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Axiphos GmbH Marketing, Trading and Consulting May 2002 On whiteness formulas Introduction After the recognition of whiteness being a increase in whiteness though. The situation special color attribute o objects much work has turns more complicated with the fact that a been invested in defining whiteness as a slight blue shade is also interpreted by the eye unique number that takes into account the as an increase in whiteness and leads to a colorimetric perception of the object under wider definition of whiteness as having a finite observation and assessment. Quantification of amount of hue and therefore emphasizing the perceived whiteness has been intimately importance of a preferred white. related to lightness levels and the absence of The use of fluorescence as a method to any hue. White pigments show however in increment both lightness and blue hue general a yellow shade originating from introduces a formidable task to elaborate a impurities (mainly Iron ions in different formula that takes into account a colorimetric oxidation stages) that has led to the terms compensation based on additive and sub- “near white” and “off-white”; these colors are ractive color mixing. really perceived under comparison with a The original idea of characterizing whiteness “preferred” white though and this is a result of through a unique number is valid today only the chromatic adaptation of the eye. after the definition of a preferred white has As a consequence first attempt to quantify been defined that is highly dependent of the whiteness have been related to the cultural group of the observer and the quantification of yellowness, lower yellowness application of the white object. From a values are considered as showing higher technical point of view formulas based on whiteness. Almost all pigments are quantified colorimetric quantities give more information in this way, remarkably also pulps and natural since they are based on psychometric raw fibers that are treated by a chemical quantities; nowadays efforts are invested in the bleaching process to lower yellowness values. definition of a whiteness subspace that The colorimetric compensation of yellowness localizes the perceived color within the color by adding a blue (or violet) dye is known as solid, though the definition of preferred white “bluing” and was quite widespread in the textile axis numbers can be then transformed into area, specially hand washing where chemical unique perceived whiteness that are strictly bleaching does not apply. The addition of a part of a color appearance whiteness model. dye leads irreversible to a loss of lightness, the Literally hundreds of whiteness formulas exist object appears grayish or duller as compared and have been applied in the past, only a with the not treated one, the compensation of selected number of them a presented and the yellow hue is interpreted by the eye as an discussed in the next sections. Primitive formulas The first attempts of describing whiteness are у() to describe the luminance factor under a based on just lightness, yellowness or given observer and illuminant. This is purely a blueness. The formula: luminance value and does not report if the observed object is bluish or yellowish (or by W Y the same token having any other hue). The formula tries to quantify whiteness just as related to lightness, normally as a relative quantity to a W B preferred white defined by a Magnesium oxide or Barium sulfate tablet, using the CIE function Axiphos GmbH Arend-Braye Str. 42, D-79540 Loerrach GERMANY http://www.axiphos.com Tel: (+49-7621) 426693 - Fax: (+49-7621) 426693 - Email: marketing@axiphos.com Page 1 of 8 tries to relate whiteness to a blue reflectance Worth mentioning is the whiteness measured defined either by the CIE function z() or to by the Leukometer of VEB Carl Zeiss, Jena, some ad hoc defined one as for example the GDR: paper brightness function B(); the relation to a MgO or BaSO4 preferred white is implicitly WI Leukometer 2 R459 R614 contained in the definition of the function B. It is clear that the formula gives always positive The use of not standardized band pass filters numbers regardless of the real color of the contributed to a loss of popularity of this type observed substrate, furthermore numbers are of formulas, specially after the introduction of not corrected by the relative amount of filter colorimeters that are based on the filters absorbed yellow light and as such it does not G (green), B (blue) and A (amber) that are take into account the action of bluing related to the CIE y(), z() and the red-portion techniques. The latter point can be corrected of the x() color-matching functions weighted by using some yellowness index that takes into by a CIE standard illuminant (normally of type account the relative amounts of blue and A). In general following relationships apply: yellow in the reflected light; first attempts to describe yellowness were based on the use of Amber filter Rx A special band pass filters as for example: Green filter