Structural color palette of disordered colloids in the Rayleigh scattering regime

TL;DR

Disordered colloidal media can produce structural colors through Rayleigh scattering, yet a comprehensive map linking microstructure to observed color across observation conditions has been lacking. The study combines Monte Carlo light-transport simulations with experimental validation on ZrO$_2$:Y$_2$O$_3$ in water and room-temperature fabrication of pigment-free solid composites (borosilicate clays and hybrid glasses) to map the structural color palette as a function of particle diameter $d$, density $\rho$, and thickness $L$, and to demonstrate pigment-free solid coloration. A key finding is the identification of parameter windows where blue appears in both diffuse transmission and reflection, and that adding absorbing species can yield blue reflectance in strongly scattering media due to differential attenuation of light paths, with the color palette governed by invariants $L/\ell_s$ and $L/\ell_a$. The work provides design rules for eco-friendly Rayleigh-color materials for art, architecture, and heritage conservation, enabling pigment-free coloration that can be tuned via microstructure and absorption.

Abstract

Structural coloration by Rayleigh scattering is widespread in nature and holds a prominent place in various art objects over a broad period of time, from ancient Chinese porcelains to Renaissance paintings. Beyond the common statement that Rayleigh scattering is the primary mechanism behind the multiple colored appearances of the sky, it appears that the relationship between material parameters and the colors that appear in different observation conditions has not been thoroughly explored so far. The present study aims to offer a general overview of the structural colors accessible by Rayleigh scattering as a function of the material parameters, alongside a scalable and environment-friendly process to realize solid composite materials with targeted structural colors in reflection and transmission. Monte Carlo light transport simulations are performed to compute the structural color palette of disordered colloids -- dielectric particles in a nonscattering matrix -- in different observation modes. In particular, we provide a range of physical parameters in which the materials exhibit the same blue color in diffuse reflection and transmission. We also show that, as may be counterintuitive at first, the addition of black absorbents to the matrix of a white (opaque) material can lead to the emergence of a blue coloration in diffuse reflection, thanks to the interplay between multiple light scattering and light absorption. These predictions are validated by optical experiments on colloidal suspensions of Yttria-stabilized Zirconia (ZrO$_2$:Y$_2$O$_3$) nanoparticles in aqueous solutions. The potential of Rayleigh-scattering materials for visual arts and design is further supported by realizing solid-state composites based on abundant materials, namely borosilicate clays and hybrid silica-based glasses, using soft chemistry at room temperature.

Figures (10)

• Figure 1: Panoply of liquid and solid Rayleigh-scattering materials, observed in different realistic conditions, often yielding different perceived colors. (a)-(b) Photographs of aqueous solutions containing 20-nm-diameter ZrO$_2$:Y$_2$O$_3$ particles with increasing particle concentrations (from left to right). (c)-(d) Photographs of borosilicate clay composite containing 20-nm-diameter ZrO$_2$:Y$_2$O$_3$ particles. (e)-(h) Photographs of a hybrid silica-based glass composite containing 20-nm-diameter ZrO$_2$:Y$_2$O$_3$ particles. (i) Photograph of a pictorial film producing a blue color without blue pigment ["Hineni" (2018) by Anne Goyer, exposed at the Pio Monte della Misericordia's museum in Naples, Italy; adapted with permission from Ref. goyer2021bleus]. Details on the dimensions and composition of the Rayleigh-scattering materials are provided in the text.
• Figure 2: Structural color palettes of nonabsorbing turbid media in the Rayleigh scattering regime. (a) Problem under consideration and definitions. Small spherical particles of diameter $d$ and refractive index $n_\text{p}$ are randomly dispersed in matrix of refractive index $n_\text{b}$ at a number density $\rho$. The turbid medium is a laterally-infinite slab of thickness $L$ and is illuminated at normal incidence. We compute and measure the ballistic transmittance, diffuse transmittance and diffuse reflectance over the visible spectrum. The diffuse components are integrated over the hemisphere. (b) Predicted structural colors for ZrO$_2$:Y$_2$O$_3$ particles in water, decomposed into ballistic transmittance (top), diffuse transmittance (middle) and diffuse reflectance (bottom), as a function of the reduced particle density $\rho L$ and particle diameter $d$.
• Figure 3: Comparison between experimental spectra (a,c,e) and numerical spectra (b,d,f) on the ballistic transmittance (a-b), diffuse transmittance (c-d) and diffuse reflectance (e-f) for samples with different particle concentrations and a sample thickness $L=10~\unit{mm}$. The solid curves with colors ranging from black to red correspond, respectively, to colloidal solutions with ZrO$_2$:Y$_2$O$_3$ mass fractions $[0.43, 0.71, 0.85, 1.1, 1.7, 4.1, 8.0, 18, 50]~\unit{wt\%}$ in the experiments and particle number densities $\rho = [1.71 \; 10^2, 2.84 \; 10^2, 3.41 \; 10^2, 4.26 \; 10^2, 6.82 \; 10^2, 1.71 \; 10^3, 3.41 \; 10^3, 8.53 \; 10^3, 3.41 \; 10^4]~\unit{\um^{-3}}$ in the numerics. The Monte Carlo simulations were performed here with no adjustable parameter, considering the finite lateral size of the cuvettes ($20~\unit{mm} \times 30~\unit{mm}$) and short-range structural correlations in the particle positions (for the highest ZrO$_2$:Y$_2$O$_3$ mass fraction of 50 wt%, the volume filling fraction is about 14.3%).
• Figure 4: Comparison of the structural colors deduced from the experimentally measured and numerically computed spectra reported in Fig. \ref{['fig:spectra-exp-vs-num']} on the wavelength range $[0.36,0.80]~\unit{\um}$. The same color variations are observed on all observation modes.
• Figure 5: Isotropic blue diffuse color with disordered colloids. (a) Range of sample thicknesses for $20~\unit{nm}$-diameter ZrO$_2$:Y$_2$O$_3$ particles in water to obtain a blue diffuse color in both transmission and reflection (shaded area) as a function of the particle number density $\rho$. The squares, labeled from a to i, indicate the nominal parameters of the samples studied in Fig. \ref{['fig:spectra-exp-vs-num']}. (b) Structural colors (already reported in Fig. \ref{['fig:colors-exp-vs-num']}) for the diffuse transmittance (top-left in each square) and diffuse reflectance (bottom-right in each square). The isotropic blue diffuse color is observed in the expected range of parameters.