were removed off the films before measuring the reflectance. Upon removal from the substrate showed a high gloss on a clean surface. The reflectance of this side of film was measured immediately after the substrate had been removed. Various film thicknesses were cast for each paint to ensure obtaining satisfactory conditions, i.e., uniform thickness of single films, accurate double thickness, and reflectance to satisfy Eq. (77). Paint film samples were all prepared from three single-pigment lacquers. Each pigmented lacquer was prepared at four different pigment 13 O ‘14 R81 (TIER) 73 volume concentrations, PVC, 1% PVC, 5% PVC, 10% PVC and 15% PVC. Samples of the mixed pigment lacquer films were prepared at 15% PVC only. The experimental values of infinite reflectance are the averages re- flectance of infinite film thickness, which were obtained by two methods: (a) patching thin films together one by one until the infinite film thick- ness was obtained; (b) casting the thin film and letting it dry and then repeatedly casting over it until the infinite thickness is obtained. It is quite difficult to cast a very thick film to obtain the infinite thick- ness with only one drawdown because of the formation of air bubbles. The first method seems likely to be the easiest way to obtain the infinite thickness film but considerable care must be exercised to avoid effects such as the rubbing effect which may give rise to incorrect measurements. The second method seems to be better, but it is quite tedious and time- consuming. We preferred the first method because of its similarity to the method of obtaining the double film thickness. To ensure the repro- ducibility of the infinite reflectance, at least four or five specimens had been used to find the average infinite reflectance for each sample. 3.2 Film Measurements In the application of K.M theory, which relates the degree of reflectance to film thickness, the equations contain two parameters K and S which characterize the optical properties of the paint film. Con- versely, these equations permit K and S to be determined from measurements of reflectance and film thickness. For the K.M single-constant theory we do not need to know the film thickness, but for the K.M two-constant theory, we must know it precisely. Without good film-thickness measure- ments, the results can be neither precise nor accurate. The second key to 74 quantitative measurements is a good evaluation of reflectance of the apint film over the balck substrate. A small error here produces a large error in S because of the way that S is calculated from the re- flectance values. 3.2.1 Measurement of Film Thickness Many methods for measuring dry film thickness are known, but only a few are suitable for all types of specimens and circumstances [120, 121]. In this study, the thickness of a free chip of paint is measured directly with a commercial micrometer caliper. ASTM Method D 374, "Thick- ness of Solid Electrical Insulation," contains instructions for calibrat- ing and using Machinist's Micrometers. A preferred procedure is set forth in ASTM Method 0 1005, "Measurement of Dry Film Thickness of Organic Coat- ings." The method should give an accuracy of i0.l mil provided the film and panel surface are not distorted. Measurements of film thickness had been made after the measurement of reflectance for each film, so that the films will not be soiled inadvertently by the micrometer. Three or four measurements were made of each film, and the average was taken as the thickness of the film. 3.2.2 Reflectance Measurements All surfaces reflect some fraction of the incident energy. The nature of the reflected energy depends not only on the type of incident beam, but also on the physical properties of the material under examination. Hence measurement of reflectance can serve to determine the optical prop- erties of materials. Measurement of reflectance spectra is not used as fully as it might 75 be because such spectra are not easily interpreted. The technique is, however, hardly novel, particularly if one is speaking about diffuse- reflectance measurement, which deals with the measurement of light re- flected in a diffuse manner (in contrast to specular or direct reflect- ance). Reflectance measurements are particularly suited for solid samples and thus provide an analytical tool in cases where samples are difficult to dissolve. Diffuse-reflectance measurement is rather widely applied in the following areas: dyestuffs, printing inks, paints and pigments, text- tiles, paper, ceramics, building materials, mining, metallurgy, and others. In this study the reflectance was measured with a Beckman DB-GT record- ing Spectrophotometer. 3.2.2.1 Spectrophotometric Components The Beckman DB-GT Spectrophotometer in this study is a single- beam/double-beam, ratio-indicating or ratio-recording instrument which can make rapid transmittance and relfectance measurements in the near-ultra- violet/visible region. The spectrophotometer is housed in a compact case with front-panel controls and a direct-reading meter. Major components include two sources, source power supplies, monochromator, beam-switching device, detector, scale-expansion and zero-suppression controls, and amplifying, regulating, and log-converting circuitry. The reflectance attachment for the Model DB-GT Spectrophotometer consists of an integrat- ing sphere equipped with a suitable end-on photomultiplier tube, of sample and reference holders, and of a lens system. The spectra were recorded on a Beckman Model 1005 Ten-Inch Laboratory Potentiometric Recorder. The optical diagram of the DB-GT Spectrophotometer is illustrated in Figure 3.1. Radiant energy from the tungsten or deuterium lamp is directed as a 76 beam by the source-selector mirror to a seven-position filter wheel. In the filter-wheel position, 200 nm to 300 nm, the entrance beam passes un- restricted. Filters mounted in the other six positions allow transmission of specific spectral regions of the entrance beam. 13 use 11 ‘\ ‘D/ 1.0’ .0, 7 l. Deuterium Lamp 7. Collimating Mirror 2. Tungsten Lamp 8. Grating 3. Source Selector Mirror 9. Condensing Lens 4. Filter 10. Sample Beam 5. Entrance Mirror 11. Reference Beam 6. Slits 12. Detector Focus Mirror 13. Photomultiplier Detector Fig. 3.1 The Optical Diagram of the 08-61 Spectrophotometer. 77 3.2.2.2 Reflectance Measurements of Films The measurement of reflectance is one of the most difficult problems of spectroscopy. Reflectance is not a simple specific property of a sample as is absorbance or polarization. Reflected light is influenc- ed by many parameters such as illuminating light, illuminating aperture, surface texture of the sample, polarisation and absorbance of the sample. The light can be reflected diffusely, specularly, or as a mixture. In applying K.M equations to the calculation of K and S coefficients, various errors may arise because of deviations of the actual experimental conditions from the ideal conditions assumed in the equations. Many of the previous workers find K and S from K.M equations by measuring Rm , Rg, R and X using Eq. (11) and Eq. (22). This method has been used successfully to obtain K and S for very weakly absorbing (nearly white) powders. But it is not a satisfactory solution to the problem for a sample of the rela- tively high absorption coefficients, for even a thin layer appears infinit- ely thick, making it difficult to determine R, R.» and X to a satisfactory degree of accuracy. Many other authors find K and S from K.M equations by measuring R0, R, R9 and X, first finding the K.M parameters a and b by using Eq. (29) and Eq. (20), then finding R“) from these parameters and the coefficients K and S are them obtained. There are some disadvantages in this method; the trouble in obtaining the K.M parameters a and b, the un- certain value of Ru, and the complicated phenomena of the substrate re- flectance, which have been mentioned before. To test the applicability of the K.M theory as far as practical con- ditions are concerned, it is first necessary to investigate how far the assumptions involved in this theory can be fulfilled, and also whether the experimental measuring conditions limit or even exclude the 78 applicability of the theory. For reasons discussed before, the equations of the K.M theory have been derived for diffuse incident irradiation. Therefore, we are limited practically to measuring diffuse reflectance only. The Beckman DB-GT Spectrophotometer is usable for these purposes. To obtain the reflectance of the sample over the ideal black substrate, most of the previous authors used black glass or various designs printed in black and white on paper such as the Morest chart. These black sub- strates, however, do not give zero reflectance. In the present study the Ultraviolet Transmitting visible absorbing filter of Corning Glass Color Filter, Violet (Corning 7-54) served as the black substrate, which gives zero diffuse reflectance almost throughout the visible region. In our technique, Ra° , together with