親愛的，我把分子變大了 (Honey, I Blew Up the Molecules)

科技大觀園 2018/07/10

左家靜｜財團法人國家實驗研究院國家高速網路與計算中心 金必耀｜國立臺灣大學化學系

親愛的，如果你把一滴水沿著一個方向切成一百份，取其中一份，再切成千分之一，然後再一次千分之一，結果就差不多是一個水分子的大小了！也就是說，一滴水大概由一萬萬萬個水分子所組成！這麼小的水分子由一個氧原子與兩個氫原子，連結而成一個略為彎曲的形狀 。

現代化學最大的貢獻便是，認識到整個物質世界，不管是一滴水、岩石、樹木、小鳥、甚至是我們自己，都是原子與分子所組成。原子的種類只有118種，最小的氫原子不到0.03奈米，而最大的銫原子也只有0.26奈米，1奈米為十萬萬分之一公尺；分子由原子結合而成，有無限種的可能組合，從最簡單的氫分子，只含有兩個氫原子，到由三十億個鹼基對組合而成的人類基因組。但不管是什麼分子，它們都是由這些極小的118種原子以特定方式構成的。

科學家發展出許多方法，可以確定分子的立體形狀與其中的原子排列，並通過模型建構，將這些分子放大到我們的尺度，用以呈現分子內原子的空間排列。球桿模型是最常見的分子模型，其中的球表示原子，桿則代表連結原子的化學鍵。不同的原子有不同的鍵結能力：氫原子只跟另一個原子鍵結，氧原子則同時能跟兩個原子鍵結，而碳原子則有兩個情形，在有的化合物中可以跟三個形成鍵結，例如石墨，而有的情形則可以跟四個原子形成鍵結，例如鑽石。球桿模型中，用不同洞數的球來表現這些不同鍵結能力的原子，描述氫原子的球只有一個洞，描述氧原子的球則有兩個洞，描述石墨與鑽石中的碳則需要兩種不同的球，一種球含有三個洞，排列為正三角形，而另一種球則含有四個洞，排列為正四面體，下圖給出DNA的球桿分子模型。

DNA的球桿分子模型（圖片來源：金必耀）

除了球桿模型外，還有許多其他製作分子模型的方式。這裡介紹用傳統串珠工藝製作分子模型的有趣方法。若使用的是圓珠子，模型中的珠子代表著分子中的化學鍵，或是特定的離子，所做出來的串珠模型，可以更真實地表現許多分子三度空間的形狀。而且選擇適當材質與顏色的珠子，這些根據奈米分子空間結構做出來的串珠模型，更像是形狀優美的科學藝術品。

串珠分子模型可以分為幾大類：（A）有機烷類分子；（B）富勒烯、奈米碳管、立體石墨烯；（C）沸石結構；（D）結構無機化學；（E）金屬串分子。

有機烷類分子

串珠模型可視為是一種「自由雲球模型」，圓珠代表電子對的空間分布，細線則給出圓珠的相互關係。實際分子的形狀是由價電子間的排斥以及價電子與原子核的吸引，兩種力量達成平衡的結構。使用珠子（斥力）與細線（引力），按分子的連結圖來進行串珠，串珠過程像是在進行一個類比演算，演算的結果是一個立體串珠模型，給出分子在三度空間的近似電子雲分布。

有機烷類分子模型。上排左至右：甲烷、螺槳烷、環丁烷。下排左至右：環己烷、異丁烷、鑽石烷。 （圖片來源：左家靜）

富勒烯、奈米碳管、立體石墨烯

在石墨烯的六邊形網絡上有規律地引進非六邊形，可以產生千變萬化的彎曲立體石墨烯。正高斯曲率由四邊形與五邊形模擬，常出現於有限的芙類分子；而七邊形與八邊形可用來模擬無限延伸、含有許多連通孔洞的三度週期立體石墨烯。

石墨烯及芙類分子的立體模型。上排左至右：碳六十（C60）、碳環（C120）、Chen–Gackstatter曲面。下排左至右：單度週期極小曲面、手性向量為(1,1)的P型三度週期極小曲面、C80@C20。 （圖片來源：左家靜）

沸石結構與晶籠水合物

數學串珠也可以用來建構許多四配位無機固體的立體模型，例如石化工業上極其重要的沸石結構、以及地球上最重要的能量來源—晶籠水合物。以沸石結構為例，矽氧四面體通過橋氧連結為各種多面體籠子，然後這些籠子進而堆疊成繽紛多彩各種樣式的三度空間週期結構。

