Absence of red structural color in photonic glasses, bird feathers, and certain beetles

Colloidal glasses, bird feathers, and beetle scales can all show structural colors arising from short-ranged spatial correlations between scattering centers. Unlike the structural colors arising from Bragg diffraction in ordered materials like opals, the colors of these photonic glasses are independent of orientation, owing to their disordered, isotropic microstructures. However, there are few examples of photonic glasses with angle-independent red colors in nature, and colloidal glasses with particle sizes chosen to yield structural colors in the red show weak color saturation. Using scattering theory, we show that the absence of angle-independent red color can be explained by the tendency of individual particles to backscatter light more strongly in the blue. We discuss how the backscattering resonances of individual particles arise from cavity-like modes and how they interact with the structural resonances to prevent red. Finally, we use the model to develop design rules for colloidal glasses with red, angle-independent structural colors.

Figure 1

Measured reflectivity spectra for three similarly prepared colloidal glasses of PMMA particles in air. Insets show photographs of the samples and diameters of the particles used to make the samples. The purple sample would appear red if not for the high reflectivity in the blue, indicated by the arrow.

Figure 2

Scattering geometry for our model.

Figure 3

Structure factor of a colloidal glass with volume fraction $\varphi =0.55$ calculated from the Ornstein-Zernike equation under the Percus-Yevick approximation.

Figure 4

Calculated reflectivity as a function of $kd$ for a photonic glass of spheres at a volume fraction $\varphi =0.55$, as calculated from Eq. (6) with the Fresnel reflection coefficient omitted. The vertical dashed lines denote the $kd$ values that correspond to the range of visible wavelengths we detect, 425 nm (blue line on the right) and 800 nm (red line on the left), when the particle size is $d=$ 334 nm.

Figure 5

(a, b) Measured (smooth lines) and calculated (dashed lines) reflection spectra of colloidal photonic glasses made of PMMA spheres. Theoretical spectra are calculated from Eq. (6) with $\varphi =0.55$ and $l= 16 \mu \mathrm{m}$. Particle diameters that best fit the measured peaks are 238 nm (a) and 334 nm (b), in good agreement with the measured particle diameters. (c) Calculated reflection spectrum for a photonic glass made from $d=334\text{−}\mathrm{nm}$ particles, including only the structure factor (thick curve, divided by 10) and only the form factor (thin curve).

Figure 6

(a) Reflection spectra from Fig. 1 plotted against the dimensionless lengthscale $kd$. Note the increased scattering at high $kd$ values (short wavelengths) for the $d=330\text{−}\mathrm{nm}$ sample. (b) Same as (a) but normalized to the single-particle form factor integrated over the detected scattering angles, ${\sigma }_{F,\mathrm{detected}}$, as defined in Eq. (7). The increased scattering at high $kd$ values disappears, indicating that it is due to the form factor. Differences in amplitude of the peaks are likely due to differences in the sample thickness $l$.

Figure 7

Resonant wavelengths of the backscattering cross-section, as calculated from Mie theory, as a function of optical diameter $n_{\mathrm{p}}d$ for a refractive index contrast $m=1.2$, which corresponds to that in our experimental system. The resonant wavelength follows the linear relation $\lambda =2n_{\mathrm{p}}d/z$ (solid lines), where $z$ is the order of the resonance. Lines correspond to different values of $z$ (top, $z=1$; middle, $z=2$; bottom, $z=3$).

Figure 8

Single-particle differential scattering cross-sections for various scattering angles as a function of wavelength for a 330-nm PMMA sphere in a colloidal glass of spheres in air at a volume fraction $\varphi =0.55$. The blue-shift in the resonance is consistent with the decrease in optical pathlength inside the sphere with decreasing angle: The longest possible pathlength $a$ is twice the diameter, and $a>b$ for any angle that differs from backscattering.

Figure 9

Calculated reflection for an inverse glass of core-shell spheres engineered to scatter most strongly in the red. The continuous red curve with the peak at 632 nm includes both the structure and the form factor, the dashed black curve only the structure factor (divided by 2), and the black continuous curve only the form factor. The vertical dashed line marks 425 nm, the low-wavelength limit in our spectral measurements. The optimal design has air cores with diameter 260 nm and silica shells with diameter 280 nm, and the particles are embedded in a silica matrix. The reflectivity peak at 632 nm is primarily due to the structure factor and determined by the shell diameter. One form-factor resonance occurs in the near-IR, boosting the structural resonance in the red, and another occurs deep in the UV (at about 150 nm), too far away from the visible regime to affect the color.