Kubelka-Munk function linearity

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. 

Fig. 1. (a) Diffuse reflectance spectra of P25 (thin line), TH (thick line), 3% [PtClJ/P25 (dashed line) and 4.0% H2[PtCl6]/TH (dotted line). The Kubelka-Munk function, F(R00), is used as the equivalent of absorbance, (b) Transformed diffuse reflectance spectra of P25 (thin line), TH (thick line), 3% [PtCl4]/P25 (dashed line) and 4.0% H2[PtCl6]/TH (dotted line). The bandgap energy was obtained by extrapolation of the linear part. 

It is usually considered more difficult to evaluate and quantify diffuse reflectance data than transmission data, because the reflectance is determined by two sample properties, namely, the scattering and the absorption coefficient, whereas the transmission is assumed to be determined only by the absorption coefficient. The absorbance is a linear function of the absorption coefficient, but its counterpart in reflection spectroscopy, the Kubelka-Munk function (sometimes also called remission2 function), depends on both the scattering and the absorption coefficient. Often, researchers list a number of prerequisites for application of the Kubelka-Munk function, but, in contrast, transmittance is routinely converted without comment into absorbance. 

Simmons (1975) compared various theories of diffuse reflectance. He introduced a modified remission function, which explains deviations from linearity when F(p) is plotted versus k. He also concluded that the Kubelka-Munk function is proportional to the absorption coefficient k as obtained from transmission measurements for "weakly absorbing samples." Unfortunately, most literature is vague in that "weak" or "strong" absorption is not specified. One value given for "weak" is F(p) < 1 (Kellermann, 1979). 

Figure 10. Linearity of Kubelka-Munk function with surface loading of [Co(neo)]. Transmittance values were obtained from diffuse reflectance UV-vis spectra taken at 654 nm. Data are shown for low loadings of [Co(neo)] on Merck silica grade 9385 and the values were converted to Kubelka-Munk units. Transmittance values < 5% were not used. The concentration of Co was determined by removal of the adsorbed complex by treatment with 1M HCl followed by colorimetric determination with ammonium thiocyanate. [Adapted from (57).]...

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 ... 

In Table 3 some essential parameters are listed, which should be referred to in a manual, since they influence the performance of a spectrometer in addition to such common parameters as range, accuracy, and reproducibility of wavelength, stray light bandpass (spectral bandwidth or slit width of the spectrometer) photometric accuracy, reproducibility, and linearity baseline flatness absorbance zero stability noise level scan speeds response times and data intervals. Furthermore possible modes of the axis are of interest absorbance, transmittance, derivative, Kubelka-Munk function [9], and the possible scaling of the axis. Most of these parameters are given in manuals, determine the limitations of the instrument, and affect each other. 

It is only partly known how far the mentioned laws for powders can be adapted to direct evaluation of the remission spectra of spots on adsorbent layers. It has been established from corresponding studies of paper chromatograms that the Kubelka-Munk function is valid only in particular regions of low concentration [372]. Jork [333 a] has found linear dependence of degree of remission on applied substance amount over relatively wide ranges of concentration. He studied substances from... 

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. 

Grayness of a fabric swatch is not directly proportional to its content of black pigment (or artificial sod). A basic formula relating reflectance to the pigment content or concentration can be appHed to the evaluation of detergency test swatches (51,99—101). In simple form, an adaptation of the Kubelka-Munk equation, it states that the quantity (1 — i ) /2R (where R is the fraction of light reflected from the sample) is a linear function of the sod content of the sample. 

The light fluxes are now linear functions of the depth coordinate z as it is predicted also by Fick s first law for steady-state diffusion without sink. For weak absorption, the equations for Td and Ro of the Kubelka-Munk formalism are also directly equivalent to the results of the diffusion approximation. Comparing Eqs. (8.22) and (8.23) with Eqs. (8.11), (8.12), and (8.14) under diffuse irradiation or under //o = 2/3, the Kubelka-Munk coefficients can be expressed by<31 34)... 

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-... 

The Kubelka-Munk theory of diffuse reflectance is a good description of the optical properties of paper. The two parameters of the theory, absorption and scattering coefficient, are purely phenomenological, but are closely related to basic properties of paper. The absorption coefficient is approximately a linear function of the chrcmgphore concentration in the paper. The scattering coefficient is related to the nonbonded fiber surface area in the paper, or the area "not in optical contact," and the Fresnel reflectivity of that surface. 

The Kubelka-Munk theory assumes a linear relation between the colorant characteristic K/S and the colorant concentration. In general, it is found that the K/S ratio of a component colorant is a nonlinear function of the concentration [3]. This means that it will not be possible to adequately describe the colorant behavior by using a linear relation. Figure 4.2a shows the linear relation that results... 

Calibration is necessary for in-situ spectrometry in TLC. Either the peak height or the peak area data are measured, and used for calculation. Although the nonlinear calibration curve with an external standard method is used, however, it shows only a small deviation from linearity at small concentrations [94.95 and fulfils the requirement of routine pharmaceutical analysis 96,97J. One problem may be the saturation function of the calibration curve. Several linearisation equations have been constructed, which serve to calculate the point of determination on the basis of the calibration line and these linearisation equations are used in the software of some scanners. A more general problem is the saturation function of the calibration curve. It is a characteristic of a wide variety of adsorption-type phenomena, such as the Langmuir and the Michaelis-Menten law for enzyme kinetics as detailed in the literature [98. Saturation is also evident for the hyperbolic shape of the Kubelka-Munk equation that has to be taken into consideration when a large load is applied and has to be determined. 

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) ... 

Further processing is also usually performed to transform the reflectance image cube to its logic (UR) form, which is effectively the sample absorbance . This results in chemical images in which brightness linearly maps to the analyte concentration, and is generally more useful for comparative as well as quantitative purposes. Note that for NIR measurements of undiluted solids the use of more rigorous functions such as those described by Kubelka and Munk are usually not required. ... 

As several researchers have shown empirically, the use of —log(reflectance) can provide, analogous to a transmittance measurement, a linear relationship between the transformed reflectance and concentration, if the matrix is not strongly absorbing as can be found for many samples studied by near-infrared spectroscopy. This issue is presented in detail below. A different approach based on a physical model was considered for UV/VIS measurements and later also applied within the mid-infrared. A theory was derived by Kubelka and Munk for a simple, onedimensional, two-flux model, although it must be noted that Arthur Schuster (1905) had already come up with a reflectance function for isotropic scattering. A detailed description of theoretical and practical aspects was given by Korttim. The optical absorption... 

A function derived by Kubelka and Munk, /(/ oo), can be used in a computer program to change the reflectance spectrum into a spectrum resembling a linear absorbance spectrum. 

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