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

Structural whiteness in natural organisms, such as butterflies and beetles, offers a sustainable alternative to traditional white pigments. Unlike pigments, structural whiteness leverages the architecture of the material itself to reflect light across a broad spectrum, often achieving this with thinner layers. Two main categories exist: uniform white (low, angle-independent reflection from disordered structures) and metallic white (shining effect from periodic architectures used for mating displays or thermoregulation). While numerical models exist for simulating the optical mechanisms of ordered and disordered structures, a comprehensive model for quasiordered structures, common in nature, has been lacking. The *Morpho theseus* butterfly, with its bright white scales covering the ventral wing region, provides an ideal case study to address this gap. Its scales effectively reflect sunlight, aiding thermoregulation in hot environments.

Previous research has explored structural whiteness in various organisms, highlighting its importance in thermoregulation, mating displays, and camouflage. Studies have categorized structural whiteness based on the light scattering intensity and angle dependence, differentiating between uniform and metallic white. Recent advancements in numerical modeling, such as the unified evolving model based on trigonometric implicit functions, have improved the efficiency of simulating optical mechanisms, particularly in ordered structures. However, a dedicated numerical model to address the challenges posed by the quasiordered structures commonly found in nature remained absent before this study.

The study utilized *Morpho theseus* and *Callophrys rubi* specimens. Optical microscopy (KEYENCE VHX-1000E), microspectrography (NOVA, Ideaoptics), and an angle-resolved spectral system (ARM, Ideaoptics) characterized the scales' optical properties. Scanning electron microscopy (SEM, TESCAN-MIRA3) and transmission electron microscopy (TEM, Bio-TEM FEI Tecnai G2 Spirit Biotwin) imaged the microstructure. TEM sample preparation involved embedding in epoxy resin, sectioning with an ultramicrotome (Leica EM UC7), and transferring sections onto copper grids. CIE 1931 normalized illuminant D65 was used to calculate tristimulus values and color representation. Finite-difference time-domain (FDTD) simulations (Lumerical FDTD Solutions) modeled the optical properties of the quasiordered structures, using chitin's refractive index (1.56n + 0.06i). Thermodynamic measurements involved placing butterfly wings on a metallic support in a vacuum chamber, using a xenon lamp (X350) to simulate sunlight and an infrared thermal imaging camera (FLIR-T630c) to monitor temperature changes.

*Morpho theseus* scales exhibit quasiordered tubular structures with diverse orientations and twining forms, resembling an early stage of gyroid structure formation observed in *Callophrys rubi*. These structures are mathematically related to the gyroid structure via the trigonometric function sin(x)cos(y) = t, a sub-element of the gyroid model. Optical simulations using the FDTD method revealed that the varied morphologies between tubular and gyroid structures caused a color mixing effect, resulting in broadband reflection in the visible light spectrum. The average reflection in the visible light spectrum reached 78.1%, significantly higher than that of *Callophrys rubi* (29.83% at 518 nm). The scales displayed low viewing angle dependence, maintaining an average reflection of 63.75% at observation angles of ±60°. Immersion in bromoform, with a refractive index close to chitin, temporarily rendered the scales semitransparent, confirming that color is due to architecture and not pigments. A numerical model was developed to characterize the quasiordered architectures, incorporating parameters for tubular form (coefficients A, B, C), period, and volume fraction (VF). Simulations showed that variations in tubular form, period, and VF led to color diversity, with a redshift in peak reflection with increasing period and increased peak reflectance with increasing VF. The anisotropic nature of the tubular structure caused reflection differences based on orientation relative to polarization direction.

The findings demonstrate a novel mechanism for achieving structural whiteness in natural organisms, driven by the quasiordered arrangement of diverse tubular structures. The mathematical correlation identified between tubular and gyroid structures provides a framework for understanding the evolution and optimization of these complex architectures. The numerical model developed can be generalized to analyze other quasiordered structures in biological and artificial systems, improving design strategies for nano-optical materials. The study’s success in combining mathematical modeling, numerical simulation, and experimental validation offers a powerful approach to investigate complex biological structures and their functions. The low angle dependence of the reflection from *Morpho theseus* scales suggests potential applications in diffuse reflection systems.

This study provides a comprehensive analysis of structural whiteness in *Morpho theseus* butterfly scales, revealing a mechanism based on quasiordered tubular architectures. The developed numerical model offers a powerful tool for understanding and designing similar structures in artificial systems. Future research could explore the evolutionary pathways leading to these complex structures and expand the model to incorporate additional factors influencing optical properties. Investigating the potential for bio-inspired design based on these findings across various applications (e.g., advanced cooling systems, camouflage technologies) is a key area for future investigation.

The study focused on a specific species of butterfly (*Morpho theseus*) and may not be directly generalizable to all organisms displaying structural whiteness. The numerical model relies on certain assumptions and simplifications, such as the refractive index of chitin and the perfect matching layer boundary conditions in the simulations. Further research should investigate the effects of environmental factors (e.g., humidity, temperature) on the optical properties of the scales and extend the numerical model to include other relevant parameters.