沸石結構（用三字母碼表示）。上排左至右：SOD、LTA、RHO。中排左至右：TSC、MOZ、ATN。下排左至右：SAS、SSF、LTL。（圖片來源：左家靜）

結構無機化學

許多無機化學分子與固體可以視為配位多面體的空間堆疊，長形的管珠特別適合用來建構此類結構，它們類似工程與建築上的空間桁架，我們可以說這些模型像是由奈米世界所啟發的空間桁架雕塑。

結構無機化學分子模型。上排左至右：星型菱形六十面體、二階馬凱多面體、多金屬氧酸Keggin結構。下排左至右：四階馬凱多面體、鈣鈦礦結構、五重對稱複合多面體。 （圖片來源：左家靜）

金屬串分子

金屬串分子是一種多金屬配位化合物，由線型排列的過渡金屬、與四條螺旋環繞在其側的有機分子所組成。很重要的是金屬串分子是一種最細小的分子電線，通電後，電流可在中間的金屬流通，而周遭的有機分子為絕緣體，可以有效地防止漏電。

十七核金屬串分子串珠模型。 （圖片來源：左家靜）

奈米世界還有很多可能的優美結構，等待著讀者去用數學串珠，將它們的立體雕塑建構出來。詳細的數學串珠技巧，以及各種我們曾經嘗試過的奈米結構，可以參考過去我們發表的工作（詳見資料來源）。

六年前為替科技部「科技大觀園」寫的一篇文章，連結全消失了。 親愛的，我把分子變大了 (Honey, I Blew Up the Molecules)

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.

數學串珠-繽紛多彩的奈米結構與幾何 (Mathematical Beading – Kaleidoscopic structures and geometries in the nano world)

Chiachin and I wrote a paper entitled "數學串珠-繽紛多彩的奈米結構與幾何" (Mathematical Beading – Kaleidoscopic structures and geometries in the nano world) for the special issue on the connection between math and art in Science Study (科學研習), a local science magazine in Taiwan. Here is the first page.

The pdf file can be found here.

科學遇見藝術─串珠分子模型的異想世界

Chia-Chin and I recently wrote a short overview manuscript entitled "Where Science Meets Art - The Fabulous World of Beaded Molecules" about beaded molecules for a meeting which will be held in Shandong, China this weekend.

Note: 由於這篇文章太晚寄出，最後未出現在論文集中。(Sept. 7, 2007) 我會另尋適當的雜誌發表。

Two articles about the construction of gyroid- and diamond-type triply periodic minimal surfaces

I wrote two articles in Chinese for the Journal, Chemistry Education in Taiwan (臺灣化學教育) last year. The pdf files have just come out:
1. 左家靜, 莊宸, 金必耀, 大家一起做多孔螺旋與鑽石型三度週期最小曲面的串珠模型（上）─立體幾何介紹，2014 臺灣化學教育, 328-335.
2. 莊宸, 左家靜, 金必耀, 大家一起做多孔螺旋與鑽石型三度週期最小曲面的串珠模型（下）─實作，2014 臺灣化學教育, 336-344.
The title can be translated as "Application of mathematical beading to carbon nanomaterials - A hands-on, collaborative approach to gyroid- and diamond-type triply periodic minimal surfaces with beads, I and II", literally. I described a simple modular approach which was developed mainly by Chern Chuang for making gyroid- and diamond-type Triply Periodic Minimal Surfaces.

Torus knot (1,2)

Kazunori showed me this beautiful torus knot (1,2) he made a few days ago. This structure can be classified as a torus knot, or more specifically a twisted torus without knot at all. The space curve that this tubular structure approximates can be described by the parametric equations for the torus knot (q=1, p=2). Therefore, it is reasonable to call it as the torus knot (1,2).

To me, this structure seems to be a perfect example to show the influence of the particular operation, Vertical Shift, described in the following papers:
1. Chuang, C.; Jin, B.-Y. Torus knots with polygonal faces, Proceedings of Bridges: Mathematical Connections in Art, Music, and Science 2014, 59-64. pdf
2. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. Comments on Structural Types of Toroidal Carbon Nanotubes, J. Chin. Chem. Soc. 2013, 60, 949-954.
3. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. On the structural rules of helically coiled carbon nanotubes, J. Mol. Struct. 2012 1008, 1-7.
Another related operation is the Horizontal shift, which is not used in this structure. Applying these two operations carefully (usually nontrivial), one can mimic the bending and twisting of many space curves in an approximate way.

