The Kubelka Munk analysis

One of the most useful applications of spectrophotometry is the qualitative and quantitive estimation of dyestuff and pigment concentrations required to match a specific colour. To do this some function of the measured reflectance is required which is linearly dependent upon the colorant concentration. This function should also be additive in a straightforward manner when colorants are mixed together in the same application. The function will need to be applied over a very narrow wavelength band at discrete intervals.

One such function that very roughly works is the simple reciprocal of reflectance. 1/R, where R is a decimal fraction and not a percentage i.e. 100% for the perfect diffuser is rated as 1. This function can be made to work a little better if a constant k, which can be derived empirically is inserted:-

Most modern match prediction techniques work upon the principles established by Paul Kubelka and Franz Munk (1931) who studied the behaviour of light in turbid media and set up differential equations which they solved by the application of calculus methods to give an exponential solution which enabled graphical solutions for opacity of pigment layers to be made. The TAPPI opacity/reflectivity chart is one such application.

The often used relationship between absorption/scatter and reflectance also evolved from this study:-

     where:

K = absorption
S = scatter
R = reflectance as a decimal fraction
= narrow wavelength band (say 0.5 to 2 nm)

and its transpose:-

In 1948 Kubelka in a work of the greatest importance gave explicit hyperbolic solutions for practically all the relationships between reflectance and opacity of colorant layers which had formerly been locked in the exponential solution. These will be dealt with later (Use of the hyperbolic solution).