Interpenetrated Boerdijk–Coxeter helix

Inspired by the work of Rinus Roelofs.
作品完成時間：2015/4

Circular helix winding around a central torus

Horibe-San just constructed another beautiful beadwork, a circular helical carbon nanotube (or circular carbon spring) winding around a toroidal carbon nanotube.

作品完成時間（約）：2015/4
作者：堀部和経

Tetrahelices - two bead models of Boerdijk–Coxeter helix

There is a helical tower based on the Boerdijk–Coxeter helix in Mito-shi, Ibaraki-ken, Japan:

Art Tower MIT

So, during my visit to Nagoya this May, I made several tetrahelices as souvenirs for my friends there. Since I brought many tubular beads with different colors to choose from, so I decided to use different color for three helical edges in each tetrahelix. In total, there should be four possible arrangements or two enantiomer pairs. But I only made two of them shown in the following picture. After I made these two tetrahelices, so many people also like to have this model, so I made two more tetrahelices before I left Japan.

Short summary of zome-type superbuckyball part I: 1D Linear and Helical C60 Polymers

Recently I've been playing with all kinds of these superbuckyballs, based on the methodology of replacing balls and struts of zometool by Ih-symmetric fullerenes and straight CNTs. Taking C60 for example, blue struts correspond to removing two atoms next to some particular C2 rotation axes. And the connecting CNTs are of chiral vector (4,0). On the other hand, yellow and red struts correspond to C3 and C5 rotation axes, respectively. In addition, due to the property of the golden field, the algebraic field of zometool, superstructures using yellow or red struts only have the possibility of being polymers of C60. This means that the number of atoms is a multiple of 60, i.e. no atoms needed to be deleted or added when constructing the superbuckyball.

Here I will briefly summarize some of the cases I've done coding with. Hopefully I'd soon come up with a short paper ready to submit to the Bridges 2013 on this topic.

First let us start with the trivial C60 dimers. As mentioned, structures with red or yellow struts cases have the atom-preserving property. The C120 isomers corresponding to joining two C60s along their fivefold and threefold axes are shown below. I should mention that they were also discussed in Diudea and Nagy's book . In particular for the C3-joined case there are two possibilities of local atomic connectivity.

C3-fused C120, case 1 (with octagons and pentagons at the interface)

C3-fused C120, case 2 (with heptagons at the interface)

C5-fused C120

For the case of blue struts (twofold rotation axes)

C2-fused C116

C2-fused C132

Some of them were already made previously by us, see here for example. But we did not realize back then this particular connection with zometool. To my knowledge, there has not been any experimental characterization of such dimers. Synthetic chemists do make C60 dimers but those are of partial sp3 characteristics, i.e. some interfacial atoms have four neighbors instead of three. Please refer to Diudea and Nagy's book for further details if you are interested.

One can come up with the one dimensional C60 chains without too much effort by enforcing periodic boundary condition. So the structure repeats itself indefinitely along the direction of polymerization. See for example below.

Also, it is one step away from constructing the 2D analog of this kind of structure.

A little bit more sophisticated extension of the above scheme is to consider helical screw symmetry. A (discrete) helical curve is defined by the angle between adjacent unit cells and the dihedral angle between next-nearest neighbors. I recommend readers of interest to play with the awesome virtual zome program vZome developed by Scott Vorthmann. You have to write Scott an email for the license of the full version of vZome. Anyway, here are some examples of helical C60 polymers.

C3-fused fourfold C60 helix

C5-fused fivefold C60 helix

Notice that if you are looking along the axes of the helices, the C60s that are four/five unit cells away lie exactly on top of each other. Curiously, this result is actually symmetry-determined, since I've tested with the relaxation scheme that does not require such symmetry. In other words, even if I optimize the geometries with full degrees of freedom of a general helix, the screw angles will still be 2*pi/4 or 2*pi/5 in the above cases.

The procedure for constructing G- and D- surfaces

Here are a few slides that show the detailed instruction for making G- and D- surfaces, which I prepared for students and teachers of TFG (Taipei) school. As I said it could be a difficult task because the gyroidal structure and D-type TPMS are complicated structures. The first bead model of a 2x2x2 G-surface took Chern and I almost five years to finally make it. Of course, I have many unfinished bead models of this structure or similar structures with different Goldberg vectors, some made by Chern and some by me, which have mistakes here or there.

