Standard Test Method for Color and Strength of Chromatic Pigments with a Mechanical Muller

SIGNIFICANCE AND USE
Color and tinting strength are the most important properties of a color pigment. This test method provides a means of testing these properties for quality control.
This test method is intended as a referee method so that such matters as the vehicle for preparing the dispersions and the white for making tints have been suggested. However, other vehicles and whites may be suitable for quality control purposes, and changes in this test method are allowed by agreement between the parties to a test.
It is assumed that the most exact comparison of mass color and tinting strength occurs when the pigment is completely dispersed. By following the procedure described in Annex A1, the conditions for achieving the maximum practical degree of dispersion with a mechanical muller may be determined. Color and strength tests should be carried out under these conditions.
The results obtained with a mechanical muller do not necessarily correlate directly with an industrial situation where different dispersing conditions exist. However, dispersion with a mechanical muller is a quick and inexpensive way of testing the color and strength of a pigment for routine quality control.
SCOPE
1.1 This test method is intended to be used to compare the color and strength of a pigment under test with a reference standard of the same type and grade.
1.2 This test method does not apply to white pigments.
Note 1—Test Method D 3022 is similar to this test method, but it utilizes a miniature sandmill rather than a mechanical muller, to disperse the chromatic pigment.
Note 2—Test Method D 332 and Test Method D 2745 are similar to this test method, but they are intended for use with white pigments, rather than chromatic pigments.  
1.3 The values stated in SI units are the preferred unit of measurement. The values given in parentheses are for information only.
1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use. Specific hazard statements are given in Section 8.

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Publication Date
31-Jan-2008
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NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: D387 − 00 (Reapproved 2008)
Standard Test Method for
Color and Strength of Chromatic Pigments with a
Mechanical Muller
This standard is issued under the fixed designation D387; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
This standard has been approved for use by agencies of the Department of Defense.
1. Scope Pigments by Reflectance Measurements
D3022 Test Method for Color and Strength of Color Pig-
1.1 This test method is intended to be used to compare the
ments by Use of a Miniature Sandmill
color and strength of a pigment under test with a reference
D3964 Practice for Selection of Coating Specimens for
standard of the same type and grade.
Appearance Measurements
1.2 This test method does not apply to white pigments.
D4838 Test Method for Determining the Relative Tinting
Strength of Chromatic Paints
NOTE 1—Test Method D3022 is similar to this test method, but it
utilizes a miniature sandmill rather than a mechanical muller, to disperse E284 Terminology of Appearance
the chromatic pigment.
NOTE 2—Test Method D332 and Test Method D2745 are similar to this
3. Terminology
test method, but they are intended for use with white pigments, rather than
3.1 Definitions of appearance terms used in this test method
chromatic pigments.
may be found in Terminology E284.
1.3 The values stated in SI units are the preferred unit of
measurement. The values given in parentheses are for infor-
4. Summary of Test Method
mation only.
4.1 Pigments are dispersed in a suitable vehicle with a
1.4 This standard does not purport to address all of the
mechanical muller. Test and standard pigments are treated
safety concerns, if any, associated with its use. It is the
identically. Opaque drawdowns are made from the dispersions
responsibility of the user of this standard to establish appro-
and compared, either visually or instrumentally, for color and
priate safety and health practices and determine the applica-
strength differences.
bility of regulatory limitations prior to use. Specific hazard
statements are given in Section 8.
5. Significance and Use
2. Referenced Documents 5.1 Color and tinting strength are the most important prop-
2 erties of a color pigment.This test method provides a means of
2.1 ASTM Standards:
testing these properties for quality control.
D332 Test Method for Relative Tinting Strength of White
Pigments by Visual Observation 5.2 This test method is intended as a referee method so that
D1729 Practice for Visual Appraisal of Colors and Color such matters as the vehicle for preparing the dispersions and
Differences of Diffusely-Illuminated Opaque Materials the white for making tints have been suggested. However,
D2244 Practice for Calculation of Color Tolerances and other vehicles and whites may be suitable for quality control
Color Differences from Instrumentally Measured Color purposes, and changes in this test method are allowed by
Coordinates agreement between the parties to a test.