the coefficients K and S, can be determined by reflectance measurements on the two films with a thickness ratio of 1:2 over the ideal black substrate. Diffuse reflectance spectra were recorded with a Beckman DB-GT Spectrophotometer equipped with a re- flectance attachment having BaSO4 as a reference. In all recording of the spectra, numerous repeated scans were made to insure the reproducibility and accuracy of the data. The instruments had been carefully calibrated and always had been checked for accuracy by measuring Standard White BaSO4, which should give 100% reflectance throughout the visible region, and by measuring the Violet filter (Corning 7-54) which should give 0% reflectance throughout the visible region. The values of reflectance of two films, one as double the thickness of the other -- R0] and R02 -- as obtained from these measurements were then checked to determine whether they satisfied the condition in Eq. (77) at every wavelength. This check can be easily made with the aid of 79 tabulated values for inspecting R01 and R02 in Table A.l Appendix I. The data for the set of films (two films with a thickness ratio 1:2) that did not satisfy this condition should be deleted. And again when the films thickness had been measured, the data for the set of films that did not give a thickness close to the ratio of 1:2 should also be deleted. This descipancy could have arisen because of the nonuniform thickness of films. The recorded reflectance spectra R0] and R02 of the valid set of films were then read out at 16 wavelengths from 400 to 700 nm at inter- vals of 20 nm. These reflectance values then were used along with film- thickness values to find the infinite reflectance Rm. and the coefficients K and S, at each wavelength for further applications. It is important to note that the reflectance values read out directly from the reflectance spectra are the values relative to BaSO4. When the absolute reflectance values or the surface correction are to be considered, the inspection of R0] and R02 should be carefully made to insure that they still satisfy the condition in Eq. (77). 3.3 Calculation All calculations of K, S, K/S, Ru, and other values from measured reflectance spectra were made on the CDC 6500 computer with programs written in FORTRAN language. These computer programs have been compiled in the Appendix II. Four different approaches dealt with the question of absolute reflection versus relative reflectance: (a) With the relative reflectance values, i.e.. with the measured reflectance values directly without any correction or interpolation. (b) With the absolute reflectance values, i.e., through Eq. (44) for each measured reflectance value, employing the absolute reflectance 80 of standard white BaSO4 as reported by Gillespie et_gl, [72]. The values for converting reflectance relative to BaSO4 to the absolute re- flectance values are tabulated in Table A.3 APPENDIX I.‘ (c) With the relative reflectance values with surface correction, utilizing (l) the Saunderson correction, computing R from Eq. (50); and (2) the Duncan method, computing R from Eq. (55). (d) With absolute reflectance values with surface correction, with both the Saunderson correction and the Duncan method. The measured re- flectance values were first converted into absolute reflectance values, and then subjected to surface correction. 3.3.1 Determination of R0° , F(Rco ), K and S for Individual Pigment From the known values of R0] and R02, the values of Rm, can be determined directly either by using Eq. (64), or first finding W and Y from Eqs. (61) and (57) and then computing Rco using Eq. (71) or Eq. (43). The same result may be obtained by first finding 0 and B from Eqs. (69) and (70) and then calculating R“, from Eq. (10). The values of R“, with the corresponding values of R0] and R02 have been tabulated in Table A.2 APPENDIX 1, which may be used to find the values of R” when R0] and R02 are known. To check the applicability of our method, the calculated reflectance values Ra, have been compared with the measured reflectance values R,,. Various samples of infinite thickness films were prepared and repeatedly measured for Ra,, and the average of all experimental values were used 81 for comparison. Note that if we started to find R0° by using absolute reflectance values or surface correction, it would be necessary to convert back to relative reflectance before comparing with the experimental re- flectance values Rm . When the values of Ron were known, F(R” ) can be found by using Eq. (72) or can be read directly from Table A.6 in APPENDIX I. The values of K and S then can be determined using Eq. (62) and Eq. (74) or other appropriate equations when film thickness was known. Computations were made at 16 wavelengths from 400 to 700 nm in 20 nm steps so that the results will be generally applicable to colored paint film. 3.3.2 TransformingyReflectance Spectra into CIE Coordinates In order to simplify the comparison of computed and experi- mental results, the reflectance data were converted to CIE chromaticity coordinates. The comparison is then stated in terms of the Hunter color difference formula. It should be noted, however, that our computations are generally applicable to any region of the electromagnetic spectrum and our application here to colored (visually) films is only a special case. The reflectance spectra are transformed into CIE coordinates by means of Eqs. (86) and (87) to find the tristimulus values X, Y, and Z, and Eq. (88) to find the chromaticity coordinates x and y. The 1931 CIE standard observer and the CIE standard source C were used. Seven sets of the tristimulus values have to be determined: one from the measured values and six sets from the six cases of calculated values, relative values, absolute values, Saunderson correction with relative 82 values, Saunderson correction with absolute values, Duncan method with relative values and Duncan method with absolute values. The measured values and the calculated values of color coordinates were then compared by transforming the CIE tristimulus values into the Hunter color coordinate space system by Eq. (89). The Hunter color- coordinate differences between the experimental and calculated values can be found by Eq. (91); calculated values minus experimental values. The color difference between experimental and calculated values were then computed by the color-difference formula, Eq. (90). Total color differ- ence AE, lightness difference AL, redness-greenness difference Aa, and yellowness-blueness difference Ab were then obtained and the results can readily be interpreted. Values of Ru, and K/S in all cases of the previous discussion may be obtained from the several Tables in APPENDIX I. The procedure for obtain- ing these values is illustrated in Figure 3.2 with each step in the dia- gram being described as follows: Step I T.l By Table A.1 in APPENDIX I, check the criterion R01 22 2 - 0R02 If R0] and R02 satisfy this criterion, then they can be used with our method and one may proceed to the next step. If they do not satisfy the criterion, that sample can not be used in this case. Step II 1.2 Find R” by Table A.2 in APPENDIX I. Step III T.6 Find K/S by Table A.6 in APPENDIX 1. 83 Using Relative Values (Using Absolute Va1ues Correction ? 1 , No I Yes STEP IV T.3 - Duncan Saunderson Correction ? STEP VIII No Yes T.5 STEP v1 1.4 1 5 [Sjtp 1' STEP 1:] 1.1 1.1 STEP I 1.1 STEP 111 ° ( T.6 STEP II STEP II STEP II ' STEP II T.2 1.2 -1-2 ' T.2 STEP V T.3 Saunderson] Duncan |STEP VIII) STEP II T.2 STEP IX T 5 1 5 STEP VII STEP 1x 1.4 1.5 , l , i _ a _ _ STEP III STEP III STEP III lSTEP IIII STEP III] T.6 ' T.6 _ T.6 7 T.6 , +1.6 Fig. 3.2 Flow Chart for Determining K/S and R00 in Various Cases by using Tables in APPENDIX 1. 84 Step IV T.3 Convert values R0] and R02 to absolute values by Table A.3 in APPENDIX I. Step V T.3 Convert values Rm obtained in absolute values back to relative values by Table A.3 in APPENDIX 1. Step VI T.4 Correct R0] and R02 values by the Saunderson correction using Table A.4 in APPENDIX I. Step VII T.4 Convert Rm obtained by the Saunderson correction back to the uncorrected values by using Table A.4 in APPENDIX I. Step VIII T.5 Correct R0] and R02 values by the Duncan method using Table A.5 in APPENDIX I. Step IX T.5 Convert Ra, obtained by the Duncan method back to the uncorrected values by using Table A.5 in APPENDIX 1. Since it is tedious to calculate K/S in cases of the Saunderson correction and the Duncan method from the measured Ra, each time it is needed, the values of K/S corresponding to different values of measured Ron for the Saunderson correction and the Duncan method are given in APPENDIX I, Table A.7 and Table A.8 respectively. Duncan has developed his method of correction from the correction equations of Saunderson [57] and Ryde [29], but his formula is misprinted in the original paper [59], and the incorrect formula has been referenced and used in some texts [46]. Duncan tabulated values of K/S obtained by his method corresponding to measured values of R,, and indicated that K/S is zero when the measured values R,, are 96% and higher. This is not true; the value of K/S is zero when R,, is 96% but then increases as the value of R“, increases as shown in Table A.8 of APPENDIX I. In most earlier work, infinite reflectance values were obtained by first determining the K.M parameters a and b. As we have noted earlier, 85 that method leads to some difficulty. In this study infinite reflect- ance values are Obtained directly without using those parameters. In many applications of K.M analysis, however, those parameters are re- quired and for those applications we have constructed Table A.9 and Table A.10 in APPENDIX I for the Kubelka-Munk parameters a and b corresponding to infinite reflectance values respectively. 