Kubelka-Munk equations

When both absorption and scattering are present, the Beer-Lambert law must be replaced by the Kubelka-Munk equation employing the absorption and scattering coefficients iC and S, respectively. This gives the redectivity... 

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 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. 

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... 

Krypton fluorocationic salts, 17 333 Krypton lasers, 14 684. See also KrF laser in laser light shows, 14 688 Krypton-xenon, purification and separation of, 17 361-362 Krypton-xenon column, 17 359, 360 Kubelka-Munk equation, 7 317-318 14 231 23 127... 

While direct, this method is the most difficult experimentally due to the diminutive nature of fiber diameters and the uncertainty involved with contact angle measurements and hysteresis. The value i ) can also be measured on flat sheets of the fiber material but due to fabric finishes and different surface properties incurred during manufacture, the surface energetics of the sheet and fiber may be very dissimilar. Therefore, the value of co8i i was determined in the following manner from detergency data. The Kubelka-Munk Equation (12-13),... 

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 normalized absorption and fluorescence spectra of Dye 1 on 200 nm AgBr are shown in Fig. 3. The absorbance was calculated from the measured reflection spectrum using the Kubelka-Munk equation. The absorption spectrum of Dye 1 in methanol solution is also given. The absorption spectrum of Dye 1 on AgBr crystals shows a typical J-aggregate character and gives a sharp and red-shifted band relative to the monomeric band. 

UV-VIS-NIR diffuse reflectance (DR) spectra were measured using a Perkin-Elmer UV-VIS-NIR spectrometer Lambda 19 equipped with a diffuse reflectance attachment with an integrating sphere coated by BaS04. Spectra of sample in 5 mm thick silica cell were recorded in a differential mode with the parent zeolite treated at the same conditions as a reference. For details see Ref. [5], The absorption intensity was calculated from the Schuster-Kubelka-Munk equation F(R ,) = (l-R< )2/2Roo, where R is the diffuse reflectance from a semi-infinite layer and F(R00) is proportional to the absorption coefficient. 

Diffuse reflection from powder sample is a complex combination of transmission, internal and external reflections, and scattering. It is dependent on the particle size, absorption and refractive indices of the studied material. The case of proper prepared powder diffuse reflection R carries the information primarily about the transmission spectrum of the sample (Willey 1976 Fuller and Griffiths 1978). The traditional method of the absorption spectra (K) calculation on the base of the diffuse reflection R is the Kubelka-Munk equation K = (1 - R)2S/2Rc, where S is the scattering coefficient, concentration of the studied material is c = 1 in our case. 

Is it a real absorption It is well known that analysis on the base of the Kubelka-Munk equation is applicable at diffuse reflection R not much less than R 30%. The case of low diffuse reflection the deviations from linearity should be taken into account. We have R is near 1%. So, we should be careful The case of strongly absorbing samples it is possible to dilute them in nonabsorbent powder, for example in KBr powder. We have not used this traditional method because were afraid of possible chemical reactions at nigh temperature treatment of the mixture of the hydrogenated SWNTs with KBr. 

Fig. 11.6 Absorption spectra K restored from diffuse reflection R by using the Kubelka-Munk equation. Spectra 1, 2, and 3 are initial nanostructures, hydrogenated and annealed at 700°C during 6 h, respectively, (a) Spectra of NFs, (b) SWNTs. Dashed curves are Drude approximation of the absorption spectra, K = A. v0 5...

FIGURE 4 Model for the derivation of the Schuster-Kubelka-Munk equation. 

Depending on the application, the Kubelka-Munk equations can be divided into two different case—single- and two-constant systems ... 

Ki and K2. The Kubelka-Munk equation is founded on the premise that once you disperse a pigment in a resin system, there is no further development. The determination of how much light enters a sample and how much exits after diffusion... 

Sample Characteristics. In many Kubelka-Munk color-matching systems the user is required to present an opaque sample to the spectrophotometer. Depending on the application, this can be done a number of ways. For coatings, the technique of cross-coating several layers of colorant until opacity is achieved is commonly used. The generated sample then has become inconsistent with the typical process thickness. Although this can add error to the formulation, it will still adhere to the limitations of the Kubelka-Munk equation [7],... 

Evaluation of soil-release effects after washing is mostly visually done by comparison with photographic standards, but also by reflectance measurements and other instnimental techniques, including microscopy. Reflectance data using the Kubelka-Munk equation correlate fairly with the oily soil content but not with residual particulate soil (which is probably partly buried within the fabric and shielded from the light path). ... 

This equation gives the ratio of absorption and scattering coefficients in the case of R diffuse reflectance in an infinitely thick, opaque layer. In the presence of a sample with A molar absorptivity and c molar concentration, the Kubelka-Munk equation takes the following form ... 

A more sophisticated solution is given by the so-called hyperbolic Kubelka-Munk equation considering a definite layer thickness d and expressing the transmitted light Tj and the reflected light / j ... 

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. 

Roo is the reflectance of an infinitely thick sample (in the near-infrared, this means an approximate 5-mm thickness and more). The theory was recently revisited by Loyalka and Riggs, ° who reinvestigated the accuracy of the Kubelka-Munk equations. They found that the coefficient k must be replaced by k = 2a with the absorption coefficient a = In(lO) ec, as derivable from Beer s law for the latter equation In(lO) = 2.303, e the molar absorptivity, and c the molar concentration. Such a dependency for k was stated earlier by other researchers when comparing more refined radiation transport theories for biomedical applications, e.g., Ref.[ l... 

Loyolka, S.K. Riggs, C.A. Inverse problem in diffuse reflectance spectroscopy accuracy of the kubelka-munk equations. Appl. Spectrosc. 1995, 49, 1107-1110. 

Ground-state absorption studies of the probe molecules adsorbed within zeolites were performed by UV-visible diffuse reflectance spectroscopy. Cationic form of the zeolite without probe molecule was used as a comparison sample. The remission function was calculated by using the Kubelka-Munk equation. Steady-state photoluminescence studies were carried out with the Hilger spectrofluorimeter. The spectra were recorded at room temperature and 77 K, respectived. 

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. 

The Kubelka-Munk equation is only valid when R corresponds to the diffuse reflectance of an opaque layer of infinite thickness, so that the background is no longer visible (i.e., the difference in intensity between the incident and reflected beams is independent of the thickness of the structure). 

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. 

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

Alderson et. al. J.S.D.C., 1961, 77, 663) modified the Kubelka-Munk equation to conform to collimated incident light, as would apply in a spectrophotometer or a colorimeter, and arrived at the equation ... 

A further development, is the Redicolour Computer (Redifon Ltd). By mathematical treatment the so-called C function is substituted for K/S of the Kubelka-Munk equation. It is claimed that by working with this C function there is improved accuracy of dye prediction, especially in heavier shades. 

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