**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

**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 Purcell^{337}
found that the fractional increase in the diffusion current due to swimming
is proportional to v_{swim}^{2} 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:

If h = 1.1 x 10^{-3} kg/m-sec,
v_{swim} = 2.5 D/R, R = 0.5 micron, D = 10^{-9} m^{2}/sec
for small molecules, then P_{d} ~ 5 x 10^{5} watts/m^{3}.
For large molecules with D = 10^{-11} m^{2}/sec, P_{d}
~ 50 watts/m^{3}. 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

Last updated on 7 February 2003