3.3.3 Computing the Reflectance of Paint Films Containing More Than One Pigment In order to test the validity of the K and S coefficients and R00 values a number of film specimens containing two or more pigments were studied. If K and S or the Kubelka-Munk function, K/S, of the indivi- dual pigments are known, then the reflectance of the mixtures can be com- puted by the following methods:- (a) A Single-constant theory. The values of the Kubelka-Munk function, K/S, of the multi-pigment system can be determined by Eq. (78) when the values of K/S of the individual pigments and their concentrations are known. The values of R0° then can be found directly from Table A.6 in APPENDIX I or by solving Eq. [11]. (b) Two-constant theory. From the known values of K and S of the individual pigments, the values K/S of the mixture can be determined by Eq. (81). The values of Ra are then determined as in the single con- stant theory. All values so computed have been compared with the measured reflect- ance values. It is important to note as mentioned earlier that if the predicted reflectance values have been calculated from the absolute values and/or surface correction, they must be converted back to relative 86 reflectance before comparing with the measured reflectance values. 3.3.4 Computing_the Color of Paint Films Containing More Than One Pigment When the predicted reflectance of the multi-pigment system is known at enough wavelengths in the visible region, the tristimulus values X, Y and Z of the system can be found by Eq. (86) or the CIE chro- maticity x, y coordinates by Eq. (88). Computed values were compared with the experimental values by first transforming the CIE tristimulus both to the Hunter color coordinate system L, a, b by Eq. (39). Then the COIOYS difference is found by Eq. (90). CHAPTER IV RESULTS AND DISCUSSION Although several samples were prepared for this study, only a few of the crucial ones will be described in detail. A number of samples have been prepared in this study only to test the method. The results discussed here are intended to show how the method works. 4.1 Results of Calculated Values ROI, of Single-pigment Lacquers The most essential data in the K.M analysis are the values of Ra because the foundation of the theory is based on the reflectance of the infinite-thickness sample. Many methods have been described by earlier workers, but there are a number of disadvantages associated with them, all of which are avoided in the method develOped for this study. Tables 4.2, 4.3, 4.4 and 4.5 show the comparison of calculated reflectance values with the average experimental reflectance values of infinite-thickness films at 15%, 10%, 5% and 1% PVC respectively, for TiO2 pigment lacquer films. These experimental and calculated reflect- ance values have been transformed into color space and compared in terms of the Hunter color-difference units which are also shown in these figures. The same comparison for some samples of Fe203 pure red iron oxide, are shown in Tables 4.6 to 4.9: Cr203, pure chromium oxide green, are shown in Tables 4.10 through 4.12. The position of these colors 87 88 600 .40 450 A “.20 ‘ --.10 l 1 l l l l l .10 .20 .30 .40 .50 .60 .70 Values of x Fig. 4.1 Chromaticity coordinates by experiment of pigment lacquer films 110 (. in area A), Fe,0 (+ in area B) and Cr203 (x in area C). Aregs A, B, C and D are Reproduced at a larger scale in Figs. 4.2, 4.3, 4.4 and 4.5 respectively. 89 .27 .28 .29 .30 .31 .32 .33 .34 .35 .36 '36 1 1 1 1 1 1 1 l '36 .35 P - .35 .33 "‘ - .33 Z’ .32 - ‘l 500 a .32 o o 0‘) 3 .31 .. -( .31 E .30 —. . .30 .29 )- 500 .1 .29 .28 '- d- .28 400 .27 l l L l l J l J .27 .27 .28 .29 .30 .31 .32 .33 .34 .35 .36 Values of x Fig. 4.2 Chromaticity coordinates of TiO2 pigment lacquer films. 0 experiment at 1 PVC 0 calculated at 1 PVC 3 experiment at 15 PVC 0 calculated at 15 PVC 90 .41 1 l 1 1 1 1 1 1 590 .40 p / \ - 39 __ 90% _ .38 - .. 80% >, 24- .37 F- - o g 36 70% 500 E o .35 1|- ‘ I- C .34 - - .33 P’ _ .32 J 1 L I I I I I .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 Values of X Fig. 4.3 Chromaticity coordinates of Fe203 pigment lacquer films. 0 experiment at 15 PVC calculated at 15 PVC C) ‘ experiment at 5 PVC £5 calculated at 5 PVC 91 I I I I I I I I 47 i- V -i .46 - ‘ .45 - AA - >5 2+- - O .44 i- ‘ § \ '3 .43 - '\\\\\ - > 8 .42 h ‘ .41 - 1 .40 - '1 .39 I I I: L I I I I Values ofx Fig. 4.4 Chromaticity coordinates of Cr203 pigment lacquer films. experiment at 15 PVC calculated at 15 PVC experiment at 5 PVC I> D; C) (I calculated at 5 PVC 92 Table 4.1 Lists of notations and abbreviations used in Tables 4.2-4.12. Symbol Definition R A value that cannot be calculated. WL Wavelength. R01, R02 The measured reflectance of two film thickness, one double the thickness of the other, on black substrate. RIM The average reflectance of infinite-thickness film obtained by experiment. RIN, RIAR The calculated infinite reflectance obtained RISR, RIASR RIDR, RIADR DR, DRA DRS, DRAS DRD, DRAD X, Y, Z SMLX, SMLY DELTA L, DELTA A, DELTA B, DELTA E from the relative and absolute reflectance values respectively. The calculated infinite reflectance obtained through the Saunderson correction by using the relative and absolute reflectance values respectively. The calculated infinite reflectance obtained through the Duncan method by using the relative and absolute reflectance values respectively. The difference between the calculated and experimental infinite reflectance using the relative and absolute reflectance values respectively. The difference between the calculated and experi- mental infinite reflectance through the Saunderson correction using the relative and absolute reflectance values respectively. The difference between the calculated and experi- mental infinite reflectance through the Duncan method using the relative and absolute reflectance values respectively. The tristimulus values in C.I.E. color system. The chromaticity coordinates x and y respectively. The notations AL, Aa, Ab and AE used in the Hunter color difference formula respectively 93 Table 4.1 Lists of notations and abbreviations used in Tables 4.2-4.12 (continued). Symbol Definition COR. Correction ABS. Using the absolute reflectance values. REL. Using the relative reflectance values. 94 0.00. ance. once. onoo. osoo. coho. coco. och {NSOO onto. c~ooo once. o—oo. case. 0000. ooc financ- oooamo «mane. woman. use n. he Nana—o uhonmo magma. o-onN. 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Noao. .-0. 0000. .-0. .~00. .050. .~00. .~00. 0000. .~00. 0000. .-0. 0000. 0000. 00.0. 0-0. 00.0. 00.0. on. 00. 02000.: 0~0..0n u>u n 0 “I 40‘s. 1010 It: O'JQJDIJ Lo “tank I... 0;. 02.0.0100 001.1 00.0 004.0 00.: 0..2 2.: x.¢ ~00 .00 an . nau:.u.:..x..s .nomcu. 00010 n .0: nautqn 105 in the chromaticity diagram are shown in Figure 4.1. The chromaticity coordinates of TiO2 pigment lacquer films both experimental and calcu- lated at l and l5 PVC are shown in Figure 4.2. and of Fe203 and Cr203 at 5 and l5 PVC are shown in Figures 4.4 and 4.5 respectively. The notations and abbreviations used in the tables throughout this chapter are indicated in Tables 4.l, 4.l3, and 4.25. Now let us consider the over-all calculated infinite reflectance values obtained using our modified theory and technique compared with the experimental values. In Table 4.2, we see that the reflectance difference at all wavelengths, except 400 nm, is less than 0.0l. .Such small differences have no real meaning since this is about the same magnitude as the experimental error. In terms of the color coordinates, a Hunter color difference AE less than 5.0 is almost imperceptible. To obtain the best results, the film thickness must be as thick as possible, but the double film thickness must not exceed the infinite film thick- ness, otherwise the K.M parameters, K and 5 cannot be found. The re- sults will be worse if the film thickness is thinner as in Table 4.5. It is important to notice also that for lower PVC, a thicker film must be used in order to obtain the best results. If a film is too thin, the measured reflectance R0] and R02 will not satisfy the criterion in Eq. (77), and the infinite film thickness reflectance cannot be determ- ined. The results for