弹性细杆的非线性力学: DNA力学模型的理论基础

作者：刘延柱

我查過，台大對面的高教書局還沒進口。

## Saturday, November 28, 2009

### Beaded Model for Supercoiled Helix

### Beaded Model for Supercoiled Helix

I spent a while searching for Dr. Ou-Yang's publications and have found a dictionary of bio-engineering with an interesting front cover:

http://books.google.com/books?id=2fOFfJNbcMcC&printsec=frontcover&dq=%E6%AD%90%E9%99%BD%E5%AE%97%E7%87%A6&hl=zh-TW&source=gbs_similarbooks_s&cad=1#v=onepage&q=&f=false

The picture is quite clear that this model is indeed made by the RAW technique that we're familiar with.

http://books.google.com/books?id=2fOFfJNbcMcC&printsec=frontcover&dq=%E6%AD%90%E9%99%BD%E5%AE%97%E7%87%A6&hl=zh-TW&source=gbs_similarbooks_s&cad=1#v=onepage&q=&f=false

The picture is quite clear that this model is indeed made by the RAW technique that we're familiar with.

## Tuesday, November 24, 2009

### Trefoil Knot

## Monday, November 23, 2009

### More photos of Beaded Trefoil Knot

Here are several photos I took when I was working on the beaded trefoil knot last weekend. It is now obvious to me why we should take such a weaving path for making beading trefoil knot. I have mentioned several difficulties before. Originally, I thought that these difficulties cannot be overcome without using several different sizes of beads. But beading can be very flexiable since we don't need to make them very tight. If we allow the structure be loose, then anything kind of graphs can be weaved in principle. Of course, the resulting structures may not be rigid or similar to what we would like to have for a fullerene.

In the following photos, one can see the intermediate structures before I started to weave the final strip of polyacene is quite flexible. This gives some rooms for us to adjust the structure, so the unavoidable deformation can be redistributed more evenly on the whole structure.

Of course, it is very important to make the first loop connected correctly. :-)

In the following photos, one can see the intermediate structures before I started to weave the final strip of polyacene is quite flexible. This gives some rooms for us to adjust the structure, so the unavoidable deformation can be redistributed more evenly on the whole structure.

Of course, it is very important to make the first loop connected correctly. :-)

## Sunday, November 22, 2009

### Beaded model of Carbon Nanotube Trefoil Knot

Trefoil knot is one of the most beautiful geometric shapes that intrigue both mathematicians and artists alike. Previously Chuang and I have shown that one can in fact tile on trefoil knot with graphitic sheet. By introducing non-hexagons on the suitable position, the resulting carbon nanotube trefoil knots can be very stable.

The problem is whether we can really build a beaded model for this new family of carbon compounds. Previously, we have tried to create a beaded model for CNTTK by weaving the girth first and then the axial direction of the tubule. Based on this strategy, we realize that the intrinsic curvature introduced by the non-hexagons is too large, so it seems that the construction of trefoil knot with beads is not possible unless large deformation is applied.

Last weekend, Chuang gave me a figure of CNTTK 504 with only one strip of polyacene is shown (see previous post). This suggested that one may weave along the axial direction first. So construction of a beaded CNTTK 504 amounts to weaving 6 strips of polyacenes. Using this weaving algorithm, one may be able to adjust the intermediate structures more easily since they are more flexible. The extra deformation that has to be introduced can then be distributed more evenly over a larger area. This is quite different from the previous weaving algorithm we adopted. The extra deformation created in the previous method is concentrated on particular segment of the tubule. The final structure may look quite distorted. Thus, the new algorithm has better potential for making a CNTTK.

This is indeed the case. I finally made one beaded CNTTK 504. The resulting beaded structure as shown in the following figure looks great. The overall shape of this structure looks quite similar to the optimized structure we obtained before. With closer inspection, we can also see the distortion away from the intrinsic curvatures created with non-hexagons.

