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

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

3.2.3.2 Diffusive Swimming

The second strategy for active diffusive intake is by swimming. Again, the nanodevice is equipped with suitable active propulsion equipment (Section 9.4) which enables it to move so as to continuously encounter the highest possible concentration gradient near its surface. Consider a spherical motile nanorobot of radius R propelled at constant velocity v_swim through a fluid containing a desired molecule for which the surface of the device is essentially a perfect sink (Section 4.2.5). Applying the Stokes velocity field flow around the sphere to the standard diffusion equation, a numerical solution by Berg and Purcell³³⁷ found that the fractional increase in the diffusion current due to swimming is proportional to v_swim² for v_swim<< D/R, and to v_swim^1/3 for v_swim >> D/R.

Diffusive intake is doubled at a swimming speed v_swim = 2.5 D/R, which for 1-micron devices is ~5000 microns/sec when absorbing small molecules, ~50 microns/sec for large molecules. The viscous frictional energy cost to drive the nanodevice through the fluid, derived from Stokes' law (Eqn. 9.73), requires an onboard power density of:

{Eqn. 3.8}

If h = 1.1 x 10^-3 kg/m-sec, v_swim = 2.5 D/R, R = 0.5 micron, D = 10^-9 m²/sec for small molecules, then P_d ~ 5 x 10⁵ watts/m³. For large molecules with D = 10^-11 m²/sec, P_d ~ 50 watts/m³. Thus the energy cost of diffusive swimming appears modest for nanomechanical systems; gains in diffusion by swimming for nanodevices will be restricted primarily by the maximum safe velocity that may be employed in vivo (Section 9.4.2.6).

In general, outswimming diffusion requires movement over a characteristic distance L_s ~ D/v_swim.³⁸⁹ For bacteria moving at ~30 micron/sec and absorbing small molecules, then L_s ~ 30 microns, roughly the sprint distance exhibited by flagellar microbes such as E. coli. For micron-scale nanodevices moving at ~1 cm/sec (Section 9.4), L_s ~ 1-100 nm for large to small molecules.

Last updated on 7 February 2003