Type I Clathrate (Weaire–Phelan)結構串珠模型

一年一度的Joint Mathematics Meetings (JMM) 不僅是數學家們交流最新研究成果的重要平台，其數學藝術展覽也同樣引人注目。在 **2017年的JMM會議** 上，一件名為 "**Bead model of Type I Clathrate (Weaire–Phelan) structure**"（Type I Clathrate (Weaire–Phelan)結構串珠模型）的藝術品，以其精巧的結構和深刻的科學內涵，吸引了參觀者的目光 。

背景介紹：籠狀化合物與Weaire–Phelan結構

為了更好地理解這件珠飾模型的意義，我們需要了解一些相關的科學概念：

• 籠狀水合物（Clathrate Hydrates）： 這是一種非化學計量的晶體化合物，由水分子和較小的客體分子（如甲烷）組成。在這些化合物中，客體分子或原子被捕獲在由氫鍵連接的水分子形成的周期性多面體籠中。
• Weaire–Phelan結構： 這是一種比之前最著名的開爾文結構（Kelvin structure）更能有效解決“開爾文問題”的結構。“開爾文問題”探討如何將三維空間劃分為具有相同體積的胞元，且總表面積最小。Weaire–Phelan結構與Type I Clathrate結構密切相關。
• Type I Clathrate結構： 這種晶體結構可以被認為是由 **十二面體** 和 **十四面體** 在三維空間中以 **1:2的比例** 密鋪而成。其中，十二面體形成體心立方排列，而十四面體填充剩餘的空間。

藝術品詳情：Bead model of Type I Clathrate (Weaire–Phelan) structure

Bead model of Type I Clathrate (Weaire–Phelan) structure

創作者：左家靜 (Chia-Chin Tsoo)與金必耀 (Bih-Yaw Jin)

• 尺寸： 20 x 20 x 20 厘米
• 材料： 木珠
• 創作年份： 2016

這件珠飾模型展示了 Type I Clathrate結構的硬球開放堆積模型。在模型中，球形的木珠代表氧原子的價電子對。較小的正氧原子核隱藏在四面體內部，在模型中並未展示 。

該模型通過串珠技術，清晰地展現了十二面體和十四面體如何在三維空間中堆積，形成Type I Clathrate的晶體結構。藝術家巧妙地利用木珠來模擬水分子的排列方式，以及客體分子被包封在這些多面體籠中的概念。

關於作者

• 左家靜 (Chia-Chin Tsoo):。
• 金必耀 (Bih-Yaw Jin): 是國立台灣大學化學系的教授，他長期以來一直對富勒烯和石墨烯等拓撲非平凡結構感興趣，並使用數學串珠的角編織技術來構建這些結構的魯棒模型。他還將串珠技術應用於構建任意sp2雜化石墨結構的近似三維曲面模型 。在本作品中，他與左家靜合作，探索了串珠技術在構建籠狀水合物模型方面的應用。

參考資料

• Chia-Chin Tsoo & Bih-Yaw Jin | 2017 Joint Mathematics Meetings | Mathematical Art Galleries: #### Statement. Type I Clathrate (Weaire–Phelan)結構串珠模型

© 2025 珠璣科學|Type I Clathrate (Weaire–Phelan)結構串珠模型

第二型氫氣水合物（Type II Hydrogen Clathrate）

第二型氫氣水合物（Hydrogen Clathrate Type II）：更大籠狀結構的幾何奧秘

除了第一型氫氣水合物，金必耀與左家靜在 2018 年 Bridges 數學藝術大會 上還展示了 第二型氫氣水合物（Hydrogen Clathrate Type II） 的串珠模型。這個模型比第一型結構更為複雜，具有更大的籠狀單元，展現了水分子如何透過幾何排列來形成穩定的氣體捕獲結構。

