Representative layer theory

The concept of the representative layer came about though the work of Donald Dahm, with the assistance of Kevin Dahm and Karl Norris, to describe spectroscopic properties of particulate samples, especially as applied to near-infrared spectroscopy. A representative layer has the same void fraction as the sample it represents and each particle type in the sample has the same volume fraction and surface area fraction as does the sample as a whole. The spectroscopic properties of a representative layer can be derived from the spectroscopic properties of particles, which may be determined by a wide variety of ways. While a representative layer could be used in any theory that relies on the mathematics of plane parallel layers, there is a set of definitions and mathematics, some old and some new, which have become part of representative layer theory.

Representative layer theory can be used to determine the spectroscopic properties of an assembly of particles from those of the individual particles in the assembly. The sample is modeled as a series of layers, each of which is parallel to each other and perpendicular to the incident beam. The mathematics of plane parallel layers is then used to extract the desired properties from the data, most notably that of the linear absorption coefficient which behaves in the manner of the coefficient in Beer’s law. The representative layer theory gives a way of performing the calculations for new sample properties by changing the properties of a single layer of the particles, which doesn’t require reworking the mathematics for a sample as a whole.

History

The first attempt to account for transmission and reflection of a layered material was carried out by Stokes in about 1860 and led to some very useful relationships. Lord Rayleigh and Mie developed the theory of single scatter to a high degree, but Schuster was the first to consider multiple scatter. He was concerned with the cloudy atmospheres of stars, and developed a plane-parallel layer model in which the radiation field was divided into forward and backward components. This same model was used much later by Kubelka and Munk, whose names are usually attached to it by spectroscopists.

Following WWII, the field of Reflectance Spectroscopy was heavily researched, both theoretically and experimentally.  The remission function, $F(R_{\infty })$, following Kubelka-Munk theory, was the leading contender as the metric of absorption analogous to the absorbance function in transmission absorption spectroscopy. 

The form of the K-M solution originally was: $F(R_{\infty })\equiv {\frac {(1-R_{\infty })^{2}}{2R_{\infty }}}={\frac {a_{0}}{r_{0}}}$, but it was rewritten in terms of linear coefficients by some authors, becoming $F(R_{\infty })\equiv {\frac {(1-R_{\infty })^{2}}{2R_{\infty }}}={\frac {k}{s}}$ , taking $k$ and $s$ as being equivalent to the linear absorption and scattering coefficients as they appear in the Bouguer-Lambert law, even though sources who derived the equations preferred the symbolism ${\frac {K}{S}}$ and usually emphasized that $K=2k$ and $S$ was a remission or backscattering parameter, which for the case of diffuse scatter should properly be taken as an integral.

In 1966, in a book entitled Reflectance Spectroscopy, Harry Hecht had pointed out that the formulation $F(R_{\infty })={\frac {k}{s}}$ led to $\log F(R_{\infty })=log(k)-log(s)$, which enabled plotting $F(R_{\infty })$ "against the wavelength or wavenumber for a particular sample" giving a curve corresponding "to the real absorption determined by transmission measurements, except for a displacement by $-log(s)$ in the ordinate direction." However, in data presented, "the marked deviation in the remission function ... in the region of large extinction is obvious." He listed various reasons given by other authors for this "failure ... to remain valid in strongly absorbing materials", including: "incomplete diffusion in the scattering process"; failure to use "diffuse illumination; "increased proportion of regular reflection"; but concluded that "notwithstanding the above mentioned difficulties, ... the remission function should be a linear function of the concentration at a given wavelength for a constant particle size" though stating that "this discussion has been restricted entirely to the reflectance of homogeneous powder layers" though "equation systems for combination of inhomogeneous layers cannot be solved for the scattering and absorbing properties even in the simple case of a dual combination of sublayers. ... This means that the (Kubelka-Munk) theory fails to include, in an explicit manner, any dependence of reflection on particle size or shaper or refractive index".

The field of Near infrared spectroscopy (NIR) got its start in 1968, when Karl Norris and co-workers with the Instrumentation Research Lab of the U.S. Department of Agriculture first applied the technology to agricultural products. The USDA discovered how to use NIR empirically, based on available sources, gratings, and detector materials. Even the wavelength range of NIR was empirically set based on the operational range of a PbS detector. Consequently, it was not seen as a rigorous science: it had not evolved in the usual way from research institutions to general usage. Even though the Kubelka-Munk theory provided a remission function that could have been used as the absorption metric, Norris selected $log(1/R_{\infty })$ for convenience. He believed that the problem of non-linearity between the metric and concentration was due to particle size (a theoretical concern) and stray light (an instrumental effect). In qualitative terms, he would explain differences in spectra of different particle size as changes in the effective path length that the light traveled though the sample.

In 1976, Hecht published an exhaustive evaluation of the various theories which were considered to be fairly general. In it, he presented his derivation of the Hecht finite difference formula by replacing the fundamental differential equations of the Kubelka-Munk theory by the finite difference equations, and obtained: $F(R_{\infty })=a{\biggl (}{\frac {1}{R}}-1{\biggr )}-{\frac {a^{2}}{2r}}$ . He noted "it is well known that a plot of $F(R_{\infty })$ versus $K$ deviates from linearity for high values of $K$, and it appears that (this equation) can be used to explain the deviations in part", and "represents an improvement in the range of validity and shows the need to consider the particulate nature of scattering media in developing a more precise theory by which absolute absorptivities can be determined."