作品完成時間（約）：2015/4
作者：堀部和経

Carbon star and other clover-shaped carbon nanotori

I recently saw a post on the clover-shaped carbon tori by my facebook friend, Tetsuaki Hirata, who is an artist from Himi, Japan and seems to be a frequent visitor of this blog. After I showed Chern about his works, Chern told me he has thought about this kind of clover-shaped carbon nanotori quite some times ago. Indeed, Chern has published a number of papers on the tubular graphitic structures. He is probably one of few people who know a lot about this kind of graphitic structures, especially on how the nonhexagons could influence the structures of a carbon nanotube. I am not surprised that he thought about this kind of structures.

1. Chuang, C.; Fan, Y.-C.; Jin, B.-Y.* Generalized Classification of Toroidal and Helical Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 361-368.
2. Chuang, C; Fan, Y.-C.; Jin, B.-Y.* Dual Space Approach to the Classification of Toroidal Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 1679-1686.
3. Chuang, C; Jin, B.-Y.* Hypothetical toroidal, cylindrical, helical analogs of C60 J. Mol. Graph. Model. 2009, 28, 220-225.
4. Chuang, C.; Fan, Y.-C.; Jin, B.-Y.* On the structural rules of helically coiled carbon nanotubes, J. Mol. Struct. 2012 1008, 1-7.
5. Chuang, C.; Fan, Y.-J.; Jin, B.-Y. Comments on structural types of toroidal carbon nanotubes, arXiv:1212.4567, 2013 submitted to J. Chin. Chem. Soc.

In the first two and the 5th papers, we talked about general structural rules of carbon nanotori and only touched helices briefly. In the next two papers, we discussed very generally how the horizontal and vertical shifts (HS and VS) can be exploited to change the direction of a straight carbon nanotube in order to obtain an arbitrary helically coiled carbon nanotubes. In Chern's Ms thesis, he also showed how to take advantage of HS and VS to create trefoil knots or torus knots in general, which was later summarized in a brief review we wrote, "Systematics of Toroidal, Helically-Coiled Carbon Nanotubes, High-Genus Fullerenes, and Other Exotic Graphitic Materials."  (Procedia Engineering, 2011, 14,  2373-2385).

Clover-shaped TCNTs are just a special class of more general curved carbon nanotubes we considered. A simple strategy is to introduce 180 twists along the tube direction (i.e. 180 degree VS) at suitable positions. I got a few nice figures of clover-shaped TCNTs from Chern the other days.

Among all these clover-shaped tori, I particularly like the five-fold carbon star.

珠璣科學(Zhu-Ji Science)

In the November issue of Science Monthly (科學月刊), I wrote the sixth article of the Zhu-Ji Science series (珠璣科學). This one is about the particular problem of "30 small balls cover a big ball" from the Japanese temple geometry.

金必耀, 左家靜, 珠璣科學─日本寺廟幾何與正十二面體串珠模型 (Zhuji Science - Japanese temple geometry and the bead model of regular dodecahedron), 科學月刊 2012, 33(11).

P, G, and D surfaces

I am planning to have a project with students and teachers of TFG (Taipei) school later this month to construct Gyroidal and D surfaces together. It could be a difficult task because the gyroidal structure is probably the most complicated bead structure Chern and I have ever made. A simple tutorial on the three-dimensional structure of a gyroidal surface and how it can be decomposed into several basic and easily weaved units seems to be useful. So I am now preparing some slides to make the project work out smoothly. Here is one of the slides about the famous P-, D- and G-types Triply Periodic Minimal Surfaces (TPMS) which I generated with matlab:

Additionally, Chern, Wei-Chi, Chia-Chin and I also have a paper jointly for the Bridges meeting last summer. Chern made the presentation. I didn't attend it, though. This paper describes the bead models of these three structures quite generally.

Chuang, C.; Jin, B.-Y.; Wei, W.-C.; Tsoo, C.-C. "Beaded Representation of Canonical P, D, and G Triply Periodic Minimal Surfaces", Proceedings of Bridges: Mathematical Connections in Art, Music, and Science, 2012, 503-506.