the colored-pigment lacquers are worse than those for white ones. This is not a result of the analytical method, however, but is related to the experimental errors of measuring reflectance. a problem which will be discussed more fully in the section on error analysis. It is well to note that a specimen of constant reflectance 106 leads to greater accuracy in the computation since the error of wave- length measurement is not a factor. Consider the Saunderson correction and the Duncan method: the Saunderson correction cannot be applied at the lower reflectance (less than 0.04 for the single-thickness film) whereas the Duncan method can- not be applied at the higher reflectance (greater than 0.96 for the double-thickness film). This is because for the measured reflectance lower than 0.04 if the Saunderson correction is applied. the corrected reflectance values turn out to be negative, which has no physical mean- ing. In the case of the Duncan method, the K.M function approaches zero as the reflectance approaches 0.96. and the K.M function increases as the reflectance increases above 0.96. as has been pointed out in a previous chapter. This will cause R0] to be greater than R02 which would have no physical meaning. It is. thus recommended that if the surface correction is to be applied, the Saunderson correction is appropriate in cases of higher reflectance and the Duncan method should be applied only in cases of lower reflectance. The advantages and disadvantages of using absolute or relative reflectance values will be discussed in the section of prediction of reflectance and color of mixtures. 107 4.2 The Coefficients K, S and K.M Function K/S In the K.M theory the scattering and absorption coefficients, 5 and K. are defined as the fraction of light scattered or absorbed per unit path length in an infinitesimally thin layer of material. A theo- retical derivation of the absorption and scattering coefficients has been given Kubelka [40], Kortum [43], and an experimental verification of this resulted reported [l22,l23]. In that derivation, it is shown that the value of the absorption and scattering coefficient depends on the angular distribution of the light in the scattering material. If all the light passing through the differential layer is collimated and is perpendicular to the layer, then the Kubelka-Munk absorption coefficient will be equal to the linear absorption coefficient for the collimated light. However, there is usually some diffused light present. Therefore, some of the light takes a longer path through the layer and the Kubelka-munk absorp- tion coefficient is larger than the linear abSorption coefficient for collimated light. If the light is perfectly diffuse, the Kubelka-Munk absorption coefficient varies as twice the linear absorption coefficient for collimated light. Later in the derivation of the K.M theory, a constant factor of 2 arises [43]. This factor of two is sometimes eliminated from being carried through the equations by redefining the scattering and absorption coefficients as S = 25 and K = 2k. This should always be kept in mind, otherwise it may lead to somewhat confusing re- sults in applying these coefficients in K.M analysis. In this study we have chosen to use the coefficients K and S, and therefore the absorption coefficient. K in this work is the Kubelka-Munk coefficient which are approximately twice the actual physical absorption coefficient of the material. 108 Table 4.13 Lists of notations and abbreviations used in Table 4.14-4.24. Symbol Definition WL Wavelength. R A value that cannot be calculated. K, KA Absorption coefficient obtained by using the relative and absolute reflectance value respectively. KS, KAS Absorption coefficient obtained through the Saunderson correction using the relative and absolute reflectance values respectively. KD, KAD Absorption coefficient obtained through the Duncan method using the relative and absolute reflectance values respectively. S, SA Scattering coefficient obtained by using the relative and absolute reflectance value respectively. SS, SAS Scattering coefficient obtained through the Saunderson correction using relative and absolute reflectance values respectively. SD, SAD Scattering coefficient obtained through the Duncan method using the relative and absolute reflectance value respectively. FR, FRA K.M function obtained by using the relative and absolute reflectance value respectively. FRS, FRAS K.M function obtained through the Saunderson correction using the relative and absolute reflectance value respectively. FRD, FRAD K.M function obtained through the Duncan method using the relative and absolute reflectance value respectively. 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They depend on the system of pig- ments and method used. It may be well to bear in mind how these coeffic- ients are obtained. In Table 4.l4 to 4.l7 the values of scattering and absorption coefficients. 5 and K, and the K.M function of TiO2 white pigment lacquer at l5%. l0%. 5% and l% PVC obtained by various methods are shown. These absorption and scattering coefficients and K.M functions obtained by various methods for Fe203 pure iron oxide red. Cr203 pure chromium oxide green. are shown in Tables 4.18 to 4.21. Tables 4.22 to 4.24 respectively. These coefficients cannot be obtained if the infinite reflectance cannot be determined. We thus see that at low reflectance values where the Saunderson correction is unapplicable, these coefficients cannot be determined by using the Saunderson correction. The same is true for the Duncan method at the high reflectance values. The scattering and absorption coefficients ought to be constant at a specific wavelength for each pigmented lacquer. But it is found that the values calculated from K.M theory for scattering and absorption co- efficients depend on the thickness of the paint film on which the reflect- ance measurements are carried out. The calculated values for K and S approach a constant value only at high film thickness. To obtain the best results. it should be remembered that the double film thickness has to be almost as thick as the thickness of the infinite thickness film. Consider roughly the effect of changes in the pigment volume con- centration. PVC. As the PVC increases the scattering coefficient in- creases whereas the absorption coefficient decreases for TiO2 white pig- ment lacquer films. Both K and S are decreased as the PVC increases for 121 Cr203 pure chromium oxide green. Now consider the results obtained by using the relative and absolute reflectance values with and without surface correction. The results obtained by using relative and absolute values are not signifi- cantly different at low reflectance but seem to differ considerably at the high reflectance. Since one of the basic assumptions of the K.M theory is that the reflectance are absolute values, neglecting to use the absolute reflectance values will result in more serious errors at high relfectivities than at low reflectivities. The absorption coefficients K obtained by using surface corrections are almost about a half of those obtained by not applying the surface correction. It seems possible to say that the absorption coefficients obtained by applying surface correction are more physically meaningful. since it is about the same as the actual physical absorption coefficient of the material. It is true. however. that these coefficients do not always give the same results because of differences in the tests and the pigment system. These include different PVC's. different vehicles and different degrees of dispersion. 4.3 Results of Predicted R_ and Color of Mixed Pigment Lacquers The mixed-pigment lacquers were prepared containing two and three single-pigment lacquers in different proportions. These were all prepared at l5% PVC. using volume as the proportion of each pigment lacquer in the mixture. The infinite reflectance of these mixtures were obtained by averaging the experimental reflectance values. The chromaticity coordin- ates by experiment of each pigment lacquer films and their mixtures are 122 ~50 I T I 1. G ..40_ 025H+756 q >, “a osow+ 223+ soc ‘ 8 ‘2ow+ R :2 20 R+ g o" ‘606:‘50N+ so R .30_ 75N+25‘R _ .20 J 1 J .20 .30 .40 .50 .60 Values of x Fig. 4.5 Chromaticity coordinations by experiment of pigment lacquer films 1102, Fe203, Cr203 and their mixtures. N for T102 R for Fe203 G for Cr203 Antecedent numbers are percentage for each component of the mixtures. 