In order to how to make this model successfully, we'd better to know the three-dimensional structures of G- and D-type surfaces a little bit. Additionally, it is crucial to know how two structures can be decomposed into several basic unit strips and how to connect these helical strips.

I am also working on an article in Chinese entitled "大家一起動手做多孔螺旋與鑽石型三度週期最小曲面的串珠模型 (A Hands-on, Collaborative Approach to Gyroid- and Diamond-type Triply Periodic Minimal Surfaces with Beads)", which describes in details the procedure to make G- and D-surfaces and also give some background information on TPMS. I might be able to finish the paper in a few days. Hopefully, I will find time to do it in English someday. But, even without detailed explanations, these slides together with other posts in this blog should already contain enough information for people who want to do it.

The first nine slides should give students a better picture of a gyroid:

In slide 10, we can see how a coronene unit corresponds to 1/8 unit cell. Important structural features of a beaded gyroid is summarized in slide 11. Then in slides 12-15, I describe how to make the basic construction unit, a long strip, which should be easy for student to make.

The remaining five slides, 16-20, use schematic diagrams to show how two slides can be combined to generate either D-surface or G-surface.

To create a 2x2x2 gyroidal surface, we need 16 strips, which can be easily done if many people work in parallel. To connect them is nontrivial, you need to follow slides 16-20 carefully. In total, there are about 5000 beads in the model.

Helically coiled carbon nanotube derived from T140

I made one more HCCNT that was derived from parent torus, T140.

The inner part should be weaved first. Here I keep the relative position of these heptagons unchanged.

The next step is to determine the HSP (Horizontal Shift Parameter) on the outer part of the torus. The pitch of the HCCNT will depend on the magnitude of HSP. For detail, check the papers mentioned in previous post.

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.

D surface constructed from four helical strips

There is another way to build a D-type triply periodic minimal surface (TPMS) with beads. Chern has told me previously that one can not only use helical strips to build G-type TPMS, one can also use exactly the same helical strips to build D-type TPMS. If one examine two helical strips carefully, one can find that there are exactly two different ways to put them together. One gives a D-type TPMS, the other one gives a G-type surface!

The following pictures are a bead model of D-surface consisting of four helical strips. Two of them are left handed, the other two are right handed. To build a D-surface, one has to put two helical strips together in an arrangement such that two neighbored strips are mirror-symmetric to each other. So the overall structure of D-surface is not chiral.

It is useful to look at other posts with the keyword helical strip, especially the one on the G-surface created by patching two helical strips with one strip shifted by half pitch.

A single helical strip

As I discussed in the previous posts, following Chern's construction scheme, I used 16 helical strips to build the overall 2x2x2 gyroical graphitic structure. Each strip contains 8 eight-bead loops, four with blue color and four with purple color as shown in the following photos.



It is interesting note that one can easily create a kink in this helical strip on purpose. A kink in a helix changes the handedness from left to right.

Helically-coiled tube in 劍湖山

I found these two interesting helically coiled macrotubes in 劍湖山 yesterday.

Example of Triple Stranded Helical Nanotube

The helix is derived from a Dnh torus (2,2,2,4) with shifting in outer rim by 2 units. Each strand contains 10 unit cells.

Helically-coiled nanotubes

Physical modeling with beads is a powerful way to get an intuition about the effect of spatial arrangement on the shapes of nontrivial carbon graphitic surfaces. But, the actual weaving of a beaded molecule is a slow process. We usually only build a few physical models. With these models in hand, it becomes much easier to figure out the general rules for construction. We then write matlab scripts with these rules. Chuang and Fan have worked out a very general algorithm to obtain an arbitrary helically-coiled nanotube that can be derived from a torus generated according to our generalized classification scheme. Here is the homologous series of helices that are derived from T240 (2,2,2,2,0,0). All of these helically-coiled models have the same inner-rim structure (D5d), but with different arrangement of pentagons in the out-rim. Whereas the girth widths of these models are quite close to each other, the pitches and grooves vary significantly for molecules with different arrangements.

