Donald J. Dahm

Representative Layer Theory for Diffuse Reflectance

The fractions of light absorbed by and remitted from samples consisting of different numbers of plane parallel layers can be related with the use of statistical equations. The fractions of incident light absorbed (A), remitted (R), and transmitted (T) by a sample of any thickness can be related by an absorption/remission function, A(R,T): A(R,T) = [(1 - R)2 - T2]/R = (2 - A - 2R)A/R = 2A0/R0. Being independent of sample thickness, this function is a material property in the same sense as is the linear absorption coefficient in transmission spectroscopy. The absorption and remission coefficients for the samples are obtained by extrapolating the measured absorption and remission fractions for real layers to the fraction absorbed (A0) and remitted (R0) by a hypothetical layer of infinitesimal thickness. A sample of particulate solids can be modeled as a series of layers, each of which is representative of the sample as a whole. In order for the layer to be representative of the properties of the individual particles of which it is comprised, it should nowhere be more than a single particle thick, and should have the same void fraction as the sample; further, the volume fraction and cross-sectional surface area fraction of each particle type in the layer should be identical to its volume fraction and surface area fraction in the sample as a whole. At lower absorption levels, the contribution of a particle of a particular type to the absorption of a sample is approximately weighted in proportion to its volume fraction, while its contribution to remission is approximately weighted in proportion to the fraction of cross-sectional surface area that the particle type makes up in the representative layer.

Bridging the continuum-discontinuum gap in the theory of diffuse reflectance

A system of equations described by Benford relate the absorption and remission properties of a layer of a material to the properties of any other thickness of the material. R, the fraction of light remitted from an infinitely thick sample, may be calculated from Benfords equations by increasing the sample thickness until the total remission converges to its upper limit. The fractions of light absorbed (a0) and remitted (r0) by a very thin layer may be similarly calculated. The relationship A(r,t) = [(1 – r)2 – t2] / r = (2 – a – 2r) a / r = 2 a0 / r0 = (1 – R)2 / R describes an Absorption/Remission Function for the material as a function of a, r and t, the fractions of light absorbed, remitted and transmitted by a specified layer. This is a more general expression than the widely used Kubelka–Munk equation, but gives results equivalent to it for the case of infinitesimal particles.

Explaining some light scattering properties of milk using representative layer theory

Milk is an example of a strongly scattering material, as its white colour indicates. For non-scattering samples, the Beer-Lambert law can be used to compute an absorption coefficient for a material and this absorption coefficient can be used to calculate or predict the absorption for a sample of any thickness of that material. However, absorption coefficients calculated for scattering samples are less directly applicable to other samples of the same material, because the processes of absorption and scattering affect each other. To overcome this, “absorbance” for a scattering sample should not be defined as {log(1/T)}, but as {-log(R+T)} or {-log(1-A)}. Interactions between absorption and scattering can be understood through consideration of a layer of single particles, here termed a “representative layer”. A reasonable approximation for the “Beers law absorbance” of a material is the {-log(1-A)} of the representative layer. Using the properties of the representative layer, the absorption and scattering properties of a sample can be understood based on the refractive index difference between the particles and the matrix, the size of the particles, the wavelength of the incident light, the concentration of the particles and the thickness of the sample. This review describes how the principles of representative layer theory can explain some of the light scattering properties of milk and examines several of the techniques used to separate the effects of absorption and scatter.

Fat Globule Size Effect on Visible and Shortwave near Infrared Spectra of Milk

Step-wise homogenisation has been applied to raw milk samples of different composition to investigate the effect of fat globule size distribution on diffuse transmission spectra in the region 400–1100 nm. Homogenisation results in significant spectral changes with two distinct phases. Initial even growth of spectral intensity across the whole spectral range, observed at lower degrees of homogenisation, was followed by a drastic fall in absorbance at the long-wave end of spectrum as the fat globules reached some critical size. Fat and protein content in the sample significantly affected the observed dependences of spectra on the applied homogenisation time. These observations have been explained as a superposition of two effects: growing fat globule density and changes in scatter nature as the particle sizes approach the light wavelengths in a corresponding spectral range. The representative layer theory has been used to illustrate the nature of the spectral effects.