第二型氫氣水合物的結構特性

第二型氫氣水合物擁有更大的籠狀結構，能夠儲存較大體積的氣體分子。它的結構由兩種籠子組成：

• 小型 12 面體籠（Dodecahedral Cage, 5¹²）： 與 Type I 相同，由 12 個五邊形組成。
• 大型 16 面體籠（Hexakaidecahedral Cage, 5¹²6⁴）： 由 12 個五邊形和 4 個六邊形組成，體積更大。

   這種結構的特點是：

   
•   更適合捕捉較大分子的氣體（如丙烷）    
•   擁有更高的空間填充能力    
•   存在於極地與深海環境中，並被視為未來的潛在氫能儲存技術

串珠模型的創新價值

這個模型延續了金必耀教授的 串珠科學建模技術，以珠子代表水分子的碳氫鍵與氫鍵，使觀眾能夠：

   
• 直觀地看到 Type II 水合物的幾何結構    
• 分辨 不同籠子的形狀與排列方式    
• 了解 如何利用這種結構來儲存氫氣與其他氣體

這種藝術與科學結合的方法，不僅讓水合物的概念變得具體可感，還為材料科學與教育提供了新的視角。

晶籠水合物與未來能源

氫氣水合物的研究對 可再生能源技術 具有重大影響：

   
•  低溫高效氫氣儲存技術，有望取代目前高壓儲氫的方法    
•   可能用於未來的太空探索，如木星的衛星 歐羅巴（Europa），可能蘊藏大量水合物    
•   應用於工業氣體分離與碳捕獲技術，提高能源利用效率

串珠模型資訊

• 尺寸： 直徑約 25 公分
• 材料： 木珠、彈性繩
• 結構特點： 重現 Type II 水合物的複雜幾何排列

更多資訊

這種結構比 Type I 更適合儲存較大分子的氣體，如丙烷。詳細內容可參閱 Bridges 2018 展覽頁面。

© 2025 珠璣科學 |第二型氫氣水合物

第一型氫氣水合物（Type I Hydrogen Clathrate）

在 2018 年 Bridges 數學藝術大會 上，金必耀與左家靜利用 串珠技術（Beading Techniques）構建了第一型氫氣水合物（Hydrogen Clathrate Type I） 的模型，將這種微觀結構以可視化的方式呈現出來。這項作品不僅是數學與藝術的結合，也讓我們能夠直觀理解水分子如何形成籠狀結構來包裹氫氣分子。

氫氣水合物是一種特殊的 籠狀水合物（Clathrate Hydrate），在高壓低溫環境下形成，由水分子透過氫鍵（Hydrogen Bonding） 排列成特定的籠狀結構，並在內部捕捉氫氣分子。第一型（Type I）水合物的結構由兩種類型的籠子組成：

• 小型 12 面體籠（Dodecahedral Cage, 5¹²）： 由 12 個五邊形組成，體積較小。
• 大型 14 面體籠（Tetrakaidecahedral Cage, 5¹²6²）： 由 12 個五邊形和 2 個六邊形組成，能容納較大分子。

串珠模型

• 尺寸： 直徑約 20 公分
• 材料： 木珠、彈性繩
• 結構特點： 以串珠方式重現水合物的氫鍵網絡

更多資訊

這種水合物在高壓低溫環境下穩定存在，被視為未來的氫氣儲存技術之一。詳細內容可參閱 Bridges 2018 展覽頁面。

© 2025 珠璣科學 |第一型氫氣水合物

A paper for the Chemical Education in Taiwan

We published a paper in Chinese for the Chemical Education in Taiwan (台灣化學教育). The title is "奈米世界的構築藝術：第一型晶籠水合物的串珠模型之結構與製作" (Architectural Beauty in Nano World: The Structures and Construction of Type I Clathrate Hydrate).
In the future, I hope I can find time to write another two manuscripts for type II and type H Clathrate Hydrates.

