**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

**9.4.2.5.2 Inclined Plane**

Another class of mechanisms for natation makes use of the
inclined plane, a basic mechanical device that can convert viscous forces into
forward motion. The simplest example is a threaded screw. In standard propeller
theory at high Reynolds number, forward thrust is proportional to the rate at
which a mass of fluid can be ejected out the rear (e.g., inertial forces). However,
at low Reynolds number, the fluid that is pushed backwards by the rotating tilted
planes does not provide thrust primarily by its inertial movement, but rather
serves as a resistive medium against which the device can push itself forward.
In the world of the nanorobot, the environment is very thick and viscous. The
motive effect is not unlike the forward motion achieved by a threaded screw
as it is screwed into a piece of wood using a screwdriver.^{3580}

The motive force and power consumption of a microscale screw
drive (Fig.
9.24) with pitch angle j and mean radius R_{screw}
may be very crudely approximated as follows. Consider a helical ribbon of width
w_{thread} and total length l_{thread} that is wrapped around
an axially-translating cylindrical body, making a pitch angle j
as measured from normal to the direction of travel of the screw body. From Stokes
law (Eqn. 9.73), a square element of that
ribbon with area w_{thread}^{2} experiences a maximum drag force
of ~6 p h w_{thread} v_{thread}.
There are n_{element} = l_{thread}/w_{thread} square
elements in the entire ribbon; neglecting flow field interactions of the elements
and of the solid center for this approximation, the maximum laminar drag force
on the entire ribbon is F_{max} ~ 6 p h l_{thread}
v_{thread}. Viscous drag is lowest at j =
0° (edge on) and highest at j = 90° (face on); a
factor of (3 - cos(2j))/4 captures the experimental
behavior of needle-shaped bodies which fall in viscous media about half as fast
sideways as they do end-on (Section 9.4.2.4), with
periodicity of p. The number of threads around the
screw is N_{thread} = l_{thread} cos(j))
/ (2 p R_{screw}) and the screw rotates at
a frequency n_{screw} = v_{thread}
/ (2 p R_{screw}), hence the total force
required to turn the screw is:

with total drag power P_{screw} ~ F_{screw}
v_{thread} = 12 p^{4} h
n_{screw}^{2} N_{thread} R_{screw}^{3}
(3 - cos(2j)) / cos(j).
To further simplify the calculation, we assume a "no slip" condition such that
each complete revolution of the screw carries the nanorobot forward by a distance
~ 2 p R_{screw} tan(j),
although more slip may occur as the thread becomes looser (e.g., at high j
and low N_{thread}). To avoid turbulence in fluid passing through the
threads, from Eqns. 9.29 and 9.65,
we must require v_{thread} << 2000 h
/ r L, a condition easily met for L ~ 1 micron devices.
Under "no slip" conditions, the velocity of forward translation is approximated
by v_{nano} ~ 2 p R_{screw} n_{screw}
tan(j), giving from Stokes law a net forward towing
force of F_{nano} ~ 6 p h R_{screw}
v_{nano} and a net mechanical efficiency e% ~ 2 cos(j)
tan^{2}(j) /[p
N_{thread} (3 - cos(2j))].

Taking R_{screw} = 1 micron, w_{thread} =
0.1 micron, N_{thread} = 1 turn, j = 60°,
h = 1.1 x 10^{-3} kg/m-sec for plasma at
310 K, and n_{screw} = 920 Hz, then v_{nano}
= 1 cm/sec, F_{nano} ~ 200 pN, v_{thread} = 0.6 cm/sec, total
power requirement is P_{screw} ~ 7.6 pW, efficiency is e% ~ 0.27 (27%),
and the pressure at the screw thread surface is p_{thread} = F_{screw}
/ (l_{thread} w_{thread}) ~ 10^{3} N/m^{2} <<
~5 x 10^{5} N/m^{2} (required to induce transient cavitation
in water at this frequency; Section 6.4.1). The outside
edge of the screw thread is blunted to minimize energy transfer to impacted
biological blood elements. A second counterrotating reverse-threaded screw mounted
coaxially doubles the motive force while reducing net viscous torque on the
natator to zero. If each screw of the screw-pair is mounted on gimbals, the
nanorobot can achieve controlled translation and rotation in any direction in
three-dimensional space; changing leading screw rotation from clockwise (CW)
to counterclockwise (CCW) enables the nanodevice to reverse direction or to
undertake more complex motions.

Another well-known instance of the inclined plane in locomotion
is the corkscrew drive (Fig.
9.25), of which the bacterial flagellum is the most familiar biological
example.^{216,581,1395,1397}
The flagellum works because of the differential viscous forces felt by thin
cylinders passing through fluid at various angles of attack (e.g., because F_{nanoN}
# F_{nanoP}; Section 9.4.2.4). The typical
bacterial flagellum is a closely-packed rigid helix ~20 nm in diameter (with
a ~3 nm flagellin protein core), and its length is almost always more than 100
times its thickness,^{338} up to 10
microns long. The bacterial flagellum is turned by a ~0.0001 pW motor that rotates
up to 300 Hz at 310 K (~15 Hz under load) and can reverse its direction of rotation
in ~1 millisec (Section 6.3.4.2). The bacterium typically
uses about 0.1% of its available metabolic energy (under growth conditions)
to run the flagellum.^{581} Forward
motion may be achieved using either planar waves or (more efficient) spiral/helical
waves. The highest swimming speed attainable by a flagellate (body length up
to 50 microns with flagella >100 microns long in eukaryotes) is ~50% of the
front-to-back wave speed of its flagellum,^{1380}
although ~20% is more typical.^{1401}
Measured swimming speeds are up to 100-200 microns/sec in various sperm species.^{1449}
The flexural rigidity of the bull sperm flagellar tail is 30 x 10^{-21}
N-m^{2} in the rigor state, and 2 x 10^{-21 }N-m^{2}
in the more flexible state in the presence of ATP;^{1451}
see Eqn. 8.6. Energy efficiency may vary widely
(Section 9.4.2.4).

Last updated on 21 February 2003