  Photonic balls are spheres tens of micrometers in diameter containing
assemblies of nanoparticles or nanopores with a spacing comparable to the
wavelength of light. When these nanoscale features are disordered, but still
correlated, the photonic balls can show structural color with low
angle-dependence. Their colors, combined with the ability to add them to a
liquid formulation, make photonic balls a promising new type of pigment
particle for paints, coatings, and other applications. However, it is
challenging to predict the color of materials made from photonic balls, because
the sphere geometry and multiple scattering must be accounted for. To address
these challenges, we develop a multiscale modeling approach involving Monte
Carlo simulations of multiple scattering at two different scales: we simulate
multiple scattering and absorption within a photonic ball and then use the
results to simulate multiple scattering and absorption in a film of photonic
balls. After validating against experimental spectra, we use the model to show
that films of photonic balls scatter light in fundamentally different ways than
do homogeneous films of nanopores or nanoparticles, because of their increased
surface area and refraction at the interfaces of the balls. Both effects tend
to sharply reduce color saturation relative to a homogeneous nanostructured
film. We show that saturated colors can be achieved by placing an absorber
directly in the photonic balls and mitigating surface roughness. With these
design rules, we show that photonic-ball films have an advantage over
homogeneous nanostructured films: their colors are even less dependent on the
angle.
