The formation mechanism of colors can be divided into two types: chemical color and structural color. Structural color, also known as physical color, is a visual effect produced by the interaction between light and the microstructure inside the material. Compared with chemical colors, structural colors have been widely studied and paid attention to by researchers due to their advantages such as resistance to photobleaching, low-temperature sensitivity, and low pollution. Tunable structural colors have good application prospects in dynamic displays, optical camouflage, and other fields, becoming a research hotspot that researchers are committed to breaking through. Self-assembled technology is an important means to achieve the structural color of photonic crystals, which is achieved by assembling monodisperse organic or inorganic particles into ordered colloidal crystals to obtain the structural color in the visible light region. Responsive photonic crystals adjust the structural color by changing the lattice spacing of photonic crystals. This method has the advantages of convenient tuning and wide tuning range and has achieved many distinctive application effects in experiments. Researchers usually prepare particles of various sizes and then test the structural colors to select particles of appropriate sizes. Although good experimental results have been achieved, this method of particle selection somewhat lacks guidance and is time-consuming and labor-intensive. A high-precision theoretical prediction model is required to guide the design of particle material and structural parameters, as well as the optimization range of tunable range.

After summarizing typical experimental measurement data and theoretical calculation data of self-assembled structural colors that have been reported, we compare and analyze the errors between the measured and calculated results. We propose a finite element method prediction model based on face centered cubic three-dimensional photonic crystals. In addition, we study the effects of parameters such as the refractive index of nanoparticles, solvent refractive index, particle diameter, and particle spacing on reflection spectra. Based on the predicted model, Fe₃O₄@SiO₂ nanoparticles of optimized size and an electrically-tuned device are prepared. The central wavelength of the reflection spectrum of the device is tested and compared with the finite element method prediction model for verification.

The calculation results of the finite element method prediction model indicate that the central wavelength of the reflection spectrum of photonic crystals red-shifts with the increase in particle refractive index and solvent refractive index. Compared with the refractive index of nanoparticles, the influence of solvent refractive index is more significant (Fig. 4). The central wavelength of the reflection spectrum of photonic crystals will red-shift with the increase in particle size (Fig. 5). The tuning range of the central wavelength of the reflection spectrum is mainly contributed by changes in longitudinal spacing, while changes in transverse spacing have a negative effect (Fig. 7). Optimized parameters are obtained by the prediction model. Nanoparticles of the optimized parameters are experimentally prepared. The tested results are well consistent with the prediction, indicating that the central wavelength of the reflection spectrum shifts in the range of 680 nm to 455 nm (Fig. 11). Compared with the analytical prediction model, our three-dimensional finite element method prediction model has higher accuracy in predicting the central wavelength of the reflection spectrum. For mono-core shell structures, the prediction error ranges of the two models are 0.49%-1.70% and 0.82%-1.49%, respectively, showing comparable performance. For core-shell structures, the prediction error ranges of the two models are 3.51%-6.11% and 0.28%-1.34%, respectively. Our three-dimensional finite element method prediction model reduces the typical prediction error value to 1/5.9 of the original value (Table 2).

We propose a finite element method prediction model for predicting the dynamic tuning characteristics of reflection spectra of self-assembled photonic crystal structures in colloidal systems. Based on this model, we calculate and analyze the effects of material and structural parameters on the tuning characteristics of the reflection spectrum. A set of self-assembled photonic crystals that can cover the entire visible spectral range are designed and optimized for material and structural parameters. Fe₃O₄@SiO₂ nanoparticles are synthesized with this optimized parameter as the target, and sandwich-structure color-changing samples are prepared. The tested results are consistent with that of the finite element method prediction model in terms of the central wavelength of the reflection spectrum. Experiments show that the finite element method prediction model can accurately predict the central wavelength of the reflection spectrum of self-assembled photonic crystals in colloidal systems. The model is simple with a wide range of applications, and the typical value of prediction error is reduced to 1/5.9 of the original value. The prediction strategy based on this finite element method prediction model helps to avoid the blind synthesis of nanoparticles, shorten the development cycle, and obtain the optimal filling coefficient to ensure the implementation of a large tuning range. The improvement research of prediction models should also focus on two aspects. 1) The self-assembled photonic crystal structure in colloidal systems may have the characteristics of short-range order and long-range disorder, allowing it to obtain very little change in reflection color at different incident angles. Therefore, finite element method models with higher accuracy prediction ability should also consider introducing perturbation variables into the crystal structure sequence, to obtain a high matching degree of the central wavelength, amplitude, and spectral width of the reflection spectrum between the prepared sample and the theoretical model at different incident angles. 2) The structural perfection and size consistency of synthesized nanoparticles need to be further improved to achieve high matching with theoretical models. The color display of high contrast also requires the nanoparticles to have better ball shape and homogeneous size.