D2745 Test Method for Relative Tinting Strength of White
5.3 It is assumed that the most exact comparison of mass
color and tinting strength occurs when the pigment is com-
pletely dispersed. By following the procedure described in
This test method is under the jurisdiction of ASTM Committee D01 on Paint
AnnexA1, the conditions for achieving the maximum practical
and Related Coatings, Materials, andApplications and is the direct responsibility of
Subcommittee D01.26 on Optical Properties.
degree of dispersion with a mechanical muller may be deter-
Current edition approved Feb. 1, 2008. Published March 2008. Originally
mined. Color and strength tests should be carried out under
approved in 1934. Last previous edition approved in 2000 as D387 – 00. DOI:
these conditions.
10.1520/D0387-00R08.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
5.4 The results obtained with a mechanical muller do not
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
necessarily correlate directly with an industrial situation where
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. different dispersing conditions exist. However, dispersion with
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D387 − 00 (2008)
a mechanical muller is a quick and inexpensive way of testing vehicle the development of tinting strength by the mechanical
the color and strength of a pigment for routine quality control. muller is influenced by the force applied, the number of
revolutions, the mass of the pigment, and the mass of the
6. Apparatus
vehicle. The conditions for obtaining the maximum tinting
strength with the mechanical muller can be determined by
6.1 Balances—(1) A balance sensitive to 10 mg and (2)an
following the procedure in Annex A1.
analytical balance sensitive to 1.0 mg.
6.2 Muller, Mechanical, equipped with ground-glass plates 9.2 If these conditions are known for a particular pigment
with a particular vehicle, or if the purchaser and seller agree
to which a variable but known force may be added in 220-N
(50-lbf) increments. The driven glass plate shall have a speed upon a particular set of conditions, there is no need to carry out
the procedure in Annex A1.
ofrotationofbetween70and120r/minandtheapparatusshall
have an arrangement for pre-setting the number of revolutions
10. Dispersion Procedure
in multiples of 50.
6.3 Rubbing Surfaces—The rubbing surfaces of the ground 10.1 Decide, by agreement or by experimentation, as dis-
glass plates shall be kept sharp by removing them from the
cussed in Section 9, the following dispersing conditions:
muller and grinding them face-to-face with No. 303 optical
10.1.1 Force applied to the muller plates;
emery, or its equivalent, and water.
10.1.2 Number of revolutions;
10.1.3 Mass of the pigment; and
6.4 Small Glass Slab or other nonabsorbent material, suit-
10.1.4 Mass of the vehicle.
able for weighing and mixing pigment pastes.
10.2 Applying these decisions, prepare a dispersion of the
6.5 Spatula—Aflexiblespatulahavinga75to150-mm(3to
reference standard pigment. Weigh onto a glass slab to within
6-in.) blade.
2 mg, the appropriate quantities of the standard pigment and
6.6 Paper Charts, white with a black band and a surface
the dispersing vehicle. Mix the pigment and vehicle together
impervious to paint liquids.
thoroughly with the spatula and transfer the mixture to the
6.7 Film Applicator, at least 75 mm (3 in.) wide with a
lower plate of the muller. Spread the mixture in a path
clearance of 100 µm (4 mils) to produce wet films about 50 µm
approximately 100 mm wide and halfway between the center
(2 mils) thick.
and rim of the lower plate, and clean the spatula as much as
possiblebywipingitontheupperplateofthemuller.Closethe
6.8 Color-Measuring Instrument, meeting the requirements
plates and carry out the mulling stages of 50 revolutions; after
of Practice D2244.
each stage collect the paste from both plates with the spatula
7. Materials and spread it around the 100-mm path on the lower plate,
wiping the spatula on the upper plate as before. When the
7.1 Reference Standard—A standard pigment of the same
mulling has been carried out for the prescribed number of
type and grade as the pigment to be tested, as agreed upon
revolutions, collect the paste and store it. Clean the glass slab,
between the purchaser and the seller.
the muller plates, and the spatula, and repeat the procedure
7.2 Vehicle—A solvent-free vehicle, such as No. 1 litho-
with exactly the same quantities of the test sample and vehicle.
graphic varnish, with 0.8 % each of cobalt and manganese
Collect the paste from this sample and store it. Clean the glass
driers (6 % types).
slab, the muller plates, and the spatula.