RY G Blue filter Rz B R700 R450 W R700 X a R X b RZ Y RY where R is the reflectance value at the Z c RZ wavelength . Further examples are the Stephansen formula: 1 b RX X Z WI Stephansen 2 R430 R670 a ac RY Y and the Harrison formula: 1 RZ Z c WI Harrison 100 R670 R430 where Observer Illuminant a b c 2° A 1.044623 0.053849 0.355824 C 0.783185 0.197520 1.182246 D65 0.770180 0.180251 1.088814 10° A 1.05719 0.05417 0.35202 C 0.77718 0.19566 1.16144 D65 0.768417 0.179707 1.073241 Within this formalism yellowness formulas take The formula of Taube: the form: WTaube G 4 G B A B W G was develop by subtracting the amount of yellowness (second term) from the index of and subsequently the Stephansen formula is: lightness and can also be expressed as: WI Stephansen 2 RZ R X WTaube 4 B 3 G and the Harrison formula is: Closely related is also the whiteness index of the ASTM: WI Harrison RZ R X 100 WI 3.388 Z 3 Y Page 2 of 8 Formulas based on a uniform color system (UCS) It was quite early recognized that yellowness formulas or those based on the relative WMacAdam Y k pc2 differences of blue and yellow light were not sufficient to describe whiteness, specially of those objects whitened through bluing where pe is the colorimetric purity and k is a techniques. The importance of lightness was constant that depends on the application, and recognized as an important contribution to the Judd formula given by: whiteness perception as with the Hunter formula: WI ( Judd ,1936) Y 6700 S 2 WHunter L 3 b where S is the distance between the sample and the preferred white in the Judd’s UCS where L and b are Hunter coordinates defined triangle. The factor 6700 is optimized for as: grading laundry white goods and may assume Y a different value for other applications. L 100 The closely related formula: Yn 0.0102 X n X WI (Coppock ) 10 Y 2 p e2 Y a 175 X Y Y n n Yn is due to W.A. Coppock and known as the Chemstrand Whiteness Scale. Further formulas based on the principle of 0.00847 Z n Y Z b 70 Y colorimetric purity are the Vaeck formula: Y n Zn WVaeck Y k E u, v Yn and (Xn,Yn,Zn) are the coordinates of the where the equivalent luminescence E(u,v) is achromatic point. The simplicity of the Hunter defined in the MacAdam UCS diagram and its formula is remarkable and takes clearly into value for a particular (u,v) must be looked up in account the importance of having high a nomogram and it defines the dominant lightness and neutral blue b values. wavelength of 472 nm as preferred whiteness Close relatives of this formula are the hue, and the formula of Anders and Daul: MacAdam formula given by: 0.5 x 0.5 x n y arctg y W Anders Daul 2 Y 1520 arctg 65 n where (xn,yn) is the coordinate of achromatic Further developments are the first Selling point for D65. formula: The fact that deviations from the neutral blue 2 WSelling 100 100 Y 2 k s 1 2 yellow axis may contribute to perceived whiteness leads to the Hunter-Judd formula: 30 with 1 Y 2 a 2 WHunter Judd 1 2 b 2 2 Y 2 YMgO Ysample 1 where a and b are Hunter coordinates as defined above. and In this respect the original Hunter formula can be generalized as: s u 2 v2 W 100 L p 2 L a 2 b 2 2 being the distance between the sample and the preferred white on the MacAdam’s UCS where Lp is the lightness of the preferred white, diagram and k is a constant. in case of MgO it assumes the value 100. A simplification of the latter is the second Selling formula: Page 3 of 8 A last formula worth mentioning is the Friele WSelling 100 100 Y k 's 2 2 formula given by: 2 2 M S WFriele A L 2 with b c Y YMgO Ysample where (A,b,c) are constants and (L,M,S) are the length of long, medium and short axis of and s defined as above and k’ is a constant color discrimination ellipsoid centered on the 6 with the typical value of 9.5 10 . preferred white. This formula is remarkable The Croes formula is given as: since it recognizes fully the importance of deviations from the neutral blue axis as contributions to perceived whiteness, WI Croes Y 13.2 Y u u n 2 v vn 2 furthermore it attempts to compensate for the different sensitivities from lightness, hue and where (u,v) are coordinates in MacAdam UCS chroma contributions. diagram and (un,vn) are the coordinates of the preferred white. Formulas considering fluorescence The extensive use of Fluorescent Whitening The question of preferred white was however Agents (FWA) to increase perceived whiteness delegated to second importance since FWA achieves the compensation of substrate manufacturers tried to develop whiteness yellowness through an additive color mixing formulas tailored to the characteristics of their process; a considerable amount of blueness products. can be introduced without loosing luminance, The formula of Stensby: on the contrary a modest lightness increase results in objects showing dazzling whites. WStensby L 3 a 3 b Depending on their chemical structure, fluorescence produced by FWAs can lead to derives from the Hunter formula and shows neutral, or to red- or green-shaded whiteness; clearly a preference for redder whites, while the existence of shade preferences