MEP (Melanophlogite, Weaire-Phelan structure, Clathrate type I)

This model is accepted for the JMM 2017. (Nov. 11, 2016)

Sodalite (Kelvin structure, or Bitruncated cubic honeycomb)

For more description of this model, see the page from the Bridge conference 2016. (Nov. 11, 2016)

Gyroidal Invinciball

A graphitic gyroid is a hyperbolic object. To make it, we need to introduce octagons at suitable positions on a graphitic sheet, which is similar to the pentagons in the spherical space such as buckyball. In some sense, we can view graphitic gyroid as a kind of "ball" in the hyperbolic space.

Students from the TFGH created this gyroidal invinciball in the hyperbolic space. Unfortunately, they used 0.6mm Nylon strings for the 12mm faceted beads. The structure is too soft to stand on its own.

Comments on type-II high-genus fullerenes

It is of interest to compare the bead model of a dodecahedron fused by 20 C60 in the previous post to high-genus fullerenes we discussed before. In fact, one can also view that bead model as a high-genus fullerene with genus=11, which is obtained by substracting one from the number of faces in a dodecahedron. The most important difference between these two types of high-genus fullerenes is the orientations of the TCNT necks relative the surface of the polyhedron. In the original high-genus fullerenes, which I will call the type-I high-genus fullerene from now on, the orientations of the TCNT necks are along the normal of the polyhedron. Here in the type-II high-genus fullerene, the orientations of the TCNT necks, which are created by fusing two neighbored C60, are lying on the surface of polyhedron (dodecahedron here) and along the directions of its edges.

Dodecahedron consisting of 20 fused C60s (still incomplete)

I am working on a beaded dodecahedral structure which is going to consist of 20 fused C60. The construction strategy is similar to certain bead models made by Mr. Kazunori Horibe. I called those models as "Carbon poti donut （波提甜甜圈）".
In the carbon poti donut, every C60 is connected to two neighbored C60 through two particular 5-fold axes. It is easy to see which two 5-fold axes I am talking about. Here I am trying to connect every C60 to three other C60 such that all 20 C60 locate on the vertices of a dodecahedron. But the bond angles generated by the trivalent C60 do not match with the angle of a dodecahedron. So the whole structure distorts quite significantly.

This model is very similar to a dodecahedron consisting of 20 dodecahedra (Clathrate cluster). I tried to make this model last month. But because each dodecahedron in this model is highly strained, I gave it up.

Four face-sharing pentagonal dodecahedra

E. A. Lord, A. Mackay, and S. Ranganathan described in their book, "New geometries for new materials", a simple cluster consisting of four face-sharing pentagonal dodecahedra arranged in a tetrahedral configuration (pp.48). Here is a bead model of this cluster.

In their book, there are more clathrate structures that one might be able to construct with beads.

Clathrate cluster

I bought some more green rice-shape beads last week and managed to finish this interesting clathrate cluster of 60 dodecahedra in the last weekend. One can still see deformation of many dodecahedra in this clathrate cluster though.

Buckyball made of 60 dodecadedra

I made this structure with beads last weekend. Still unfinished. The finished structure should have 60 dodecahedra arranged like a buckball. One has two ways to interpret this structure:
1. If every dodecahedron represents a carbon atom, we have a standard C₆₀.
2. If we still use beads to represent CC bonds, then we have a giant molecule, C₇₅₀. In this molecule, 450 carbon atoms are sp³ hybridized or tetra-valent and 300 atoms are sp² hybridized or trivalent. But I suspect these sp² hybridized carbon atoms are not energetically favorable, so it is better to have hydrogen atoms connected to these sp2-carbons. Then we get C₇₅₀H₃₀₀!



The bead model of this structure (see here) might be first constructed by Emilie. She asked me to comment about this structure in my blog long time ago (I couldn't find the exact location though).

I decided to make one from beads after I saw the same structure made by a toy designer, Dick Esterle, who actually invented this kind of toys, in the Bridges conference last week.

two more pictures of an sp3 structure

sp3 hybridization

The structure shown in the right of following figure has demonstrate that one can, in general, make 3-D structrures that contain the sp3 bonding. Of course, it is harder to weave this type of structures.

From beaded fullerene

Chuang made them.