Obtaining material absorption properties from remission spectra of directly illuminated, layered samples

The effective linear absorption coefficient, K, obtained in transflection from directly illuminated samples made up of layers of thickness d, may be approximately related to the linear absorption coefficient, k, of the material making up the layers through the following empirical equations: k = [(1 – ½ exp(–2Kd)] K and K = [(1 + exp(–4kd)] k. The fractions of incident intensity absorbed or remitted by one layer may be modeled by assuming that that the light that moves through a sample has both the characteristics of directed and diffuse radiation.

Separating the effects of scatter and absorption using the representative layer

The physical interpretation of the absorbance observed in a non-scattering sample is straightforward; it is a simple function of the concentration of the analyte(s) and its (their) ability to absorb light. In a scattering sample, the phenomena of absorption and scattering affect each other. Consequently, the measured “absorbance” is more difficult to interpret and is not suitable for direct comparison to an “absorbance” obtained from a non-scattering sample. This paper describes a strategy for separating the effects of scatter and absorption. The amount of light absorbed by a sample can be determined by measuring both the amount of light remitted and the amount transmitted by the sample. Using the mathematics of plane parallel layers, it is possible to model the sample as a series of “layers” of any thickness and calculate the absorption, remission and transmission for each of these hypothetical layers. The absorption computed for a layer having a thickness of one particle, which we term the “representative layer”, can be used to benchmark the absorbance that would be observed from the sample in the absence of scatter.

Illustration of failure of continuum models of diffuse reflectance

Continuous models are widely used to model diffuse reflection from particulate samples. They are embodied in various theories, including diffusion theory, the equation of radiation transfer, and Kubelka-Munk. Using a simple system, it is shown that such models do not simultaneously predict the fractions of incident light that are transmitted directly and diffusely. Discontinuous models do have the capability to do so, but have a variety of limitations as well.

Speaking Theoretically … Understanding Confusing Phenomena in Remission Spectra

T ake a look at Figure 1. What can you say about the relationship of the two samples? I imagine that you would say that Sample 1 (spectrum in bold) has a larger particle size than Sample 2, and that there is more of the material with characteristic absorptions around 1750 nm (marked by large arrow) in Sample 2. We’ll discuss first why we like your answer, and then we’ll tell you why you’re wrong. We are going to try to use a few different approaches in the hope of finding one you can relate to.

Comparison of the Use of Volume Fractions with other Measures of Concentration for Quantitative Spectroscopic Calibration Using the Classical Least Squares Method

Since the commercial development of modern near-infrared spectroscopy in the 1970s, analysts have almost invariably used units of weight percent as the measure of analyte concentration, due largely to the historical precedent from other analytical methods, including other spectroscopic techniques. The application of the CLS algorithm to a set of binary and ternary liquid mixtures reveals that the spectroscopic measurement sees the sample differently; that the measured absorbance spectrum is in fact sensitive to the volume fraction of the various components of the mixture. Because there is not a one-to-one relationship between volume fraction and other measures of analyte concentration, nor is the relationship linear, this has important implications for the application of both the CLS algorithm and the various other, more conventional, calibration algorithms that are commonly used.

Relation of representative layer theory to other theories of diffuse reflection

There is currently no single theoretical treatment of diffuse reflectance data that is definitive and applicable to all cases. Continuous and discontinuous mathematical approaches have each been developed, and each has merits and limitations. Discontinuous approaches usually incorporate the feature of modeling a sample as a series of distinct layers, parallel to each other and perpendicular to the incident beam. The Representative Layer Theory has been proposed as a mechanism for modeling a particulate sample as a series of layers, enabling one to use the discontinuous mathematics. This paper outlines the Representative Layer Theory, compares and contrasts it with other theories of diffuse reflectance and presents examples that illustrate the strengths and weaknesses of the various theories.