1. Combined absorption and scattering(KubelkaMunk analysis) Most opaque coloured objects illuminatedbywhite lightproduce diffusely reflectedcoloured radiation by the combined processes of lightabsorption and light scattering.1COMPILED BY TANVEER AHMED

2. KubelkaMunk analysis Consider the simplecase of a light beampassing verticallythrough a very thin pigmented layer ofthickness dx in a paintfilm (Figure 1.28). Weconsider separately the downward (incident)and upward (reflected)components of theincident light beam, assuming that theabsorption coefficient isrepresented by K andthe COMPILED BY TANVEER AHMEDscattering2 coefficient by S. 3. At the same time, the The downward flux (intensity I)upward flux (intensity J) is:is: decreased by absorption = decreased by absorption = kJ dxKI dx decreased by scattering = decreased by scattering = SJ dxSI dx increased by backscatter = increased by backscatter = + + SI dx (from the radiationSJ dx (from the radiationproceeding downward),proceeding upwards, of which J summarised by Eqn 1.21:is the intensity), which is summarised by Eqn1.20:3 COMPILED BY TANVEER AHMED 4. KubelkaMunk analysis Solution of these differential equations dependson the boundary conditions applied, but in the absence of scattering (S = 0) leads tothe LambertBouguer law for the downward flux.4 COMPILED BY TANVEER AHMED 5. KubelkaMunk analysis For an iso-tropically absorbing andscattering layer of infinite thickness (or at least so thick that the backgroundlayer reflection is negligible), it leads tothe widely used KubelkaMunk expression (Eqn 1.22):where R = Jo/Io is the reflection factorat the surfacefor a sample of infinite thickness.5 COMPILED BY TANVEER AHMED 6. KubelkaMunk analysis The K, S and K/S values provide the colour technologist with functions which, in principle, are additive And linearly related to concentration of dyes and pigments in solid substrates. For example, for a dyed substrate where the scattering is attributed entirely to the textile substrate and therefore does not vary with dye concentration [D], we have a particularly simple form of concentration dependence (Eqn 1.23):6 COMPILED BY TANVEER AHMED 7. KubelkaMunk analysis where Kf and Kd are the light absorptioncoefficients for the fibre and dye respectively, at the wavelength of measurement, and Sf is the scattering coefficient of the fibre atthe same wavelength. dye concentration [D],7 COMPILED BY TANVEER AHMED 8. KubelkaMunk analysisLimitation Although thisrelationship has certainlimitations (for example, when dealing with highly exhausting aciddyes on wool and when takingmeasurements near thewavelength of maximum absorption),good linearity isobserved(Figure 1.29; the rawdata from which this plot isderived is shown in8 Figure 1.31).COMPILED BY TANVEER AHMED 9. KubelkaMunk analysis MAJORLimitation Some of the major limitations to the KubelkaMunk typeof analysis are that it deals with diffuse monochromatic radiation and handles only two fluxes (diffuse light travelling upwards or downwards) through a homogeneous absorbing and scattering medium. The light loss through edges is thus neglected, as are the surface and the totally internally reflected components of the incident light beam. Other assumptions such as the uniform distribution of the dyes or pigments, and the lack of interactions between them, are also not realised. Such factors lead to a nonlinearity of the KubelkaMunk function when measured over wide concentration9 COMPILED BY TANVEER AHMEDranges.