|Figure 1. A diamond surface with a
six-membered ring attached.
The upper three carbon atoms (dark gray) are shown with missing
bonds where the bracket would be extended. The lower carbon atoms
are shown likewise, where the diamond crystal would be extended.
The free surfaces are terminated with hydrogen atoms (white) save
for three embedded nitrogens (light gray) included to avoid the
need for crowded hydrogens. The illustrated structure was minimized
using the MM2 potential energy function (Chem3D Plus implementation).
Three methods for designing the bridge come to mind. The first
is to build the bridge atom by atom and "search" for
the proper configuration. This is much like a computer program
for playing chess. Placing an additional atom on the end of the
structure is like making a move in chess. One wants it to be a
step toward the solution, but one can't tell if it is the right
step, except by trying it. Only after more atoms are added (more
chess moves are made), can we tell whether the bridge matches up
at the far end (whether these moves lead to a better chess
position). If not, one must take back those moves and try others.
In the absence of a good predictive theory, this kind of search
takes a tremendous amount of computation, just as chess playing
programs do. Without this, designing a new bridge from scratch
every time one has a specific need does not look like such a good
The second method is to design a "universal" structure that has length and angle adjustments. This would be a flexible structure with many two-position adjustment points. These might be chains that could be shortened by one atom, or atoms with different bond lengths that could be substituted. By changing which adjustment points were set to "long" and which were "short," the length of the whole structure could be varied by small amounts. Such a structure might have some disadvantages. It would have to be large in order to get sufficient variability. It is unlikely that it could be made very stiff without being so large as to dwarf the device it was meant to hold. We have not been able to think of any good structures that avoid these problems, and this area is still open for innovation.
The third method is to design hundreds of short, strong molecular brackets and then classify them by offset and angle. After each arrangement of atoms is designed, a program computes its detailed shape, and the results are stored in a dictionary. The designer uses the dictionary to choose the proper bracket to support the device in its proper place. To choose the right bridge from the catalog, we first imagine the diamond crystal extended up past the mounting ring we are trying to secure. For the "number one" atom on the mounting ring, we find its location within a unit-cell of diamond crystal. We also note the angle in 3-space of a vector that expresses the orientation of the ring. We then look up the position and angle in the dictionary to find the closest match. We find an entry for a known bracket and the (x,y,z) offsets to each of its three attachment points in the diamond lattice. The dictionary tells the designer what bracket to use and where in the diamond lattice it will attach. The bracket is free standing, attaching to the diamond crystal with just three bonds (Fig. 1). In the final design, the diamond only comes as close to the device as the offset says to, and the bracket spans the remaining distance.
To design a family of brackets, we begin with a stack of six-membered carbon rings. Such stacks are found within the structure of hexagonal diamond (lonsdaleite) and are a strong, compact structure (Fig. 2). Each ring has three covalent bonds to the ring below and three to the ring above. This gives good stiffness. A barrel-like stack of six-membered rings is straight, so we must introduce some variation to make it bend. One way is to use seven-membered rings. Each seven-membered ring has three attachments above and three below. The seventh atom distorts the ring in some direction. A second seven-membered ring on top of the first has six different places where the seventh atom can interrupt the ring. The many combinations of seventh atoms on different levels give a range of combined twists, bends, and offsets from the normal lattice. All extra carbon bonds that hang out of the structure are capped with hydrogen.
