Researchers have introduced a simple yet powerful new rule based on Rayleigh scattering theory that accurately links the absorption behavior of composite media, like aerosols or colloids, to the properties of their nanoparticle constituents.
Improved Insight into Light Absorption by Small Particles
Understanding how light interacts with particles suspended in air or liquids, such as aerosols or colloids, is essential in environmental science, optics, and nanotechnology. A new study by Hans Moosmüller and Justin B. Maughan of the Desert Research Institute in Reno, Nevada; Prakash Gautam from Pratt Community College in Pratt, Kansas, and Christopher M. Sorensen from Kansas State University in Manhattan, Kansas, introduces a fresh approach to a classic optical problem. The team derived a new “Rayleigh mixing rule” that precisely relates the imaginary part of the refractive index, responsible for absorption, of a composite medium to the properties of the small particles suspended within it (1).
Published in the Journal of Quantitative Spectroscopy and Radiative Transfer, the study simplifies how scientists can model or determine the absorption characteristics of nanoparticle-laden materials when particle sizes fall within the Rayleigh regime, where particles are much smaller than the wavelength of the incident measurement light used (1).
A new rule based on Rayleigh scattering theory that accurately models the absorption behavior of aerosols or colloids © Dimitrios-chronicles-stock.adobe.com
A Simpler Path Using Rayleigh Scattering
In their derivation, the researchers utilized Rayleigh scattering theory to develop an intuitive and straightforward relationship between a composite material’s absorption coefficient and the complex refractive index of the suspended particles. According to Moosmüller and co-authors, the Rayleigh mixing rule enables scientists to work backward—using measured absorption spectra of the medium to estimate the imaginary part of the particle's refractive index (1).
This is especially valuable when dealing with modern nanoparticle colloids, many of which are now commercially available and used in fields ranging from optics to drug delivery (1).
Spectroscopy and Refractive Index Calculations
At the core of the study is a spectroscopic approach to understanding how suspended particles absorb light. When particles are in the Rayleigh regime (that is, much smaller than the wavelength of incident light), scattering effects become negligible, and absorption becomes the dominant interaction. The study shows how the absorption cross-section scales linearly with the imaginary component (κ) of the refractive index, modified by a derived factor ξ(m)—a key function of the complex refractive index (1).
This factor, ξ(m), quantifies how the real and imaginary parts of the particle’s relative refractive index influence absorption. The authors show that in certain optical conditions, ξ(m) can significantly change absorption behavior, sometimes by more than a factor of 20—making it a critical consideration for accurate quantitative modeling based on absorption (1).
Comparing to Other Mixing Rules
To validate the new approach, the Rayleigh mixing rule was compared to traditional effective medium theories, including the Maxwell Garnett and Bruggeman rules. In the limit of low particle volume fractions—a common case in real-world colloids and aerosols—the Rayleigh rule agrees with both, giving it additional theoretical credibility (1–3).
However, the Rayleigh rule’s elegance lies in its simplicity and direct dependence on measurable quantities like absorption spectra and known real refractive indices of suspending media (for example, air, water). For spectroscopists, this offers a powerful tool to estimate elusive particle properties from straightforward extinction or absorption measurements (1–3).
Implications and Future Use
Because the absorption coefficient can be measured with standard spectroradiometers across the solar spectral range, this new mixing rule opens efficient, non-invasive characterization techniques for nanoparticles. The researchers also note that while shape effects exist, the dependence on morphology is weak in the Rayleigh regime, broadening the method’s applicability across different particle types (1).
Ultimately, the new Rayleigh mixing rule serves both as a practical measurement tool and a theoretical bridge between nanoscale optical physics and real-world spectroscopic applications.
References
(1) Moosmüller, H.; Maughan, J. B.; Gautam, P.; Sorensen, C. M. A Mixing Rule for Imaginary Parts of Refractive Indices of Aerosols or Colloids in the Rayleigh Regime. J. Quant. Spectrosc. Radiat. Transf. 2025, 331, 109254. DOI: 10.1016/j.jqsrt.2024.109254
(2) Haspel, C. Some Insights on the Interaction of Light with Atmospheric Aerosol Particles Comprised of Disordered Materials. In Springer Series in Light Scattering: Volume 10: Direct and Inverse Problems of Light Scattering Media Optics; Springer: 2024; pp 179–212. DOI: 10.1007/978-3-031-66578-3_4
(3) Markel, V. A. Maxwell Garnett Approximation in Random Media: Tutorial. J. Opt. Soc. Am. A 2022, 39 (4), 535–544. DOI: 10.1364/JOSAA.450850
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