More measurements on C80

I have perfomed more measurements of length for weaving C₈₀.

The scaling factor are calculated as follows
80:1 105/96=1.09
80:2 107/96=1.11
80:3 106/96=1.10
80:4 108/96=1.13
80:5 105/96=1.09
80:6 105/96=1.09

By the way, if anyone would like to build the seven isomers of C₈₀, here are the spiral codes for all of them
80:1 1 7 9 11 13 15 28 30 32 34 36 42
80:2 1 7 9 11 13 18 25 30 32 34 36 42
80:3 1 7 9 11 14 22 27 30 34 36 38 40
80:4 1 7 9 11 14 23 28 30 33 35 37 39
80:5 1 7 9 12 14 20 26 28 32 34 39 42
80:6 1 7 10 12 14 19 26 28 32 34 39 42
80:7 1 8 10 12 14 16 28 30 32 34 36 42

The resulting beaded C₈₀'s are given in the figure below.

Supplementary Constraint in Using Spiral code

Is the sequence as given by the spiral code enough for the building of beaded fullerene? Well, not really, you need to have some extra conditions to get the work done. For instance, the spiral code for the dodecahedron is 1 2 3 4 5 6 7 8 9 10 11 12, which gives the spiral sequence 555555555555 of 12 pentagons. Here I will give the detailed procedure for weaving a dodecahedron based on this sequence.

The weave of our beaded molecules can be done easily with stiff thread along. I found that there is no need to use needles. So the supply we need is just beads and a wheel of lines for threading. With a simple counting, we need at least 30 (20*3/2 = 30) beads (4mm) and 50 cm of stiff thread (0.3 mm). Work clockwise or counterclockwise 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 4 beads into one end of line in your right hand. Pass the other end of thread (thread in your left hand) to the nearest bead in the previous group. Cross the thread in you left hand back through the last bead you just added and pull tight. A new 5-bead group should result.

Step 3: Repeat step 2 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.

When repeating the step 2, the number of beads need to be added to the thread in your right hand and also the beads in the previous group need to stitched though by the thread in you left hand can change. An experienced beader can figure out this number easily as the beading process continues. Chemists may take the advantage their chemical knowledge to decide how many beads in the neighbor group you need to stitch through with the thread in your left hand. Basic criteria is that you need to stitch through all of the beads (chemical bonds) belong to the same carbon atoms.

Simple Method to Generate Colored beaded fullerene?

Almost all of the artworks I posted are colored beaded fullerenes which may or may not have holes on their surface, depending on the types of fullerenes I want to create. As I have shown last time that the spiral code is in fact a convenient recipes for systematically constructing a beaded fullerene (spheroidal shape). But this is only for one-color beaded structure. It seems that there is no easy method to do colored beaded structures. I am still trying to figure out a simple method which can generate an arbitrary colored pattern on the beaded work based on the spiral code with some extra instruction on the color pattern.

A simple estimate of scaling factor for the minimum length of thread

I made some simple measurements in order to get an estimate for the length of thread I need. The results for three fullerenes are

Isomer 50:24 theoretical length, 60 cm; measurement: 68 cm; scaling factor s = 68/60=1.1
Isomer 50:43 theoretical length, 60 cm; measurement: 69 cm; scaling factor s = 68/60=1.2
Isomer 80:7 theoretical length, 96 cm; measurement: 106 cm; scaling factor s = 68/60=1.1

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The scaling factor is about 1.1-1.2. I guess this result should hold for other fullerene too, at least for me. But this estimate could vary for other people. Those people who can make an even tighter structure are going to have a smaller s. On the contrary, measurements on the loose structure will lead to larger scaling factor. I suspect that structures with scaling factor around 1.4 are already very loose.

Since we need some extra length about 20 to 30 cm to tight up the final beaded molecule, so the total length of thread to use for building a fullerene with N carbon atoms can be chosen as

L = 2*(3/2)*N*d*s + K = s*3*N*d + K
where s = 1.1 to 1.2, K = 20 to 30 cm and d (= 4mm) is the diameter of the bead.

However, I think it would be interesting that we can perform more measurements to get a better estimate for s.