|Figure 2. A stack of four six-membered carbon rings. 'D' indicates the three bonds to the diamond substrate. The top ring attaches to the device being supported. Hydrogen atoms attached to the two middle rings are not shown. (The structure appears to be curving slightly to the left. It should be completely straight, and we are looking for the bug in our software.)|
Figure 3 shows a typical two-layered bracket with a hexagonal mounting ring on each end. Even a structure of just two layers can have quite a bit of twist and offset. The structure is compact and stiff, with three or more covalent bonds at each cross-section. Here are the major ways that a normal stack of six-membered carbon rings can be varied to make brackets for cataloging:
The computer program to build the catalog proceeds as follows:
Enumerate all the possible brackets using the above rules,
starting with the shortest first. For each bracket, compute its
shape using a molecular mechanics program. The most important
aspect of its shape are the three bonds coming out of the
mounting ring on each end. With one end attached to a diamond
lattice, we compute the offset and angle of the ring on the other
end, and enter it into the catalog. Since computing the shape of
the bracket is the hard part, we save time by making catalog
entries for the mirror image of the bracket, the bracket upside
down, and the bracket attached to a vertical face of the diamond
|Figure 3. The same structure with a seventh atom inserted in two of the rings. The top ring is rotated, displaced sideways, and tilted. Thousands of such variations will be be classified in a catalog according the location of their top ring. The designer selects the bracket that matches the location of the part he wishes to support.|
Not every structure we compute will become an entry in the
catalog. When many brackets reach the same place and angle, we
only want the shortest and stiffest one. The catalog will be made
to a certain spatial and angular resolution. If we try to find
one entry for every 0.154 Å (a tenth of a carbon-carbon bond
length), then number of position points in a unit will be around
1029. For each of these, we need a variety of angles. Since bonds
can bend much more easily than they can change length, an angular
accuracy of plus or minus 10 degrees may suffice. Accounting for
all the spherical symmetries, we need 66 different angle entries
per approximate position, derived from as few as 4250 bracket
designs. (A single bracket may be entered into the table in as
many as 16 different ways.) The shape of many more than 4250
brackets will have to be computed to get a sufficient variety of
angles and locations. It will be interesting to see how clumpy
the distributions of brackets is, and to see if there are any
regularities that will allow us to predict the shape of an
It is possible that reaching the full diversity of the catalog will require putting too many layers in the bracket. Such a bracket would be too long and floppy to be of much use. If this is true, all brackets with more than a certain number layers will be designed with thick bases. Imagine the thick base as a short bracket made from three parallel hexagonal tubes. It is short and stiff. On top of this is a normal one-tube bracket. The richer structure of the thicker bracket allow it to have many more variations per layer, making a diverse set of shapes easier to generate.
If the designer is not happy with the spatial and angular resolution he finds in the catalog, he can pull a few tricks. The device he is building is likely to be anchored at several places. If one of those anchors is at a slightly wrong place, he can pick the other anchors to push the structure back in the right direction. Likewise, slightly wrong angles can be pitted against each other to give a correct final position. Such a mildly strained structure should work just fine.
To begin the project, we selected an existing molecular mechanics program. Programs that compute the shapes of molecules come in a variety of speeds. The structures we are simulating contain nothing but the atoms and bonds of locally-unremarkable organic molecules. We are not studying unstable transition states in chemical reactions, so we don't need "molecular orbital" programs that model the quantum mechanics of electron clouds. Instead we used a "molecular mechanics" program that treats each chemical bond as a spring with a certain resting length. Additional springs handle the desire of an atom to keep its bonds at certain angles to each other. By using only forces between the centers of atoms, this program can go very fast. The program we selected is STRFIT3 by Martin Saunders and Ronald Jarret of Yale University, which gives results closely approximating those of the classic MM2 program . Around this we are building programs to generate the brackets and enter them in the catalog after their shape is known.
This system is implemented in Digitalk Smalltalk/V Mac on an accelerator-assisted Macintosh II. After we have verified that STRFIT3 is producing shapes that agree with known molecules, we intend to run the system every night and build a catalog of nanomechanical brackets.
The interesting thing about this project is considering design problems in a world in which angles can be varied but lengths cannot, with lengths and flexible angles like those found in real molecules. The catalog we build now will probably not be the one used when nanostructures are actually built. By the time fabrication technology is available, designers will want to use the latest modeling programs and the fastest computers to rebuild the catalog with high accuracy. By creating the tools to build a catalog today, we can get a glimpse of the techniques and pitfalls of designing mechanical structures in which 'every atom is in its place.'
Martin Saunders and Ronald Jarret, "A New Method for
Molecular Mechanics," Journal of Computational
Chemistry, Vol. 7, No. 4, 578-588 (1986).
Ted Kaehler is a computer scientist who spent his sabbatical from Apple Computer working with the Foresight Institute. He and Foresight would like to thank Martin Saunders of Yale for allowing us to use the program STRFIT3 and for his additional help. Ted's participation was funded by the Restart Program of Apple Computer, Inc.
[Editor's note: For current information, visit Ted Kaehler's home page at http://www.webPage.com/~kaehler2/.]
From Foresight Update 10, originally published 30 October 1990.
Foresight thanks Dave Kilbridge for converting Update 10 to html for this web page.
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