7.3 White Tinting Paste—Awhite paint compatible with the
NOTE 4—The most common sources of error in this procedure are
dispersion vehicle, such as 57 parts of rutile titanium dioxide
inaccurate weighing, incomplete transfer of the pigment and vehicle
dispersed in 43 parts of the vehicle described in 7.2.
mixture, and contamination of the plates by previous samples.
NOTE 3—Because the choice of vehicle and white tinting pigment may
11. Masstone Color Procedure
affect the results, they should be agreed upon between the purchaser and
the seller.
11.1 Draw down a portion of the test and standard pastes in
juxtaposition on a paper chart over a vacuum-drawdown plate
8. Hazards
or other suitable plane surface with the film applicator. Make
8.1 Some pigments are potentially toxic and therefore
sure that the coating is opaque.
should be handled with care. Obtain specific precautions from
11.2 Immediately compare the colors visually while still
the manufacturer or supplier.
wet, using Practice D1729, and record the results. Set the
8.2 Manysolventsandpaintvehiclespresentexplosion,fire,
drawdowns aside in a dust-free area to dry. When dry, repeat
and toxicity hazards, so they should accordingly be handled
the visual color difference evaluation and record the results.
with care. Again, obtain specific precautions from the manu-
See Practice D3964.
facturer or supplier.
11.3 If desired, evaluate the color difference instrumentally
9. Dispersing Conditions using Practice D2244, and report the color difference in units
as agreed upon between the purchaser and seller.
9.1 The conditions for dispersing the pigment on the me-
chanical muller should be such that the maximum tinting
NOTE 5—Wet color difference evaluations may not agree with dry color
strength is developed. For each pigment and each dispersing difference evaluations because of such phenomena as flooding and
D387 − 00 (2008)
flocculation. In the case of a difference between the wet and dry
13.4 If desired, the relative tinting strength of the sample
evaluations, the purchaser and the seller should agree upon which
pigment can be calculated from instrumental measurements
condition is the standard.
using the following equation:
NOTE 6—Color difference measurements of wet paints may require a
2 2
special adapter to protect the instrument from fouling. Because color TS 5 1 2 R /2R / 1 2 R /2R T
@~ ! # @~ ! # ~ !
` ` u ` ` s
difference-measuring instruments differ widely in their design, the user
where:
may have to develop his own adapter.
TS = tinting strength of test pigment,
12. Tint
...


This document is not anASTM standard and is intended only to provide the user of anASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation:D387–86(Reapproved 1994) Designation:D387–00(Reapproved2008)
Standard Test Method for
Color and Strength of ColorChromatic Pigments with a
Mechanical Muller
This standard is issued under the fixed designation D387; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (e) indicates an editorial change since the last revision or reapproval.
This standard has been approved for use by agencies of the Department of Defense.
1. Scope
1.1 This test method is intended to be used to compare the color and strength of a pigment under test with a reference standard
of the same type and grade.
1.2 This test method does not apply to white pigments.
NOTE 1—Test Method D3022 is similar to this test method, but it utilizes a miniature sandmill rather than a mechanical muller, to disperse the
colorchromatic pigment.
NOTE 2—Test Method D332 and Test Method D2745 are similar to this test method, but they are intended for use with white pigments, rather than
color pigments.
1.3are similar to this test method, but they are intended for use with white pigments, rather than chromatic pigments.
1.3 ThevaluesstatedinSIunitsarethepreferredunitofmeasurement.Thevaluesgiveninparenthesesareforinformationonly.
1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory
limitations prior to use. Specific hazard statements are given in Section 78.
