[I tried to send this last week, but it never appeared, so I'm sending
On the Extropians list, there was discussion of whether static friction
could or would be a problem in the bearing designs in Nanosystems. James
Donald on that list argued that by assuming that van der Waals forces
between objects could be calculated by summing over pairs of interactions,
the book was overlooking higher-order effects which would increase static
friction to the point where bearings would not rotate.
Eric Drexler disagreed with Donald's argument, and suggested that he
undertake a specific analysis of a bearing with 19 atoms in the inner ring
and 27 atoms in the outer ring, with a coaxial, coplanar geometry, as
illustrated in Nanosystems figure 10.14.
I made an attempt to analyze this kind of ring system using some of
the formulas and models from Nanosystems. I was interested in whether
compliance on the part of the inner ring would increase the barrier heights
significantly above those calculated and graphed in Nanosystems, which I
believe assume that the inner ring is perfectly rigid.
Assume you have an inner ring of 19 atoms, 27 in the outer. Assume the
distance between the rings is .2 nm, a common value in section 10.4 of
Nanosystems. I couldn't find a clear statement of the ring diameter, but
I took it to be about 2 nm. I don't think my results are too sensitive to
this value. (The 34/46 bearing by Ralph Merkle in figure 10.17 has a
"somewhat ill defined" radius of about 3.5 nm.)
Drexler's Vvdw formula 3.8 with Hydrogen interfaces (as is assumed in most
of 10.4) has an H-H energy minimum at about 0.3 nm. Since the bearing
presses the H's as closely as 0.2 nm we can see that the forces between
the inner and outer rings will be mostly repulsive.
I calculated the Vvdw at closest approach for a pair of H's (0.2 nm) from
eqn 3.8 to be 14.4 maJ or 1.4e-20 J. As an inner H moves around the ring
it will seek an energy minimum halfway between two outer H's at about
.3 nm from each This produces Vvdw of -.76 maJ. The difference between
these two is 14.4 - (-.76) or 15.2 maJ, 1.5e-20 J.
If the inner ring is rigid, then at any given time some of the atoms will
be close to outer H's and some will be in the more energetically favorable
positions between H's. These factors will tend to cancel out as the ring
rotates through different positions, as Drexler's analysis shows in figure
10.14, with barrier heights of less than 1e-32 J. (The actual value is off
the bottom of Drexler's chart, far below the level of roundoff error, and
looks like it could easily be many orders of magnitude less than this.)
However, if the inner ring is not perfectly rigid, its atoms may shift
to line up with the spaces between the atoms in the outer rings, due to the
considerable energy savings of 15 maJ to do so. If this happens, then as
the ring rotates atoms will "pop" (Drexler's term) from one inter-atomic
space to the next.
As I understand it from his posting on the Extropians list, Ralph Merkle
has shown that even when such local energy minimization is taken into
effect, the potential energy as a function of rotational position is
periodic with period GCD(m,n)/mn. In this case it is 1/19*27 or 1/513 of a
revolution, just over 1/2 a degree.
What could happen in this situation, then, is that at each rotation of
1/2 degree, one of the inner atoms will "pop" over to the next inter-
atomic gap. Then at the next rotation of 1/2 degree, another inner atom
will pop over. These pop-overs will occur in a regular pattern which
depends on the number of atoms in the inner and outer rings. In the
present example, where the number of atoms in the two rings are relatively
prime, only one pop-over will occur per revolution step of 1/2 degree.
Each time this pop-over occurs, one of the inner atoms must move from a
low-energy state, through the high-energy state, and back to a low-energy
state. This will therefore represent a barrier height of the difference
between these two states, or about 15 maJ as calculated above.
In this case, then, if the inner ring is sufficiently compliant to allow
these pop-overs to occur, static friction barrier heights of 1.5e-20 J
may exist. This is greater by over 12 orders of magnitude than Drexler's
calculation of less than 1e-32 J assuming a perfectly rigid inner ring.
Obviously it will be desirable to make the inner ring as rigid as possible
in order to keep static friction low. But I think this analysis demonstrates
that the amount of static friction is very dependent on the degree to which
the rings are compliant and able to locally minimize energy. It would be
good to see an analysis of exactly how much compliance would be expected
in actual ring designs in order to determine whether the increased barrier
height is large enough to be a concern.