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

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

9.4.2.4 Force and Power Requirements

Consider a spherical nanorobot of radius R_nano that is falling at uniform velocity v_nano through an incompressible (Newtonian) fluid of viscosity h. If the Reynolds number N_R << 1 (Eqn. 9.65), then Stokes¹³⁷³ found that the total force of resistance imparted on the sphere by the fluid is given by:

{Eqn. 9.73}

also known as Stokes law. This result applies only in the case of a single sphere in an infinite expanse of homogeneous viscous fluid, and neglects all inertial terms. If the fluid container is finite in size, or if there are other spheres in the neighborhood, or if inertial forces cannot be entirely ignored because N_R >~ 1 (e.g., R_nano = 1 micron, v_nano > 1 m/sec in water), then the equation must be corrected.¹³⁷⁴ Dynamic effects due to sphere oscillation, sudden release from rest, or variable speed also require corrections,¹³⁷⁴ and there are other corrections for neighboring spheres.¹³⁷⁵ Most of these corrections are of order near-unity for micron-scale objects.

Eqn. 9.73 provides a useful approximation of the motive force required to drive a spherical nanorobot through blood plasma. Assuming a perfectly efficient drive mechanism, the power requirement³³⁷ is at least:

{Eqn. 9.74}

For example, a force of F_nano = 200 pN and a power of at least P_nano = 2 pW are required to drive a R_nano = 1 micron spherical nanorobot at v_nano = 1 cm/sec through blood plasma at 310 K with h = 1.1 x 10^-3 kg/msec. These formulas may also be used to estimate nanodevice velocity, given F_nano or P_nano.

Of course, drive mechanisms are not perfectly efficient. Propulsion efficiency is often poor for objects that swim at low Reynolds numbers (e.g., typically N_R ~ 10^-3 for flagellates).³⁵⁷⁸ A sphere driven by a helical propeller (e.g., flagellar propulsion; Section 9.4.2.5.2) of arbitrary length may have a propulsion efficiency as low as e% ~ 0.01(1%).³⁸⁹ Propulsive efficiency for organisms with spherical heads 10-40 times larger than their flagellar radius (optimum shape), using helical flagellar beats, ranges from e% = 0.10-0.28 (10%-28%),¹³⁷⁷ requiring a propulsive input power of ~P_nano/e% ~ 7-20 pW in the previous example. Other drive mechanisms may prove more efficient.

Force requirements for nonspherical nanorobots are similar in magnitude. For example, consider a cylinder of radius R_nano and length L_nano translating uniformly in a direction normal to its axis. Lighthill^1367,1376 shows that the force (applied at the midpoint) required to move this object at velocity v_nano is given by:

{Eqn. 9.75}

for L_nano >> R_nano.³⁶³ (If the perpendicular force is applied at a distance c from one end of the cylinder, then the L_nano² term in Eqn. 9.75 is replaced by 4 c (L_nano - c) when c >> R_nano.) Eqn. 9.75 does not hold for a perpendicular force applied near the ends of the cylinder. Also, a constant F_nano actually produces a slowly varying velocity field along the cylinder length; for a midpoint-applied force, v_nano is at maximum at the midpoint, but the percentage variation over most of the cylinder is not large.³⁶³ Experiments show that needle-shaped bodies fall in viscous media about half as fast sideways as they do end-on;¹³⁷⁸ that is, the force on a long cylinder translating parallel to its axis is F_nanoP ~ (1/2) F_nanoN. The power requirement is then calculated as in Eqn. 9.74.

Interestingly, there seems to be a sharp minimum size limit of ~0.6 microns for free-swimming foraging microbes, below which size locomotion has no apparent benefit.³⁵⁸¹ This theoretical conclusion is supported by the observation that the smallest 97 genera of motile bacteria have a mean length of 0.8 microns, whereas 18 of 94 nonmotile genera are smaller.³⁵⁸¹

Last updated on 21 February 2003