Cellulose, collagen, and chitin are created by living creatures. They are very strong and impact-resistant. They retain the shape and heal by themselves. They support the cells and hold them together, allowing the organisms to maintain their shape without falling apart. These biomolecules form crystalline-amorphous periodic nanostructure. Strings of nanostructure are then spirally bundled into fibers. The helical microstructure makes biological tissues elastic and resilient, while exhibiting the optical properties of chiral liquid crystals.

We use polarized optical microscopes and electron microscopes to analyze the structure of biological tissues and observe how bio-liquid crystals undergo a series of phase transitions and phase separations to grow and form the shape. Our long-term goal is to study liquid crystal phase transitions in living creature to understand whether they drive the growth and pattern formation.

Bio chiral liquid crystals assemble into photonic crystals by themselves, which have brilliant saturated metallic colors. The shiny metallic shells of beetles and the gorgeous wings of butterflies are typical natural photonic crystals. They are non-toxic, edible, and long-lasting when dried, making them ideal for use as pigments, public displays, color-changing sensors, drug delivery capsules, preserved foods, clothing fibers, and building materials.

Biomaterials are generated by nature and degrade naturally, and can be used as bioenergy after being discarded. As long as we manage the farms and forests properly and keep production and consumption in balance, these materials are renewable and sustainable.

生物與食物中的液態晶體

纖維素、膠原蛋白、甲殼素是生物製造出來的物質。它們很強韌，耐衝擊，會自我修復，支撐包裹著生物的細胞，使生物能維持住形狀，不會軟爛，不會隨隨便便就散形崩裂。這些生物分子會形成晶體-非晶體奈米結構，交錯排列，串成一串。一串一串的奈米結構再螺旋捆綁成一束一束的纖維。這樣的結構使生物細胞組織柔軟又強韌。連帶地，螺旋狀的纖維展現出手性液晶的光學特徵。

我們使用偏光顯微鏡和電子顯微鏡分析生物組織的結構，觀察生物液晶如何進行一連串的相變與相分離，生長成形。我們的長期目標是研究生物中的液晶相變，了解它們是否驅動生物生長，使生物長出形態。

生物手性液晶會自己組裝成光子晶體，顯色濃郁飽和，帶有金屬色澤。金龜子閃亮的甲殼和蝴蝶絢麗的翅膀是非常典型的天然光子晶體。它們無毒，可以食用，乾燥後保久不腐壞，非常適合做色素、公共顯示器、變色型感測器、藥物輸送膠囊、保久食物、衣料纖維、和建材。

生物材料自然生成，自然降解，廢棄後可以做生質能源。只要我們妥善管理農場和林場，讓生產和消耗保持平衡，這些資源就能永續循環利用。

Polarizing optical microscope (POM) is a powerful tool to record the birefringence color of a material. The orientation of the molecules or the fibers (director field) can be derived based on the spectrum of the birefringence color. POM is often for mineral identification, liquid crystal phase identification and topological defect identification.

Microscope lovers, science youtubers and even artists love to observe the biological tissues, starch grains or soap film under POM, mainly for the beautiful colors and patterns. Lots of artworks are published on the internet. In fact, the colors are not just about beauty, and they are useful to derive the direction of the polymer molecules or the fibers. The birefrinegnece can be from the molecule itself. Cellulose and starch arrange themselves into periodic crystal and amorphous nano structures. They show strong structural birefringence, too.

We collected plant samples, including grass stems, leaf skins, bark, thorns, fruit skin and starch in root vegetables. We observed these samples under a POM to check if they show significant birefringence colors. The techniques of liquid crystal phase identification and topological defect identification (Schlieren texture, Grandjean texture, focal conic domains, etc.) are applied to the biological samples, and the structure of the tissue is derived in three dimensions.

Our long-term goal is to study liquid crystal phase transitions and topological defect self-assembly in creature and plants to understand whether they drive the growth and structure formation.

偏光顯微術

偏光顯微鏡可以觀察紀錄樣本的雙折射色。雙折射色的顏色（頻譜）和亮度分佈可以用來推測分子或纖維的指向，重建分子/纖維在2維甚至3維空間中如何排列。偏光顯微術很常用來鑑定礦物寶石、辨識液晶相、辨識拓樸缺陷。

顯微鏡愛好者和藝術家很愛用偏光顯微鏡觀察生物組織、澱粉顆粒、肥皂水，因為圖案非常美麗，看起來很神秘。很多攝影和藝術創作發表在youtube和instagram上，形成分享風潮。事實上，這些圖案不只是美麗。如果能運用光學與物理知識，這些偏光顯微鏡圖片可以用來推測高分子或纖維的指向。有些分子自帶雙折射性。纖維素和澱粉會結晶，會自發形成晶體/非晶體交錯層疊像千層蛋糕一樣的奈米結構，也會展現出明顯的雙折射色。

