Designing angle-independent structural colors using Monte Carlo simulations of multiple scattering

Significance Structural color comes from interference of light scattered from a nanostructure. Disordered nanostructures have structural colors that are independent of viewing angle, similar to dyed materials. Unlike dyes, structural colors resist fading and can be broadly tuned, making them useful for many applications. However, making a nanostructure with a given color is challenging because there are so many tunable parameters. Furthermore, applications such as cosmetics or displays require specific component materials. To solve this design problem, we develop a model that quantitatively predicts the color for given experimental parameters. We then use optimization to determine the parameters required to make a target color under specific constraints. This approach makes it possible to engineer structural color for many different applications. Disordered nanostructures with correlations on the scale of visible wavelengths can show angle-independent structural colors. These materials could replace dyes in some applications because the color is tunable and resists photobleaching. However, designing nanostructures with a prescribed color is difficult, especially when the application—cosmetics or displays, for example—requires specific component materials. A general approach to solving this constrained design problem is modeling and optimization: Using a model that predicts the color of a given system, one optimizes the model parameters under constraints to achieve a target color. For this approach to work, the model must make accurate predictions, which is challenging because disordered nanostructures have multiple scattering. To address this challenge, we develop a Monte Carlo model that simulates multiple scattering of light in disordered arrangements of spherical particles or voids. The model produces quantitative agreement with measurements when we account for roughness on the surface of the film, particle polydispersity, and wavelength-dependent absorption in the components. Unlike discrete numerical simulations, our model is parameterized in terms of experimental variables, simplifying the connection between simulation and fabrication. To demonstrate this approach, we reproduce the color of the male mountain bluebird (Sialia currucoides) in an experimental system, using prescribed components and a microstructure that is easy to fabricate. Finally, we use the model to find the limits of angle-independent structural colors for a given system. These results enable an engineering design approach to structural color for many different applications.