Kubelka-Munk relation

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. 

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... 

The effect on the gold particle size characteristics of calcination in air could be followed with a combination of UV-visible diffuse reflectance spectroscopy, monitoring the evolution of the surface plasmon absorbance, and TEM. The well established Kubelka-Munk relation for the diffuse reflectance of non-translucent films of fine powders gives the following for each wavelength ... 

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. 

This analytical approach is difficult to apply to individual pigments because physical data relating to refractive index, dispersion curves and the absorption curves in the solid state are not available. A colligative approach, based on the Kubelka-Munk analysis which characterises pigments by only two constants, an absorption and a scattering coefficient, has been applied with considerable success to the computation of the proportions of pigments in mixtures needed to match a given colour. Much of the book Colour physics for industry is devoted to this topic [37]. 

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... 

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... 

The Kubelka-Munk theory relates q(A) to scattering, absorption, and film thickness (scattering coefficient S, absorption coefficient K, film thickness h). 

Equations (3) and (4) are formally identical with the earlier Kubelka s hyperbolic solutions of differential equations for forward and backward fluxes (11), although the Chandrasekhar-Klier and Kubelka s theories start from different sets of assumptions and employ different definitions of constants characterizing the scattering and absorption properties of the medium. In Kubelka s theory, the constants a, b, and Y are related to the Schuster-Kubelka-Munk (SIM) absorption K and scattering S coefficients as... 

The empirical modeling element indicates an increased emphasis on data-driven rather than theory-driven modeling of data. This is not to say that appropriate theories and prior chemical knowledge are ignored in chemometrics, but that they are not relied upon completely to model the data. In fact, when one builds a chemometric calibration model for a process analyzer, one is likely to use prior knowledge or theoretical relations of some sort regarding the chemistry of the sample or the physics of the analyzer. For example, in process analytical chemistry (PAC) applications involving absorption spectroscopy, the Beer s Law relation of absorbance vs. concentration is often assumed to be true and in reflectance spectroscopy, the Kubelka-Munk or log(l/P) relations are assumed to be true. 

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 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. 

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]

Argyle et al. (2003, 2004) introduced a method to determine the average valence of Al203-supported VOx species under the conditions of propane ODH catalysis. First, calibration measurements were made the catalyst was reduced for various periods of time in H2 at 603 K, and then the amount of 02 required to fully restore the UV-vis spectra was measured by mass spectrometry. Spectra of the fully oxidized sample were recorded to generate the background. These relative reflectance spectra were converted by applying the Kubelka-Munk function and then the intensity in the range 1.5-1.9 eV was related to the extent of... 

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... 

The DR spectrum of a dilute sample of "infinite depth" (i.e., up to 3 mm) is usually calculated with reference to the diffuse reflectance of the pure diluent to yield the reflectance, Ri. RiA is related to the concentration of the sample, c, by the Kubelka-Munk (K-M) equation ... 

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... 

Kubelka and Munk developed a theory describing the diffuse reflectance process for powdered samples, which relates the sample concentration to the scattered radiation intensity. The Kubelka-Munk equation is as follows ... 

The Kubelka-Munk theory relates the extinction coefficient to the reflection. In the simplest case, it is assumed that light is only scattered in two directions in the incident and in the backward direction for an incident ray normal to the surface of the test sample. Also, both incident light and emitted light are diffuse. According to Kubelka and Munk, then. 

In diffuse reflectance spectroscopy, there is no linear relation between the reflected light intensity (band intensity) and concentration, in contrast to traditional transmission spectroscopy in which the band intensity is directly proportional to concentration. Therefore, quantitative analyzes by DRIFTS are rather complicated. The empirical Kubelka -Munk equation relates the intensity of the reflected radiation to the concentration that can be used for quantitative evaluation. The Kubelka-Munk equation is defined as ... 

The optical measmements of diffuse reflectance are dependent on the composition of the system. Several theoretical models have been proposed for diffuse reflectance, which are based on the radiative transfer theory, and all models consider that the incident hght is scattered by particles within the medium. The most widely used theory in photometric sensors is the Kubelka-Munk theory, in which it is assumed that the scattering layer is infinitively thick, which may, in practice, be the case with the chemical transducers utilized in photometric sensors. The absolute value of the reflectance R is related to the absorption coefficient K and the scattering coefficient S by the equation... 

The usual Lambert-Beer Law that is the basis of solution spectrometry is not valid for densitometry because both absorption and scattering of radiation occur during direct zone measurement on a layer. The Kubelka-Munk equation is usually used to relate signal intensity and zone concentration (weight per zone) for the reflectance (absorption) mode of densitometry ... 

This theory relates to the direct observation of the scattering of a single particle as compared to the Kubelka-Munk theory, which relates to multiple scattering between particles. Mie theory also takes into account the absorption, which the particle may also exhibit. 

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