Triply-Stranded helix (TEM)

A few years ago, prof. L. P. Hwang discovered an interesting new family of the self-organized triply-stranded helically coiled nanoropes. These nanorope can be roughly classified into to two types according to their pitch angles. For instance, the pitch (p) and diameter (2r) of the type I nanoropes are found to be around 150 nm and 80 nm as shown below:



The width of girth is about 20 nm. It seems that the TEM figure shows many ankles similar to pentagons in the beaded model I posted in the previous blog.

By the way, just as a reminder. I remember when prof. Hwang first saw this TEM picture, he thought this was a nanobraid. He told me about this result. And asked me whether I can give an explanation to this phenomenon or not. Then I took a few hours to learn how to make a braid. After I went home that night and showed my daughter the TEM picture. Asked her what that was. She replied immediately, "That is a braid, of course." I spent some times try to find out a possible mechanism, but without success. Why? That is because the projection of a braid on the cross section of a braid is a lazy eight. Mathematically, this can be generated by a 3D curve with the parametrized form: (sin w t, cos 2 w t, p t). Note that the oscillation frequency along the y-direction is twice of that along x-direction. Why? Why not with the same frequency? If this is the case, we should have a triple-stranded helix. What kind of physical force can generate a lazy eight curve? That is beyond my imagination. Instead, I start to question the original thought of prof. Hwang. It is not hard to find that the TEMs for either a braid and a tripled-stranded helix are very similar. So result of TEM can be interpreted by both structures. Only further experiments can justify which structure is correct. With help from a technician at the TEM laboratory, we knew we can take advantage of the stereo-TEM technique to get TEM pictures at two different angles. Then reconstruct the 3D structure by merging these two pictures. The result supported the triple-stranded helix. Of course, this does not mean that the story is over. It is still an interesting problem to work out a possible mechanism for the growth and stability of this kind of triple-srtanded helices.

Su, C.-J.; Hwang, D. W.; Lin, S.-H.; Jin, B.-Y.*, Hwang, L.-P.* Self-organization of triple-stranded carbon nanoropes Phys. Chem. Comm. 2002, 5, 34. DOI: 10.1039/B110151J and pdf (open access!)

Helically Coiled Carbon Nanotube with large pitch angle and groove width

I made this helicoid today. This one has pretty large pitch angle and smaller girth width. The motivation for this structure is from T240 carbon innertube, i.e. compound IV1. This helicoid has the same inner-rim structure as that of carbon innertube, T240, I made before. The arrangement of pentagon in the outer region still some relationship to T240/IV1. It is the simplest possible combination that can lead to a helical structure I can think of. The groove of this helicoid is very large comparing with previous helicoid. In fact, it is possible to put the second helicoid with the same structure to create a double stranded helically coiled carbon nanotube. The number of units in a pitch seems to be smaller, only about 4 unit cells. Of course, I am not exactly sure whether the simulation based on more sophisticated force-field will produce the same result or not. I suspect the deviation will be small.

Concerning pitch angles and groove widths of helically coiled carbon nanotubes

In the previous post, we have seen the good agreement for the geometries obtained from the beaded and computer-aided models respectively. A more general question to ask is how the pitch angle and groove width of a helically coiled carbon nanotube depend on the relative arrangement of pentagons and heptagons. I believe that the agreement of these two parameters could be used as a stringent test for the quality for the force field of the beaded representation. To perform this test, we need to work out the general classification scheme for the helically coiled carbon nanotubes. And the geometry optimization using force field such as Tersoff potential is needed. Of course, it is necessary to construct more different types of helically-coiled carbon nanotubes with beads.

Beaded Helicoid

Finally, I almost have a beaded helicoid. Although it is still in progress, the overall structure of the helix can be seen easily. Before I started to do this one, I still have some doubt about whether the resulting structure will look close to the real helically coiled carbon nanotube or not. This is because at the beginning stage, the beaded structure is so flexible and does not seem to work. But when the one-fourth of the first pitch is done, the structure is already very stable. This is really amazing. Now I have one pitch of the helicoid finished and still have some more to go.




By the way, the structure of helicoid is essentially derived from the twisted type of toroids. The twisted type of toroid is not stable for the small toroidal systems. But if we change the cyclic boundary condition, which apparantly induces very large strain energy in the twisted type toroid, and let the two ends of the tube go freely, then we will have the helicoid.

Of course, there is an additional constraint to be satisfied in order to avoid the intersection of the tube. Then not every twisted toroid can generate a stable helicoid.