**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

**9.3.1.2 Nanocilium Manipulators****
**

Equally complex nanomechanical ciliary systems could probably
be designed. But for simplicity, consider a nanociliary manipulator patterned
more closely after the muscular trunk of the elephant or tentacle of the squid^{1231}
and based only loosely on the biological cilium. The model consists of a cylindrical
central shaft of radius r_{shaft} and length l_{shaft}, the
tip of which is bonded to one or more conduited shaft-attached control cables
(radius r_{cable}) whose lengths and tensions can be predictably altered,
causing the shaft to flex. A cable pulled with a force F_{cable} applies
a transverse deflection force to the nanocilium shaft of approximate magnitude
F_{deflect} ~ (r_{cable}/l_{shaft}) F_{cable}
for small deflections. The force required to deflect the terminus of a simple
cantilevered rod is F_{deflect} = 3 E_{shaft} I_{shaft}
d / l_{shaft}^{3}, where E_{shaft} is Young's modulus
and I_{shaft} = p r_{shaft}^{4}
/ 4 is the second moment of area (also known as the moment of inertia of the
area) for a shaft of circular cross-section.^{364}
Solving for the deflection gives:

and a deflection angle q_{shaft}
= tan^{-1} (d / l_{shaft}). The shaft begins to buckle when
F_{cable} exceeds the Euler force:^{364}

The maximum force that can be transmitted through a diamondoid
cable of circular cross-section and tensile strength^{536}
T_{str} = 1.9 x 10^{11} N/m^{2} with r_{cable}
= 1 nm is F_{max} = p r_{cable}^{2}
T_{str} = 600 nN. (For hollow cylinders of inside radius r and outside
radius R, r_{shaft}^{4 }should be replaced by (R^{4 }-
r^{4}) in Eqn. 9.44.)

For a large nanocilium, we take r_{shaft} = 50 nm,
l_{shaft} = 1000 nm, and E_{shaft} ~ 10^{8} N/m^{2}
(~tendon strength^{521}). At maximum
deflection where F_{cable} = F_{max} ~ 600 nN, then d ~ 400
nm, q_{shaft} ~ 22°, and bending stiffness
k_{shaft} = F_{deflect} / d = (3 p
/ 4) (E_{shaft} r_{shaft}^{4} / l_{shaft}^{3})
~ 0.001 nN/nm. Maximum lateral force that may be applied by the nanocilium tip
is F_{deflect} ~ 0.6 nN; setting this force equal to the Stokes law
force (Eqn. 9.73) gives a maximum nanocilium
velocity through water of ~10 cm/sec. If cable tension is uniformly distributed
over a shaft endplate of radius r_{shaft}, the maximum mechanical pressure
is p_{shaft} = F_{max} / p r_{shaft}^{2}
= 8 x 10^{7} N/m^{2} < E_{shaft}. Shaft buckling
commences at F_{buckle} ~ 5 nN with d = 3 nm and q_{shaft}
= 0.2°; classical positional variance of the tip due to thermal noise^{10}
is Dx = (kT / k_{shaft})^{1/2} ~
2 nm, very close to the minimum deflection at buckling. If the control cable
may be moved at v_{cable} ~ 1 m/sec, then maximum nanocilium operating
frequency n_{max} ~ v_{cilium} /
r_{shaft} = 20 MHz; however, power dissipation is at least P_{cilium}
~ F_{max} r_{shaft} n_{cilium}
= 30 pW at an operating frequency n_{cilium}
= 1 KHz, plus up to ~60 pW of Stokes frictional dissipation (Eqn.
9.74) at the maximum ~10 cm/sec tip velocity.

For a very small nanocilium, we take r_{shaft} = 10
nm, l_{shaft} = 100 nm, and E_{shaft} = 10^{10 }N/m^{2}
where F_{cable} = F_{max} ~ 600 nN, then d = 25 nm, q_{cable}
= 14°, k_{shaft} = 0.2 nN/nm, F_{deflect} = 6 nN, p_{shaft}
= 2 x 10^{9} N/m^{2} < E_{shaft}, F_{buckle}
= 78 nN, Dx = 0.1 nm, maximum n_{max}
~ 100 MHz and P_{cilium} = 6 pW at n_{cilium}
= 1 KHz. Maximum Stokes velocity in water is ~10 m/sec, but even at ~1 cm/sec
the Stokes power dissipation is an additional ~6 pW.

Coiling the cable around the flexible shaft permits twisting motions to be imparted to the manipulator. Attachment of at least three control cables (at 120° circumferential intervals) allows full spherical angular coordinate steering of tip position. Adding additional control cable triplets on successive connection rings spaced along the length of the shaft allows independent addressing of intermediate shaft segments to permit the shaft to flex into a variety of sinuous shapes with some additional control of total radial extension. Two orthogonal pairs of tensioned cables attached on either side to ball joints separating each segment gives a more complex but highly redundant and highly controllable articulated manipulator (Fig. 9.3).

Last updated on 20 February 2003