Kubelka-Munk reflectivity function

The color of kraft lignin as compared with the untreated Bjorkman milled wood lignin (spruce) is demonstrated by the curves for the Kubelka-Munk reflectance function F (R ) vs. the wavelength (Figure 1). The shoulder at around 500 mis particularly interesting since it is in the visible region. Sodium borohydride reduction of kraft lignin causes a certain... 

Figure 3.9 shows the reflectance spectra of unbleached and peroxide-bleached TMP from black spruce. Reflectance is the ratio of the intensity of reflected to incident light, and is thus mathematically analogous to the transmittance of the Beer-Lambert Law. Unlike transmittance, however, reflectance is not easily rendered to a quantity proportional to chromophore concentration. Diffuse reflectance is related to chromophore concentration by the Kubelka-Munk remission function, Equation... 

The reflectance spectra were recorded in the geometry R0,d (4) in the frequency range 5000-40,000 cm-1 in a previously described apparatus (4, 6). The absorbance is represented by the logarithm of the Schuster-Kubelka-Munk (SKM) function, F(RX) = (1 — RX)2/2RX, where Rx is the reflectance measured against a white standard. Since the comparison of the reflectance spectra with the transmission spectra of complexes in molecular sieves revealed that the scattering coefficient is a constant independent of the wavelength (4), the logarithm of the SKM function is, but for an additional constant, a correct representation of absorbance. 

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. 

Ultraviolet-visible (UV-vis) diffuse reflectance spectra of supported WOx samples and standard W compounds were obtained with a Varian (Cary 5E) spectrophotometer using polytetrafluoroethylene as a reference. The Kubelka-Munk function was used to convert reflectance measurements into equivalent absorption spectra [12]. Spectral features of surface WOx species were isolated by subtracting from the W0x-Zr02 spectra that of pure Z1O2 with equivalent tetragonal content. All samples were equilibrated with atmospheric humidity before UV-vis measurements. 

To continue the derivation, the next step is to determine the variation of the absorbance readings starting with the definition of absorbance. The extension we present here, of course, is based on Beer s law, which is valid for clear solutions. For other types of measurements, diffuse reflectance for example, the derivation should be based on a suitable function of T that applies to the situation, for example the Kubelka-Munk function for diffuse reflectance should be used for that case ... 

In the diffuse reflectance mode, samples can be measured as loose powders, with the advantages that not only is the tedious preparation of wafers unnecessary but also diffusion limitations associated with tightly pressed samples are avoided. Diffuse reflectance is also the indicated technique for strongly scattering or absorbing particles. The often-used acronyms DRIFT or DRIFTS stand for diffuse reflectance infrared Fourier transform spectroscopy. The diffusely scattered radiation is collected by an ellipsoidal mirror and focussed on the detector. The infrared absorption spectrum is described the Kubelka-Munk function ... 

Opaque minerals like iron oxides are frequently examined in the reflectance mode - and usually give diffuse reflectance spectra. Reflectance spectra provide information about the scattering and absorption coefficients of the samples and hence their optical properties. The parameters of reflectance spectra may be described in four different ways (1) by the tristimulus values of the CIE system (see 7.3.3) (2) by the Kubelka-Munk theory and (3) by using the derivative of the reflectance or remission function (Kosmas et al., 1984 Malengreau et ak, 1994 1996 Scheinost et al. 1998) and, (4) more precisely, by band fitting (Scheinost et al. 1999). 

The Kubelka-Munk function (f (r)), the remission function, is often used to relate diffuse reflectance spectra to absorption and scattering parameters. This function is the ratio of the absorption, k, and the scattering, s, coefficient and is related to the diffuse reflectance, r, by... 

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-... 

From this expression (Kubelka Munk function) it follows that, within the range of validity of the theory, q,/ depends only on the ratio of the absorption coefficient to the scattering coefficient, and not on their individual values. The equation has been most useful where reflectance measurements are used to obtain information about absorption and scattering (e.g., in textile dyeing, thin layer chromatography, and IR spectroscopy). 

Present data illustrate the technique for an in situ determination of surface areas. Related methods had been applied primarily to the study of site distributions in clay minerals, particularly by Russian workers (66), and they were used by Bergmann and O Konski in a detailed investigation of the methylene blue-montmorillonite system (3). In fact, changes in electronic spectra arising from surface interactions received sufficient attention in the past to warrant their review by A. Terenin (65). Most of these investigations involved transmittance spectra but new techniques in reflection spectrophotometry and applications of the Kubelka-Munk relation have facilitated the quantitative evaluation of spectra in highly turbid media (35, 69, 77). Thus, in agreement with the work of Kortiim on powders and anhydrous dispersions (31, 32, 33), our results demonstrate the applicability of the Kubelka-Munk function... 

Fig. II. UV-Visible reflectance spectra (Kubelka-Munk function vs. wavenumber) of (a) silicalite-1 (b) TS-1. (From Boccuti et al., 1989.)...

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. 

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. 

To emphasize the similarities and differences between spectroscopy in transmission and reflection, both configurations are described in the following sections. The absorbance and the Kubelka-Munk functions are derived. 

The motivation for transforming reflectance data into the Kubelka-Munk function is to obtain a representation of the absorption spectrum of the sample, which also allows one to relate intensities directly to concentration. It is sometimes debated as to whether the transformation should be performed, as described below. 

As long as 0 < p < 1 (deviations may occur in experiments), the conversion of reflectance into the Kubelka-Munk function can be performed, just as absorbance can be calculated from transmittance for 0 < t < 1 (and it usually is without any further consideration). However, the proportionally to the concentration may not hold true. As applies for the Lambert-Beer law, the Kubelka-Munk function is a limiting case for the range of weak absorption. For low reflectance (transmittance), the Kubelka-Munk function (absorbance) will not be proportional to the concentration of the absorbing species, and quantification is not possible. 

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). 

Klier (1972) deduced that for 0.6 < p < 1, which corresponds to 0.13 > F(P) > 0, the Kubelka-Munk absorption coefficient should be nearly proportional to the true absorption coefficient. Deviations from the proportionality up to a factor of two occurred for lower reflectance values. In the range p > 0.6, the Kubelka-Munk function should be nearly proportional to the absorber concentration. Through comparison with the radiative-transfer equation formulated by Chandrashekhar (1960), Klier related the phenomenological coefficients to the true absorption and scattering coefficients a and [Pg.142]

FIGURE 6 Comparison of the different appearance of band areas depending on representation. (A) reflectance or transmittance, (B) absorbance, and (C) Kubelka-Munk function. 

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