123 shown in Figure 4.5. The calculated infinite reflectance and color coordinate space system compared with the experimental values are shown in Tables 4.26 to 4.32. Only the relative and absolute reflectance values without surface correction can be used for calculating infinite reflectance of every mixed-pigment lacquer films. This is because the single-pigment lacquer used in this study, T102, gives a very high re- flectance, hence the Duncan correction cannot be applied. and the colored-pigment lacquers, Fe203 red and Cr203 green. give a very low re- flectance at some wavelengths. and hence the Saunderson correction cannot be used. The results from the use of absolute reflectance values are not significantly different from those obtained using the relative values. This may result from the fact that the absolute reflectance values of the standard white BaSO4 are too high. Using relative reflectance values seems to be adequate in practical use. Consider the two color-mixture theories used. The single-constant theory give the better results than the two-constant theory. This is not a result of the theoretical advantage. however. but is related to the experimental errors of measuring and working procedure. a problem which will be discussed more fully in the error-analysis section. It can be seen from Tables 4.26 to 4.32 that the two-constant theory cannot be used without surface correction. The results obtained by using the two-constant theory without surface correction are greatly different from the experimental values. It can be seen from the figures that the results obtained by using two-constant theory with surface correction, where applicable, give the better results. It is sometimes argued that the single-constant theory is not 124 Table 4.25 Lists of notations and abbreviations used in Tables 4.26-4.32. Symbol Definition NL R REX RINM I, RINM 2 RNA 1, RNA 2 RMS 1, RMS 2 RMAS l, RMAS 2 RMD l, RMD 2 RMAD l, RMAD 2 ONE CONST. TWO CONST. REL., ABS. DUN. Wavelength. The value that cannot be calculated. The infinite reflectance of the mixture obtained by experimentation. The calculated infinite reflectance values of the mixture using single constant and two-constant theory with relative reflectance value respectively. The calculated infinite reflectance values of the mixture using single constant and two-constant theory with the absolute reflectance value respectively. The calculated infinite reflectance values of the mixture using single constant and two-constant theory with the relative reflectance value respectively. The calculated infinite reflectance values of the mixture using single constant and two-constant theory through the Saunderson correction with the absolute reflectance value respectively. The calculated infinite reflectance values of the mixture using single constant and two-constant theory through Duncan method with the relative reflectance value respectively. The calculated infinite reflectance values of the mix- ture using single constant and two-constant theory through Duncan method with the absolute reflectance values respectively. Single constant theory. Two-constant theory. Using relative and absolute reflectance value respectively. Duncan method. 125 Table 4.25 Lists of notations and abbreviations used in Tables 4.26-4.32 (continued). Symbol Definition SAUND. Saunderson correction. w TiO2 pigment lacquer. R Fe203 pigment lacquer. G Cr203 pigment lacquer. 126 Omh.rmN no mmapwnt mxb $3; 4:29.333 E: 1:. 33; 33.3.... no zofiiaxxou 0N3 candy a m a a a a a a a a a a souuh.on c0—oo.m_o o~€o~.o~ ~0m~o.~ no¢€h. m0ecm.~l a a m z a m m .z m a a a nommh.on chm0h.m~o omn0~.o~ 0hm~o.N ~o~¢h. cowem.~l u (pdwo m (hauo 4 (wduo buzz ampzar Zn wuzmawmuua z 1 a x . ... a a. . a. .. mm.~. m ..m~. ..MO. 0-:. «#0:. 000m. 550%. wn_~. aura. can . 000 . a x z 1 m e . z m . z . .mom. ”mm” «mm~. wwww. «05:. nhmc. 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This might be true. however. it seems likely that the single-constant theory is adequate for practical purposes. that is to say. that the errors in a predicted formulation that are due to the two-constant theory are about the same size or more as the errors in making up a formulation in practice. It may be well to bear in mind the following quotation in from the pioneering researcher whom utilized the K.M theory, resulted for the widespread use of this theory: Deane B. Judd [l24]. "The complete truth is usally too complicated to be of industrial interest. You can go broke with the truth. if it comes in a complicated package. Each worker must seek a compromise between simplicity and accuracy." It is not suggested here that a practical color-mixture formula should drop the two-constant theory whenever one wants to predict the color of a pigment dispersion. There will no doubt always be cases where the simpler single-constant theory is adequate. What is suggested is that in cases where the single-constant theory is found to be too inaccurate. the two- constant theory should be thoroughly examined. 4.3.1 Application of the Theory in the Ultraviolet Begion The K.M analysis is generally concerned with propagation and interactions of light with matter, i.e., scattering and absorption, not only in the visible spectrum, but also in any regions of the spectrum. The most important restriction on the K.M analysis is that it applies only to a single wavelength at a time. It is not specifically restricted to any re- gion of the electromagnetic spectrum. Although applications of K.M analysis 134 commonly are in the visible region, it is possible. for specific purposes. to extend the range into the ultraviolet or infrared regions. There is, however. a limitation concerning particle size effects. To reflect light of given wavelength efficiently, a particle must posses a diameter approximately one half of that specific wavelength or larger [126]. If a particle is small, compared with the wavelength, a ray of light can simply go around it without being refracted. The Beckman DB-GT Spectrophotometer, which also is capable of record- ing reflectance spectra in the ultraviolet region, was used to obtain re- flectance spectra in this region for testing the formulas and method developed in this study. Since the pigmented lacquer films of Ti02, Fe203 and Cr203 which were used for our measurements in the visible region, have reflectances which are too low in the ultraviolet region to be of any use, it was necessary to find other pigments. PEMCO Virgo Blue Stain and FERRO Low Infrared reflectance red pigments were found to have sufficiently high reflectances. The experimental and calculated values for the infinite reflectance of both the individual pigments and their mixtures were obtained in the ultraviolet region at 300 nm and 350 nm. The calculated results are in good agreement with the experimental results as shown in Table 4.33. It is clear that for the case of appropriate particle sizes and sufficiently reflectances, the K.M theory can be applied in the ultraviolet regions giving agreement between experimental and calculated results as good as that obtained in the visible region. 135 Table 4.33 Comparison of computed and experimental infinite reflectance values in ultraviolet region of PEMCO Virgo Blue Stain and FERRO Low Infrared reflectance red lacquer pigment films. Pigment lacquer wL R01 R02 R00 R00 computed experiment . 300 .025 .026 .02604 .026 BLUE 350 .070 .07l .071015 .071 300 .l98 .20l .20105 .202 RED 350 .274 .305 .30937 .307 50 RED 300 .045926 .053 ... 50 BLUE 350 .ll434 .l05 80 RED 300 .0852 .081 4. 20 BLUE 350 .lBl79 .150 33 RED 300 .036577 .047 + 67 BLUE 350 .094965 .098 136 4.4 Error Analysis There are two types of errorzsystematic and random. A syste- matic error will cause the value of a measurement to be displaced a fixed amount from the value of the quantity measured, therefore, a syste- matic error is a constant error. Factors which cause systematic errors include the ambient environment. the calibration of the measurement equip- ment, the measurement technique employed, and the observer. Instrumental errors depend on the instrument used and the manner in which it is being used. A random error has random causes and will cause several independent measurements of a quantity to be distributed randomly about their average value. The cause of the random errors is the uncontrollable and unnoticed fluctuation of the many influencing factors of the measurement. An important source for limitations in accuracy of the calculated values is the repeatability of the lacquer pigment films. There may be several reasons for this, and most are connected with the inaccuracy of working procedures and insufficient apparatus. The predicted result can, therefore, be valid only if the lacquer pigment films are obtained under the same conditions. The unavoidable errors for both experimental and calculated values will be now discussed in the following subsection. 