Tori without latitudes?

Previously, I posted a picture of a beaded torus belonging to another family of toroidal carbon nanotubes which I claimed latitude coordinates do not exist. Therefore, I thought a new classification scheme is needed for this family of compounds. I wondered how long it will take before we figure out the algorithmic rule for them. Since this family also contains important systems of great interests, such as armchair-type toroidal nanotubes. It is imperative that we need to work out the systematic generating rules. Fortunately, it didn't take much time for Chern Chuang to show that our generalized classification scheme is indeed general enough to cover this kind of toroidal system. Now we can probably claim all existing toroidal carbon nanotubes have well-defined latitude coordinates. The details will be discussed in his thesis. Here are two beaded models he just constructed.

球中柱

Prof. Mou said the other day that his group discovered an interesting structure which looks like a sphere with a hole puncturing through the north and south poles(球中柱). Topologically, this structure is equivalent to the toroidal carbon nanotube, since both of these types of structures have genus one (one hole). The general classification scheme of toroidal carbon nanotube that Chuang, Fan, and I developed can generate this kind of structure easily. Basically, what we need to do is to increase the separation between heptagons, and keep pentagons close to equator at the same time. The resulting structure should look like the ball with a long and thin hole (球中柱). The following figure generated by Chuang should be close to 球中柱. I wish I can find some people to make a beaded model for this compound. More than 1000 beads are needed! Probably I need to think about how to create an automatic weaving (beading) machine to do the job. Then there is no need to ask our sophomores make beaded nanotori in the service course II (服務課程 II) as our dept. head suggested.



Prof. Mou also mentioned that the long and thin hole puncturing through the north and south poles is a mathematically singularity. Naive thinking might suggest that we should put heptagons close to the region of singularity in order to simulate the negative Gaussian curvature; however, in the case of carbon hole-in-a-sphere, we have to have an equal number of pentagons far away from the sigularity, but lying around equator to satisfy the Euler-Poincare theorem V-E+F=2-2g=0, where g=1 is the genus of this system.

Torus without latitudes

Until now, all of the toroidal systems we have described and constructed can be classified according to the arrangement of pentagons and hexagons in different positions of lattitudes. In other word, the latitudes are well defined. Is there any toroidal system not in this family? The anwser is yes. The beaded molecule shown in the following figure (created by Chuang) is an example for toroids without latitudes (discovered by Ihara). Through the Goldberg transformation, a whole family of compounds can be generated. In particular, the five arms of the compounds can be elongated by inserting more hexagons in them, we can thus obtain the armchair-typed toroidal carbon nanotubes.
Unfortunately, I don't know if it is possible to generalize our classification scheme to include this type of compounds yet.

Another interesting beaded structure by Chuang

Although similar structure like this fullerene with two holes (genus 2) has been reported before, it is nice to see the corresponding beaded model (by Chuang Chern again).

Mobius Carbon Tori

Prof. Mou asked me about the possiblity of making Mobius-like toroidal structures. The answer is yes, it is possible. The strain energy might be large though. Here is some discussions I wrote in my group blog on April 7, 2007.

=(REPOST)===============================================================
A few months agos, I have a discussion with Chuang on the toroidal carbon nanotubes consisted only hexagons. I mentioned that there are several possible ways to patch the two ends of a nanotube into a circular torus. Most people have discussed the simplest situation that does not allow the twisting of the nanotube. If the nanotube is either armchair or zigzag, we can view the resulting torus as consisting of many parallel armchaired or zigzagged strips.

But if the nanotube is either chiral or twisted zigzag (armchair), the situation is quite interesting from topological point of view. Firstly, if we loose the patching constraint (boundary condition), there could be many different tori that we can obtain. Secondly, the more striking thing is that the whole torus may only contain one strip, which is similar to the Mobius strip. This also remind me of the algorithm Fan used to generate the carbon nanotube by imagining that SWNT is made of polyacetylene. （I should ask Fan to comment on this point.）

Yesterday I read a section of Pickover's "This Mobius Strip" on the Mobius prismatic doughnut, in which Pickover has also noticed the same problem in different context.

Now there is no doubt that we should go ahead and look at the electronic structure, physical and optical properties for this kind of Mobius carbon tori with different boundary conditions.

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.

Detailed Weaving Procedure for C60

Here I will describe the complete procedure for constructing a beaded C60 based on a tabular formulation as shown in the following table, which is quite popular in the oriental craft society. As I mentioned in the last December, the technique required for creating a beaded fullerene is the so-called "Right Angle Weave" (RAW). The whole weaving process can be viewed as a sequence of elementary steps. A loop consisted of n beades is formed in each elementary step. This is probably what the RAW meant. Then repeatly applying the RAW to generate a sequence of loops according to suitable spiral codes, a resulting network will automatically correspond to the correct fullerene structure.

We will use a column in the table to describe the details of building of a particular n-bead group. The second row of each column gives instruction on the number of beads in the neighboring groups that should be passed through from the thread in your hand, the third row the number of beads that should be added into the two ends of the thread, and, finally, the last row tell us how to form an n-bead group by crossing right thread into the last bead in your left thread.

The weave of our beaded molecules can be done easily with stiff thread alone and there is no need to use needles. So the only supply required is just beads and a wheel of lines for threading. With a simple estimate we can figure out that we need at least 90 beads and 110 cm of stiff thread (for 4mm beads). Work clockwise or counter-clockwise according to the spiral code.

