**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

**9.4.2.4 Force and Power
Requirements**

Consider a spherical nanorobot of radius R_{nano}
that is falling at uniform velocity v_{nano} through an incompressible
(Newtonian) fluid of viscosity h. If the Reynolds
number N_{R} << 1 (Eqn. 9.65),
then Stokes^{1373} found that the
total force of resistance imparted on the sphere by the fluid is given by:

also known as Stokes law. This result applies only in the
case of a single sphere in an infinite expanse of homogeneous viscous fluid,
and neglects all inertial terms. If the fluid container is finite in size, or
if there are other spheres in the neighborhood, or if inertial forces cannot
be entirely ignored because N_{R} >~ 1 (e.g., R_{nano} =
1 micron, v_{nano} > 1 m/sec in water), then the equation must be
corrected.^{1374} Dynamic effects
due to sphere oscillation, sudden release from rest, or variable speed also
require corrections,^{1374} and
there are other corrections for neighboring spheres.^{1375}
Most of these corrections are of order near-unity for micron-scale objects.

Eqn. 9.73 provides a useful
approximation of the motive force required to drive a spherical nanorobot through
blood plasma. Assuming a perfectly efficient drive mechanism, the power requirement^{337}
is at least:

For example, a force of F_{nano} = 200 pN and a power
of at least P_{nano} = 2 pW are required to drive a R_{nano}
= 1 micron spherical nanorobot at v_{nano} = 1 cm/sec through blood
plasma at 310 K with h = 1.1 x 10^{-3} kg/msec.
These formulas may also be used to estimate nanodevice velocity, given F_{nano}
or P_{nano}.

Of course, drive mechanisms are not perfectly efficient. Propulsion
efficiency is often poor for objects that swim at low Reynolds numbers (e.g.,
typically N_{R} ~ 10^{-3} for flagellates).^{3578}
A sphere driven by a helical propeller (e.g., flagellar propulsion; Section
9.4.2.5.2) of arbitrary length may have a propulsion efficiency as low as
e% ~ 0.01(1%).^{389} Propulsive efficiency
for organisms with spherical heads 10-40 times larger than their flagellar radius
(optimum shape), using helical flagellar beats, ranges from e% = 0.10-0.28 (10%-28%),^{1377}
requiring a propulsive input power of ~P_{nano}/e% ~ 7-20 pW in the
previous example. Other drive mechanisms may prove more efficient.

Force requirements for nonspherical nanorobots are similar
in magnitude. For example, consider a cylinder of radius R_{nano} and
length L_{nano} translating uniformly in a direction normal to its axis.
Lighthill^{1367,1376
}shows that the force (applied at the midpoint) required to move this object
at velocity v_{nano} is given by:

for L_{nano} >> R_{nano}.^{363}
(If the perpendicular force is applied at a distance c from one end of the cylinder,
then the L_{nano}^{2} term in Eqn.
9.75 is replaced by 4 c (L_{nano }- c) when c >> R_{nano}.)
Eqn. 9.75 does not hold for a perpendicular
force applied near the ends of the cylinder. Also, a constant F_{nano}
actually produces a slowly varying velocity field along the cylinder length;
for a midpoint-applied force, v_{nano} is at maximum at the midpoint,
but the percentage variation over most of the cylinder is not large.^{363}
Experiments show that needle-shaped bodies fall in viscous media about half
as fast sideways as they do end-on;^{1378}
that is, the force on a long cylinder translating parallel to its axis is F_{nanoP}
~ (1/2) F_{nanoN}. The power requirement is then calculated as in Eqn.
9.74.

Interestingly, there seems to be a sharp minimum size limit
of ~0.6 microns for free-swimming foraging microbes, below which size locomotion
has no apparent benefit.^{3581 }This
theoretical conclusion is supported by the observation that the smallest 97
genera of motile bacteria have a mean length of 0.8 microns, whereas 18 of 94
nonmotile genera are smaller.^{3581}

Last updated on 21 February 2003