Kubelka-Munk reflection

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... 

Kubelka-Munk reflectance arises from the assumption that the coefficients K and S are the same for forward and reverse flux. From a many-flux analysis it can be concluded that the angular distribution of the forward and reverse flux is not the same [2]. 

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 ... 

The nature and the distribution of different types of Fe species in calcined (C) and steamed (S) samples were investigated by means of UV-vis spectroscopy. UV-vis spectra of Fe species were monitored on UV-vis spectrometer GBS CINTRA 303 equipped with a diffuse reflectance attachment with an integrating sphere coated with BaS04 and BaS04 as a reference. The absorption intensity was expressed using the Schuster-Kubelka-Munk equation. 

Fig-1 Absorption spectra, obtained through the Kubelka-Munk transformation of diffuse reflectance spectra, of indolinonaphthospiropyran adsorbed onto silica gel. Spectra are shown for coverages of (A) 2.35, (B) 9.49, (C) 34.2, and (D) 46.7 /ig/m2. (Data adapted from Ref. 12.)... 

The Kubelka-Munk theory treats the diffuse reflectance of infinitely thick opaque layers [4], a situation achieved in practice for UV/VIS spectroscopy through the use of powder path lengths of at least several millimeters. In this instance, the Kubelka-Munk equation has the form... 

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 ... 

The tinctorial strength in white reductions is thus quantitatively defined by the Kubelka-Munk relation between the spectral absorption coefficient K and the spectral scattering coefficient S in which [ refers to the reflection of a completely opaque layer. The ratio K/S is proportional to the tinctorial strength. 

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 ... 

Spectra of solid samples are usually recorded in the units of reflectance (R) or percent reflectance (%/ ), which is analogous to percent transmittance in that reflectance equals the ratio of the reflected radiation to the incident radiation. With diffuse reflectance, the reflected signal is attenuated by two phenomena absorption (coefficient k) and scattering (coefficient s). Lollowing the Kubelka-Munk theory, these two coefficients are related to the reflectance of an infinitely thick sample, by... 

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).

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