**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

**4.3.4.2 Nanopendulum
Orientation Sensing**

A nanopendulum may also be employed for orientational sensing.
As one example of a large class of related devices, consider a simple rigid
spherical nanopendulum of radius r and mass m, with a large hemispherical bob
free to swing through the entire solid angle about a central pivot point (Fig.
4.5), with a gimballed housing to allow active avoidance of the pivot support
beam. The nanopendulum bob moves along a spherical surface located far enough
from a concentric spherical sensor grid of radius R = r + z_{sep} to
minimize van der Waals interactions (Sections 3.5.1
and 9.2), with bob and sensor surfaces electrically neutral
or uniformly mutually repulsive. Much like the pin cushion receptor model (Section
3.5.7.4), the sensor grid consists of a spherical array of diamondoid sensor
rods Dx ~ 1 nm wide that are rapidly and simultaneously
extended into the evacuated sensor cavity to determine bob position during a
measurement cycle of duration t_{meas}. Radial sensor rod velocity should
exceed typical tangential bob velocity by at least 1-2 orders of magnitude.
For the example given below, a sensor stud on a 500-nm bob that is turning at
v_{turn} ~ v_{thermal} = 3.6 mm/sec passes a ~1 nm sensor rod
in ~280 nanosec; a sensor rod displacing ~1 nm in the radial direction in ~1
nanosec travels at v_{rod} ~ 1 m/sec. This nanopendulum may serve both
as an orientation sensor and as a rotation rate sensor, using principles similar
to vestibular (semicircular canal) mechanics.^{449}

If a simple nanopendulum spherical sensor housing with bob
initially at rest in a uniform vertical gravity field g is rotated through the
smallest detectable angle q_{min} (radians)
around the center pivot, then ignoring friction, the bob's position over the
sensor grid moves a distance Dx, hence

For Dx = 1 nm and r = 500 nm, Dq_{min}
~ 2 milliradians (~0.1 deg). For a practical sliding interface (e.g., between
bob and sensor surface) of contact area S and turning velocity v_{turn},
Drexler^{10} estimates frictional dissipation,
primarily driven by shear-reflection drag and band-stiffness scattering, as
P_{drag} ~ k_{1} v_{turn}^{2} S, where k_{1}
~ 400 kg/m^{2}-sec, hence the energy dissipated between the bob and
the nearest sensor rod (the one interacting with the bob) is E_{drag}
~ k_{1} v_{turn} S Dx. Bob gravitational
energy E_{bob} = m g r sin Dq_{min}
~ m g r Dq_{min} for small displacements.
The bob swings freely at the minimum displacement if E_{bob}/E_{drag}
>1. If m = 10^{-15 }kg, r = 500 nm,
and S = 1 nm^{2}, then v_{turn} < 2.5 m/sec or w
< 49 megaradians/sec for the bob to swing freely (vs. v_{thermal}
= 3.6 mm/sec (Eqn. 3.3) and w_{res}
~ 9900 rad/sec for this pendulum).

The bob-up/bob-down energy differential is ~2 m g r = 9.8
zJ ~2.3 kT for m = 10^{-15} kg and r = 500 nm. To distinguish opposite
orientations (Dq ~ p)
requires N_{meas} ~ (kT / 2 m g r)^{2} ~ 1 measurement for this
sensor. To distinguish Dq_{min} ~ 2 milliradians
requires averaging N_{meas} ~ (kT / E_{bob})^{2} ~ 190,000
independent measurements (of ~0.002 kT/measurement) or t_{meas} ~ 190
microsec using a sensor rod clock cycle of ~1 nanosec. Thus, after subtracting
out the effects of translation displacements as determined by linear accelerometers
(Section 4.3.3), a nanodevice can measure angular displacements
in its orientation to within ± Dq_{min} in
a measurement time t_{meas} ~10^{-4} sec. With sensor rods moving
at ~1 m/sec, energy dissipation for each rod sliding in its housing is ~0.01
zJ/cycle (Section 10.2.1). If only ~100 test rods need
be triggered to continuously track the moving stud, then a sensor rod clock
cycle of ~1 nanosec implies a continuous nanopendulum power dissipation of ~1
pW; additional drag power loss between stud and contacting rod is only P_{drag}
~0.0004 pW.

Many other configurations of this basic device are readily conceived and may be more volume-efficient, including nested spheres or ovoids, two-dimensional rocking disk sections for in-plane rotational measurements (three devices required for monitoring all angular degrees of freedom in a rigid rotator), and multiaxial circumferential tracks or tubes containing rolling balls or sliding plugs. J. Logajan suggests that a pair of linear three-dimensional translation accelerometers should be able to unambiguously extract the 3 translation and 3 rotation components of all possible acceleration vectors. Rotation about any axis will result in distinctly different mass displacements while accelerated translation along any axis will result in identical displacements.

Last updated on 17 February 2003