© 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 squid¹²³¹ 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_shaft³, where E_shaft is Young's modulus and I_shaft = p r_shaft⁴ / 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 q_shaft = tan^-1 (d / l_shaft). The shaft begins to buckle when F_cable 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 T_str = 1.9 x 10¹¹ N/m² with r_cable = 1 nm is F_max = p r_cable² T_str = 600 nN. (For hollow cylinders of inside radius r and outside radius R, r_shaft⁴ should be replaced by (R⁴ - r⁴) in Eqn. 9.44.)

For a large nanocilium, we take r_shaft = 50 nm, l_shaft = 1000 nm, and E_shaft ~ 10⁸ N/m² (~tendon strength⁵²¹). 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_shaft⁴ / l_shaft³) ~ 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_shaft² = 8 x 10⁷ N/m² < 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 noise¹⁰ 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 = 10¹⁰ N/m² 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 10⁹ N/m² < 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