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
184.108.40.206 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 220.127.116.11.) 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 rcable, length Lcable, and density r, with shear modulus G and working stress sw. 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 sw and maximum strain smax = sw/G, traveling at vshear ~ (G/r)1/2, the transverse wave velocity in the cable medium. The maximum power passing through the cable is PAC = p rcable2 DE vshear, where energy density DE ~ G smax2 = sw2 / G, hence:
limiting maximum cable operating frequency to ~vshear/Lcable or:
In the DC case, the maximum surface speed of a spinning cable is the flywheel bursting speed vburst ~ (sw / r)1/2 and the maximum shear stress is ~sw. 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 PDC ~ p rcable2 sw vburst, so:
limiting maximum cable operating frequency to ~vburst / 2 p rcable or:
Hence PDC/PAC ~ (G / sw)1/2 ~ 7, taking G = 5 x 1011 N/m2 and sw ~ 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 qmax at the other end, acquires at maximum stress an energy Ecable = (1/2) ktorsion qmax2 where ktorsion = p G rcable4 / 2 Lcable; setting Ecable = p rcable2 Lcable DE gives qmax = 2 Lcable sw / rcable G ~ 4 radians for an Lcable = 1 micron and rcable = 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., nDC <~ nACm.
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 ~rcable; 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 ~rcable 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 Ipower (W/m2) through a medium of heat capacity CV and thermal conductivity Kt overheats the immediate environment in a time toverheat ~ Lcable CV DT / Ipower, but thermal equilibration time tEQ ~ Xbio2 CV / Kt where Xbio is a characteristic thermal conduction path length (e.g., minimum-size biological elements of size Xbio ~ 1 micron). Requiring for safety that toverheat > tEQ, then Ipower < (Kt Lcable DT / Xbio2), 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) DE, giving at maximum frequency nACt a power flux of IAC = nACt Lcable sw2 / 2 G, hence:
For a DC cable in the intermediate thermal scenario, power flux is IDC = sw vburst = 2 p nDCt rcable sw, hence:
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, Kt = 0.623 W/m-K for water at 310 K, and Xbio = 1 micron.
An AC cable carrying a power throughput of p rcable2 IAC = 1000 pW (near-peak nanorobot requirement) with rcable = 5 nm and Lcable = 2 microns is thermally limited to an operating frequency of nACt <~ 60 KHz (<< nACm ~ 6 GHz), giving a power density of IAC / Lcable = 6 x 1012 W/m3. The same cable in DC mode is thermally limited to an operating frequency of nDCt <~ 40 KHz (<< nDCm ~ 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 rcable2 IAC = 100 watts (human basal requirement) with rcable = 4 microns and Lcable = 0.4 meter is mechanically limited to an operating frequency of nACm <~ 30 KHz (< nACt ~ 60 KHz), again giving a power density of IAC / Lcable = 6 x 1012 W/m3. The same cable in DC mode is thermally limited to an operating frequency of nDCt <~ 10 MHz (< nDCm ~ 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 18.104.22.168) 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