© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

6.4.3.4 Gear Trains and Mechanical Tethers

Lengthy nanoscale gear trains may transmit power over great distances because mechanical efficiency for steric nanogear pairs may be ~99.997% (Section 6.3.2). Thus the initial mechanical input power only falls to 94% after passing through a sequential gear train of 2000 units, a 10 micron long train assuming each gear is ~5 nm in diameter.

Torque power may also be transmitted via a long rotating cable inside a fixed tubular sheath, somewhat resembling an automobile speedometer cable or a dentist's drill (~10-100 Hz). (Reciprocating cables are briefly treated in Section 7.2.5.4.) Two modes of power transmission are readily distinguished -- first, a twist-and-release or "AC" strategy, resulting in the propagation of a torsion wave which looks locally like a shear wave, and second, a constant-twist or "DC" strategy, in which a driver end turns the cable at a constant rate (up to the bursting velocity) and the cable is maintained at constant torque (up to the shear strength limit). In the following analysis, we consider a cylindrical transmission cable of radius r_cable, length L_cable, and density r, with shear modulus G and working stress s_w. The author thanks J. Soreff for helpful clarifications.

The AC case may be thought of as the propagation of a shear sound wave with maximum shear s_w and maximum strain s_max = s_w/G, traveling at v_shear ~ (G/r)^1/2, the transverse wave velocity in the cable medium. The maximum power passing through the cable is P_AC = p r_cable² D_E v_shear, where energy density D_E ~ G s_max² = s_w² / G, hence:

{Eqn. 6.46}

limiting maximum cable operating frequency to ~v_shear/L_cable or:

{Eqn. 6.47}

In the DC case, the maximum surface speed of a spinning cable is the flywheel bursting speed v_burst ~ (s_w / r)^1/2 and the maximum shear stress is ~s_w. Ignoring the minor complication that a real cable cannot withstand maximum static shear and maximum tangential velocity simultaneously, the maximum power passing through the cable is approximated by P_DC ~ p r_cable² s_w v_burst, so:

{Eqn. 6.48}

limiting maximum cable operating frequency to ~v_burst / 2 p r_cable or:

{Eqn. 6.49}

Hence P_DC/P_AC ~ (G / s_w)^1/2 ~ 7, taking G = 5 x 10¹¹ N/m² and s_w ~ 10¹⁰ N/m² for a diamondoid cable, so the initial conclusion is that the DC strategy allows transmission of ~1 order of magnitude higher mechanical power than the AC strategy, for a given cable size and material. A cable, secured at one end and twisted through a maximum angle q_max at the other end, acquires at maximum stress an energy E_cable = (1/2) k_torsion q_max² where k_torsion = p G r_cable⁴ / 2 L_cable; setting E_cable = p r_cable² L_cable D_E gives q_max = 2 L_cable s_w / r_cable G ~ 4 radians for an L_cable = 1 micron and r_cable = 10 nm diamondoid cable.

The AC cable may suffer transmission losses due to shear wave radiation from an oscillating torque.¹⁰ However, this radiation can be suppressed by imposing the torque between the cable and a coaxial sheath, or by imposing opposite torques on a pair of cables, both going from the same transmitter to the same receiver. In either case, no net torque need be imposed on the surrounding medium; even startup transients may be cancelled in the latter case. Thermally induced transient irregularities may still interact with the moving cable to radiate some power, but this remains a minor effect. A rotating cable suffers small additional transmission losses due to frictional interactions with the jacket in which it is encased, but drag power losses are much less than shear wave radiation losses in most nanoscale applications,¹⁰ so transmission efficiency is very nearly 100%. In some cases, there may be an additional strain limit imposed by supercoiling. Interestingly, double-stranded free DNA in vivo behaves much like a cable under torsional stress, with one negative superhelical turn per 100-200 base pairs (34-68 nm of "cable"); torsional stress is relieved by unwinding the double helix.⁹⁹⁷

However, thermal safety of in vivo mechanical power tethers is also of paramount concern. Under severe braking, loading or jamming of a cable, significant heat may be released. For a cable in vivo, a temperature rise of 63 K at the outer jacket surface (e.g., in contact with body fluids) boils water. Even a DT > 10 K may constitute an unacceptable risk in a conservative nanomedical design, as this is more than enough to trigger biological responses, for example by heat-shock proteins (HSPs). (Activation of HSPs in some cases may be beneficial to human health, but conservative nanomedical design requires minimizing all such unplanned side-effects.)