The problem is whether we can really build a beaded model for this new family of carbon compounds. Previously, we have tried to create a beaded model for CNTTK by weaving the girth first and then the axial direction of the tubule. Based on this strategy, we realize that the intrinsic curvature introduced by the non-hexagons is too large, so it seems that the construction of trefoil knot with beads is not possible unless large deformation is applied.

Last weekend, Chuang gave me a figure of CNTTK 504 with only one strip of polyacene is shown (see previous post). This suggested that one may weave along the axial direction first. So construction of a beaded CNTTK 504 amounts to weaving 6 strips of polyacenes. Using this weaving algorithm, one may be able to adjust the intermediate structures more easily since they are more flexible. The extra deformation that has to be introduced can then be distributed more evenly over a larger area. This is quite different from the previous weaving algorithm we adopted. The extra deformation created in the previous method is concentrated on particular segment of the tubule. The final structure may look quite distorted. Thus, the new algorithm has better potential for making a CNTTK.

This is indeed the case. I finally made one beaded CNTTK 504. The resulting beaded structure as shown in the following figure looks great. The overall shape of this structure looks quite similar to the optimized structure we obtained before. With closer inspection, we can also see the distortion away from the intrinsic curvatures created with non-hexagons.

### A new picture of CNTTK 504

Chuang gave me a new matlab fig file of CNTTK 504 last weekend, which shows more clearly on how a CNTTK is composed of six strips of polyacenes (with pentagons and octagons). In order to see the change in orientation this strip colored in red, one seem to have to use two different projections. I am not sure if one can do better than this. For instance, is it possible to use just one projection to give the relative arrangement of these polygons?

## Monday, November 16, 2009

### Beaded icosahedrons

Beaded models for icosahedron are harder to make. This is because each vertex has five edges connected to it, so one has to face a pentavalent coordination. It is very difficult, if not impossible, to make a pentavalent coordinate with spherical beads. But if want use beads with long aspect ratio, stable and beautiful icosahedral structures can still be made.

### Mobius Short

In addition to standard way of creating one sided surface like Mobius band through twisting the strip, another way to create a one sided surface is the Mobius short. Here we use a T-shape band to create the one sided surface without twisting. The following photos show the beaded Mobius short I constructed last weekend.

## Wednesday, November 11, 2009

### Length of fishing thread

A liitle bit weaving technique is given here. This is important especially when one want to weave a long strip of beads (or equivalently zigzag carbon nanoribbon, ZGNR) with a width that consists of one or two hexagons, or more generally, odd or even numbers of hexagons. When we want to make a Mobius ZGNR, we will meet the situation like this. Basically, we need to weave a long strip first and then connect both ends in a particular way.

If we want to weave a long strip of ZGNR with a width of one hexagon, corresponding to a polyacene, we should observe that the length of fishing threads in both hands are running out at equal rate. It is easy to understand this by checking the following figure (a).

However, if we want to weave ZGNR with a width of two hexagons as shown in the bottom of the following figure (b), we should easily see that the one end of thread runs out much faster than the other end. It is also easy to understand why that happen by looking at the figure.

Of course, one can avoid this problem by following different weaving path. (see (c) of the following figure) I usually don't like to use weaving path (c). This is because weaving path (c) has the problem that it is much harder to make the structure constructed by this path tight.

If we want to weave a long strip of ZGNR with a width of one hexagon, corresponding to a polyacene, we should observe that the length of fishing threads in both hands are running out at equal rate. It is easy to understand this by checking the following figure (a).

However, if we want to weave ZGNR with a width of two hexagons as shown in the bottom of the following figure (b), we should easily see that the one end of thread runs out much faster than the other end. It is also easy to understand why that happen by looking at the figure.

Of course, one can avoid this problem by following different weaving path. (see (c) of the following figure) I usually don't like to use weaving path (c). This is because weaving path (c) has the problem that it is much harder to make the structure constructed by this path tight.