Spiral code for creating fullerenes

As I have mentioned that the details of the beading procedure for creating a fullerene molecule describable by a spiral should be completely determined by the sequence of pentagons and hexagons. Once this sequence is given, then we can just carry out the weaving process by making the 5 or 6-bead group using the RAW according the recipies. Sound simple, right. But we still need to know the sequence in order to make the fullerene we intend to create. Fortunately, the complete list of all possible isomers for fullerenes in the range C₂₀ to C₅₀ and isolated-pentagon isomers of fullerenes in the range C₆₀ to C₁₀₀ are tabulated in Fowler's "An Atlas of Fullerenes". Instead of giving the whole sequence of 5- and 6-gons. Fowler also gives another simplified notation for the sequence of 5- and 6-gons in the spiral. Since there are exactly twelve pentagons in a fullerene, and the others are hexagons, therefore we only need to know the positions for the pentagons in the spiral.
Here I will illustrate his notation with two simple examples. The first one is an isomer of C₈₀, Fowler's 80:7 isomer (the 7th isomer out 7 isolated-pentagon isomers of C₈₀), with the spiral code, 1 8 10 12 14 16 28 30 32 34 36 42. According this spiral code, I have to make a pentagon first, then 6 hexagons, and then a pentagon, hexagon, pentagon, and so on, in a clockwise spiral. Finally a C₈₀ is created. In the process of making this fullerene, I don't need to worry about the connectivity or geometry, the resulting beaded C₈₀ is in good agreement with the actual geometry of C₈₀ due to the repulsion between different beads.

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Another one I made is isomer 50:24 with spiral code 1 2 3 4 7 12 17 22 24 25 26 27, which is an isomer of C₅₀. The shape of this isomer looks like a bean or cocoon.

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Minimum length of thread needed

The typical procedure to weave a beaded spheroid is to start from a 5- or 6-bead group, then use the right-angle weave technique to extend outward by adding more 5- or 6-bead groups aroud the central (first)-bead group in a spiral fashion. Whenever a 5-bead group is introduced, negative surface curvature is created, and thus the surface would be curved inward. According the Euler theory, to close a spheroid, 12 pentagons are needed. Thus along the path of spiral, we can put any number of hexagon we want, but a total number of 12 pentagons are needed to get a closed spheroid. Of course, if the sequence is arbitray, many of the structures are going to have very high energy. When a spheroidal fullerene can be contructed in this way, the thread will go through each each exactly twice, the amount of thread needed will simply s*2*N*d, where s is the scaling parameter to take the part of thread outside the beads into account.

According to the work by the British chemist, Fowler, there exist a class of fullerene system which can not be described by this way. An example is tetrahedral C380 as shown in his book entitled "An Atlas of Fullerenes". For this kind of system, we probably have to use longer lenth of thread because some beads will be stitched more than two times.

Minimum length of thread needed to weave a fullerene

What is the minimum length of thread you need to build a stable beaded fullerene. By stable I mean the beaded fullerene should satisfy VSEPR principle everywhere. Probably not many people working on beadings worry about this kind of problem. I am not sure how hard the problem is in general. But the lower bound for the minimum length is quite easy. Since every bead has to be stitched at least twice to get a stable structure, therefore the minimum length must be greater than 2*N*d, where N is the number of beads and d the the diameter of bead. Better estimation can be found if the ratio of the size of hole and diameter of bead is taken into account. Furthermore, if the part of thread outside the beads is included, slight improvement can be obtained. This is just simple exercise in geometry, which can be worked in principle. We can also use empirical approach by performing a real experiment to find out the actual length used, then divide this number by the theoretical minimum 2*N*d, the scaling factor.

More complicated issue is that the lower bound with the scaled factor included is in general not the minimum length. It is possible to finish the beaded molecule, some beads may always be stitched through more than twice! This problem is not easy to explain. I will talk about this later.

Force Field of Beaded Molecules

After a few days of thinking, I now realize that almost all of physical models commonly used for constructing the structure of molecules do not really use the microscopic force field of true molecules. The microscopic force field of a molecule is nothing but the repulsion between the electron pairs.The structure of physical model built from traditional balls and sticks does not obtain its structural stability from the repulsion between sticks. There is no interaction between these sticks in fact. As far as I can think of, our beaded molecule is probably the only physical model that really use the similar kind of pairwise hard-sphere repulsion to get the three-dimensional structure of the molecule. This is also the reason why the geometry of the beaded fullerene is in such a good agreement with that of the true molecule.

Right now, this approach only works for sp²-hybridized atoms. I guess it is possible to use beads to represent the repulsion interaction in the sp³-hybridized atoms, but it is probably not possible to generate the sp-type repulsion in the beaded network.