In the most optimistic scenario, power is immediately shut off the instant a fault is detected, instantly converting a DC cable into a half-cycle AC cable under relaxation. To prevent stored energy from causing damage in the event of a heat pulse, a DC cable should never be operated faster than the fastest physically equivalent AC cable, e.g., n_DC <~ n_ACm.

In the most pessimistic scenario, the cable jams at a single pointlike defect and then radiates the entire power flux from a sphere of radius ~r_cable; the practical effect is to preclude power cables altogether, because total cable power could not be allowed to exceed the heat flux from a droplet of water of radius ~r_cable whose temperature had been raised by DT.

An intermediate, yet conservative, scenario allows that the entire power flux (not just the energy already stored in the cable) must be dissipated, but that the whole length of the cable may be employed as the radiator during a dissipative event -- as, for example, if power continues to be transmitted after the fault but the cable jacket retains physical integrity. In this scenario, a cable carrying power flux I_power (W/m²) through a medium of heat capacity C_V and thermal conductivity K_t overheats the immediate environment in a time t_overheat ~ L_cable C_V DT / I_power, but thermal equilibration time t_EQ ~ X_bio² C_V / K_t where X_bio is a characteristic thermal conduction path length (e.g., minimum-size biological elements of size X_bio ~ 1 micron). Requiring for safety that t_overheat > t_EQ, then I_power < (K_t L_cable DT / X_bio²), which defines considerably more restrictive operating frequency limits for cables.

For an AC cable in the intermediate thermal scenario, the energy density per cycle is ~(1/2) D_E, giving at maximum frequency n_ACt a power flux of I_AC = n_ACt L_cable s_w² / 2 G, hence:

{Eqn. 6.50}

For a DC cable in the intermediate thermal scenario, power flux is I_DC = s_w v_burst = 2 p n_DCt r_cable s_w, hence:

{Eqn. 6.51}

Adopting the intermediate scenario, the general conclusion is that very small cables tend to be thermally limited, while very large cables are both thermally and mechanically limited. In the following configurations, we assume a diamondoid cable with DT = 10 K, r = 3510 kg/m³ for diamond, K_t = 0.623 W/m-K for water at 310 K, and X_bio = 1 micron.

An AC cable carrying a power throughput of p r_cable² I_AC = 1000 pW (near-peak nanorobot requirement) with r_cable = 5 nm and L_cable = 2 microns is thermally limited to an operating frequency of n_ACt <~ 60 KHz (<< n_ACm ~ 6 GHz), giving a power density of I_AC / L_cable = 6 x 10¹² W/m³. The same cable in DC mode is thermally limited to an operating frequency of n_DCt <~ 40 KHz (<< n_DCm ~ 70 GHz) with the same power density. If thermal limits were ignored, the cable could carry 0.2 milliwatts in AC mode and 1 milliwatt in DC mode.

Similarly, an AC cable carrying a power throughput of p r_cable² I_AC = 100 watts (human basal requirement) with r_cable = 4 microns and L_cable = 0.4 meter is mechanically limited to an operating frequency of n_ACm <~ 30 KHz (< n_ACt ~ 60 KHz), again giving a power density of I_AC / L_cable = 6 x 10¹² W/m³. The same cable in DC mode is thermally limited to an operating frequency of n_DCt <~ 10 MHz (< n_DCm ~ 100 MHz) with the same power density. If thermal limits were ignored, the cable would still carry only 100 watts in AC mode but could carry up to 700 watts in DC mode.

The refractive index profile of a sufficiently wide (Section 6.4.3.2) rotating diamond cable could be engineered, and might possibly be made sufficiently transparent, to also act as a photonic channel simultaneously conveying optical power or communications signals.

Last updated on 18 February 2003