Achieving structural white inspired by quasiordered microstructures in *Morpho theseus*

The study investigates how quasiordered micro/nanostructures in Morpho theseus wing scales produce structural whiteness and contribute to thermal regulation. Structural whiteness in nature is categorized into uniform white (angle-independent, often disordered) and metallic white (angle-dependent shine from periodic architectures, common in insects). While periodic structures have been modeled effectively, quasiordered or disordered architectures—prevalent in nature and critical for cooling—lack suitable numerical models. Building on recent trigonometric implicit-function frameworks that connect diverse photonic structures, the authors aim to establish a numerical model that captures quasiordered features and to elucidate their optical mechanisms and thermal benefits in M. theseus.

Background literature highlights structural whiteness across organisms and its functions in thermoregulation, courtship, and camouflage. Prior work has used high-refractive-index pigments as a contrast to eco-friendly structural whiteness. A unified evolving model based on trigonometric implicit functions has been proposed to bridge various photonic microstructures and improve simulation efficiency. Gyroid minimal surfaces (notably in Callophrys rubi) have been mathematically described and are a benchmark for ordered, angle-dependent coloration. However, robust modeling of quasiordered natural structures remains underdeveloped, motivating the current work.

Specimen characterization: Morpho theseus and Callophrys rubi specimens were sourced from Shanghai Dieyu Corporation (China). Optical images were obtained using a KEYENCE VHX-1000E microscope (halogen lamp 12 V, 100 W). Reflectance spectra were measured with a NOVA microspectrograph (Ideaoptics). Micro-angle-resolved spectra were recorded with an ARM system (Ideaoptics), spot diameter 30 µm, angle range −60° to +60° in 1° steps. SEM (TESCAN-MIRA3) was used after sputter coating (Q 150T ES plus). Cross-sections were imaged by TEM (FEI Tecnai G2 Spirit Biotwin). TEM sample preparation: Scales were embedded in epoxy at 23 °C, polymerized at 60 °C for 48 h, sectioned by ultramicrotome (Leica EM UC7), and transferred to copper grids. CIE color processing: Using CIE 1931 illuminant D65, XYZ tristimulus values were computed from simulated reflectance R(λ), converted to RGB, and visualized in MATLAB R2016a. FDTD optical simulations: Conducted in Lumerical FDTD Solutions with structures built via MATLAB R2016a. Chitin refractive index set to 1.56 + 0.06i. A plane-wave source at normal incidence with polarization along x-axis was used. Periodic boundaries in x and y and perfectly matched layers in z were applied. Mesh convergence was tested. The numerical model f(x,y,z) uses thresholds: outside the range of t is air; inside is chitin. Structural parametrization bridges tubular and gyroid forms using trigonometric implicit functions. The gyroid is described by sin(x)cos(y) + sin(y)cos(z) + sin(z)cos(x) = t, and tubular elements by sin(ω1 x)cos(ω1 y) = t_tub. A generalized form A·sin(x)cos(y) + B·sin(y)cos(z) + C·sin(z)cos(x) = t (t ≥ a) was used to tune morphology (A,B,C) between pure tubes (1,0,0) and gyroid (1,1,1). Period (unit cell size) and volume fraction (VF) were varied. Thermodynamic measurements: Wings were mounted on a metallic support in a vacuum chamber with foam to minimize conduction and fixed with specimen needles. Scales were removed for control tests using an alcohol-soaked swab. A xenon lamp (X350) simulated solar irradiation with an electronic switch. Wing temperature changes were recorded by an IR thermal camera (FLIR-T630c) through a ZnSe window (high mid-IR transmittance). A bromoform immersion test (n ≈ 1.56) was used to index-match chitin to verify structural, not pigment, origin of whiteness.

• Morphology: M. theseus scales contain quasiordered tubular architectures with varying orientations, helix degrees, and twining; they resemble intermediates between simple tubes and gyroid structures.
• Mathematical linkage: The tubular subelement sin(x)cos(y) is a constituent of the gyroid equation, enabling a unified numerical description via coefficients (A,B,C) tuning from (1,0,0) tubes to (1,1,1) gyroid.
• Optical performance (experiments):
  • M. theseus shows broadband, high-intensity visible reflectance with an average reflectivity of 78.1%, about 4× higher than gyroid domains in C. rubi.
  • C. rubi gyroid exhibits a narrow reflectance peak at 518 nm with 29.83% intensity.
  • Reflectance variance: M. theseus 2.75 vs. C. rubi 9.42.
  • Angle-resolved: At observation angles up to ±60°, M. theseus maintains high reflectance, averaging 63.75% in the visible, indicating low angle dependence.
  • Index-matching test: Immersing a single M. theseus scale in bromoform (n ≈ 1.56) rendered it semitransparent with disappearance of color pixels; upon evaporation, reflectivity recovered to ~78%, confirming structural origin of whiteness.
  • Volume fraction of chitin in disordered tubular domains averages 10.24%.
• Structural statistics: The unit-cell periods (cubic sizes) range from 326 to 1063 nm; VF ranges from 5.00% to 16.22%.
• Optical simulations (FDTD):
  • Anisotropy: Tubes perpendicular to polarization yield high transmission (average reflectivity <5%); parallel alignment produces reflection peaks (623–670 nm), with peak narrowing and increased saturation as helicity increases.
  • Period dependence: Increasing period induces a uniform redshift of the reflectance peak. For state (0,1,0), peak position shifts 311→1038 nm as period increases 300→1000 nm; peak width broadens 56→195 nm. Peak amplitudes remain approximately 64% (0,1,0), 58% (1,1,0), and 36% (1,1,1).
  • VF dependence: Increasing VF (e.g., 5%→16%) generally increases peak reflectance and redshifts peak position; for morphology (2/3,1,1/3), peak rises from 46.44% to 59.4% at 668 nm with peak broadening and suppression of a lower-intensity ~400 nm peak.
• Thermal function: Thermodynamic experiments show the white scales reduce wing temperature under direct illumination, evidencing a cooling function.

The work demonstrates that the broadband, angle-robust structural white of Morpho theseus arises from a distribution of tubular microstructures spanning morphologies between simple tubes and gyroid helices. By embedding the tubular subelement within a generalized trigonometric implicit-function model, the authors bridge ordered gyroid photonics with quasiordered natural architectures. Optical experiments and FDTD simulations jointly show that morphological diversity (coefficients A,B,C), period, and volume fraction create a spectrum of narrowband colors at the microdomain level; spatial mixing over many domains yields high, broadband white reflectance macroscopically. The low material volume fraction and strong visible reflectance deliver efficient passive cooling, supporting biological thermoregulation and suggesting utility for engineered radiative-cooling or camouflage surfaces. The model offers a principled route to tune reflectance spectra via structural parameters while preserving high brightness and low angle dependence.

This study identifies and models the quasiordered tubular microstructures in Morpho theseus scales that generate structural whiteness and enhance thermal management. By formulating a numerical framework that continuously connects tube-like and gyroid architectures using trigonometric implicit functions, the authors replicate measured optical features and clarify how morphology, period, and volume fraction govern spectral responses. Experimentally, M. theseus exhibits high average visible reflectance (~78%), weak angle dependence (maintaining ~63.75% up to ±60°), and thermally beneficial cooling under illumination. The methodology provides a concise tool for analyzing quasiordered structures and can inform the design of nano-optical materials for efficient cooling, camouflage, and photothermal conversion. Future applications are implied in optimizing artificial materials guided by these structure–property relationships.