Super Buckyball as a Molecular Sculpture

I have written an article about super buckyball, Super buckyball as a molecular sculpture - its structure and the construction method (分子雕塑─超級珠璣碳球的結構與製作), in Chinese recently. I guess it will appear in the next issue of CHEMISTRY (The Chinese Chemical Society, Taipei) (化學季刊), a local chemistry journal in Taiwan. I tried to describe the construction method of the super buckyball in details in this paper. Also, in the reference 1 of this paper, I commented on how this paper was inspired by the Horibe's works, particularly, the idea of fusing many C60s into structured super fullerenes, which is the way I understand many of his beautiful models. I wrote it in Chinese because I hope local high-school students in Taiwan can read the paper more easily and reconstruct the model as a school activity.

分子雕塑 ⎯ 超級珠璣碳球的結構與製作

金必耀

臺灣大學 化學系

摘要:串珠是最適合用來建構各種芙類分子模型的材料,珠子代表芙類分子中的碳碳鍵,珠子的硬殼作 用正好模擬微觀芙類分子內的化學鍵作用。本文將介紹以模組化方式,讓許多對基本串珠模型建構有一 定認識的人,親手一起協同製作大型的超級芙類分子模型,非常適合作為中學化學與立體幾何教育的活 動,所製作的巨型模型不僅是一個為微觀分子模型,更可以說是一件具有科學含意的雕塑藝術品。

Super Buckyball as a Molecular Sculpture − Its Structure and the Construction Method

Bih-Yaw Jin

Department of Chemistry, Center of Theoretical Sciences and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan

ABSTRACT

Mathematical beading can be exploited to construct faithful physical model of any fullerene. The hard sphere interactions among different beads effectively mimic the ligand close packing of carbon-carbon bonds in fullerenes. Here we show a simple modular approach for students to build complicated graphitic structures together. Particularly, we describe in details the structure of the so-called super buckyball, which consists of sixty fused buckyballs, and our hands-on experience in making its bead model by the students of the Taipei First Girls High School collaboratively.

珠璣科學(Zhu-Ji Science)

I have written four articles in Chinese about beading and science for the local science magazine "Science Monthly (科學月刊)" now.

1. 金必耀、左家靜, 珠璣科學—八字編與正多面體（Figure eight stitch and regular polyhedra）, 科學月刊 2012, 33(1), 72-73.
2. 金必耀、左家靜, 珠璣科學—串珠碳六十（Bead model of Buckyball）, 科學月刊 2012, 33(3), 230-233.
3. 金必耀、左家靜, 珠璣科學—串珠碳七十與含端蓋碳管（C70 and end-capped carbon nanotubes）, 科學月刊 2012, 33(5), 386-388.
4. 金必耀、左家靜, 珠璣科學—串珠碳八十與螺旋碼（Spiral codes and seven isomers of C80）, 科學月刊 2012, 33(7), 556-558.

In the next article, I will talk about the connection of bead model to the valence sphere model of chemical bonding.
5. 珠璣科學 ─ 化學鍵與串珠價球模型 (Chemical bonding and the bead valence sphere model)

Truncated octahedron

C60 and extended C168 are unique because neighbored nonhexagons in them are separated by exactly one carbon-carbon bond. Is there any other graphitic structure with the similar property? The answer is yes. Chern and I have written an article on the carbon nanotori and nanohelices with this property a few years ago.

Chuang, C; Jin, B.-Y.* “Hypothetical Toroidal, Cylindrical, Helical Analogs of C60.” J. Mol. Graph. Model. 2009, 28, 220-225.

Of course, it is easy to see that there are another four Archimedean solids with this property if we allow nonhexagons to be squares or triangles. They are truncated octahedron (see the following photo), truncated cube, truncated tetrahedron, and truncated dodecahedron. Note that C60 is the truncated icosahedron. So all five truncated Platonic solids belong to this class.

Nice catalog for the 2012 JMM Math art exhibition

I got a very nice catalog for the 2012 Exhibition of Mathematical art of JMM from Chern today. Here are two pictures, one for the cover page and the next one for our beadworks on TPMS.

Beadworks from the viewpoint of a chemist

The difference that Mr. Horibe and I see these bead models is probably due to our different backgrounds. Mr. Horibe is trained as a math teacher, while I am a theoretical chemist. Even though I am acquainted with some basic mathematics for my own researches, I emphasized the chemical implication of these bead models as a method to realize nanoscaled molecules and materials. For instance, I recognized how to correctly interpret these bead models microscopically immediately after I my first buckyball with beads was done. And I also fully appreciated the power of mathematical beading as a new, and powerful method to make arbitrary fullerenes and, later on, further extended the method to arbitrary molecules.