is the formula of Berger: illustrated by the formula: W Berger Y a Z b X 220 (G B) 100 G 2 WI (C 429) 100 G 0.242 B 2 with originally due to Hunter but modified to give a a b neutral white preference, or the formula 2° observer 3.400 3.895 10° observer 3.448 3.904 L 3 b 10 Y 21 Y Z WI (CDML) shows a preference for green whites as well Y the formula of Croes: that shows a blue white preference. Due to the additive nature of the process it was WI Croes RY RZ R X readily recognized that linear formulas could be built for measuring perceived whiteness in Much of the development of linear whiteness any of the colorimetric spaces: formulas was done by Ganz, who formulated a general formula as: W B G A k1 W L b a k2 WGanz Y P x0 x Q y 0 y W Y x y k3 where the values of the parameters determine since they can be regarded as variations of the the hue preference as seen from the table: same theme; this represents also a first rationalization of existing formulas, since most hue preference of them can fit in one of the listed formulas, red neutral green allowing a classification of origins and P -800 +800 +1700 preferences. Q +3000 +1700 +900 Page 4 of 8 the sample, daylight conditions were chosen where (x0,y0) is the coordinate of the as reference for work on whiteness achromatic point for the D65 illuminant. determination and modern formulas are strictly At this stage it was clearly recognized the valid for D65. importance of the amount of UV acting onto Modern formulas: general linear forms Starting point is the Roesch color solid as The isoleukai for Y=100 and d= 470 nm depicted in the figure, under following consist of fairly parallel and equally spaced flat conditions: curves that can be approximated by straight Illuminant D65 lines; this is the case for example for the Dominant wavelength for neutral formulas of Berger and Stensby but the lines whites 470 nm have different slope because of their different preference for greenish or reddish whites and The plane with the said dominant wavelength is controlled by the angle in the figure. describes (blue) colors with the same spectral To set up the whiteness formula following purity at different levels of color saturation S, parameters must be determined numerically: the colors perceived as white will lie within a The gain of whiteness with increasing limited region on this plane; perpendicular to saturation ∂W/∂S this plane are the ones corresponding to hues. The impact of lightness on whiteness Curves with same whiteness are called ∂W/∂Y isoleukai, these are defined by the whiteness The impact of the hue on whiteness formula (regardless of its general form), it must ∂W/∂H be remarked that an isoleuke contains the same values for whiteness W but different values for hue (or shade deviation). isoleuke whiteness axis achromatic W point S dominant wavelength 420 nm In general the following relationship holds: W W tan H S Page 5 of 8 and each whiteness point is characterized by W cos P S cos the slope of the isoleukai: W S W sin W Y Q S cos and their angle with respect to the line with where η is the angle between the line with d= d= 470 nm: 470 nm and the x axis, and (x0.y0) is the 45 arctg W S W H achromatic point for D65. The next step is the evaluation of the shade W S W H deviation or tint by the formula: TGanzGriesser m x0 x n y 0 y It must be remarked that the numerical value of (∂W/∂S) sets the extension of the whiteness scale and is closely related to the amount of where cos UV present in the illuminant; this is a direct m consequence of the presence of fluorescence BW resulting from excitation of the FWA and inherent to the nature of the formula. and With these definitions following linear formula can be written down: sin n BW WGanz D Y P x0 x Q y 0 y and is the angle of the perpendicular to the where line with d= 470 nm and the bandwidth BW is a constant related to the sensitivity of the eye W to distinguish different shades of white. D Y The Ganz and Ganz-Griesser formulas The formulas are expressed by: (x0.y0) = (0.313795,0.330972) WGanz Y P x0 x Q y 0 y The numerical scale sets up following threshold values: TGanzGriesser m x0 x n y 0 y threshold for undistinguishable whiteness sample pairs: 5 Ganz whiteness points and describe whiteness and shade deviation threshold for undistinguishable shade (tint) for daylight D65. The formula parameters deviation in sample pairs: 0.5 Ganz- adopt following numerical values: Griesser points D= ∂W/∂Y = 1 The instrument for conducting the ∂W/∂S= 4000 measurements must be equipped with a dominant wavelength is 470 nm (η= device to regulate the amount of UV falling 48.18154° and α=41.81852°) onto the samples, this amount must be set φ= 15° (light preference for greenish (and maintained) to an amount similar to that whites) encountered in daylight in order to obtain BW= 0.0008 reliable whiteness data. Some problems arise because small unavoidable physical Under these conditions following parameters differences among instruments result in large are calculated: discrepancies in measured whiteness numbers; for this reason