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The Kubelka-Munk theory, ^[1] ^[2] devised by Paul Kubelka^[3] and Franz Munk, is a fundamental approach to modeling the appearance of paint films. As published in 1931, the theory addresses “the question of how the color of a substrate is changed by the application of a coat of paint of specified composition and thickness, and especially the thickness of paint needed to obscure the substrate”.

In the paper, “fundamental differential equations” are developed using a two-stream approximation for light diffusing through a coating whose absorption and remission (back-scattering) coefficients are known. The total remission from a coating surface is the summation of: 1) the reflectance of the coating surface; 2) the remission from the interior of the coating; and 3) the remission from the surface of the substrate. The intensity considered in the later two parts is modified by the absorption of the coating material. The concept is based on the simplified picture of two diffuse light fluxes moving through semi-infinite, plane parallel layers, one proceeding "downward" and the other simultaneously "upward"

While Kulbeka entered this field through an interest in coatings, his work has influenced workers in other areas as well. In the original paper, there is a special case of interest to many fields is "the albedo of an infinitely thick coating". This case yielded the Kubelka–Munk equation, which describes the remission from a sample composed of an infinite number of infinitesimal layers, each having $a 0$ as an absorption fraction, and $r 0$ as a remission fraction. The authors noted that the albedo of an infinite number of these infinitesimal layers is "solely a function of the ratio of the absorption and back-scatter (remission) constants $a 0 / r 0$ , but not in any way on the absolute numerical values of these constants". (The equation is presented in the same mathematical form as in the paper, but with symbolism modified to reduce confusion.)

R_{\infty }=1+{\frac {a_{0}}{r_{0}}}-{\sqrt {{\frac {a_{0}^{2}}{r_{0}^{2}}}+2{\frac {a_{0}}{r_{0}}}}}

While numerous early authors had developed two constant similar equations, the mathematics of most of these was found to be consistent with the Kubelka-Munk treatment.^[4] Others added additional constants to produce more accurate models, but these generally did not find wide acceptance.

Due to its simplicity and its acceptable prediction accuracy in many industrial applications, the Kubelka-Munk model remains very popular. However, in almost every application area, the limitations of the model have required improvements. Sometimes these improvements are touted as extensions of Kubelka-Munk theory; sometimes as embracing more general mathematics of which the Kubelka-Munk Equation is a special case; and sometimes as an alternate approach.

Paints

In the work of Kubleka and Munk, there are several special cases important to paints that are addressed. The theory was used to give a mathematical definition of "hiding power". Hiding power is the ability to hide the surface of an object. The hiding power of a coating measures its ability to obscure a background of contrasting color. Hiding power is also known as opacity or covering power.^[1]

Paper Coatings

The Kubelka–Munk theory is also used in the paper industry to predict optical properties of paper, avoiding a labor-intensive trial-and-error work approach. The theory is relatively simple in terms of the number of constants involved, works very well for many papers, and is well documented for use by the pulp and paper industry. If the optical properties (e.g., reflectance and opacity) of each pulp, filler, and dye used in paper-making are known, then the optical properties of a paper made with any combination of the materials can be predicted. If the contrast ratio and reflectivity of a paper are known, the changes in these properties with a change in basis weight can be predicted.^[5]

Color

Early practitioners, especially D R Duncan,^[6] assumed that in a mixture of pigments, the colors produced in any given medium may be deduced from formulae involving two constants for each pigment. These constants, which vary with the wave-length of the incident light, measure respectively the absorbing power of the pigment for light and its scattering power. The work of Kubelka and Munk was seen as yielding a useful systematic approach to color mixing and matching. By resolving the Kubelka-Munk equation for the ratio of absorption to scatter, one can obtain a "remission function" ^[7]: $F(R_{\infty })={\frac {(1-R_{\infty })^{2}}{2R_{\infty }}}={\frac {a_{0}}{r_{0}}}$ . We may define $K$ and $S$ as absorption and back-scattering coefficients, which replace the absorption and remission fractions $a 0$ and $r 0$ in the Kubelka–Munk equation above. Then assuming separate additivity of the absorption and coefficients for each of $i$ components of concentration $c_{i}$ : ${\frac {(1-R_{\infty })^{2}}{2R_{\infty }}}={\frac {a_{0}}{r_{0}}}={\frac {K}{S}}={\frac {\Sigma (c_{i}K_{i})}{\Sigma (c_{i}S_{i})}}\approx {\frac {\Sigma (c_{i}K_{i})}{S}}$

For the case of small amount of pigments, the scatter $S$ is dominated by the base material, and is assumed to be constant. In such a case, the equation is linear in concentration of pigment.^[4]

Spectroscopy

One special case has received much attention in diffuse reflectance spectroscopy, that of an opaque (infinitely thick) coating, which can be applied to a sample modeled as an infinite number of infinitesimal layers. The two steam approximation was embraced by the early practitioners. ^[8] ^[9] There were far more mathematics to choose from, but the name Kubelka-Munk became widely regarded as synonymous with any technique that modeled diffuse radiation moving through layers of infinitesimal size. This was aided by the popular assumption was that the Kubelka-Munk function (above) was analogous to the absorbance function in transmission spectroscopy.