Collaborating with Chern, now a graduate student of chemistry in MIT, we quickly became familiar with topological and geometric aspects of certain graphitic structures. Of course, Chern Chuang plays a critical role. He is very good at geometry, especially polyhedra and minimal surfaces. He also likes to make all kind of mathematical models since he was a high school student. He worked out the necessary mathematics for describing many interesting graphic structures systematically for his Master degree thesis. At the beginning, his focus was on the carbon nanotori with a goal to find out their general structural rules. We believe we have a good understanding of them now. The main findings are summarized in two papers:

1. Chuang, C.; Fan, Y.-C.; Jin, B.-Y. Generalized Classification of Toroidal and Helical Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 361-368.
2. Chuang, C; Fan, Y.-C.; Jin, B.-Y. Dual Space Approach to the Classification of Toroidal Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 1679-1686.

We also submitted a paper entitled “Comments on Structural Types of Toroidal Carbon Nanotubes” to Chemistry European Journal last year. In this paper, we summarized the carbon nanotori in thirteen canonical achiral structures, which are related to each other through three types of geometric maniputations, namely rim rotation, horizontal shift, and generalized Stone-Wells transformations. Unfortunately, the paper was rejected because of the criticism we had on the paper by F. Beuerls et al.

We quickly realized that we could systematically create quite generally a large family of graphitic structures using carbon nanotori as building blocks. This includes helically coiled carbon nanotori (carbon helices for simplicity), carbon torus knots, high-genus fullerenes, singly-, doubly-, and triply periodic minimal surfaces. Of course, one can also have all kind of variations of pseudo-periodic minimal surfaces by using heptagons instead of octagons. Part of this endeavor is summarized in "Systematics of Toroidal, Helically-Coiled Carbon Nanotubes, High-Genus Fullerenes, and Other Exotic Graphitic Materials" (Procedia Engineering, 2011, 14, 2373-2385) by Chuang, C., Fan, Y.-C. and me. These graphitic structures can be succinctly summarized as follows:

MATH imagery of American Mathematical Society

AMS MATH imagery: Beaded Fullerene of Schwarz's D Surface.

Construction procedure of C60 (Japanese translation)

Prof. Sonoda made a nice Japanese translation of the detailed construction described in the supporting information of the article, Molecular Modeling of Fullerenes with Beads, J. Chem. Edu 2012 , 89 (3), 414–416.", by Prof. Cuccia and me.



Also, here is the original English version of the first page, the next two pages are the same.

Japanese translation of 珠璣科學─串珠碳六十

Prof. Inoue (井上吉教) of Hikone prefecture university and one of his Chinese students kindly translated my article "珠璣科學─串珠碳六十" (The Science of Beading - Beaded C60) with Dr. Tsoo for the March issue of the "Science Monthly (科學月刊)" magazine.



(New post, scanned images 6/12/2012)


Prof. Inoue also made a few fullerenes as shown in the following pictures. I didn`t check them very carefully. Possibly, one of them is the icosahedral C80 and another is the cylindrical shape C84. There seems　to be a fullerene belonging to D_3h point group.

Helically coiled carbon nanotube derived from torus 120

I made another HCCNT (Helically coiled carbon nanotube) derived from the parent molecule, carbon nanotorus with 120 carbon atoms yesterday.

The construction of this carbon helix is quite straightforward. First we should know that this structure can be decomposed into six strips. To simplify the weaving process, one should start from the inner part of HCCNT.

To make a helical tube, one still need to finish the remaining two strips. Particularly, we need to be careful about the relative position between two neighbored pentagons. The systematic way to generate a whole family of HCCNTs from a parent TCNT is based on the concept of horizontal shift parameters (HSP). By applying a suitable HSP, one can create a whole family of HCCNTs.
The details of structural rules of HCCNTs can be found in the following three papers we published:

Chuang, C.; Fan, Y.-C.; Jin, B.-Y.* Generalized Classification of Toroidal and Helical Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 361-368.
Chuang, C; Jin, B.-Y.* Hypothetical toroidal, cylindrical, helical analogs of C60 J. Mol. Graph. Model. 2009, 28, 220-225.
Chuang, C.; Fan, Y.-C.; Jin, B.-Y. On the Possible Geometries of Helically Coiled Carbon Nanotubes J. Mol. Struct. 2012, 1008, 1-7.


In fact, Chern made a bead model of the same structure a few years ago. But in the Bridges conference held in Pecs, Hungary, I met Laura Shea and gave that model to her as a souvenir. Since then, both Chern and I didn't make any new model of helically coiled carbon nanotubes.