我們採集植物的莖、葉片、樹皮、刺、果皮、和根莖中的澱粉，用偏光顯微鏡觀察它們，檢查它們是否有呈現明顯雙折射色，辨認它們的液晶相和拓樸缺陷。

我們的長期目標是研究生物中的液晶相變和拓樸缺陷自我組裝，了解它們是否驅動著生物生長、使生物長出型態。

A phase transition is the change of inner order of a material. The material transit from one inner order to another one when the temperature, pressure, and/or the concentration meet the transit point. For example, the phase transition of water is among solid, liquid, and gas w. r. t. the temperature. Additional to the well-known liquid, liquid crystal has many subtle levels of inner order in a fluid medium. The inner order is determined by the shape or the electron configuration of the constituent of the material. The transition between the phases plays crucial roles in growth and forms.

Topological defect arises during phase transitions. When the combination and the arrangement satisfy the topological conservation laws, the defects assemble into ordered structure by themselves. The new order is topologically protected, self-retained, and self-healing.

Our research focus on the rules of the right combination, and if the topological defects drive the formation of natural patterns and optical texture. We want to understand whether topological defects are involved in the growth and formation of cellular tissues, insect shells, colloids, etc. We hope to use this knowledge to create biomimetic optoelectronic components and medical materials.

液態晶體中的拓樸缺陷

物質中原子分子排列的方式決定了物質的相（phases)，例如：固態、液態、或氣態。當溫度、壓力、濃度改變，原子分子排列的方式改變，物質的狀態改變，稱為相變。許多物質的液態不是只有一種狀態，它的分子排列整齊度有很多個層次。它是液體會流動會變形，又有固態晶體的光學特徵，稱為液晶。分子的電子組態和形狀會決定液晶相。相變會促成這些分子組裝起來，長出結構。

在物質相變的時候，拓樸缺陷就會冒出來。它們會帶著週圍的原子分子跑，自己組裝起來，形成有週期有秩序的結構。拓樸缺陷有哪些組合、最終會形成什麼結構，遵守拓樸守恆定律。因為有守恆定律保護，這些結構很有彈性，可以自己維持形狀不會潰散，受擾亂也會自己癒合恢復原狀。

我們研究這些拓樸缺陷如何組裝，遵守著哪些規則。我們想要了解拓樸缺陷是否有參與木質組織、昆蟲甲殼、膠體⋯生長成形。我們希望能運用這些知識製作仿生光電元件和醫藥材料。

Jieh-Wen Tsung*, Ya-Zi Wang, Sheng-Kai Yao, and Shih-Yu Chao

Applied Physics Letters, vol. 119, 121906, 2021

Featured article

Creation of a topological defect array in liquid crystals has been a notable focus in recent years, because the defect array can be utilized as precision optics, templates of self-assembled microstructures, and elastomer actuators. So far, the defect arrays are created intuitively by trial and error. Systematic rules to arrange defects into stable long-ranged arrays are in demand. A model of two-dimensional square and hexagonal defect array was developed based on previous experimental results. The model is generalized for defect crystals and quasicrystals in this research. A crystal is the periodic repetition of a unit cell. A stable defect crystal must have minimum free energy, and the arrangement of the defects must obey the topological conservation laws. By solving the Euler–Lagrange equation of the director field of a unit cell and by integrating the topological rules into the boundary conditions, the director field of a defect crystal can be easily obtained. A large variety of defect crystals and quasicrystals are derived. The lattices are rectangular, triangular, square, pentagonal, and hexagonal. The defects can be either radial or azimuthal (vortex-like). The nematic and vector orders are both considered. The collection of defect crystals is presented here as a catalog for the designers.

