**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

**4.3.4.3 Nanopendulum
Tachometry**

The smallest angular velocity detectable using a simple nanopendulum
is given approximately by w_{min} ~ Dx
/ r t_{meas} = 2 milliradian/sec (~0.1 deg/sec) for Dx
= 1 nm, r = 500 nm, and t_{meas} = 1 sec. The largest detectable angular
velocity w_{max} ~ 2 p
/ t_{meas} if t_{meas} is the minimum time required to count
one complete rotation of the sensor; the smallest t_{meas} ~ 2 p
/ w_{max} ~ 100 nanosec for w_{max}
~ 49 megaradians/secor ~5% of the bursting speed (Eqn.
4.17) for an r = 1 micron spherical diamondoid nanodevice.

What rotational displacements will normally be encountered
by an in vivo medical nanorobot? From Eqn. 3.2,
a 1-micron device experiences instantaneous Brownian rotations of w_{B}
= (kT / 4 p h R^{3} t)^{1/2}
radians/sec, where t = (MW_{kg} / 32 p
kT N_{A }c_{H2O} R^{4})^{1/2}, the mean time
between molecular collisions with the device (mostly by water molecules), from
Eqn. 4.3, where c_{H2O} ~ 3.3 x 10^{28}
molecules/m^{3}. For R = 0.5 microns and T = 310 K, t
~ 6 x 10^{-20} sec/collision and w_{B}
~ 7 x 10^{9} rad/sec, with each displacement of ~ t
w_{B} = 4 x 10^{-10} radian, none of which are detectable.
However, the net of all Brownian rotational displacements is one rotation every
16 sec, ~0.4 radian/sec (Section 3.2.1).

The largest measurable angular displacements in vivo are caused
by tumbling due to differential shear forces in the bloodstream. Normal vessel
wall shear rates in physiological bloodflow range from 100-1400 radians/sec
in the larger arteries to 500-4000 radians/sec in the smaller arteries and capillaries.^{386}
The measured rolling velocity of white cells during vascular "tank tread" locomotion
(Section 9.4.3.6) on venule walls is ~40 micron/sec,
or ~8 radians/sec, which may be encountered by nanodevices employing cytocarriage
(Section 9.4.7). The human vestibular mechanism has
a frequency response range of 0.048-260 radians/sec.^{449}

Thus, the maximum range of physiologically relevant rotation
rates is 10^{-2} to 10^{4 }radians/sec. Since the natural harmonic
frequency of a simple nanopendulum is w_{res}
= (g/r)^{1/2} = 4400 radians/sec for r = 500 nm, the possibility of
forced oscillations and resonance need be faced only when measuring the highest
likely physiological rotation rates, in which case the addition of a damping
force F_{damp} ~ m v_{turn} w_{res}
during resonance events should eliminate the problem. F_{damp} for critical
damping peaks at ~mg ~ 0.01 pN for the m = 10^{-15} kg, r = 500 nm case
(v_{turn} ~ 2 mm/sec).

Last updated on 17 February 2003