## Tuesday, November 10, 2009

### Beaded Mobius strip with three 180 degree twists

Once one got the idea to make a beaded Mobius strip, it is then straightforward to construct different kind of Mobius bands at next level of complexity.

Here is a simple beaded Mobius strip with three 180 degree twists, that I just made in the Starbucks. Similar to the standard Mobius strip with one 180 twist, this one is also a non-orientable surface. Of course, one need to weave a longer strip before one can glue two ends together. I believe that the shape of beaded model of this kind of Mobius strip should be similar to the optimal shape obtained by van der Heijden again.

An interesting little book on the Mobius strip by Pickover.

Here is a simple beaded Mobius strip with three 180 degree twists, that I just made in the Starbucks. Similar to the standard Mobius strip with one 180 twist, this one is also a non-orientable surface. Of course, one need to weave a longer strip before one can glue two ends together. I believe that the shape of beaded model of this kind of Mobius strip should be similar to the optimal shape obtained by van der Heijden again.

An interesting little book on the Mobius strip by Pickover.

### Other beaded Mobius Strip

With some googling, I found that the beaded Mobius strip has been done in different ways before as shown in the following site:

http://www.instructables.com/id/Geek_chic_Moebius_strip_earrings/

Here is the mobius strip earrings Davisjan designed:

My design is very different in the sense that I have put an emphasis on the equilibrium shape of a beaded Mobius strip originating from the hard-sphere repulsion among three trivalent beads. That's why the shape of my design has such a close resemblance to the optimal shape obtained in van der Heijden's theory.

http://www.instructables.com/id/Geek_chic_Moebius_strip_earrings/

Here is the mobius strip earrings Davisjan designed:

My design is very different in the sense that I have put an emphasis on the equilibrium shape of a beaded Mobius strip originating from the hard-sphere repulsion among three trivalent beads. That's why the shape of my design has such a close resemblance to the optimal shape obtained in van der Heijden's theory.

### The Shape of Beaded Mobius Strips

Although I know that one can form a mobius strip from a graphene nanoribbon for some time. I have never thought about making one by using beads. Part of the reason is that I am quite fond of making beaded fullerenes that contain nonhexagons. I have the belief that the structural stability of beaded fullerenes come from these nonhexagons and the hexagons play only a minor role in the stability.

We know that a carbon nanoribbon mobius strip (CNR-MS) can be form by twisting the strip 180 degrees and then glue two opposite ends of strip together. The CNR-MS consists of hexagons completely. Since there is no any nonhexagon in the CNR-MS, one might suspect that the corresponding beaded model may not have a stable equilibrium shape. But in fact this is not true. Due to the bending of hexagonal sheet made from the beads, the resulting beaded Mobius strip has a very stable equilibrium shape.

In the following figure, I show two beaded Mobius strips I made yesterday. These two Mobius strips have very stable equilibrium structures. This is different from the Mobius band made by a long strip of paper or belt. Of course, it is much easier to make a mobius band out of these materials. The resulting strips usually can adopt more conformations. One can easily distort the structure in more different conformations. So it is usually not easy to see the most stable conformation for the Mobius strip made by paper or belt.

Beaded Mobius strips are different since they have large hard-sphere repulsion among different beads when the corresponding surface is bended. Therefore the beaded mobius strip can adopt a more unique equilibrium shape in the absence of external pressure.

Compare the beaded mobius strip with the shape published by G. H. M. van der Heijden in Nature Material two years ago. One can see the strong resemblance between the two shapes as shown in the following figure. In my opinion, the similarity is not surprising since the beaded structures are inextensible and have their energy mainly from bending deformation, which is exactly the type of Mobius strips van der Heijden is considering about. The two beaded Mobius strips in the following figure do not have exactly the same ratio between length and width as the Mobius strip van der Heijden showed in his paper. I suspect that if I use similar ratio, for instance a longer length, the resulting shape of the beaded Mobius strip will even be closer to van der Heijden's.