In the field of Infrared Spectroscopy, it was common to prepare solid samples by finely grinding the sample with potassium bromide. This led to a situation analogous to the described in the section just above for pigments, where the analyte had little effect on the scatter, which was dominated by the KBr. In this case, the assumption of the function being linear with concentration was reasonable.

However, in the field of Near-infrared spectroscopy, the samples are generally measured in their natural (often particulate) state, and deviations from linearity at higher at higher absorption levels were routinely observed. The Kubelka-Munk function was all but abandoned in favor of "log(1/R)". A more general equation, called the Dahm equation, was developed,^[10] along with a scheme to separate the effects of scanter from absorption in the log(1/R) data.^[11] In the equation, $R_{n}$ and $T_{n}$ are the measured remission by and transmission through a sample of $n$ layers, each layer having absorption and remission fractions of $a$ and $r$ .

A(R,T)={\frac {(1-R_{n})^{2}-T_{n}^{2}}{R_{n}}}={\frac {(2-a-2r)a}{r}}

In other areas of spectroscopy, there are shifts away from the the strict use of Kubelka-Munk as well.^[12]

Semiconductors^[13]

References

^ ^a ^b Kubelka, Paul; Munk, Franz (1931). "An article on optics of paint layers" (PDF). Z. Tech. Phys. 12: 593–601.
^ "Kubelka-Munk Theory - an overview". www.sciencedirect.com. Retrieved 22 February 2021.
^ Kubelka, Paul. "Autobiographical Sketch: Paul Kubelka".
^ ^a ^b Wendlandt, Wesley (1966). Reflectance Spectroscopy. New York: Interscience.
^ Bajpai, Pratima (2018). Biermann's Handbook of Pulp and Paper (Third Edition); Volume 2: Paper and Board Making; Chapter 11 - Optical Properties of Paper. Elsevier Inc. pp. 237–271.
^ Duncan, D. R. (1940). "The colour of pigment mixtures". Proc Physical Soc (London). 52: 380.
^ Judd, D B (1963). Color in Business, Science, and Industry (2 ed.). New York: John Wiley & Sons, Inc.
^ Schuster, Aurhur (1905). "Radiation through a foggy atmosphere". Astrophysical Journal. 21 (1): 1–22. Bibcode:1905ApJ....21....1S. doi:10.1086/141186.
^ Kortüm, Gustav (1969). Reflectance spectroscopy Principles, methods, applications. Berlin: Springer. ISBN 9783642880711. OCLC 714802320.
^ Dahm, Donald (1999). "Representative Layer Theory for Diffuse Reflectance". Applied Spectroscopy. 53 (6): 647–654. Bibcode:1999ApSpe..53..647D. doi:10.1366/0003702991947298. S2CID 96885077.
^ Dahm, Kevin (2013). "Separating the Effects of Scatter and Absorption Using the Representative Layer". Journal of Near Infrared Spectroscopy. 21 (5): 351–357. Bibcode:2013JNIS...21..351D. doi:10.1255/jnirs.1062. S2CID 98416407.
^ Alcaraz de la Osa, R.; Iparragirre, I.; Ortiz, D.; Saiz, J. M. (March 2020). "The extended Kubelka–Munk theory and its application to spectroscopy". ChemTexts. 6 (1): 2. doi:10.1007/s40828-019-0097-0.
^ Makuła, Patrycja (2018). "How To Correctly Determine the Band Gap Energy of Modified Semiconductor Photocatalysts Based on UV–Vis Spectra". J. Phys. Chem. Lett. 9(23): 6814–6817.

This optics-related article is a stub. You can help Wikipedia by expanding it.

[:1-1] Kubelka, Paul; Munk, Franz (1931). "An article on optics of paint layers" (PDF). Z. Tech. Phys. 12: 593–601.

[scidir-2] "Kubelka-Munk Theory - an overview". www.sciencedirect.com. Retrieved 22 February 2021.

[3] Kubelka, Paul. "Autobiographical Sketch: Paul Kubelka".