4.4.l Errors of the Measured Reflectance Spectra The following are errors which could be caused in the reflecte ance spectra. (a) Measurement errors due to the instrument and instrumental cali- bration. In this study the Beckman DB-GT Spectrophotometer with the Beckman Model l005 Ten-Inch Laboratory Potentiometric Recorder was used. Under the best calibration; the error of the DB-GR Spectrophotometer by 137 recorder output is 1 0.5% with reference to the reflectance. This means that the error is higher at the higher reflectance than the lower one. Besides the systematic error of the instrument, the error in setting the wavelength is the most serious cause especially in cases of the inclined spectra. Only a few nanometers error in wavelengths may cause much error in reflectance values at the steep portion of the spectra. This will have the result that calculated reflectance values at the region near 400 nm for Ti02 and at various wavelengths of the colored lacquer pigment films will be less accurate than the other portion of the white one. The other cause of error is the readability in the graph paper recorded by the recorder. This error will cause a large percentage of error at low re- flectance. (b) Errors caused by specimens. Besides the inaccuracy of working procedures for the pigmented lacquer that casues some difficulty with reproducibility of the results, the preparation of films is the other cause of inaccuracy. It is not possible to prepare films of reasonably uniform thickness for all the lacquer pigments used. the double film thickness is not really double. This will casue inaccuracy in R0], R02 and of course the calculated R0,. There are many other environmental factors that may be encountered as sources of errors in specimens. One of them is relative to the standard reference. One general statement is always true: the greater is the deviation between the spectral distribution of sample to standard, the more likely is the measured ratio of sample to standard to be in error. In conclusion these errors may be classified into three types: (l) the nature of pigmentation system and films prepared for measurement. (2) the operational characteristics of the instrument being used and (3) faulty analytical techniques. A number of specimens should be prepared 138 for measurements and the average values should be used to ensure the reproducibility. It is found that starting from the same lacquer pig- ment in preparing films for measurement is more reproducible than start- ing from the different lacquer pigment of the same formula. The error in reflectance measurement leads to the invalid results. 4.4.2 Errors in the Determinations of K/S, S and K The value of K.M function, K/S, obtained with the technique modified in this study depends on the accuracy of R.’ as can be seen from Eq. (ll), where Rco depends on the accuracy of R01 and R02. As pointed out earlier that the best results could be obtained when the thickness of double film thickness is almost as thick as of the infinite thickness film. The thinner the film the more inaccurate result will be obtained. There are several possible sources of error in determinations of the absorption and scattering coefficients, K and S, S is dependent on the accuracy of R0], R02 and film thickness, X. The error resulting from the thickness, X, of the sample was caused by the nonuniform thickness of the film, the working procedure in measuring film thickness and the instru- ment being used. In an attempt to offset these possible sources of error in the measurement of X, at least five measurements of each film thickness were made and averaged. However, an error of about 5-l0% was still possible for the individual values of X. The error is even worse for the case of colored lacquer pigment films which have to be so very thin that the value of X is not large compared with variation in the accuracy of the instrument. The important thing to remember is that the absorption and scattering coefficients will tend to be constant only if the double film thickness approaches the infinite film thickness. 139 4.4.3 Sources of Error in Predictinngeflectance and Color of the Mixed Pigment Lacquer Films Errors occurred both in the experimental and calculated re- flectance values. For the experimental reflectance values. errors may be caused by the working procedure in mixing and the proportion of each individual pigment lacquer contained in the mixture. For the calculated reflectance values, there are several sources of error in each individual value as has been discussed previously. It is well to compare the results of the single theory and two-constant theory, since there are more sources of error for two-constant theory than the single one. This is eSpecially true for the values of film thickness. The accuracy of the micrometer used in this study is about i 0.1 mil with careful use. This is signifi- cantly large compared to the film thickness as which are of the order of 5 mils. This will cause quite a great error for the individual values of K and S, which explains why the single constant theory gives a more accurate result than two-constant theory, for the single-constant theory is not concerned with the thickness of films. 4.5 Conclusions Several useful conclusions can be drawn from this study. First, the modified theory and method, as developed here are shown to give re- liable estimates of infinite reflectance without the difficulties that arise in using the other methods. The second conclusion grows out of the observation that the coefficient K obtained directly from K.M theory with- out any corrections are almost twice the coefficient K obtained by applying the surface correction. As it has been mentioned in the section 4.2 that the K.M coefficients, K are approximately twice the true phenomena, this 140 seems likely to be true that the actual physical absorption coefficient can be obtained by using surface correction. In other applications, such as radiative heat transfer, if the K.M analysis is to be used, it is strongly recommended that the surface correction must be made. The third conclusion, this study describes not only the use of the new de- veloped formulas and method to predict the infinite reflectance of films containing a single pigment, wavelength by wavelength, but also the ex- tension of these results to mixed-pigmented films and color prediction. The best results in formulation using K.M analysis in the system studied will be obtained for color prediciton of mixtures if the single constant theory is used; and if the two constant is used, the surface correction must be used. Two constant theory cannot be relied upon, however, in the formulation of a system containing the white pigment. The fact that the two constant theory failed in this case does not mean that the theory has no value. This form is used in industry, and it is very successful in the case of mixtures of colored pigments and also at low values of PVC (less than 5% PVC) [67,78,125]. Our choice of experimental PVC values for tne mixture was dictated by the requirements that many paints are formulated at levels of PVC about this value and upwards to or beyond the critical PVC. When the PVC is so high that the pigment particles are not separated by distances comparable with their own sizes, the mixing laws, Equations (79) and (80), no longer predict the proper K and S for use in the K.M theory [67]. The system used in this study pro- vided a severe test for the K.M theories, firstly the mixtures containing white pigment TiO2 which the single constant is preferred. Secondly the lacquer pigments of 15% PVC were used for the mixtures which are too high for the proper predicted K and S. It is the achievement of these 141 conditions, plus a knowledge of the exact thickness of the films, which makes application of the two constant theory inapplicable for this study. The fourth conclusion, it has been demonstrated that the formulas and method developed here can be used to predict infinite reflectance in other regions of the spectra, specifically the ultraviolet. Finally, it is possible to say that the single constant theory-with or without surface correction-is adequate for practical use. In summary, the modified theory and method, as developed in this study have been shown to give reliable estimates of reflectance and color both for the single and mixed pigment lacquer films. The calculated re- sults were found to be in good agreement with the experimental determina- tions obtained in our laboratory. 10. 11. 12. 13. 14. 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Hemmendinger, Official Digest 37, No. 487, 895 (1965). G. Wyszecki and W. S. Stiles, Color Science, Concepts and Methods, Quantitative Data and Formulas, JOhn W1ley & Sons Inc., NIYT"l967. A. C. Hardy, Handbook of Colorimetry, Cambridge, Mass., The Technology Press 1936. Committee on Colorimetry, Optical Society of America, The Science of Color, New York, Thomas Y. Crowell Co.. 1953. P. J. Bouma, Physical Aspects of Colours, An Introduction to the Scientific Study of Colour Stimull’and'COTour SensatIOns, 2nd Edition, St.Martin‘s Press Inc., New York, 1971. W. D. Wright, The Measurement of Colour, Adam Hilger, London, 4th Edition 1969. ASTM Standards, Part 27, D2244-68, 390 (1975). David L. MacAdam, Color Measurement and Tolerances, Official Digest December 1965, pp. 1847-1531. Gunter Wyszecki, Recent Developments on Color-Difference Evaluations, in J. J. Vos. L. FT 0. Friele and W. 1.. WaTraven: Colorlle'fric's, AIC/Holland, c/o Institute for Perception TNO, Soesterberg-1972, pp. 339-379. Henry Hemmendinger: Development of Color Difference Formulas, JPT, Vol. 42, No. 542, 1970, pp. 132. National Coil Coaters Association Committee on Color, Visual Examples of Measured Color Difference, Reported by the Technical Section, Cemmittee on Color, Philadelphia (1972). R. S. HUNTER: Photoelectric Color Difference Meter, JOSA 48, 1958, pp. 985-995. E. I. duPont de Nemours & Company, Du Pont ELVACITE Acrylic Resins Properties and Uses, 1970, p. 33. Oil & Colour Chemists' Association, Paint Technolo Manuals, part two Solvents, Oils, Resins and Orlers, Chapman and Hall Ltd., London, 1961, p. 200. ()il & Colour Chemists' Association, Paint Technolo Manuals, part six, Pigments, Dyestuffs and Lakes; Chapman and Eall Ltd., London, 1966. p. 8. 