Absorption and scattering properties of nanoparticles in an absorbing medium: modeling with experimental validation

TL;DR

This study develops and validates a framework that connects the absorption and scattering properties of nanoparticles in an absorbing medium to the measurable reflectance and transmittance of thin-film composites. By combining the two-flux Kubelka–Munk theory with Saunderson correction and Mie theory for particles in absorbing media, the authors extract and compute the coefficients $K$ and $S$ from experimental transmittance $T$ and reflectance $R$, and from theory using an effective refractive index and a conceptual-sphere radius $R_{cs}$. They show that for TiO$_2$ nanoparticles in PMMA, the scenario with $R_{cs}=d/2$ best reproduces the observed $K$ and $S$ spectra, illustrating strong attenuation effects due to the surrounding medium and PMMA absorption in the UV. The validated model also enables prediction of the overall optical properties ($T$, $R$, $\mathcal{A}$) of other composites, such as air-bubble/PMMA, highlighting its potential as a fast design tool for radiative cooling materials and related applications.

Abstract

Absorption and scattering properties of nanoparticles immersed in an absorbing medium are essential in understanding the overall properties of composites and in designing materials with expected functionalities. In this paper, we establish a model based on both Kubelka-Munk theory and Mie theory that links the absorption and scattering properties of individual particles with the reflectance and transmittance spectra of its thin-film composite, supported by detailed experiments. Thin films consisting of TiO$_2$ nanoparticles embedded in PMMA are fabricated on glass substrates using spin-coating and then peeled off to form standalone samples for spectroscopy measurements. By using the Kubelka-Munk theory in combination with the Saunderson correction, the absorption $K$ and scattering $S$ coefficients of multiple nanoparticles are extracted from the measured transmittance and reflectance. On the other hand, the absorption $K$ and scattering $S$ coefficients are the sum of absorption and scattering cross-sections of individual particles, which are calculated from the Mie theory specified for particles in an absorbing medium, with the scattering $S$ coefficient further modulated by the anisotropy factor $g$. The effect of the particulate medium is incorporated through an effective refractive index. The overall model is validated by matching well between the $K-S$ coefficients extracted from experimental data and theoretical calculations. This agreement provides deep insight into the significant attenuating effect of absorption and scattering on each particle due to the surrounding medium. The validated model of nanoparticles immersed in an absorbing medium can be used to obtain preliminary results for materials designed in a number of applications, such as radiative cooling.

Figures (13)

• Figure 1: The connection between light scattering by an individual particle in a medium and overall optical properties of a thin-film composite based on Mie theory, radiative transfer equations, and Kubelka-Munk theory. The two-way mapping shows the possibility of component analysis from the spectroscopy measurements, following the purple arrow direction, and the prediction of composite functionality from the calculation of the scattering properties of individual particles in the reversed direction (red arrow).
• Figure 2: Schematic model of for light path in thin-film composite: (a) Optical paths of light passing through the thin film, where the light is partly absorbed and partly scattered by the composite components. For a given incident light, the total transmittance $T$ (thick purple arrows) passing through the thin-film backside (z = $d$) and total reflectance $R$ (thick red arrows) at the thin-film frontside (z = 0) are measurable in spectroscope experiments with an integrating sphere. The dependence of $T$ and $R$ on the composite components is described in detail in section \ref{['sub:KM']} (equations \ref{['eq:Tkm']}-\ref{['eq:R']}). Using the equations, the absorption $K_{\rm{exp}}$ and scattering $S_{\rm{exp}}$ coefficients are inverted from the measured $T$ and $R$. (b) Given a number of uneven particles per unit volume $N_{\rm{p}}$, the theoretical values $K_{\rm{theo}}$ and $S_{\rm{theo}}$ are defined as the function of absorption $C_{\rm{abs}}$ and scattering $C_{\rm{sca}}$ cross-sections of individual particles, weighted by the particle size distribution $\rho_(a)$ and anisotropy factor $g$ as detailed in section \ref{['sub:KM']} (equations \ref{['eq:K_sim']}-\ref{['eq:S_sim']}). (c) Theoretical model for the absorption $C_{\rm{abs}}$ and scattering $C_{\rm{sca}}$ cross-sections of individual particles. Each TiO$_2$ particle with the refractive index $(n_{\rm{p}}, \kappa_{\rm{p}})$ dispersed in PMMA medium with $(n_{\rm{h}}, \kappa_{\rm{h}})$ is assumed to be in a conceptual sphere shown by the dashed line with a radius $\mathcal{R}_{\rm{cs}}$ and an effective refractive-index $(n_{\rm{eff}}, \kappa_{\rm{eff}})$, detail in section \ref{['sub:Mie']}
• Figure 3: Images of thin-film composites of PMMA with different concentrations of TiO$_2$: 0%, 0.5%, 1%, and 5% in (a) for series M15 and in (b) for series M55. The sample size is about 2.3 $\times$ 2.3 cm$^2$. The contrast between samples and green (grey) background gradually changes with the increase in particle concentrations.
• Figure 4: UV-Vis-NIR spectra of thin-film composites. The total transmittance $T$ is in (a) and (d), and the total reflectance $R$ is in (b) and (e) of series M15 and series M55, respectively. While, the absorption $\mathcal{A}$ is calculated by $\mathcal{A} = 1 - T - R$ in (c) and (f). In each series, there are four samples with different mass concentrations: 0% in black, 0.5% in green, 1% in red, and 5% in purple.
• Figure 5: (Color online) $K_{\rm{exp}}$-$S_{\rm{exp}}$ coefficients extracted from measured data: series M15 in (a) and (b), series M55 in (d) and (e). The $K_{\rm{exp}}$-$S_{\rm{exp}}$ values are determined by inverting equations \ref{['eq:Tkm']}-\ref{['eq:R']} in the wavelength range from 0.2 to 2.5 $\mu$m with three mass concentrations of 0.5% in green, 1% in red, and 5% in purple, respectively. The uncertainty of $(K_{\rm{exp}}+S_{\rm{exp}})$ in gray area determined by equation \ref{['eq:errorSK']} is presented in (c) for series M15 and in (f) for series M55.