[:0-4] Wendlandt, Wesley (1966). Reflectance Spectroscopy. New York: Interscience.

[5] Bajpai, Pratima (2018). Biermann's Handbook of Pulp and Paper (Third Edition); Volume 2: Paper and Board Making; Chapter 11 - Optical Properties of Paper. Elsevier Inc. pp. 237–271.

[6] Duncan, D. R. (1940). "The colour of pigment mixtures". Proc Physical Soc (London). 52: 380.

[7] Judd, D B (1963). Color in Business, Science, and Industry (2 ed.). New York: John Wiley & Sons, Inc.

[Schuster-8] Schuster, Aurhur (1905). "Radiation through a foggy atmosphere". Astrophysical Journal. 21 (1): 1–22. Bibcode:1905ApJ....21....1S. doi:10.1086/141186.

[Kortuem-9] Kortüm, Gustav (1969). Reflectance spectroscopy Principles, methods, applications. Berlin: Springer. ISBN 9783642880711. OCLC 714802320.

[DahmJ1-10] Dahm, Donald (1999). "Representative Layer Theory for Diffuse Reflectance". Applied Spectroscopy. 53 (6): 647–654. Bibcode:1999ApSpe..53..647D. doi:10.1366/0003702991947298. S2CID 96885077.

[11] Dahm, Kevin (2013). "Separating the Effects of Scatter and Absorption Using the Representative Layer". Journal of Near Infrared Spectroscopy. 21 (5): 351–357. Bibcode:2013JNIS...21..351D. doi:10.1255/jnirs.1062. S2CID 98416407.

[alcaraz-12] Alcaraz de la Osa, R.; Iparragirre, I.; Ortiz, D.; Saiz, J. M. (March 2020). "The extended Kubelka–Munk theory and its application to spectroscopy". ChemTexts. 6 (1): 2. doi:10.1007/s40828-019-0097-0.

[13] Makuła, Patrycja (2018). "How To Correctly Determine the Band Gap Energy of Modified Semiconductor Photocatalysts Based on UV–Vis Spectra". J. Phys. Chem. Lett. 9(23): 6814–6817.

