The Beer-Lambert law, a cornerstone of analytical chemistry, describes the attenuation of light in absorbing, non-scattering media. In homogeneous solutions, absorbance increases linearly with concentration, enabling straightforward quantitative analysis. However, when applied to solid samples, particularly those with rough interfaces or particulate structures, this linearity breaks down. Forward and backward scattering alter photon paths, causing deviations from the expected linear relationship between absorbance and concentration. This limitation has historically hindered the precise compositional analysis of solid mixtures, forcing researchers to rely on empirical or qualitative methods that lack rigorous theoretical backing.

To overcome these limitations, the study utilizes diffuse reflectance spectroscopy with an integrating sphere, which collects scattered light from all directions to yield the sample’s diffuse reflectance. The Kubelka-Munk function then maps this reflectance to absorption and scattering coefficients. By combining the Kubelka-Munk function with the principle of mixture additivity, the researchers derived an equation that quantitatively describes absorbance as a function of mass fraction for solids. More importantly, they identified a specific regime at small mass fractions where the Kubelka-Munk function increases linearly with the mass fraction, providing a reliable calibration axis for composition analysis.

The experimental validation involved measuring diffuse reflectance spectra for nine inorganic semiconductors, including TiO2, ZnO, and BiVO4, diluted with BaSO4 at various mass fractions. The results confirmed that while absorbance versus mass fraction plots showed clear non-linear deviations, the Kubelka-Munk function versus mass fraction plots exhibited linear behavior at low loadings. For eight of the studied semiconductors, this linearity persisted up to a mass fraction of approximately 0.2, while for BiVO4, it held up to 0.1. This linear relationship was found to be robust across different sample morphologies, although the slope of the calibration curve varied depending on the specific morphology, reflecting the influence of scattering on optical properties.

The study further demonstrated the practical utility of this linear calibration method by applying it to the quantitative composition analysis of semiconductor mixtures. Using Bi2O3/TiO2 and BiVO4/TiO2 systems as examples, the researchers showed that the additive principle holds for the Kubelka-Munk function in spectral regions where the diluent’s absorption is minimal. By exploiting the linear relationship, they were able to determine the mass fractions of the individual components in these mixtures with close agreement to the preparation values. This approach provides a simple, rapid, and non-destructive strategy for determining the composition of solid mixtures, which is particularly valuable for the development of heterojunction photocatalysts.

The influence of sample morphology on the calibration linearity was also examined. By synthesizing Bi2O3 and BiOCl samples with different morphologies, such as microrods and microspheres, the researchers observed that while the linear relationship remained valid, the slopes of the calibration curves changed significantly. This finding underscores the importance of considering sample morphology when applying the calibration method. The slope differences were more pronounced for BiOCl, suggesting that the assembly of constituent units, such as nanoplates forming microspheres versus random aggregates, has a stronger impact on the optical response than particle shape alone. This insight highlights the need for recalibration when the morphology or preparation history of a sample changes.

The theoretical basis for the linear relationship is rooted in the approximation that at small mass fractions, the contribution of the reference material’s absorption and scattering coefficients to the total Kubelka-Munk function remains constant. This allows the function to be expressed as a linear combination of the semiconductor’s mass fraction and a constant term. The study provides a rigorous derivation of this relationship, filling a gap in the literature where previous observations of linear dependencies were often empirical or qualitative. By establishing a clear theoretical justification, the authors have provided a foundation for the broader application of the Kubelka-Munk function in quantitative solid-state analysis.

The implications of this work extend beyond photocatalysis to any field requiring the precise characterization of solid mixtures. The ability to accurately determine the composition of scattering semiconductors can facilitate the comparison of materials prepared with different component ratios and help optimize the composition for enhanced performance. The method imposes no constraint on the inter-component mass ratio, provided that the mass fractions fall within the linear regime, which can be readily achieved by adjusting the diluent. This flexibility makes the calibration method a versatile tool for researchers working with a wide range of solid-state materials.

In conclusion, the study presents a significant advancement in the spectroscopic analysis of scattering semiconductors. By establishing a linear calibration method based on the Kubelka-Munk function, the authors have provided a rigorous and practical approach for the quantitative determination of composition in solid mixtures. The method’s robustness across different morphologies and its theoretical underpinning make it a valuable addition to the toolkit of materials scientists. As the demand for optimized photocatalytic and other solid-state materials continues to grow, this calibration method will likely play a crucial role in accelerating the development of new materials with tailored properties.

The findings of this study highlight the importance of revisiting fundamental spectroscopic principles in the context of complex solid-state systems. By addressing the limitations of traditional methods and providing a new, reliable calibration approach, the researchers have opened new avenues for the precise characterization of materials. This work not only enhances our understanding of light-matter interactions in solids but also provides a practical solution to a common analytical challenge, paving the way for more accurate and efficient material development in the future.

Subject of Research: Linear calibration of scattering semiconductors via Kubelka-Munk function

Article Title: A linear calibration method for scattering semiconductors using the Kubelka-Munk function

Article References: Zhao, S., Luo, L., Zhang, F., Jiang, B., Jaroniec, M., & Zhang, L. (2026). A linear calibration method for scattering semiconductors using the Kubelka-Munk function. iScience, 29(11), Article 117932. https://doi.org/10.1016/j.isci.2026.117932

Image Credits: AI Generated

DOI: 10.1016/j.isci.2026.117932

Keywords: Kubelka-Munk function, diffuse reflectance spectroscopy, semiconductor characterization, Beer-Lambert law, photocatalyst composition, solid-state spectroscopy, optical calibration, material analysis, linear, calibration, method, scattering