On the Shape of Beaded Molecules

According to our experience, the major 3D structural features of the beaded molecules are very close to the optimized geometry obtained from more sophisticated force-field simulation. We find it is possible to give an explanation for this surprising result based on the VSEPR (Valence Shell Electron Pair Repulsion) theory as described in any General Chemistry textbook.

As I have commented before that beaded molecules are the bond-representation of the corresponding fullerene molecules. The beads are in fact C-C double bonds, instead of carbon atoms. This can be a little bit confusion to people who are familiar with the typical ball-and-stick model. When I shew these beaded molecules to my colleagues, many said that the structures are not correct. This is because at the first sight most people will view these beads as carbon atoms. However, they are not. Even though the bond-representation is a little inconvenient from traditional point of view, we believe that it is worthy to take the beaded-representation seriously since switching viewpoint is quite easy and, more importantly, the resulting giant beaded fullerenes are so aesthetically beautiful and faithful in its 3-D shape. We will show that the reason why the beaded molecules can have such a good structure is exactly due to this very bond-representation.

VSEPR theory accounts for the molecular shape for AX_n by assuming that the repulsion between the electron pairs surrounding the central atom A causes these pairs to be oriented as far as possible. Hence in order to minimize the interaction between the two chemical bonds in the AX₂, resulting shape becomes linear. When there are three chemical bonds around the central atom, the minimum structure will adopt a planar trigonal shape.
In the beaded molecules we constructed, there are only three kind of n-bead groups, i.e 5-, 6-, and 7-bead groups. Since two neighbored groups can only share one common bead, there are always three n-bead groups meet with each other by sharing one common bead with each other. Therefore these three common beads will form a close-pack triangle, representing exactly the three CC bonds around the central carbon atom. When the thread through these three beads is stretched outward strongly, the beads will be forced to stay as flat as possible just like situation described in the VSEPR.

Now we can understand why our beaded molecules can simulate the real toroidal compounds. It is exactly because the beads represent the chemical bonds instead of atoms, so that the repulsion among the beads in our beaded molecules can mimic the physical force that gives rise the structural features in the fullerenes. This is the reason why the 3D structures of our beaded molecules are so similar to those of the real molecules.

Right angle weave

I am not sure if there is a need to discuss the basic weaving technique I used in constructing these 3-D structures. After reading the relevent pages on beadings by googling through the internet, I realized that I have been unknowingly using the so-called "Right Angle Weave" repeatedly in building all these beaded molecules.

This weaving technique is quite straightforward. Since there are already many discussions this technique. I am not going to talk too much about this. Here is good site to look at this technique:

http://www.beadwrangler.com/samplers/raw1/making_raw_samples1.htm

Two conformations of the upper branch

In the process of weaving either branches of the toroid, it is not hard to notice that the beads on the edge form a large pentagon. It is understandable since there are exactly ten heptagons arranged in a petagonic pattern in the inner-rim. The extension of of the inner-rim outward by weaving more beads into the structure make the pentagon pattern even more vivid.

If we only introduce hexagons, the resulting structure is going to be like a funnel on both side of the inner-rim. But if five pentagons are introduced like the one the following picture, the place where the petagon is located will have a positive intrinsic curvature.

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The structure now have a bistable potential. By which I mean the five hexagons surrounding each pentagon can have two relatively stable spatial positions. We can easily flip the hexagons as shown in the previous picture into the one like the following picture. From this conformation, we can create the final toroidal structure.

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Detailed procedure of contruction of achiral T240 with pentagons located at edge positions

1. Creation of inner rim:

I create the inner rim in three steps this time:

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Upper branch

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Side view of upper branch

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Lower branch

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Finished work

Finish up by connecting both branches at suitable position.

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Ten isomers of T240

Here are ten isomers of T240 I currently made. I guess there are at least two more high-symmetry T240 in this family.
You can see the two empty positions in the picture are for these two isomers.

It is worthy to note that there are three chiral structures in these 10 tori. If included, there are 15 isomers in total for T240.

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Strongly distorted tori

Here is an example I get a strongly distorted tori which is derived from a parent torus with D_5d symmetry, even though the correspondent connectivity graph contains correct number of pentagons and heptagons. The Euler formula is also satisfied.

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I also have another example derived from D_5h symmetry. Here I simply moved pentagons' position one unit, I am unable to connect upper and lower branches completely.

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These examples tell us that even though the number of distinct tori grows exponentially, most of them are highly strained and strongly distorted away from toroidal shape.