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@@ Line 1: / Line 1: @@
 {{Merge|Diffuse reflectance spectroscopy#Paul Kubelka|discuss=Talk:Kubelka-Munk theory#Proposed merge of Diffuse reflectance spectroscopy#Paul Kubelka with Kubelka-Munk theory|date=February 2021}}
 {{More citations needed|date=February 2021}}
-The '''Kubelka-Munk theory''', <ref>{{cite journal
+The '''Kubelka-Munk theory''', <ref name=":1">{{cite journal
 | last1   = Kubelka
 | first1  = Paul
@@ Line 11: / Line 11: @@
 | volume  = 12
 | pages   = 593–601
+| url     = http://www.graphics.cornell.edu/~westin/pubs/kubelka.pdf}} </ref> <ref name="scidir">{{cite web |title=Kubelka-Munk Theory - an overview |url=https://www.sciencedirect.com/topics/engineering/kubelka-munk-theory |website=www.sciencedirect.com |access-date=22 February 2021}}</ref> devised by Paul Kubelka<ref>{{cite web |last1=Kubelka |first1=Paul |title=Autobiographical Sketch: Paul Kubelka |url=http://www.graphics.cornell.edu/~westin/pubs/kubelka-autobio.html}}</ref> and Franz Munk, is a fundamental approach to modeling the appearance of paint films.  As published in 1931, the theory addresses “the question of how the color of a substrate is changed by the application of a coat of paint of specified composition and thickness, and especially the thickness of paint needed to obscure the substrate”.
-| url     = http://www.graphics.cornell.edu/~westin/pubs/kubelka.pdf}} </ref>
-<ref name="scidir">{{cite web |title=Kubelka-Munk Theory - an overview |url=https://www.sciencedirect.com/topics/engineering/kubelka-munk-theory |website=www.sciencedirect.com |access-date=22 February 2021}}</ref>
-devised by Paul Kubelka<ref>{{cite web |last1=Kubelka |first1=Paul |title=Autobiographical Sketch: Paul Kubelka |url=http://www.graphics.cornell.edu/~westin/pubs/kubelka-autobio.html}}</ref> and Franz Munk, is a fundamental approach to modeling the appearance of paint films.  As published in 1931, the theory addresses “the question of how the color of a substrate is changed by the application of a coat of paint of specified composition and thickness, and especially the thickness of paint needed to obscure the substrate”.
 In the paper, “fundamental differential equations” are developed using a [[two-stream approximation]] for light diffusing through a coating whose absorption and remission ([[back-scattering]])  coefficients are known.  The total [[Diffuse reflectance spectroscopy#remission|remission]] from a coating surface is the summation of: 1) the reflectance of the coating surface; 2) the remission from the interior of the coating; and 3) the remission from the surface of the substrate.  The intensity considered in the later two parts is modified by the absorption of the coating material.  The concept is based on the simplified picture of two diffuse light fluxes moving through semi-infinite, plane parallel layers, one proceeding "downward" and the other simultaneously "upward"
@@ Line 24: / Line 22: @@
 == Paints ==
-In the work of Kubleka and Munk, there are several special cases important to paints that are addressed.  The theory was used to give a mathematical definition of "hiding power". Hiding power is the ability to hide the surface of an object. The hiding power of a coating measures its ability to obscure a background of contrasting color. Hiding power is also known as opacity or covering power.
+In the work of Kubleka and Munk, there are several special cases important to paints that are addressed.  The theory was used to give a mathematical definition of "hiding power". Hiding power is the ability to hide the surface of an object. The hiding power of a coating measures its ability to obscure a background of contrasting color. Hiding power is also known as opacity or covering power.<ref name=":1" />
 == Paper Coatings ==
@@ Line 30: / Line 28: @@
 == Color ==
-Early practitioners, especially D R Duncan,<ref>{{cite journal |last1=Duncan |first1=D. R. |title=The colour of pigment mixtures |journal=Proc Physical Soc (London) |date=1940 |volume=52 |page=380}}</ref> assumed that in a mixture of pigments, the colors produced in any given medium may be deduced from formulae involving two constants for each pigment. These constants, which vary with the wave-length of the incident light, measure respectively the absorbing power of the pigment for light and its scattering power. The work of Kubelka and Munk was seen as yielding a useful systematic approach to color mixing and matching.  By resolving the Kubelka-Munk equation for the ratio of absorption to scatter, one can obtain a "remission function" <ref>{{cite book |last1=Judd |first1=D B |title=Color in Business, Science, and Industry |date=1963 |publisher=John Wiley & Sons, Inc. |location=New York |edition=2}}</ref>:<math>F(R_\infty) = \frac{(1-R_\infty)^2}{2R_\infty} </math> .  We may define <math>K </math>and <math>S </math> as absorption and back-scattering coefficients, which replace the absorption and remission fractions {{math|''a''<sub>0</sub>}} and {{math|''r''<sub>0</sub>}} in the Kubelka–Munk equation above.   Then assuming separate additivity of the absorption and coefficients for each of <math>i </math> components of concentration <math>c_i </math>:    <math>\frac{(1-R_\infty)^2}{2R_\infty} = \frac{K}{S} = \frac{\Sigma(c_i K_i)}{\Sigma(c_i S_i)} \approx \frac{\Sigma(c_i K_i)}{S} </math>