the Ganz-Griesser P = -1868.322 formulas are applied with instrument-specific Q = -3695.690 parameters calculated with the aid of proper m = -931.576 calibrated samples. The formulas are n = 833.467 expressed as: Page 6 of 8 WGanz Y P x Q y C calibrated instrument; this procedure leads to satisfactory results when inter-instrumental comparison (specially shade deviation values) TGanzGriesser m x n y k are mandatory, though to the price of non- transferable parameters. where the values of (P,Q,C) and (m,n,k) are not universal and apply only for the specific The CIE formulas The formulas are expressed by: As with the Ganz formula, the instrument for conducting the measurements must be WCIE Y 800 x0 x 1700 y 0 y equipped with a device to regulate the amount of UV falling onto the samples and this amount must be set (and maintained) to an amount TCIE 900 x0 x 650 y 0 y similar to that encountered in daylight in order to obtain reliable whiteness data. and describe whiteness and shade deviation The CIE equations are object of a norm issued (tint) for daylight D65. The formula parameters by the CIE and adopted by many institutions adopt following numerical values: like ISO, Tappi, AATCC, DIN, ASTM, etc. Strictly speaking the CIE formulas are valid D= ∂W/∂Y = 1 only for illuminant D65 and for UV amounts ∂W/∂S= 1800.36 similar to daylight, however some institutions dominant wavelength is 470 nm (η= allow to use the CIE formulas in conjunction 48.18154° and α=41.81852°) with illuminants different than D65. As shown φ= 16.6173° (light preference for above the value of ∂W/∂S determines the greenish whites) scaling of the whiteness values and it is closely BW= 0.000901 related to the amount of fluorescence excited (x0.y0) = (0.313795,0.330972) from the FWA; there has been no study about the behavior of the isoleukai for other The numerical scale sets up following illuminants. Recently the ISO has extended the threshold values: CIE formulas to be used in conjunction with the illuminant C (introducing the term “indoor threshold for undistinguishable whiteness”), on the grounds that although the whiteness sample pairs: 2.3 CIE amount of UV differs notably from that of whiteness points daylight, the coordinates of the achromatic point do not differ much. While probably the threshold for undistinguishable shade isoleukai are not too distorted compared with deviation in sample pairs: 0.2 CIE those of daylight, the assumption remains to shade points be proved true. _____________ In a later work Ganz gave the CIE equations in CIE-L*a*b* space as: WCIE L*a*b* 2.41 L* 4.45 b * 1 0.0090 L* 96 141 .4 TCIE l *a*b* 1.58 a * 0.38 b * These are a generalization of the originally postulated linear formulas. Performance of linear whiteness formulas One must not forget that linear formulas were Another problem arising from the linear form is developed for fluorescent whites, they perform that the formulas are “open”, any sample, even fairly well for medium to high whiteness levels, a colored one will show certain degree of but the linear approximation starts breaking whiteness; proper assessment requires down for very high whiteness values or for assistance form the human observer. While medium to strong shaded samples. For limits for whiteness have been postulated, for samples with low content of fluorescence or example: especially for just bleached materials they do samples are white if -20 < W Ganz < not give reliable data and one should switch to (8*Y-490) a yellowness formula to obtain proper samples are white if 40 < W CIE < (5*Y- assessment. 280) Page 7 of 8 the validity of these limits is highly dependent on the observer and they fail specially with if W CIE > 5*Y-275 heavy shaded samples. Recently Uchida has proposed corrective W PCIE 2 P 2 W terms to the CIE formula as follows: if 40 < W CIE< 5*Y-275 where W WCIE 2 TCIE 2 ____________ Pw 5 Y 275 800 U V 100 Y x 0.82 1700 R T 100 Y y 0.82 and 2° observer 10° observer U 0.2761 0.2742 V 0.00117 0.00127 R 0.2727 0.2762 T 0.0018 0.00176 Conclusion Hundreds of whiteness formulas have been published or applied in different fields over the last 70 years and under a variety of conditions. Certainly the field of whiteness has evolved also during all those years, imposing new additional challenges to the developed formulas, the instrumental data has however stayed back and failed to provide enough reliable data to conduct quantitative and conclusive studies. Whiteness perception, although occupying a small amount of the total color solid, is still a psychochromatic phenomena attached to three quantities as any other color, this is a hint that a proper description must be based on three quantities that describe fully the perceived whiteness. Still the quest for the “most beautiful white” remains open, but it is a recognized fact that its definition depends on cultural background of the observer group and application of the object, a specialization of whiteness formulas based on absolute principles seems possible and represents truly the goal of the next developments in this area. Page 8 of 8