Molecular tetrapod!

Based on the Lenosky's work on the carbon nanostructure derived from periodic minimal surface (as told by Chuang Chern), I found that it is possible to make carbon nanotetrapod. Although I soon discovered (through googling) that this kind of tetrapod has been studied by a Korean group a few years ago (PRL 2003), I still found this kind of structures intriguing.

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A bag of beaded donuts

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Some more photos for the making of D5h T240 a

The part for the making of inner-rim is the same as that of D_5h T240 b.

The making of the out-rim

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Finished work:

This is the finished beaded structure for the D_5h T240 a. This is such a beautiful molecule, different from all of the D_5d toroids I made before. It doesn't take me a long time to realize the five pair of pentagons located in the outer-rim can be subjected to the Stone-Wales transformation to get the D_5h T240 b shown before.

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The effect of slightly different procedure.

Here is an example I tried to finish step 2a (extension of inner-rim outward) as mentioned in the previous post first without connecting the inner-rim into a cylinder. Since there are two many beads in the outer-rim, the resulting structure shows interesting ripples. I found that it is quite difficult to weave using this approach. I made many mistakes, so I have to do many tries before I can get the right structure.

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The details of making D5h T240 b

After posting so many beaded toroids, I will describe briefly the procedure to make a D_5h T240 b.

Of course, there are many different ways to reach a particular toroid. Among many different ways of weaving, I find it most convenient to separate the whole process into two phases. The first phase is to construct the inner rim with suitable arrangement of heptagons. Then after the inner rim is finished, we can start to weave the outer rim, which is the second phase of whole procedure. The details of weaving can vary each time, though.

Phase one: the weaving of the inner rim

a. Note that in order to make a D_5h toroids, I have to put heptagons in the eclipse position.

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b. Repeat the previous steps to get the whole ribbon of inner rim. There are ten heptagons in total.

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c. Connect two sides of ribbon to finish the inner-rim structure.

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Phase two: the weaving of the outer-rim

a. Extend the inner rim by one more strip of hexagons for both the upper and lower branches. Note that the pattern of large pentagons formed on both branches.

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b. Connect the upper and lower branches gradually.

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c. Finish up the whole toroid.

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Tiny T160 Toroid

T160 is one of the first few tori I made with 4mm beads. Its inner core has the same arrangement as those of the T200 and T260 I mentioned before. The pentagons on outer rim are located at the mid-edge positions. It may have stable isomers with pentagons at vertex postion.

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Chiral T240 belonging to C5 point group

I have managed to make the chiral isomer of T240 (D_5h) by shifting the pentagons on the outer rim by one bead. Unfortunately, I was unable to put the beads into a complete torus due to the huge strain energy for this structure. The stable chiral forms of T240 originating from the isomer with D_5h point group do not exist! It is quite amazing it is possible to get a first order approximation for the geometry and stability of carbon spheroids or toroids by using molecular beadings. But the price to pay is that you need to spend several hours on the beadings for a single structure.

Previously, I have made two D5h T240 which can be converted into each other by Stone-Wales transformation. But, according to Ihara's paper, there seems to be another kind of T240 isomer which can be generated by moving ten pentagons on the outer rim to the mid-edge position (i.e. shifting 36 degree about z-axis). I am not sure about the stability of this structure. But since the outer-rim pentagons and inter-rim heptagons are arranged in the staggered pattern, I suspect the strain energy for this type of structure is large.

Carbon Toroids with D5h Symmetry

Up to now, all of the toroids I made and posted here belong to either D_5d or C₅ point groups. Molecules with D_5d symmetry have mirror symmetry. So they are achiral compounds. But those molecules with C_5d symmetry are chiral.

In addition to these types of compounds, there exists the third kind of toroids with D_5h symmetry. Since they have inversion symmetry, molecules belongs to this catogory are optically inactive. I have made two beaded toroids of this kind this weekend. Both contain 240 carbon atoms.

Interestingly, these two are connected to each other by the Stone-Wales transformation.

D_5h T240 a

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D_5h T240 b

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Geometry generated by Chuang.

Another kind of T240

There are many different types of T240. Previously, the three isomers I have shown all have the same inner rim configuration, but with different arrangement in the pentagon distribution on the outer rim. Here I give another structure of T240 with different inner rim configuration. I.e. the heptagons have different spatial locations.

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Side view:

From The Beaded Mo...