Diffuse Reflectance and Transmittance: Mathematical Treatments in Spectroscopy
The analysis of scattering materials in absorption spectroscopy requires specialized mathematical frameworks to account for how light interacts with a medium. Unlike clear samples, scattering materials redirect light in multiple directions, making simple transmittance measurements insufficient. To solve this, researchers have developed various treatments, most of which rely on the concept of plane parallel layers—dividing a sample into discrete, flat sections to model the movement of light.
Most of these models utilize a two-flux (or two-stream) approximation, which assumes that light travels either in the same direction as the incident beam or in the opposite direction. Depending on the specific application, some treatments require measuring both remitted (reflected) and transmitted light, while others assume the sample is "infinitely thick," meaning no light passes through it.
ไม่มีภาพประกอบ
Key Facts
- Plane Parallel Layers: The foundational concept of dividing samples into layers to calculate light interaction.
- Two-Flux Approximation: A simplification assuming light moves only forward or backward.
- Kubelka-Munk Theory: A widely used model for coatings based on absorption and back-scatter constants.
- Remission: A term often used interchangeably with reflectance in the context of diffuse scattering.
- Scatter-Corrected Absorbance: A calculated value that remains proportional to sample thickness, removing the interference of scattering.
Foundational Theories and the Stokes Formulas
George Gabriel Stokes is credited with establishing the fundamental principles of spectroscopy. In 1862, he published formulas to determine the amount of light remitted and transmitted from a "pile of plates." While he used summations of geometric series, his results are expressed as continuous functions, allowing them to be applied to fractional numbers of plates.
The Stokes Equations
Stokes defined the relationship between transmission (T), remission (R), and absorption (A) for n layers using specific variables (Ω, Ψ, and Δ) derived from the reflectance (r) and transmittance (t) of a single layer. These formulas provide a rigorous basis for understanding how light behaves as it passes through multiple interfaces.
Radiative Transfer and the Kubelka-Munk Model
In 1905, Arthur Schuster expanded these concepts by addressing the equation of radiative transfer, which describes how radiation propagates through a medium affected by absorption, emission, and scattering. He specifically looked at "diffuse illumination," where radiation enters layers at all possible angles.
The Kubelka-Munk Theory
Published in 1931 by Paul Kubelka and Franz Munk, this theory focuses on the optics of paint. It posits that an infinitesimal layer of coating absorbs and scatters a constant portion of light. The Kubelka-Munk equation describes the remission from a sample composed of an infinite number of these infinitesimal layers, focusing on the ratio of the absorption fraction (a₀) to the remission fraction (r₀).
Deane B. Judd later refined this by defining a "remission function," which served as a measure of absorption, sometimes called "pseudo-absorbance." This allowed researchers to relate percent reflectance from an infinitely thick sample to the ratio of absorption and scattering coefficients.
Parametric Approaches and the Benford Equations
During the 1920s and 30s, researchers at General Electric developed instruments to record spectral data as "% Reflectance." In 1946, Frank Benford introduced parametric equations that provided results equivalent to the Stokes formulas but used fractions to express reflectance and transmittance.
The Benford equations allow for the calculation of ART (Absorption, Remission, Transmittance) fractions for samples of varying thicknesses (e.g., n+1, d/2, or 2d). A critical addition by Paul Kubelka in 1954 was the consideration of inhomogeneous layers, noting that the direction of illumination matters when layers are not uniform.
Advanced Solutions and Modern Refinements
Giovanelli and Chandrasekhar
In 1955, Ron Giovanelli published exact solutions to the radiative transfer equation for semi-infinite ideal diffusers. These solutions, simplified by the work of Subrahmanyan Chandrasekhar, utilize the albedo of single scatter (the fraction of radiation lost by scattering) and the H-integral to determine total reflectance.
The Contributions of Gustav Kortüm and Harry Hecht
Gustav Kortüm's 1969 book, Reflectance Spectroscopy, dominated the field for two decades, particularly in DRIFTS and NIR Spectroscopy. He emphasized the distinction between specular (mirror-like) reflection and diffuse reflection. Harry Hecht later synthesized these theories, suggesting that layers should be interpreted as the mean particle diameter of the sample rather than being infinitesimally small.
The Dahm Equation and Scatter Correction
In 1994, Donald and Kevin Dahm introduced numerical techniques to calculate remission and transmission. They developed the ART function, which remains constant for a sample composed of any number of identical layers. This function provides a bridge between the Kubelka-Munk remission function and the Kortüm-Schuster equation for isotropic scatter.
The Dahms also developed a method for calculating scatter-corrected absorbance. By iteratively applying Benford's equations to simulate thinner samples, they identified a point where the absorbance becomes constant, providing a value that is truly proportional to the sample thickness.
Summary of Mathematical Treatments
| Treatment/Theory | Primary Focus | Key Assumption/Feature |
|---|---|---|
| Stokes Formulas | Pile of plates | Geometric series; continuous functions |
| Kubelka-Munk | Infinitesimal layers | Ratio of absorption to back-scatter constants |
| Benford Equations | Parametric fractions | Recursive calculation for varying thicknesses |
| Giovanelli/Chandrasekhar | Semi-infinite diffusers | Exact solutions using albedo and H-integrals |
| Dahm Equation | ART Function | Numerical approach to scatter-corrected absorbance |
Frequently Asked Questions
What is the difference between specular and diffuse reflection?
Specular reflection is mirror-like reflection from a smooth surface, governed by different laws than diffuse reflection, which occurs when light is scattered in many directions by a rough or particulate surface.
What is the two-flux approximation?
The two-flux approximation is a mathematical simplification where all light in a scattering medium is assumed to travel either in the same direction as the incident beam or in the exact opposite direction.
Why is the Kubelka-Munk theory important for coatings?
It provides a way to relate the reflectance of a coating to its absorption and scattering coefficients, allowing for the prediction of color and opacity based on the ratio of these constants.
What is scatter-corrected absorbance?
It is a calculated absorbance value that removes the effects of light scattering, ensuring that the resulting numerical value is proportional to the thickness of the sample, similar to how absorbance works in non-scattering liquids.
How does the ART function differ from the remission function?
The ART function (Absorption/Remission/Transmittance) is a more general numerical tool that remains constant for any number of identical layers and approaches the remission function in the case of infinitesimal layers with zero absorption.