Munk relationship, Kubelka

Dahm and Dahm [22] have shown that both log(l/R) and the Kubelka-Munk transformation are nonlinear functions of concenfrafion. The log(l/K) transformation is nonlinear because the absorbance coefficient is considered the additive sum of the absorbance coefficients of all absorbing species in the sample, and it does not consider that scattering varies as a function of wavelengfh. The nonlinearity of the Kubelka-Munk relationship is due... 

The concentration of dye on a fabric after a given dyeing time can be determined by three methods (1) measurement of the decrease in dye concentration in solution with time by ultraviolet-visible spectroscopy, (2) determination of the dye concentration on the fabric dyed for a given time by dye extraction and ultraviolet-visible spectroscopy, or (3) by measurement of the reflectance spectra of the dyed fabric followed by application of the Kubelka-Munk relationship in which K/S. (K/S is the scat-... 

The intensity of the emitted fluorescence In is, therefore, directly proportional to the amount of substance applied a This relationship is much simpler than the Kubelka-Munk function and always leads to a linear calibration curve passing through the origin If this is not true then interference is occurring [5]... 

The term on the left side of Eq. (4) is often termed the remission function (or the Kubelka-Munk function), and it is frequently denoted by f(Rx). Equation (4) indicates that a linear relationship should exist between f(Rx) and the sample absorption. 

This reflectance spectrum can be related to concentration by converting it to an absorbance-like spectrum by using either the Kubelka-Munk or the log(l/R) relationships ... 

If over the region of interest, the scattering coefficient hardly varies with wavelength, the shapes of the remission spectrum and the absorption spectrum should be very similar. The relationship between the remission function and the reflectance spectrum is shown in Figure 7.2 left, and the Kubelka-Munk functions of the different iron oxides are illustrated in Figure 7.2, right. 

Fig. 7.2 Left Relationships between diffuse reflectance (r), the specular reflectance (R) and Kubelka-Munk function (f(r)) of maghemite. Right Kubelka-Munk function of various Fe oxides (Strens. Wood, 1979, with permission).

Diffuse reflectance R is a function of the ratio K/S and proportional to the addition of the absorbing species in the reflecting sample medium. In NIR practice, absolute reflectance R is replaced by the ratio of the intensity of radiation reflected from the sample and the intensity of that reflected from a reference material, that is, a ceramic disk. Thus, R depends on the analyte concentration. The assumption that the diffuse reflectance of an incident beam of radiation is directly proportional to the quantity of absorbing species interacting with the incident beam is based on these relationships. Like Beer s law, the Kubelka-Munk equation is limited to weak absorptions, such as those observed in the NIR range. However, in practice there is no need to assume a linear relationship between NIRS data and the constituent concentration, as data transformations or pretreatments are used to linearize the reflectance data. The most used linear transforms include log HR and Kubelka-Munk as mathemati-... 

Kubelka-Munk theory assumes linear relationships for KIS versus concentration and K/S versus thickness as well as the validity of the additive theory ... 

Because the incident monochromatic light is absorbed, reflected, and scattered by the opaque layer material, the theoretical relationship between amount of absorption and amount of substance does not follow the simple Beer-Lambert law that is valid for solutions. The Kubelka-Munk equation is the most accepted theoretical relationship for TLC, but its use is not necessary because of the ability of densitometer software to handle empirical nonlinear regression functions. 

In fact, since the underlying assumptions of the Kubelka-Munk equation are experimentally not fully satisfied, the relationship between the sample concentration and the KJS value slightly deviates from the linear relationship, especially at high analyte concentration. 

Some typical results are presented in Figs. 2-4. The reflectance in a typical wide range calibration is shown in Fig. 2. The similarity to the hyperbolic Kubelka-Munk function is obvious. The theory predicts an increase in reflectance with increasing layer thickness or decreasing particle size. The relationship between the... 

With systems that measure reflectance, the relationship between reflectance and the glucose concentration is described by the Kubelka-Munk equation ... 

ModiHed Kubelka—Munk Optical Treatment of Photon Flux in a Coating and Its Relationship to UV Curing as Measured with a UV NlOl Cure Tester... 

A number of attempts have been made to describe both theoretically and practically diffuse reflectance and scattering functions to enable a linear relationship to be established between absorbance (A), expressed as logio(l/Ii) where R is the reflectance, and molecular concentration. Perhaps the commonest relationship encountered is that ascribed to Kubelka and Munk, who established nine assumptions and 16 variables. These can be simplified to the Kubelka-Munk function, namely ... 

The Kubelka-Munk function /(/ ) gives a relationship between the diffuse reflectance the absorption coefficient k, and the scattering coefficient s. 

The Kubelka-Munk (K-M) model is applied as a linearization function to signals with scattering and absorptive characteristics as often encountered in diffuse reflectance. This relationship is given as follows (from V. P. Kubelka and F. Munk, Z. Tech. Physik 12, 593,1931) ... 

In order to convert property values from raw property data to cluster values, property operator mixing rules are required (Shelley El-Halwagi 2002 Eden et al. 2002). The property relationships can be described using the Kubelka-Munk theory (Biermann 1996). According to Brandon (1981), the mixing rules for objectionable material (OM) and absorption coefficient (k) are linear, while a non-linear empirical mixing rule for reflectivity has been developed (Willets 1958). 

According to this equation, I is directly proportional to the amount of substance applied (a). This is a much simpler relationship than the Kubelka-Munk equation for absorption (above). 

For diffuse reflectance spectroscopy the Kubelka-Munk function, f Roo), is most appropriate [128, 129]. The K-M theory indicates that linear relationships of band intensity vs. concentration should result when intensities are plotted as the K-M function f Roo) = k/S, where k is the absorption coefficient and S is the scattering coefficient (cfr. Chp. 1.2.1.3). The use of the K-M equation for quantitative analysis by diffuse reflectance spectroscopy is common for measurements in the visible, mid-IR and far-IR regions of the spectrum. Measurement of scattered light (ELSD) allows quantitative analysis. 

The theories of diffuse reflection spectroscopy have been summarized by Griffiths and Dahm [1]. None of these theories allows R(y) to be converted to a parameter that varies linearly with concentration over a wide range, but the Kubelka-Munk theory has provided the simplest and most useful parameter for practical measurements in the mid-infrared. Kubelka and Munk [2,3] derived a relationship between the reflectance of a sample at infinite depth, Roo(v), and its absorption coefficient, fe(v), and scattering coefficient, (v). (A sample whose DR spectrum does not change as it is made thicker is regarded as being at infinite... 

InstmmentaHy, both the hiding power and tinting strength can be determined from the amount of the incident light reflectance of coated white and black substrates. Relationships derived from Kubelka and Munk theory (6) are appHed in actual calculations. 

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