2. Referenced Documents
2.1 ASTM Standards:
D332 Test Method for Relative Tinting Strength of White Pigments by Visual Observation
D1729 Practice for Visual EvaluationAppraisal of Colors and Color Differences of Diffusely-Illuminated Opaque Materials
D2244Test Method Practice for Calculation of Color Tolerances and Color Differences from Instrumentally Measured Color
Coordinates
D2745 Test Method for Relative Tinting Strength of White Pigments by ReflectiveReflectance Measurements
D3022 Test Method for Color and Strength of Color Pigments by Use of a Miniature Sandmill
D3964 Practice for Selection of Coating Specimens for Appearance Measurements
E97Test Method for Directional Reflectance Factor, 45-deg 0-deg, of Opaque Specimens by Broad-Band Filter Reflectometry
This test method is under the jurisdiction of ASTM Committee D-1D01 on Paint and Related Coatings, Materials, and Applications and is the direct responsibility of
Subcommittee D01.26 on Optical Properties.
Current edition approved Aug. 26, 1986.Feb. 1, 2008. Published October 1986.March 2008. Originally published as D387–34T.approved in 1934. Last previous edition
D387–81.approved in 2000 as D387–00.
ForreferencedASTMstandards,visittheASTMwebsite,www.astm.org,orcontactASTMCustomerServiceatservice@astm.org.For Annual Book of ASTM Standards
, Vol 06.01.volume information, refer to the standard’s Document Summary page on the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
D387–00 (2008)
D4838 Test Method for Determining the Relative Tinting Strength of Chromatic Paints
E308Practice for Computing the Colors of Objects by Using the CIE System 284 Terminology of Appearance
3. Summary of Test Method
3.1Pigments are dispersed in a suitable vehicle with a mechanical muller. Test and standard pigments are treated identically.
Opaque drawdowns are made from the dispersions and compared, either visually or instrumentally, for color and strength
differences. Terminology
3.1 Definitions of appearance terms used in this test method may be found in Terminology E284.
4. Summary of Test Method
4.1 Pigments are dispersed in a suitable vehicle with a mechanical muller. Test and standard pigments are treated identically.
Opaque drawdowns are made from the dispersions and compared, either visually or instrumentally, for color and strength
differences.
5. Significance and Use
4.1Color5.1 Color and tinting strength are the most important properties of a color pigment.This test method provides a means
of testing these properties for quality control.
4.2This5.2 This test method is intended as a referee method so that such matters as the vehicle for preparing the dispersions
and the white for making tints have been suggested. However, other vehicles and whites may be suitable for quality control
purposes, and changes in this test method are allowed by agreement between the parties to a test.
4.3It5.3 Itisassumedthatthemostexactcomparisonofmasscolorandtintingstrengthoccurswhenthepigmentiscompletely
dispersed. By following the procedure described in Annex A1, the conditions for achieving the maximum practical degree of
dispersion with a mechanical muller may be determined. Color and strength tests should be carried out under these conditions.
4.4The5.4 Theresultsobtainedwithamechanicalmullerdonotnecessarilycorrelatedirectlywithanindustrialsituationwhere
different dispersing conditions exist. However, dispersion with a mechanical muller is a quick and inexpensive way of testing the
color and strength of a pigment for routine quality control.
5.6. Apparatus
5.1
6.1 Balances—(1) A balance sensitive to 10 mg and (2) an analytical balance sensitive to 1.0 mg.
5.2
6.2 Muller, Mechanical, equipped with ground-glass plates to which a variable but known force may be added in 50-lbf
(220-N)220-N (50-lbf) increments. The driven glass plate shall have a speed of rotation of between 70 and 120 r/min and the
apparatus shall have an arrangement for pre-setting the number of revolutions in multiples of 50.
5.36.3 Rubbing Surfaces—The rubbing surfaces of the ground glass plates shall be kept sharp by removing them from the
muller and grinding them face-to-face with No. 303 optical emery, or its equivalent, and water.
5.4
6.4 Small Glass Slab or other nonabsorbent material, suitable for weighing and mixing pigment pastes.
5.5
6.5 Spatula—A flexible spatula havinga3to 6-in. (75 to 150-mm) blade.
5.6—A flexible spatula having a 75 to 150-mm (3 to 6-in.) blade.
