Kubelka–Munk theory

In <a href="/facts/Optics/p0vq4S1T">optics</a>, the Kubelka–Munk theory devised by Paul Kubelka and Franz Munk, is a fundamental approach to modelling the appearance of <a href="/facts/Paint/Q176AsIG">paint</a> 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". The mathematical relationship involves just two paint-dependent constants.
In their article, differential equations are developed using a <a href="/facts/Two-stream_approximation/0znHxkvS">two-stream approximation</a> for light diffusing through a coating whose absorption and remission (<a href="/facts/Back-scattering/C79g5QLB">back-scattering</a>) coefficients are known. The total <a href="/facts/Diffuse_reflectance_spectroscopy/IGfRaCua">remission</a> from a coating surface is the summation of: 

<ol><li>the reflectance of the coating surface;</li>
<li>the remission from the interior of the coating; and</li>
<li>the remission from the surface of the substrate.</li></ol>
The intensity considered in the latter 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, with one flux proceeding "downward", and the other simultaneously "upward".
While Kubelka entered this field through an interest in coatings, his work has influenced workers in other areas as well. In the original article, there is a special case of interest to many fields is "the <a href="/facts/Albedo/TJDRdKEB">albedo</a> of an infinitely thick coating". This case yielded the <a href="/facts/Diffuse_reflectance_spectroscopy/IGfRaCua">Kubelka–Munk equation</a>, which describes the remission from a sample composed of an infinite number of infinitesimal layers, each having a0 as an absorption fraction and r0 as a remission fraction. The authors noted that the remission from an infinite number of these infinitesimal layers is "solely a function of the ratio of the absorption and back-scatter (remission) constants a0/r0, 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 article, but with symbolism modified.)

R
          
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              a
              
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              r
              
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                  a
                  
                    0
                  
                  
                    2
                  
                
                
                  r
                  
                    0
                  
                  
                    2
                  
                
              
            
            +
            2
            
              
                
                  a
                  
                    0
                  
                
                
                  r
                  
                    0
                  
                
              
            
          
        
        .
      
    
    {\displaystyle 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 similar two-constant equations, the mathematics of most of these was found to be consistent with the Kubelka–Munk treatment. 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.

Kubelka–Munk theory open-in-new

Kubelka–Munk theory