1*1. G. Volz, Advanced Methods of Testin the O tical Pro erties of Pi nts, Progress in Organ1c Coatings, 3 - , I SEQUOIA S. A.. Lausanne, p.4. .‘ _. _.-..-_.-J'! J 1 . 101. 102. 103. ‘104. 105. '106. 107. 108. 109. 110. 1'11 . 112. 113, 114. 115. 116. 117. 148 Technical Data Sheet, PA 68, 5-65 5M; Technical Bulletin 2-5-26, Unitane OR-56O Titanium Dioxide, PA-41-5M-11, 163, American Cyanamid Company, Pigments DiViSion, Wayne, New Jersey. Tech. Report: Calcium Products, Pfizer Minerals, Pigments 8 Metals Division, TR-23-5, 1972. Charles R. Martens, Technology of Paints, Varnishes and Lacquers, Reinhold Book Corporation a subsidiary of Chapman-Reinhbld,‘lfic., New York, 1968, p. 495. W. M. Morgans, Outlines of Paint Technology, Griffin, London 1969. (fig T. C. Patton, Paint Flow and Pigment Dispersion, Interscience, 1 N. Y. 1964. R. I. Ensminger, Pigment Dispersion in Synthetic Vehicles, Off. ' Digest, Jan. 1963, p.71. G. D. Parfitt, Dispersion of Powders in Liquids with special reference to pigments, Elsevier Publishing—Company Limited, London, 1969. P. Nylen/E. Sunderland, Modern Surface Coatings, Interscience ' Publishers a division of John Wiley & Sons Ltd., London, 1965. Willy Herbst, The Dispersion of Orgenic Pigments in Printingplnk and Paint S stems ’With Various 1 es of Dis ersin E ui ment, Progr. 0rg. Coatings, 1(1972773) 207. W. W. McCarthy, Pigment Dispersions by Sonic Techniques, Off. Digest, 37 (1965) 1650. R. H. Schiesser, Yiu-Kwan Lui, W. D. Schaeffer and A. C. Zettlemoyer, Pi ment DiSpersion in High Speed Impeller Mixers, Off Digest, March 1962, pp. 265-285. V. T. Crowl, Dispersion, Flocculation, Flotation. in Pi ments, edited by D.lPetterson, Elsevier PUBlishing Co. Ltd., Eondon 1967. 1973 Annual Book of ASTM STANDARD, Part 21, D1210-64, p. 185. F. B. Stieg, The Influence of PVC on Paint Properties, Progr. Org. Coatings 1(1973) 351-373. D. 5. Newton, JOCCA, 45 (1962) 180. 11. K. Asbeck, Some New Approaches to the Concepts of Critical Pigment Volume Concentration, Off. Digest, 36(1964)529. Pi. A. Gardner and G. G. Sward, Paint TestingyManua1,l3th Ed., .ASTM Special Technical Publication 500, Philadelphia, 1972, Chapter 4.1, pp. 251-259. 149 '118. E. I. Stearns, The Practice of Absorption Spectrophotometry, Wiley-Interscience a DiVision of John Wiley 8 SONS, New Ybrk, 1969, p. 302. ‘119. The Annual Book of ASTM Standards, Part 21. ‘120. G. G. Sward, Measurement of Film Thickness, in H.A. Gardner and G. G. Sward, Paint Testifig—Manual, 13th'Ed., ASTM Special Technical Publication 500, Philadelphia, 1972, Chapter. 4.2. '121. G. Kampf, A Survey of Methods for Measuring the Thickness of Paint Films,lProgr. Org. Coatings 1(1973) 335-350. ‘122. L. F. Gate, The Determination of Light Absorption in Diffusing. Materials py a Photo Diffusion MBdél, J) Phys. DizlAppl. Phys., 4 4,104911971). '123. B. J. Brinkworth, On the Theory of Reflection by Scattering and Absorbing Media, J. Phys. D.: Appl. Phys., 4.1105 (1971). “124. D. B. Judd, Introductory remarks at Williamsburg, Va. meeting, TAPPI, Technical Association of the Pulp and Paper Industry, TAPPI, Vol. 49, No. 4, 1966, p. 110A. 125. C. I. Gandhi and G. C. Williams, Light Transmittance of Pi mented Film as an Index of Hiding_Power, Official Digest, Sept. 965, Vol. 37*part 2, P. 1111} 126. C. R. Martens, Emulsion and Water-Soluble Paints and Coatings, Reinhold Publishing Corporation, New York, 1965, p.68. 127. H. D. Headley, Private Comnunication. APPENDIX I TABLES This APPENDIX consists of 10 Tables: 1 'Table A.1 Table for inspecting R0], R02. In this table the minimum 1.: values of R0] for a given Rozthat is the limitation of the 5 Table A.2 Table A.3 Table A.4 Table A.5 formulas and method developed in this study are listed. From the known values of R01 and R02, the values of R0° can be found directly from this table. The relative reflectance values which were measured relative to standard white BaSO4 have been converted to the absolute reflectance values. The absolute reflectance values of standard white BaSO4 used in this calculation at 400, 420, 440 460, 480, 500, 520, 540, 560, 580, 600, 620, 640, 660, 680, and 700 are 0.995, 0.999 0.999, 0.999, 0.999, 0.998, 0.998, 0.998, 0.998, 0.998, 0.998, 0.998, 0.998, 0.998, 0.998 and 0.997 respectively [72]. The reflectance values with the Saunderson correction are listed corresponding to the measured reflectance values. In this calculation the values of k1 = 0.04 and k2 = 0.6 were used. The reflectance values corresponding to the Duncan method are listed. The values of k1 = 0.04 and k2 = 0.555 were used in 150 'Table A.6 'Table A.7 'rable A.9 151 computing for this table. The Kubelka-Munk functions. and Table A.8 are the tables of the Kubelka-Munk function corresponding to measured values Rm incorporating the Saunderson and Duncan correction respectively. These two tables can be used to find the Kubelka-Munk function with surface correction directly. and Table A.10 are the values of the Kubelka-Munk parameters a and b corresponding to R . These two parameters are necessary for some applicagions of K.M theory, such as in computing hiding power. 152 663. com. 30». moon or». now. on). «no. ~66. one. . .mm 66:. o6m. :63. ~69 oNo. 6o). 6o). moo. 6o1. dom. 5 .66 mm». 66». no». mmo. no. mom. 6o). ooo. 33:. ~36. 6 .6o as». om). um). 36). 66. an». our. 6m6. mm:. mm». . .om .w». 6.». ca». 0.6. 366. no). 6.». as). 666. no). a .66 6o». .66. 6.x. 66o. own. 36.. m6n. can. woo. 6L3. . .3? hnu. 6ao. ~16. cor. 666. 020. :66. 66». ~66. Jon. . .67 6pc. oco. :co. moo. con. won. Lon. moo. one. mnr. 5 .66 omn. oat. ~16. mox. mom. ~3.. can. c6n. 6n. n66. 5 .Ar man. 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PROGRAM TABLE, this program has been used to construct Tables A.l, A.2, A.3, A.4, A.5, A.6, A.7, A;8. and A.9 and A.l0. Only formula statements and some descriptive statements have to be changed for each table. 2. PROGRAM PREDICT, this program has been used for calculating R“, K, S, K/S. The difference between experimental and calculated reflectance values and also in terms of color coordinates. 3. PROGRAM MIXTURE, this program has been used for K.M analysis in cases of predicting the reflectance and color of the mixture. 182 10 lb (5 J0 “Av PRU‘1P‘A04 oUéoI6057o ' U ---T14c---LOAn MUUh --Ll--Ld ----- I run LUAUtH l NUNMQL I83 TuVLt CUL 0:00 FIN V3.0'P360 UPT=1 PHUUKAM IADLL(INPUTOUUIVUI) DIMLNDIUN NH(IUU)¢HA(IUleU)QRNIIOOvll) INILUtP NH PRINT 11 ll £0T1?Tf1H1o//////////966X9“TAdLE A. 2“//) H P l FUKMAT(5QXQ°TAULE FOR INDPLLTINU K019 HUZ“//o~8lo°(MlNIMUM VALUtb ABFlkgllCOPdtbPuNUING TO Given «ua)°//) K N u I“ POHMAT(JBX9°(HLHCtNl K04)“v«HAo°(PEKCtNI R02)“/914X9”ROZ°92H “92X. K“U.U“98Xo“0.1°08X9°U.Z“92A9“U.J’odxv90.“998X990.b°vdlv“0.0998A990. X7°QdKQ°U.h°9dKO“Uogoo7X0°HUC°9CH “CZKOgooUOde9°00IOOZXO9002QOZAO. XU.J“06AO°U.9‘odAv“U.D°9CX9°U.O°9&X0“O.7998X99Uooqul9°0.9°//9IQKOZ X8(£H°°)cblodh(dfi°“)) ”U 6 1:19100 .- :VAH 'U ill-l" II o—b— v—f' 11:0 Cleo :~ \p—p—op—o AVG C C .0 1C ”C H O A I ‘- v LAAAC AH—hfiT “4 H N*U‘ 1D}- ZIC I A ”00" I Cur N‘FJTVIIH “‘5‘ e FUHM tNU (7 UV Ill/Al FwA IAuLLD l -HnUuHAM----AuUHc>>- --LAbcLEC---CUMMUN----AOUHE>3— Iutht beltflfi AUUtHn LulPTLa LleA :IL) UUulUU . OIUOUd thLMbG 010002 CUNLCIU UIUbUd OCOO .-.—hr— L u U .— thh ----- ----- END OF MAP ---------- CDC 6500 FTN V3.0-P380 0PT=I 184 PREDICT rRCGRAM .I! IAUcOQ.’ A I. I I .I S 93 I 93 0 CI 0 S R R A I L BA 325220 F . o / R . V 32 I3A33N o E I I . I (H SIRIII o I o I . E I 9 IHFOD CI 0 I I I I I C ’2 OIIRH93 E ZI : R I AI I A .33 ‘AKZFH7IO .0 I . I I). A T 9TI.C91.99LQ, 0 .fihI I I .\ Re .I C 3A lfr()iflu( N .J.A o t. N on R E II (292.). 9A A .I85 7 I I o) O L MR S3533)E E=T3 I I R ZI I F 159.6(AIIAVD S oIS. X "K O (i. I E RIORDIHDIN N I IORE E Q I IA I R 02.) vIRDHLI 0 ./ :IS o D o I TI A IJZIR QADA O I / I) I N ZI I RR I E «((32 QII 9 VI .I. S OI a OI II I QI R V 3KI3I2RII9 A N IR I = I2 R S. I I IADI23 029I N 0 03 .0 Is TO I . o O T. R CASJII3IE G R IADSI I R9 0 I2 I A AICIISZIYU I C R220 90 Q) I II I L BZRHDA3SL9 S I $..A 3: SI I I/ I E ZJOQZIIAMI E M Err—c II..) .I I 5 AI A R VIIIRRDRSQ. D 3 o 0“ XI»... II N E ZI I IbZZQOAD ’I H .II EQU IR I U RI H H 2 933III OIL E 8 E: : S OCL II R L O I” I T 3III2.CRI9U L 9 II NIA 29 I A I. G I IabAJ...) 92ID P S IAUO IXV R o G V 29 0 H RJAKIIIJX 9 H 0 :II E 92 O ”I I. II AICADSZIL) A 0 INDS CE )I L E «I A N )) BHunflAafbHu, S .1 AVIA}. $Nr» 271 A C \Ii )o 0 ‘11 VIF QIHZIHRJI F I‘RI. 