(Note that the figure in the upper right is taken from a paper in Nature Material by van der Heijden, which is about the equilibrium shape of an inextensible Mobius strip. There might be some copyright problem.)

Construction of a beaded Mobius strip based on hexagonal network is straightforward. One first construct a long strip of zigzag honeycomb lattice as shown in the upper right figure. When the strip is long enough, then we can twist the strip by 180 degrees and weave these two opposite ends together with beads. To make a more stable structure, one can use the remaining fishing thread to go through beads located on the edge. Note that there is only one edge, not two edges, for the Mobius strip. Once we do that, the resulting Mobius strip will be very stable.

We know that a carbon nanoribbon mobius strip (CNR-MS) can be form by twisting the strip 180 degrees and then glue two opposite ends of strip together. The CNR-MS consists of hexagons completely. Since there is no any nonhexagon in the CNR-MS, one might suspect that the corresponding beaded model may not have a stable equilibrium shape. But in fact this is not true. Due to the bending of hexagonal sheet made from the beads, the resulting beaded Mobius strip has a very stable equilibrium shape.

In the following figure, I show two beaded Mobius strips I made yesterday. These two Mobius strips have very stable equilibrium structures. This is different from the Mobius band made by a long strip of paper or belt. Of course, it is much easier to make a mobius band out of these materials. The resulting strips usually can adopt more conformations. One can easily distort the structure in more different conformations. So it is usually not easy to see the most stable conformation for the Mobius strip made by paper or belt.

Beaded Mobius strips are different since they have large hard-sphere repulsion among different beads when the corresponding surface is bended. Therefore the beaded mobius strip can adopt a more unique equilibrium shape in the absence of external pressure.

Compare the beaded mobius strip with the shape published by G. H. M. van der Heijden in Nature Material two years ago. One can see the strong resemblance between the two shapes as shown in the following figure. In my opinion, the similarity is not surprising since the beaded structures are inextensible and have their energy mainly from bending deformation, which is exactly the type of Mobius strips van der Heijden is considering about. The two beaded Mobius strips in the following figure do not have exactly the same ratio between length and width as the Mobius strip van der Heijden showed in his paper. I suspect that if I use similar ratio, for instance a longer length, the resulting shape of the beaded Mobius strip will even be closer to van der Heijden's.

(Note that the figure in the upper right is taken from a paper in Nature Material by van der Heijden, which is about the equilibrium shape of an inextensible Mobius strip. There might be some copyright problem.)

Construction of a beaded Mobius strip based on hexagonal network is straightforward. One first construct a long strip of zigzag honeycomb lattice as shown in the upper right figure. When the strip is long enough, then we can twist the strip by 180 degrees and weave these two opposite ends together with beads. To make a more stable structure, one can use the remaining fishing thread to go through beads located on the edge. Note that there is only one edge, not two edges, for the Mobius strip. Once we do that, the resulting Mobius strip will be very stable.

## Monday, November 9, 2009

### Ih-Symmetric C380 : The Smallest Stable 11-Genus Graphenoid (A Repost from byjingroup blog)

This is the smallest possible 11-Genus Ih-symmetric fullerene while its stability is comparable to the usual TCNT C120 we acquainted with. The MOPAC AM1 optimization is still on progress. This kind of molecule is made from puncturing 12 identical holes, which I prefer to term them as (D5d-symmetric) necks, between a pair of concentric dodecahedron and icosidodecahedron, both tiled properly with graphene patches.

However, I think that this molecule is of particular interest because it is really small and in the mean while stable. The geometric features, as in the TCNT case, are controlled by four parameters. But in contrast to the TCNT case, these four parameters put huge constraint on the output molecule's stability. Which is to say, only a few of the parameter combinations under certain number of atoms are of practical interest, as compare to the large variety of TCNT isomers. In particular, I think this could be the only stable high-genus fullerene under 500 atoms.

