Nanomedicine, Volume I: Basic Capabilities

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 squid1231 and based only loosely on the biological cilium. The model consists of a cylindrical central shaft of radius rshaft and length lshaft, the tip of which is bonded to one or more conduited shaft-attached control cables (radius rcable) whose lengths and tensions can be predictably altered, causing the shaft to flex. A cable pulled with a force Fcable applies a transverse deflection force to the nanocilium shaft of approximate magnitude Fdeflect ~ (rcable/lshaft) Fcable for small deflections. The force required to deflect the terminus of a simple cantilevered rod is Fdeflect = 3 Eshaft Ishaft d / lshaft3, where Eshaft is Young's modulus and Ishaft = p rshaft4 / 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:

{Eqn. 9.43}

and a deflection angle qshaft = tan-1 (d / lshaft). The shaft begins to buckle when Fcable exceeds the Euler force:364

{Eqn. 9.44}

The maximum force that can be transmitted through a diamondoid cable of circular cross-section and tensile strength536 Tstr = 1.9 x 1011 N/m2 with rcable = 1 nm is Fmax = p rcable2 Tstr = 600 nN. (For hollow cylinders of inside radius r and outside radius R, rshaft4 should be replaced by (R4 - r4) in Eqn. 9.44.)

For a large nanocilium, we take rshaft = 50 nm, lshaft = 1000 nm, and Eshaft ~ 108 N/m2 (~tendon strength521). At maximum deflection where Fcable = Fmax ~ 600 nN, then d ~ 400 nm, qshaft ~ 22°, and bending stiffness kshaft = Fdeflect / d = (3 p / 4) (Eshaft rshaft4 / lshaft3) ~ 0.001 nN/nm. Maximum lateral force that may be applied by the nanocilium tip is Fdeflect ~ 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 rshaft, the maximum mechanical pressure is pshaft = Fmax / p rshaft2 = 8 x 107 N/m2 < Eshaft. Shaft buckling commences at Fbuckle ~ 5 nN with d = 3 nm and qshaft = 0.2°; classical positional variance of the tip due to thermal noise10 is Dx = (kT / kshaft)1/2 ~ 2 nm, very close to the minimum deflection at buckling. If the control cable may be moved at vcable ~ 1 m/sec, then maximum nanocilium operating frequency nmax ~ vcilium / rshaft = 20 MHz; however, power dissipation is at least Pcilium ~ Fmax rshaft ncilium = 30 pW at an operating frequency ncilium = 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 rshaft = 10 nm, lshaft = 100 nm, and Eshaft = 1010 N/m2 where Fcable = Fmax ~ 600 nN, then d = 25 nm, qcable = 14°, kshaft = 0.2 nN/nm, Fdeflect = 6 nN, pshaft = 2 x 109 N/m2 < Eshaft, Fbuckle = 78 nN, Dx = 0.1 nm, maximum nmax ~ 100 MHz and Pcilium = 6 pW at ncilium = 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