+Early practitioners, especially D R Duncan,<ref>{{cite journal |last1=Duncan |first1=D. R. |title=The colour of pigment mixtures |journal=Proc Physical Soc (London) |date=1940 |volume=52 |page=380}}</ref> assumed that in a mixture of pigments, the colors produced in any given medium may be deduced from formulae involving two constants for each pigment. These constants, which vary with the wave-length of the incident light, measure respectively the absorbing power of the pigment for light and its scattering power. The work of Kubelka and Munk was seen as yielding a useful systematic approach to color mixing and matching.  By resolving the Kubelka-Munk equation for the ratio of absorption to scatter, one can obtain a "remission function" <ref>{{cite book |last1=Judd |first1=D B |title=Color in Business, Science, and Industry |date=1963 |publisher=John Wiley & Sons, Inc. |location=New York |edition=2}}</ref>:<math>F(R_\infty) = \frac{(1-R_\infty)^2}{2R_\infty} = \frac {a_0}{r_0}  </math> .  We may define <math>K </math>and <math>S </math> as absorption and back-scattering coefficients, which replace the absorption and remission fractions {{math|''a''<sub>0</sub>}} and {{math|''r''<sub>0</sub>}} in the Kubelka–Munk equation above.   Then assuming separate additivity of the absorption and coefficients for each of <math>i </math> components of concentration <math>c_i </math>:    <math>\frac{(1-R_\infty)^2}{2R_\infty} = \frac {a_0}{r_0}  = \frac{K}{S} = \frac{\Sigma(c_i K_i)}{\Sigma(c_i S_i)} \approx \frac{\Sigma(c_i K_i)}{S} </math>
 For the case of small amount of pigments, the scatter <math>S</math> is dominated by the base material, and is assumed to be constant.  In such a case, the equation is linear in concentration of pigment.<ref name=":0">{{cite book|last1=Wendlandt|first1=Wesley|title=Reflectance Spectroscopy|date=1966|publisher=Interscience|location=New York}}</ref>
 == Spectroscopy ==
+One special case has received much attention in [[diffuse reflectance spectroscopy]], that of an opaque (infinitely thick) coating, which can be applied to a sample modeled as an infinite number of infinitesimal layers.  The two steam approximation was embraced by the early practitioners. <ref name="Schuster">{{cite journal |last1=Schuster |first1=Aurhur |title=Radiation through a foggy atmosphere |journal=Astrophysical Journal |date=1905 |volume=21 |issue=1 |pages=1–22 |doi=10.1086/141186|bibcode=1905ApJ....21....1S }}</ref> <ref name="Kortuem">{{Cite book |title=Reflectance spectroscopy Principles, methods, applications |last=Kortüm |first= Gustav |date=1969 |publisher=Springer |isbn=9783642880711 |location=Berlin |oclc=714802320}}</ref>  There were far more mathematics to choose from, but the name Kubelka-Munk became widely regarded as synonymous with any technique that modeled diffuse radiation moving through  layers of infinitesimal size.   This was aided by the popular assumption was that the Kubelka-Munk function (above) was analogous to the absorbance function in transmission spectroscopy.
-One special case has received much attention in diffuse reflectance spectroscopy, that of an opaque (infinitely thick) coating, which can be applied to a sample modeled as an infinite number of infinitesimal layers.
-<ref name="alcaraz">{{cite journal |last1=Alcaraz de la Osa |first1=R. |last2=Iparragirre |first2=I. |last3=Ortiz |first3=D. |last4=Saiz |first4=J. M. |title=The extended Kubelka–Munk theory and its application to spectroscopy |journal=ChemTexts |date=March 2020 |volume=6 |issue=1 |pages=2 |doi=10.1007/s40828-019-0097-0}}</ref>
+In the field of [[Infrared Spectroscopy]], it was common to prepare solid samples by finely grinding the sample with potassium bromide.  This led to a situation analogous to the described in the section just above for pigments, where the analyte had little effect on the scatter, which was dominated by the KBr.  In this case, the assumption of the function being linear with concentration was reasonable.
+However, in the field of [[Near-infrared spectroscopy]], the samples are generally measured in their natural (often particulate) state, and deviations from linearity at higher at higher absorption levels were routinely observed.   The Kubelka-Munk function was all but abandoned in favor of "log(1/R)".  A more general equation, called the Dahm equation, was developed,<ref name="DahmJ1">{{cite journal |last1=Dahm |first1=Donald |title=Representative Layer Theory for Diffuse Reflectance |journal=Applied Spectroscopy |date=1999 |volume=53 |issue=6 |pages=647–654 |doi=10.1366/0003702991947298|bibcode=1999ApSpe..53..647D |s2cid=96885077 }}</ref> along with a scheme to separate the effects of scanter from absorption in the log(1/R) data.<ref>{{cite journal |last1=Dahm |first1=Kevin |title=Separating the Effects of Scatter and Absorption Using the Representative Layer |journal=Journal of Near Infrared Spectroscopy |date=2013 |volume=21 |issue=5 |pages=351–357|doi=10.1255/jnirs.1062 |bibcode=2013JNIS...21..351D |s2cid=98416407 }}</ref>    In the equation, <math>R_n</math> and <math>T_n</math> are the measured remission by and transmission through a sample of <math>n</math> layers, each layer having absorption and remission fractions of <math>a</math> and <math>r</math>.
+:<math>A(R,T) = \frac {(1-R_n)^2-T_n^2}{R_n} = \frac {(2-a-2r)a}{r}</math>
+In other areas of spectroscopy, there are shifts away from the the strict use of Kubelka-Munk as well.<ref name="alcaraz">{{cite journal |last1=Alcaraz de la Osa |first1=R. |last2=Iparragirre |first2=I. |last3=Ortiz |first3=D. |last4=Saiz |first4=J. M. |title=The extended Kubelka–Munk theory and its application to spectroscopy |journal=ChemTexts |date=March 2020 |volume=6 |issue=1 |pages=2 |doi=10.1007/s40828-019-0097-0}}</ref>
+==Semiconductors<ref>{{Cite journal|last=Makuła|first=Patrycja|date=2018|title=How To Correctly Determine the Band Gap Energy of Modified Semiconductor Photocatalysts Based on UV–Vis Spectra|journal=J. Phys. Chem. Lett.|volume=9(23)|pages=6814–6817}}</ref>==
 ==References==