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II. II I SITE AIAPRO A N I» o .2 TEI OK. QUICUI O M II II YIa 5:... ..LA LIIIIDR s E opLIIZRO I d.JSIEII2/. «J .H UOUOI CU§IJIKJ QUI AHRRIIMI §EIEII U9 IIIIRD.) $R2HIIO .0 I AHH/HHAHI oI ..r A o. ./ IORZIZAIIX9 SAAAC6T3 IDOHo. I0“ FIIHSJBS ITRIo-I R QGIToloII/Io SOIII DC 9393 .56 CI H BUBF. .5 o IUQAQQ CA 121.43. 22 00 E/ oRII/ ...r ILLIII THIIIIII I 91.17. EIIIIIIAJIZ AXYZSOHS NICA Eh IOUHRRHE. .N :: HIDIO III 5. :HIIIIUIUOEAO III ...LTIII R ZAIIJZM- I2 ’L//IIIIIIF QCI OS 0 .00 OI CBIII. 05E 9 II TING 01. III ITIIAN o o. ISIII IHLIuCC... D.:AHIJCHHR 3AMU3 I“ 6H 9 9 O 0.... O’DUIE 9 (CLIAI OIOHOI 9 9 T FT; 0 OI II II 02I_3I\IIII IezIIzo U. .AIABIT 9H 0.2 OIH.F(.AI9 IAOOOOOUIJ:bOAutEOFIIZ/OOI.:II: odeK)IZ.N)I190RHRKII):)IA529 )(IP MIICOI QDIVMLIR /SOOOOI0I.IIOIoprIUIIIIIOfiJnUI: ..I II4 -I.HCI._DHOIISIehu_. : :CI...II:HHI:23UI= :: ASCI‘CISPIA hYEN.IIIIT./EIE QQTCUU. (3?. I Tzscd:UII III...CII9 :RI: I. .IIIIEAIIII: ..JI IN III) 0.. “$232 93 C 9 96 A A .0K 411er ATAIA IIOOI XXPDIPCIIIZI \ IOJASIIIHCIAIII .IO XIIIYI GE OJIJ IIIII FEAAHKJSUMOMIIOFDM IIDHNHNNUUDQbIII CE UICHI2IIIIIEUIIICHIIHIIIIIEbRIII OM..II\HI2\J)1.¢9.ITIAMAAHAJ\AMH 2 ..A.HEIIARIRIRAAAIAOA no I INOIoARI (U AAA (AR-MAIIA DUIC). (Jr) Hl/CIHJAJJINAfact. cuLUKOKLUHFFhUHUHuttEIIIIU NH C FIoR(.KUN ldC FIIIHAIKUN FIZC PD‘JKQSISI OXInurUPP PRFRFFDFRFTIIREFPFRHRHPHRD I... R IRIFSIAII RRR IPOPFSAAGI IHPR XXXXXXv/XX X X X X 0 0 OIO 20 0 0 I I I 2 nu 0 46 AU 0 I. 2 O i; 7. 2 I 2 a S I I I I 2 2 C C C C C C C S 0 S 0 S 0 S 0 S 0 S 0 5 0 S I I 2 2 3 .J 4 A S S 6 .O 7 7 185 CDC 6500 FIN V3.0“P380 0PT=I PREDICT FRCGRAM Ilifi‘ ‘ ‘ Lang I R E I R E I. I I U I I U I. S / L 0 / L II 2 I A 7. I A “>1 R 0 V R 0 V AI . I . I IS I . E I . E RA I I c I I c O I II I N II I N 0R SI I A DI I A ZI Id 5 T I2 0 T I. R. In C R. I. C I. 9. R E .0. R E re 0) . L O) O L RI ZI I F ZI I F 0’ II I E (I I E 5, IS I R ID I R .I RR I E RR I E I“ OI R V QI R I I. So I I So I U S. .. o T .. o L AI I2 I A I2 I o 21 II I L II I 5 RI I/ I E I/ I B 95 SI 5 R DI D A IA RII R E RI R E OR CAI I H 04 I H OI IOI G I IOI G I II I. 265 0 aOI 0 II I. 9 II L H 0 II L H II I“ 9 IR A 7| Q I A T. II S. IIoIa I IIDIo I AA 1A2 AquII W IDII u I). SIG III oII II IIIR II RR SRO 35R.Sa NII “DRGDZ N 09 II 3..-LI oIO OII “IIRJIO 015be O OI RoIHu III HQSRQ ISOQI TII OI oI/I TI?— OI oS/I .ISO OI I II T.4/21 CRRI TokoZI C 66I SUIb A I.IOI EQOIA I..9I EU335A565A 05505 DOIIR c O OZTIZI/I. O ORIIRIII 0. N77 . RI. IIUI A22.C()((1I C442 DIDDRO NooIEIIZI.D UII/HRDHHoI D//IIoIooI/Io DoIIITQTRS .ZEIIIanqédaIII. :ULI)??U...I NooCoSIII IHIIDNooOASIII IAoRRZNIIZR IIZoI.$IS3ITIZZII20.oUIDAISIIIRIIIII KIII:OI,\4.D3O HRHKIII:O.IU$QO 11:: ..KII... CI =II : HI 23:96: : :CI: II :«HI: “CEIIIICSSII ES)(I):R) (N)))ED)(I):R) (HAIIIEAAII HCIb(I)_IOXIIIIHCIDIII.IOXT1(((HCC(I ISRSbIIS OUDUD IDHDUIID DGIAAA IIAA FRO”) S 0 / i I FIIIHSIKUN I.CC FIIIHUIK‘UNNFIeC FFIZIIHAIKUNUIBC FFIEIIRAIKONLOHOHIOU// OALUUIRIASIIIOIRIASIII“RIASIII°I THE ABSOLUTE REFLECTANCE VALUES 0 / gRIASIIII'RSTBAIIII - 6 (i) 2)) ( 4 ((1, RlAb IIOO IooI I II TSSI bOID CbSUAbbDA oIR ESSAI A2 RooZRUIZR RooHETIHo 077.T I.) (SIIS C22.IU()2 SAoIA AQZRGDIo AIDIR NooIC AIO IRISF AII/ IlU) R./Ao C//)E0RAI NIIIH O OILI UIIITQTRD OGSIS : AAAGoooI ISAIA IEIZBNIIZR :QIISSIHRRRIIIII I:Rb=r\_OT= : :KII: O. II:AI55 IIICDDII III II I..UIIICLA AII RIIRIOXNIIIHCCII SSIISIEIDDDCPRDO AAS/A DSAAA IIAA “ALOGIRIADIII‘IRIADIII‘RIADIII'I ADIIII’RSIBAIIII I I I II. I \II III IIIII IOIID DAOIA IRIDF R./Aa G 0’1) 7IIRI ZIIo I Q. DID oSAIA nu 2 I. :I I‘ : OIISbIo I:RD:698£3/ARRIORAE:XIXR II: AI673UIIIADI 09 pLIEHE : IzN T IAIXOMF D:CI RIIRIOXIITATRIAIXOEONINI DDIIDIEIINHNIHMOICRZIIII AAD/A DINIHIOIR/O (III-IA ,N 0“, IO 80’ 3I FB/ III /I O0 /9I AOXE /IR IBNS 8/9C NI OI Elk/I obIR II:I IBQCE : OI OI ...? OHU Ix 'IIXOH.PL OzINNZIF A I OI OO O OIOLV I9 III-59 vAIH ILI=:I29IC LflRIIUoo C “DIOOROHGN IOIIIONSNA 9x OII OIAIT OIOIIXRIRC IRORFSSAGI RRR IRORFSDAGIIIRRR IIRRIRFSoAGI RRR IIRRIRFSoAGIHCPFPNIFA/ X X X X S X I 3 1U“ 3 3 4 4 3 3 4 4 C C C C C 0 5 0 b 0 S 0 8 8 9 9 C 0 I l l 1 x x xx I 5 S S S S C C 5 0 S 0 I a 2 3 I I I I X 135 IbI 661 66 140 [’1‘ N OI OH 02I3A IIN II IIIA OFRFH DIDID XX XX 0 0 S 3 4 I I I 5 CC 5 0 5 4 .3 5 I I I 186 CDC 6500 PIN vs.o-Pseo 0pr=1 PREDICT PROGRAM 7894 1111 7 8 9 4 0000 TTTT O nVlO AVA G)G(G)G( O) 0‘ O) 0‘ XRX: XRX: E:E\IE:E) U)DI.H.J)UI. 160 165 \I ) )\II)I \I)II(I( ((RWVDRD MNASADA 1717.1111 RODHRRRRE : : : : = : : U )))))))N 1111111111 0 .9 o. , c. 9 OT! 12345.07” ((((l‘(l\0 RRRRRRRC 14 0 l 2 170 =0 )) ): )) ZD(L) A A A S 0 I f L I. § 0 M $N 9)), 039 QLLLL (510 l(l\l\l\ 2L0 5 9 Y. ._ 2 : XOODEO 3000 EL)OIEVAYZU1\IOOO $0 L: 0......N L111 10()0N\I))IOI\= : .— 7:4EL3ILLLTTE))) LYONUUFDDUOOH((( MXDIXDIXYZCGIXYZ )=IOE-SO XYZ 01C 3 00 21 S O S 0 7 U 8 9 1 1 1 l 195 o l 9 N. \I \I \l \l \I F 3 / U! 2 3 4 5 6 S 9 a) R3 E E E E E E ) 8/ E! U D U D 0 U I F/ T. 9 9 9 9 9 A S 10 U 2 3 4 5 6 R XF T L L L L L D 0 26 NI. 8 8 B B B E ( 21 .15 D nu D nu D R ’9 O. 0 9’9)! 9 9 0 S /N R1 CO 2/3/4)5)6 : A {9 09 NX (5(S(/(/‘) , E .1 .9 Eo LoLoL/L/L/ l H 0: 90 R1 AOAOASASA/ ‘ .1 XA E9 DIOIUoDoDS E H E! ]R F9 OFOFQOCOOO D I C) ’0 FY )4)Q)l)l)0 I N1 ’0 [LL 293’4F5F6l W .E( 4! DH (X‘X‘Q‘Q‘F s ’ RS 9X S LbLbLOLOLh 5/ ER 8.... ) R9 0 OUQDXDXUQ E/ FD F9 9 09 059596$69X \I U. Fl‘ 6/ G Ll \’ \I O) O) ’\l 9)b 0 2 L5 19 1Q 0 06 l aU3OQ55509 \l = 9 A5 D) 9 o 9 c 9 "(1(1( 0‘. 0‘5 \I) \I 9 VS... N xa E 6 Q’VI/VIFYFVIOYOVO o L) 10 \l N E O 25- c OX/L/LbLbLlLlLrU (L l\ L 0K Cl 96 A XL/MSN 9M OMFMFMI Yl\ L 1‘ EC N:\:Ol P 7M9505K5X55555F (VI 8 L TI AINAQ s 25‘... '0’1019"995 T‘ D 8 CH T’QRX 99 )1) Q)9)G )0), RT! 0 IT C)1.U..C R O 9A1F203u‘ Q 05 0.0x OR 6 0 F7129 9 U EXTI\:DI\..DI\SI\L(SI\“ so 2 EE Ll\I 90 O T7Lx QXEKEXEXUX O /b s 9 RT FA 9x0 L A QLLXLULULHLALO )/ 9 pl ER).IA.H 0 N9 HUM—IMLMLM M M o ’\y \l N R01 90 C 129 5 gAaDAS 0.3 05L L) o L Ev... ((/O 09 9 96 9V 9V 9R 9R ’L (L 5 l\ HF F 9014 0 F R 9X) \I \I )U)U\IR Y‘ 0 L TN D’on 0 0X3162E3E4C5C0 .Z : O A I NHBI 099(N(V(T( ( (o ’5. \I T D G S 90F. 5 C OOZIZIZUZNZNZN L7) 1 0 NF "1(6/ M 9 B 9R 9T. 9L 90 ’0 90 (A) ( 0 2 I0 R:914 R EY )U)A)U)S)S)C XBL L G I:\I6 R El) 9 o E 09 AIS/“L1454R5Rb a.‘ A )110 AE TQNXB T IOT(A(EIB(E(E(N 2 . VI 0 )1(() PC \I 93F OXLYEYRYAYOYDYA O)l\ L‘LLL MN N111 0.0 N Cgrr. QM 9 9 ’N ON 0C 0LT. OLAB( 0A 1(291 I 9 '0) )¥_)E\IU)U)N 1(R OV..L CT IRIRs o 9901Y25354A5AbU (YU Sal—rt - ))D O C i 0 90x 6 xx ’(B(U(U(:D(\D(D 6(r3 M 0)LIL(\ 9E “(’91 Q BOXXOXQIAQ XQXOXO 55.! '\IL(\I\T XL H9199 H 39399999999999, .0 o ZLtLLH OanVlO‘vAG Ola/x Ooxoxoxoxoxox 7 .0?» :(LABQES/Eblwalug39/6911M61341415‘1b1 17IUOLEV:=SUI(RI(1R(Al‘l‘2A2(Z(2(2(2(2( ===N= D:)):N T T DIR T T! T T T T T 1 LLLTISIL((L.INMHNHN ’Mo NMNM/TNMNMNMN (((N( ((LL(NIRTI.RI)R91.HTAH.OLIHI.H.1HI LLLOLUFLABLLURU RORNOXROR‘UIEHUROVUH‘ ABVCDDIDDDCCPFRPFP vFlPFpFIDPFPFPFP X X X XX 0 80 0 0 0 0 0 0 0 U 4. 35 S 6 7 8 9 In 2 3 1 1 l 2 2 2 1 l 260 FCHM PHYN 250 FORM PRIN 260 FORM 200 O 210 215 220 225 230 PROGRAM 235 240 245 250 260 265 270 275 MAP trhuwufixzp‘ 01“. 35. 25 ---11M£---Lo FHA LUADER FRcDILT Y Tim: m ... C(‘XUO FUN!- U3 putCFTILOZ (fiqxfihabrrc U vu—«"txmc pm O-u'fi'D I I I I I I I I I I m 2 187 PREDICT CDC 6500 FTN V3o0-P380 OPT=1 PRINT 270vXI7)o¥I7192I719SHLXI71OSMLYI7IODLITIoDALITIIDbLITIQDEITI 270 fOkMATéIXQ'DUNCAN COR. AuS.° cax.5F10.5.6x.4r10.S//) P91 280 IOQHAT(//16X0’FIGo So COMPARISISN 0F CALCULATED VALUES IN TERMS x UF NCFLECTANCL AND COLOR FUR VARI ?? CA?E§ HI H°IIOZ7X9°THE AVERA KG? EXPERIM%NTAL VALUCS 0F INFINITE LM. H CKN 55 FOR XA VC' PRINT IlOoICODEIoICODEZvICODE3 PRINT 18090 PRINT 1301(HLI1101=10N1 PRINT 290 290 FUdWATIIXI* ‘ USING RELATIVE VALUES’I/I PRINT 3109((AK(I)91:19N19ISII)v1=1qN)v(FR(1191=19N11 310 £3?W?I§§39'K“94X9Ibibohl/ 1X9“5°Q“X916F804/lvIXQ’FR'O3X916F8o4/l/1 I u‘\ 320 FURHAT(5KQ o’oo USING ABSOLUTE VALUES°//) PRINT 3 09((AKAII)oI=loN)v(bA(I)91:1 N)9(FRA(I)91=1 330 ngfiATfiiéoo*KA°o3X916F8.“//9IX9°SA “03X 916F8.4//9IXO'FRA’92X916F8.41 « BQOXEESTAEII/le'... USING THE SAUNDERSON CORRECTION WITH RELATIVE VAL / PAINT 3SO.((AKS(I).I=19N)o(SS(I)oI=loN)v(FRS(I)9I=1vN)) 350 ggjnATéésq°K$°03X916F8o4/II1X9°SS°oJX91618.4/lo1X0°FRS’92X916F8041 -L 3boxfig§§e;§l//IX.QOO USING THE SAUNDERSON CORRECTION WITH ABSOLUTE VAL PR NT 370 ((AKASI ) =1N) (SAS( RASI I- JTUXFOAMAI(1X:*KA$“92§910F8:QI;IIA9°B SATZXOIbégoklls119:}RA59v1X916F8 .41// PRINT 350 380 FORMATIIXQ"9 USING DUNCAN METHO‘ WITH RELATIVE VALQ559/l1 39o FUQHAT(11.0KD*.3x.16+8.41/elx.°5 °.Jx.1o 5.4/Iqlxv°FHD¢c2xcl6F8.4) SSINT 2989((AKUIIIv1319N)c(bO(I)oI=10N)9(fRU(II91=ION11 «10 Foodnt((MDTTTAAA OOFHUU 00(T008 OITI19IK‘K IrUIIII/ICKI ' 9((1‘ bl‘ O 9691‘ 9) 0 “5T BERLIOU E HEFLECTANCE 0F SINGLE AND DOUBLE FILM THICKNESS ON 0 M L I F D C T N E M G l p O D FII N UNN U '9 0 311 R 5:: U EII K N99 C KI) Ab CIIHUU I) (E ) 1 0 O C R K S 9 O I U S I N L 2 I R A R I O V o 2 I! I H I E a c I C 0 II 1 N O I R A S ( ( T l S l o C H 5.00 9 I L 0 7E: = o 9 F S o 0)) 2) K E 2 9:99 (2 I . o II36 T. 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