In the literature, only the Terrones group have had studied this kind of HG fullerene. But I have never noticed that if they ever described the general rule of how the molecules are constructed. Beside knowing exactly how the construction rule works in their cases, the experience we learned from TCNT construction scheme allows me to develop a trick of building HG fullerene with far fewer number of atoms than them, which I shall describe in later posts.

BTW, I think this molecule is the best suited one for making its beaded molecule. I have made two of them, here's one:

Viewed along a C5-axis,

Viewed along a C3-axis,

The size compare to D6d TCNT C144,

(Posted on byjingroup blog by Chern Chuang, Wednesday, November 5, 2008)

### Beaded Virus

In all of the high-genus fullerenes, the structures of inner parts are covered by an outer sheet. It is difficult to see the organization of beads inside. In the following few photos, I gave the inner-layer structures of a high-genus fullerene at several different stages of construction. The first figure shows the HG-fullerene with only inner part. It is astonishing that this molecule has such a resemblance to a virus.

With a careful examination, one can see the inner part consists essentially of twelve necks arranged on the surface of a dodecahedron. More importantly, these necks are nothing but the inner-rim of a toroidal carbon nanotube with 120 carbon atoms (T120), which is my favorite molecule. So a simple counting indicates that this structure should have 12*10=120 heptagons in all necks. To weave this structure, I have used about 3 fishing threads with a length about 3.5 m. Of course, by a carefully designed weaving sequence, one can in principle use only one fishing thread to make the whole structure.

With a careful examination, one can see the inner part consists essentially of twelve necks arranged on the surface of a dodecahedron. More importantly, these necks are nothing but the inner-rim of a toroidal carbon nanotube with 120 carbon atoms (T120), which is my favorite molecule. So a simple counting indicates that this structure should have 12*10=120 heptagons in all necks. To weave this structure, I have used about 3 fishing threads with a length about 3.5 m. Of course, by a carefully designed weaving sequence, one can in principle use only one fishing thread to make the whole structure.

From Beaded Virus |

From Beaded Virus |

From Beaded Virus |

From Beaded Virus |

From Beaded Virus |

From Beaded Virus |

### Collection of High-Genus Fullerenes

Up to now, I guess that Chuang has constructed 11 high-genus fullerenes. The following figure shows eight of them. I have given high-genus dodecahedrons (d), (e), and (f) away as gifts, though. All five high-genus dodecahedrons in the figure are different in the sense that they have completely different arrangement of heptagons.

(a) High-genus tetrahedron.

(b) High-genus cube.

(c) (d) (e) (f) (g) High-genus dodecahedron.

(h) High-genus icosahedron.

I don't have beaded model for high-genus octahedron. Maybe I will construct one when I have time.

(a) High-genus tetrahedron.

(b) High-genus cube.

(c) (d) (e) (f) (g) High-genus dodecahedron.

(h) High-genus icosahedron.

I don't have beaded model for high-genus octahedron. Maybe I will construct one when I have time.

## Sunday, November 8, 2009

### Some photos from another store carrying gemstore beads

I have never used gemstone beads to create beaded fullerenes. I find it too expensive. Chuang have constructed many beaded fullerenes with gemstone beads though. I have posted some of them on this blog. You can find them though the label, "gem stone".

I went to the Yan-Ping North Road (延平北路) this weekend. On this street and nearby neighborhood, you can find a lot of interesting stores. I took a few pictures outside several stores close to Mamabear (小熊媽媽). I figured they may not be happy if I took pictures inside the stores.

The following pictures were from "Crystal Supermarket" (水晶超市). This store carries a wide varieties of gemstone beads including cat's eye stones, opal, crystal, and many other gem stones which I do not know.

The prices for these stones are expensive to me. Typically, I need several hundreds of beads for a small fullerene and thousands for larger structure such as high-genus fullerenes.