6.6 Paper Charts, white with a black band and a surface impervious to paint liquids.
5.7
6.7 Film Applicator, at least 3 in. (75 mm) wide with a clearance of 4 mils (100 µm) to produce wet films about 2 mils (50 µm)
thick.
5.8Color-Measuring Instrument,meetingtherequirementsofMethodD2244atleast75mm(3in.)widewithaclearanceof100
µm (4 mils) to produce wet films about 50 µm (2 mils) thick.
6.8 Color-Measuring Instrument, meeting the requirements of Practice D2244.
6.
7. Materials
6.1
7.1 Reference Standard—Astandard pigment of the same type and grade as the pigment to be tested, as agreed upon between
the purchaser and the seller.
6.2
7.2 Vehicle—Asolvent-free vehicle, such as No. 1 lithographic varnish, with 0.8% each of cobalt and manganese driers (6%
types).
6.3
D387–00 (2008)
7.3 White Tinting Paste—A white paint compatible with the dispersion vehicle, such as 57 parts of rutile titanium dioxide
dispersed in 43 parts of the vehicle described in 6.27.2.
NOTE 3—Because the choice of vehicle and white tinting pigment may affect the results, they should be agreed upon between the purchaser and the
seller.
7.8. Hazards
7.1Some8.1 Somepigmentsmaybearepotentiallytoxicandthereforeshouldbehandledwithcare.Obtainspecificprecautions
from the manufacturer or supplier.
7.2Many8.2 Many solvents and paint vehicles present explosion, fire, and toxicity hazards, andso they should accordingly be
handled with care. Again, obtain specific precautions from the manufacturer or supplier.
8.9. Dispersing Conditions
8.1The9.1 Theconditionsfordispersingthepigmentonthemechanicalmullershouldbesuchthatthemaximumtintingstrength
is developed. For each pigment and each dispersing vehicle the development of tinting strength by the mechanical muller is
influenced by the force applied, the number of revolutions, the mass of the pigment, and the mass of the vehicle. The conditions
for obtaining the maximum tinting strength with the mechanical muller can be determined by following the procedure inAnnex
A1.
8.2If9.2 Iftheseconditionsareknownforaparticularpigmentwithaparticularvehicle,orifthepurchaserandselleragreeupon
a particular set of conditions, there is no need to carry out the procedure in Annex A1.
9.10. Dispersion Procedure
910.1 Decide, by agreement or by experimentation, as discussed in Section 89, the following dispersing conditions:
910.1.1 Force applied to the muller plates;
910.1.2 Number of revolutions;
910.1.3 Mass of the pigment; and
910.1.4 Mass of the vehicle.
910.2 Applyingthesedecisions,prepareadispersionofthereferencestandardpigment.Weighontoaglassslabtowithin2mg,
theappropriatequantitiesofthestandardpigmentandthedispersingvehicle.Mixthepigmentandvehicletogetherthoroughlywith
thespatulaandtransferthemixturetothelowerplateofthemuller.Spreadthemixtureinapathapproximately100mmwideand
halfway between the center and rim of the lower plate, and clean the spatula as much as possible by wiping it on the upper plate
ofthemuller.Closetheplatesandcarryoutthemullingstagesof50revolutions;aftereachstagecollectthepastefrombothplates
with the spatula and spread it around the 100-mm path on the lower plate, wiping the spatula on the upper plate as before. When
the mulling has been carried out for the prescribed number of revolutions, collect the paste and store it. Clean the glass slab, the
muller plates, and the spatula, and repeat the procedure with exactly the same quantities of the test sample and vehicle. Collect
the paste from this sample and store it. Clean the glass slab, the muller plates, and the spatula.
NOTE 4—The most common sources of error in this procedure are inaccurate weighing, incomplete transfer of the pigment and vehicle mixture, and
contamination of the plates by previous samples.
10.Masstone Color Procedure
10.1
11. Masstone Color Procedure
11.1 Draw down a portion of the test and standard pastes in juxtaposition on a paper chart over a vacuum-drawdown plate or
other suitable plane surface with the film applicator. Make sure that the coating is opaque.
