**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**10.1.2.2 Mechanical
Oscillators**

The most accurate macroscale mechanical clock was the Shortt
pendulum clock, the primary timekeeper at the Greenwich and U.S. Naval observatories
until the late 1940s. The Shortt clock consisted of a pendulum swinging freely
in near-vacuum at constant temperature, driven by another identical clock slaved
to the first; frequency stability was Dn / n
~ 4 x 10^{-8}.^{1696}

In the nanoscale realm, Drexler^{10}
proposes a n_{osc} ~ 1 GHz mechanical clock
intended to drive a nanomechanical computer CPU (Section
10.2.1). In this clock, a DC electrostatic motor (Section
6.3.5) turns a crankshaft that converts rotary motion into sinusoidally
oscillating linear motions of a drive rod. A cam surface on the drive rod forces
a follower rod into up or down positions, generating regular clock pulses with
intervals determined by the position of the follower with respect to the mean
position of the ramp on the cam surface. Clock stability is limited by the stability
of the crankshaft rotation rate. The proposed DC electrostatic motor has radius
R_{motor} = 195 nm, rim speed v_{rim} = 1000 m/sec, and output
power P_{motor} = 1.1 microwatt, giving a developed torque of t_{motor}
~ 2 x 10^{-16} N-m. From Eqn. 4.23,
the minimum detectable force (e.g., to establish feedback control, such as a
Watt governor) for a 195-nm force sensor with 99% reliability is at most ~0.4
pN, giving a minimum detectable torque of t_{min}
= 8 x 10^{-20} N-m. At constant torque, variation in crankshaft angular
velocity Dw / w (= Dn
/ n) ~ t_{min}
/ t_{motor} = 4 x 10^{-4}.

Numerous alternative mechanical oscillators are readily conceived:

A. *Tuning Fork* -- A diamondoid tuning
fork with tines of length L_{fork} = 100 nm, half-thickness R_{fork}
= 10 nm, density r_{fork} = 3510 kg/m^{3}
and Young's modulus E_{fork} = 1.05 x 10^{12} N/m^{2}
has a natural vibrational frequency of:^{1697,1698}

** _{}**
{Eqn. 10.1}

In 1998, Sandia National Laboratories began offering a product line of electrostatically-driven ~1 MHz polysilicon "tuning fork" microresonators.

B. *Helical Spring* -- A helical spring
with spring constant k_{s} = 10 N/m and mass m_{spring} ~ 2
x 10^{-19} kg (~40-nm-edge diamond cube) has a natural vibrational frequency
of:

** _{}**
{Eqn. 10.2}

ignoring gravity and spring mass.

C. *Circular Membrane* -- A flexible,
thin circular membrane of radius R_{membrane} = 100 nm, uniform density
r_{membrane} = 3510 kg/m^{3}, and
thickness h_{membrane} = 10 nm, if clamped at its boundary and stretched
by a tension F_{membrane} (the force per unit length anywhere in the
membrane), has characteristic frequencies of vibration of :^{1698}

where F_{membrane} ~ 100 N/m for a 10-nm thick diamondoid
sheet stretched to near a conservative working stress of ~10^{10} N/m^{2},
and b is a constant of order unity, derived from
the roots of Bessel functions for various diametral and circular vibrational
nodes.

D. *Torsional Pendulum* -- Another
oscillator is the torsional pendulum, wherein a diamondoid rod of radius R_{rod}
= 35 nm, length L_{rod} = 1900 nm, density r_{rod}
= 3510 kg/m^{3}, and shear modulus G_{rod} = 5 x 10^{11}
N/m^{2}, is clamped at one end and the other end twists around the longitudinal
axis in vacuo at a characteristic oscillation frequency:^{1164}

** _{}**
{Eqn. 10.4}

If the end of this rod twists through an amplitude DX,
then the time averaged power loss due to shear radiation^{10}
is P_{rad} = p R_{rod}^{6}
DX^{2} G_{rod}^{3/2} / 256
L_{rod}^{6} r_{rod}^{1/2}
~ 26 pW for DX __<__ 95 nm, a rod strain of
DX / L_{rod} <~ 5% and an oscillator power
density of D_{rod} = P_{rad} / p
R_{rod}^{2} L_{rod} ~ 4 x 10^{9} watts/m^{3}.
The stored torsional energy is E_{rod} = (1/2) k_{torsion} q^{2}
= p G_{rod} DX^{2}
R_{rod}^{2} / 4 L_{rod}, so the characteristic decay
time for the oscillation in the exemplar system is:

giving plenty of access time for clock resonance driver mechanisms.
Aside from exogenous noise sources, which can be considerable in magnitude but
may be filtered out by good design, one important endogenous source of uncertainty
in n_{osc} is the elastic longitudinal displacement
(DL) for the end of a thermally excited rod that
randomly alters rod length, slightly changing the frequency. For bulk diamond
at T ~ 300 K, and ignoring entropic contributions which become important only
in longer, narrower rods than in this example, DL
~ 10^{-16} L_{rod}^{1/2} / 2 R_{rod} (Tables
5.8 and 5.16 in Drexler^{10}). Since n_{osc}
~ L_{rod}^{-1}, then Dn / n
~ DL / L_{rod} and so:

for k_{elast} ~ 2.5 x 10^{-33 }m^{3},
L_{rod} = 1900 nm, and R_{rod} = 35 nm. Another systemic source
of uncertainty is rod length variation due to the slowly temporally- and spatially-varying
ambient human body temperature. Given a coefficient of volume expansion b_{D}
= 3.5 x 10^{-6 }K^{-1 }for diamond,^{567}
a typical variation of DT ~ 3 K per t_{circ}
~ 60 sec circulation time for bloodborne nanorobots, and a thermal recalibration
time of t_{recalib} ~ 1 sec (to correct for temperature dependence),
then Dn / n ~ t_{recalib}
DT b_{D} / t_{circ}
~ 2 x 10^{-7}. J. Soreff notes that yet another limitation on oscillator
accuracy is the 1/Q resonance width, which is a function of the design details.

Clock accuracy may be defined by a time measurement error
over a single observation cycle (N_{obs} = 1) of Dt_{error}
= t_{actual} - t_{clock}, where t_{actual} = (n_{osc})^{-1}
and t_{clock} = (n_{osc} (1 + Dn
/ n))^{-1}. Time measurement errors over
multiple cycles (e.g., N_{obs} = n_{osc}
t_{obs} > 1), spanning a total observation time t_{obs} between
clock recalibrations, are randomly distributed around zero but do not sum to
zero, constituting instead a random walk with a maximum excursion of:

This assumes that the errors are uncorrelated -- e.g., for
thermal fluctuation errors, individual observation times must lie at least t_{EQ}
apart, where t_{EQ} is the time required for thermal equilibration with
the environment (Eqn. 10.24). For the exemplar
oscillator with Dn / n
~ 10^{-6} at n_{osc} = 1 GHz, accumulated
t_{error} ~ 1 nanosec during a continuous (uncalibrated) observation
time of t_{obs} ~ 1000 sec.

Last updated on 23 February 2003