In 1982, Gerry Birth convened a meeting of experts in several areas that impacted NIR Spectroscopy, with emphasis on diffuse reflectance spectroscopy, no matter which portion of the electromagnetic spectrum might be used. This was the beginning of the International Diffuse Reflectance Conference. At this meeting, was Harry Hecht, who may have at the time been the world's most knowledgeable person in the theory of diffuse reflectance. Gerry himself took many photographs illustrating various aspects of diffuse reflectance, many of which were not explainable with the best available theories. In 1987, Birth and Hecht wrote a joint article in a new handbook, which pointed a direction for future theoretical work.

In 1994, Donald and Kevin Dahm began using numerical techniques to calculate remission and transmission from samples of varying numbers of plane parallel layers from absorption and remission fractions for a single layer. Using this entirely independent approach, they found a function that was the independent of the number of layers of the sample. This function, called the Absorption/Remission function and nick-named the ART function, is defined as: $A(R,T)\equiv {\frac {(1-R_{n})^{2}-T_{n}^{2}}{R_{n}}}={\frac {(2-a-2r)a}{r}}={\frac {a(1+t-r)}{r}}=2F(R_{\infty })={\frac {2a_{0}}{r_{0}}}$ . Besides the relationships displayed here, the formulas obtained for the general case are entirely consistent with the Stokes formulas, the equations of Benford, and Hecht's finite difference formula. For the special cases of infinitesimal or infinitely dilute particles, it gives results consistent with the Schuster equation for isotropic scattering and Kubelka–Munk equation. These equations are all for plane parallel layers using two light streams. This cumulative mathematics was tested on data collected using directed radiation on plastic sheets, a system that precisely matches the physical model of a series of plane parallel layers and found to conform. The mathematics provided: 1) a method to use plane parallel mathematics to separate absorption and remission coefficients for a sample; 2) an Absorption/Remission function that is constant for all sample thickness; and 3) equations relating the absorption and remission of one thickness of sample to that of any other thickness.

Mathematics of plane parallel layers in absorption spectroscopy

Using simplifying assumptions, the spectroscopic parameters (absorption, remission, and transmission fractions) of a plane parallel layer can be built from the refractive index of the material making up the layer, the linear absorption coefficient (absorbing power) of the material, and the thickness of the layer. While other assumptions could be made, those most often used are those of normal incidence of a directed beam of light, with internal and external refection from the surface being the same.

Determining the A,R,T fractions for a surface

For the special case where the incident radiation is normal and the absorption is negligible, the intensity of the reflected and transmitted beams can be calculated from the refractive indices {η₁ and η₂} of the two media, where {r} is the fraction of the incident light reflected, and {t} is the fraction of the transmitted light: $r={\frac {(\eta _{2}-\eta _{1})^{2}}{(\eta _{2}+\eta _{1})^{2}}}$

$t={\frac {4\eta _{1}\eta _{2}}{(\eta _{2}+\eta _{1})^{2}}}$ , with the fraction absorbed being zero ( $a$ = 0 ).

Expressing particulate mixtures as layers

If one wants to use a theory based on plane parallel layers, optimally the samples would be describable as layers. But a particulate sample often looks a jumbled maze of particles of various sizes and shapes, showing no structured pattern of any kind, and certainly not literally divided into distinct, identical layers. Even so, it is a tenet of Representative Layer Theory that for spectroscopic purposes, we may treat the complex sample as if it were a series of layers, each one representative of the sample as a whole.

Definition of a Representative Layer

In a representative layer,

1)  the void fraction in the representative layer is the same as in as in the sample.

2)  the volume fraction of each kind of particle in the layer is the same as its volume fraction in the sample as a whole. (The volume fraction of a particle type is its volume fraction divided by the total volume of the sample.)

3) the surface area fraction of each kind of particle will be the same in the layer as in the sample.  (Surface area fraction is defined analogously to volume fraction except voids have no surface to consider. The surface area for each individual type of particle must be computed, and totaled for all particles.)

It is convenient to think of a representative layer as one particle thick, because if the absorption and scattering properties of the layer are determined by combining the properties of the individual particles, it is simpler to combine them if the layer is nowhere more than one particle thick.

List of principle symbols used

Where a given letter is used in both capital and lower case form (r, R and t ,T ) the capital letter refers to the macroscopic observable and the lower case letter to the corresponding variable for an individual particle or layer of the material. Greek symbols are used for properties of a single particle.

• $a$ – absorption fraction of a single layer
• $r$ – remission fraction of a single layer
• $t$ – transmission fraction of a single layer
• A_n, R_n, T_n – The absorption, remission, and transmission fractions for a sample composed of n layers
• α – absorption fraction of a particle
• β – back-scattering from a particle
• σ – isotropic scattering from a particle
• $k$– absorption coefficient defined as the fraction of incident light absorbed by a very thin layer divided by the thickness of that layer
• $s$ – scattering coefficient defined as the fraction of incident light scattered by a very thin layer divided by the thickness of that layer