10.2
11.2 Immediately compare the colors visually while still wet, using Practice D1729, and record the results. Set the drawdowns
asideinadust-freeareatodry.Whendry,repeatthevisualcolordifferenceevaluationandrecordtheresults.SeePracticeD3964.
10.311.3 If desired, evaluate the color difference instrumentally using MethodPractice D2244, and report the color difference
in units as agreed upon between the purchaser and seller.
NOTE 5—Wetcolordifferenceevaluationsmaynotagreewithdrycolordifferenceevaluationsbecauseofsuchphenomenaasfloodingandflocculation.
In the case of a difference between the wet and dry evaluations, the purchaser and the seller should agree upon which condition is the standard.
NOTE 6—Color difference measurements of wet paints may require a special adapter to protect the instrument from fouling. Because color
difference-measuring instruments differ widely in their design, the user may have to develop his own adapter.
11.Tint Color Procedure
11.1
12. Tint Color Procedure
12.1 Determinebycalculationtheamountofwhitepigmentpastethatmustbeaddedto0.5gofthecolorpigmentpastesothat
D387–00 (2008)
the mixture contains 1 part of dry color pigment to 10 parts of dry white pigment. For stronger or weaker pigments this ratio may
be adjusted accordingly, for example, 1:20 or 1:5, respectively.
11.212.2 Weigh 500 6 2 mg of the standard color pigment paste onto a glass slab. Then weigh the amount of white pigment
pastedeterminedin11.112.1,andplaceitnexttothecolorpigmentpasteontheglassslab.Thoroughlymixthetwopastestogether
with the spatula until a uniform color is observed.
112.3 Prepareatintmixtureofthetestcolorpigmentpasteandthewhitepigmentpasteonaseparateglassslabbytheprocedure
described in 11.212.2.
11.4Draw12.4 Drawaportionofthetestandstandardtintpastesdowninjuxtapositiononapaperchartasin10.111.1.Evaluate
the color difference visually as in 10.211.2 and, if desired, instrumentally as in 10.311.3. Clean the spatula blade and glass slabs.
12.13. Calculation of Tinting Strength
12.1If13.1 If the colors of the test tint paste and the standard tint paste are visually the same, the tinting strength of the test
pigment is equal to that of the standard pigment, and the relative tinting strength of the test pigment is 100%. However, if the test
and standard colors are not the same, the difference may be due to either tinting strength or hue (shade).
12.213.2 To determine the relative tinting strength of the test pigment, repeat the operations of Section 1112, but this time use
an amount of the test pigment paste that is estimated to give the closest color match to the standard pigment paste. Repeat this
procedure until satisfied that the closest color match has been obtained.At this point any residual color difference between the test
and the standard pigments is attributed to a shade difference, rather than a strength difference. Note and record this shade
difference.
123.3 Calculate the relative tinting strength of the test pigment by dividing the mass of the standard paste by the mass of the
test paste used to obtain the closest color match; multiply by 100 to express the result in percent.
12.4If13.4 Ifdesired,therelativetintingstrengthofthesamplepigmentcanbecalculatedfrominstrumentalmeasurements(see
Test Method E97 or Practice E308) using the following equation:
2 2
TS 5[~1 2 R ! /2R # /[~1 2 R ! /2R # ~T!
` ` u ` ` s
where:
TS = tinting strength of test pigment,
R = measured reflectant factor (as a decimal),
`
T = assigned tinting strength of standard, usually 100%, and subscripts “ u” and “s” refer to the test and standard pigments,
respectively.
12.4.1For
13.4.1 To determine R use the lowest reflectance value measured in the range 420 to 680 nm with a spectrophotometer or
`
abridged spectrophotometer. This method of calculation is not valid when any tristimulus value or filter colorimeter reading is
substituted for R unless it can be demonstrated that with such a substitution the function (1−R ) /2R still varies linearly with
` ` `
concentration for the pigments being tested over the range of concentrations of interest. Because the use
...

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