拓樸缺陷晶體型錄

本研究推導出向列型液晶中可以穩定存在的拓樸缺陷陣列。我們把拓樸缺陷當做原子，放進適當的晶格中，尋找可以讓拓樸缺陷晶體穩定存在的排列組合。

關鍵參數為：

- 晶格結構：長方形、三角形、正方形、五角形、六角形

- 缺陷拓撲荷：±1/2、±1、±2

- 缺陷形狀：放射狀或渦旋狀

- 液晶的內在秩序：向列型或向量型。

這些晶體滿足以下條件：

- 液晶的指向場是Euler-Lagrange Equation的解，擁有最小自由能

- 總拓樸荷為0，因此可以維持晶體的結構穩定存在。

我們的長期目標是做出一套拓樸缺陷晶體指向場的型錄。設計者可以直接把指向場線用光配向、刮痕、電極圖案等方式印／刻在基板上為液晶配向。

Jieh-Wen Tsung*

Liquid Crystals, vol. 48, no. 9, pp. 1295-1380, 2021

Selected as the front cover

Orientation of nematic liquid crystal (NLC) around a topological defect is expressed by the equation, ψ=sϕ+ϕ0, where ψ, s, ϕ and ϕ0 are the director spatial phase, topological charge, azimuthal angle and a spatial phase shift, respectively. Conventionally, defect array is modelled by simply adding the ψ fields of all the defects. The resultant array is a combination of the defects imposed by design (ψs) and the defects due to the lattice geometry (ψg). However, defect arrays generated in homeotropic confinement show hyperbolic hedgehogs (ψh) between the ψs's, which are out of the scope of the conventional model. In this research, two-dimensional defect ‘crystals’ with various lattice structures (square and hexagonal), various defect shapes (radial or circular) and various lattice constants (pixel size) are generated by pixelated patterned electrodes in homeotropic NLC cells. The results verify the two selection rules, 1) The total s must be zero, 2) ϕ0 must be a constant throughout the array, and a new model composed of ψs, ψg and ψh is established. Calculation proves that the selected modelled array has minimum free energy. The ψs-ψg-ψh model is versatile and easy to apply, which benefits the design of new topological defect arrays.

拓樸缺陷如何組裝起來形成"晶體"

拓樸缺陷為何/如何自發性地排列成有規律的結構，是拓樸軟物質領域最重要的研究主題。

本研究使用有圖案的電極抓取液晶中的拓樸缺陷，把拓樸缺陷排列成正方形陣列與六角形陣列。我們使用偏光顯微術辨識拓樸缺陷，從雙折射色和亮度推測液晶分子的指向。

實驗結果顯示，缺陷排法遵守拓樸守恆定律，且自由能最小化，缺陷陣列可以大範圍穩定存在。如果不遵守拓樸守恆定律或自由能沒有最小化，缺陷會移動、變形、湮滅。

本研究基於實驗結果，推導出新的拓樸缺陷陣列的模型，是topological soft matter領域第一套有步驟有條理的模型。

Ya-Zi Wang , Sheng-Kai Yao, Jing-Kai Chou and Jieh-Wen Tsung*
Liquid Crystals, DOI: 10.1080/02678292.2023.2179118, February, 2023

Topological defect arrays in liquid crystal have great potential for use in smart windows, displays, gratings, spatial light modulators, and optical vortex generators, making them an emerging optical material. Liquid crystal has birefringence and has been used for decades to tune the polarization state of light, resulting in technologies such as displays, lenses, and phase modulators. Defects are discontinuities or displacements in the liquid crystal that can scatter and diffract light. The defect arrays are topologically protected, self-retained, and self-healing, making them ideal for use in durable optical devices. Since the liquid crystal is fluid and the molecules align with an electric field, the optical properties are highly tunable. 

A large-scale defect cluster in nematic liquid crystal (NLC) looks like fog (scattering) on rippling water (lens imaging) with glittering sparkles (diffraction). The deformed NLC around the defects act as microlenses, creating lens imaging. The light scattered by the defect core results in a bluish powdery haze. When exposed to a strong light beam, the defect array produces glittering rainbow-colored starbursts, which are the diffraction of light. Well-aligned periodic defects show significant lens imaging and diffraction, while disordered crowded defects show the scattering of light. 

To produce these defects, vertically aligned NLC cells with pixel electrodes were fabricated. The key parameters for creating the defects are the electrode shape (pad, annulus, crossed strips, porous electrode), the array structure (square or hexagonal), and the pixel size (ranging from 5 μm × 5 μm to 200 μm × 200 μm). The location and types of the defects are identified using a polarized optical microscope.

The appearance of the defect array can be accurately described by analyzing the two-dimensional (2D) Fourier transform of the defect distribution, which is the spatial frequency spectrum. The 2D Fourier transform of the optical path difference results in the 2D diffraction pattern. Annulus and porous electrodes create the densest irregular defects, the haziest texture, and the most comfortable visual experience.

水感、霧感、閃亮感的物理原理

軟物質中拓樸缺陷會自發性地組裝起來，形成新的結構。

這些結構受拓樸守恆定律保護，有彈性，有韌性，受扭曲會自動恢復原狀，破裂後會自動癒合。

光通過含有拓樸缺陷的介質，會散射、繞射、折射，拓樸缺陷可以切換光子的角動量子數。

善加運用拓樸缺陷，可以製作視覺光電元件和光量子運算元件。

本研究的主題是拓樸缺陷造成的視覺效果。

向列型液晶中的拓樸缺陷團簇看起來像就像水面漣漪上（透鏡成像）漂浮著霧氣（散射），閃閃發光（繞射）。

排列整齊的缺陷會有顯著的透鏡成像和繞射。

無序擁擠的缺陷會凸顯出光的散射霧化。