Step 1: Using one end of the line. String on 5 beads into the thread, letting them fall to the center of line. Take the other end of the line and cross it back through the last bead. Pull tight to form the first group with 5 beads. If necessary, reposition it toward the middle of the line.
Step 2: Add another 5 beads into one end of line in your left hand. Cross the thread in your right hand back through the last bead you just added and pull tight. A 6-bead group should result.
Step 3: Pass the other end of thread (thread in your right hand) to the nearest bead in the previous group. Add another 4 beads into one end of line in your left hand. Cross the thread in you right hand back through the last bead you just added and pull tight.
Step 4: Repeat step 3 twice. Now we have the first five bead groups done. Note that one more hexagon is needed to finish up the first layer of five hexagons surrounding the central core.
Step 5: Pass the thread in your right hand to the two nearest beads in the second bead group. Add another 3 beads into one end of line in your left hand. Cross the thread in you left hand back through the last bead you just added and pull tight.
Follow each step in Table 1, until the whole sequence of spiral code is finished, the beaded molecule will appear.

As you continue working on the beaded molecule, you will notice that it tends to curve slightly when a new pentagon is added. This is because a positive Gaussian curvature will be created whenever a pentagon is introduced. According to the Euler theorem, there will be 12 pentagons in a spheroid. When repeating the step 2, the number of beads need be added to the thread in your right hand and also the beads in the neighboring groups need to be stitched through by the thread in you left hand can change. An experienced beader can figure out this number easily as the beading process continues. You may take the advantage of your chemical knowledge to decide how many beads in the neighboring group you need to stitch through with the thread in your left hand or right hand at later stage of beading. Basic criteria is that you need to stitch through all of the existing beads belong to the same carbon atoms.

Although the spiral code is a convenient recipe for systematically constructing a beaded fullerene with the spheroidal shape, we found that this is only for monochrome beaded fullerenes. It seems that there is no easy method to generate a prescribed colored pattern on the beaded molecule based on the spiral code. The best way is probably to use the tabulated format as shown Table 1.



Table 1. The complete procedure for constructing a beaded C60 in a tabular format
Steps 1 2 3 4 5 6
RT[a] 0 0 1 1 1 2
LT[b] 5 5 4 4 4 3
Cross[c] 5 6 6 6 6 6

Steps 7 8 9 10 11 12 13 14 15 16
RT[a] 1 1 2 1 2 1 2 1 2 2
LT[b] 4 3 3 3 3 3 3 3 3 2
Cross[c] 6 5 6 5 6 5 6 5 6 5

Steps 17 18 19 20 21 22 23 24 25 26
RT[a] 1 2 2 2 2 2 2 2 2 3
LT[b] 4 3 3 3 3 3 3 3 3 2
Cross[c] 6 6 6 6 6 6 6 6 6 6

Steps 27 28 29 30 31 32
RT[a] 2 3 3 3 4 5
LT[b] 3 2 2 2 1 0
Cross[c] 6 6 6 6 6 5

[a] The number of beads passed through for the thread in your left hand. [b] The number of beads added into the thread in your right hand. [c] Form an n-bead group by crossing the thread in your right hand back through the last bead you just added and pull tight.

Two TEM images taken from 7° apart for a type I nanorope

We can distinguish helix from braid by using the stereo TEM. The figure above shows two TEM images with the tilted angle 7° apart. Using these two stereograms, we can see the 3-D appearance of the sample object, and thus determine the relative depths of three strands in the nanorope. By superimposing the two images of stereo pair shown in the previous figure, the entwisting nanorope in this figure is indeed a right-handed triple helix instead of a braid.

More T120 and T120 Keychains

I have made many T120 keychains, many of them have been given away to my colleague or friends as souvenirs. For me, it typically takes an hour to create a T120 beaded fullerene. I have lost count about how many T120s I have made in the last few months.

Carbon nanosprings

These beautiful beaded helices are actually springs. With so many beaded nanosprings in hand, it may not a bad idea to do some physical experiments to get spring constants for these nanospring. Then apply dimensional analysis to get spring constants for the real carbon nanosprings.

Beaded Helicoids with smaller girth

Two more beaded helicoids with probably the smallest girth widths created by Chern Chuang (莊宸).

Beaded Helicoids

A comparison of the two helicoids I made:


I saw today on Chuang's desktop two more carbon nanohelicoids with smaller girth's width. The pitch angles of both helicoids are larger than 45 degree just like the second one I made.

Schwartzite P with 4 unit cells

Finally, thanks for Chuang's effort, now we have this amazing P-type Schwartzite that contains four unit cells. With this model, we can see very clearly the detailed structures of four unit cells connected by necks with negative Gaussian curvatures. The neck is in fact the inner-rim of a D4h torus bearing a strong resemblance to the type V torus which has the D5h symmetry.

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 （完成圖）

Chuang Chern helps me finish this beautiful beaded model for carbon nanohelix. There are approximately two pitches in this helicoid and twenty heptagons and pentagons. The inner rim structure is exactly the same as the type V torus. Each pitch contains about ten heptagons, suggesting that the tori with the five-fold rotational symmetry are more stable than tori with six-fold rotational symmetry.

The molecular geomtry on the right is created based on the 3D coordinate generated by a guy in Britain. Compare these geometries, I am happy to see that the pitch angle of the physical modeling constructed by beads is in good agreement with the computer-based molecular modeling.

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.