I went to the Yan-Ping North Road (延平北路) this weekend. On this street and nearby neighborhood, you can find a lot of interesting stores. I took a few pictures outside several stores close to Mamabear (小熊媽媽). I figured they may not be happy if I took pictures inside the stores.

The following pictures were from "Crystal Supermarket" (水晶超市). This store carries a wide varieties of gemstone beads including cat's eye stones, opal, crystal, and many other gem stones which I do not know.

The prices for these stones are expensive to me. Typically, I need several hundreds of beads for a small fullerene and thousands for larger structure such as high-genus fullerenes.

### Longuet-Higgins' Rhombo blocks

When reading the book, "King of Infinite Space", a bibliography about the Canadian mathematician Donald Coxeter, a magnetic toys called Rhombo blocks mentioned in pages 251-252 caught my eye. According to the description in the book, it is some kind of toys very similar to the one, "ball of whacks", I bought a few months ago.

M. Longuet-Higgins was the inventor of Rhombo blocks. I checked Rhombo blocks after I got accessed to the internet (see the picture below) this morning. Indeed, the magnetic design of this toy is probably similar to the "ball of whacks". I guess that the difference is the shape between the two mainly. I am curious about who invented this kind of design first.

Again the Rhombo block is quite expensive, though.

One can also build a Kepler ball (Rhombic triacontahedron) using Rhombo blocks. Based on the following picture, it seems to me that only 10 blocks are needed.

M. Longuet-Higgins was the inventor of Rhombo blocks. I checked Rhombo blocks after I got accessed to the internet (see the picture below) this morning. Indeed, the magnetic design of this toy is probably similar to the "ball of whacks". I guess that the difference is the shape between the two mainly. I am curious about who invented this kind of design first.

Again the Rhombo block is quite expensive, though.

One can also build a Kepler ball (Rhombic triacontahedron) using Rhombo blocks. Based on the following picture, it seems to me that only 10 blocks are needed.

## Saturday, November 7, 2009

### Dodecahedron with crystal beads

In the store next to Mamabear, I found many beaded dodecahedrons, mainly for decoration, made with more expensive crystal beads. I believe that people have known how to make spherical beaded structures in the ancient time. The first person who discovered how to do it has probably been lost in history for a long time.

### A few pictures of Mamabear (小熊媽媽)

I took many pictures of Mamabear, a local store in Taipei with many selections of beads, this morning.

Incidentally, they also sell beads for making a beaded football (in this case, a C80 indeed) in Mamabear. People seem to know how to make C60, C80, and other cage-like structures (such as the beaded pineapple as shown in the next photo) with beads for a long time.

Incidentally, they also sell beads for making a beaded football (in this case, a C80 indeed) in Mamabear. People seem to know how to make C60, C80, and other cage-like structures (such as the beaded pineapple as shown in the next photo) with beads for a long time.

### Beaded Model for Scherk's Singly Periodic Minimal Surface

It is just amazing! I met Chuang unexpectedly while I was waiting for the bus home yesterday evening. Chuang told me he has figured out how to construct graphitic sheet with the structure of the Scherk's singly periodic minimal surface (SPMS) and has posted it on my group blog as shown in the following figure. It is indeed a beautiful structure. In some sense, we can view this structure as an intersection of two graphene sheets. Instead of a straight line for the intersection, here one need a chain of octagon loops to mimic this intersection.

Apparently, Chuang has worked out this when he has a shooting training in the army. I hope that he can still shoot the target in the training course.

This SPMS structure is not complicated. It is quite straightforward to make a beaded model for this kind of graphitic structure. It took me several hours to make it today.

Apparently, Chuang has worked out this when he has a shooting training in the army. I hope that he can still shoot the target in the training course.

This SPMS structure is not complicated. It is quite straightforward to make a beaded model for this kind of graphitic structure. It took me several hours to make it today.

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