**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

**3.2.3.1 Diffusive Stirring**

The first strategy for active diffusive intake is local stirring.
For this, the nanodevice is equipped with suitable active appendages used to
manipulate the fluid in its vicinity. Transport by stirring is characterized
by a velocity v_{a}, the speed of the appendage, and by a length L_{a},
its distance of travel, which together define a characteristic stirring frequency
n_{stir} ~ v_{a}/L_{a} sec^{-1}.
Movement of molecules over a distance L_{a} by diffusion alone is scaled
by a characteristic time ~L_{a}^{2}/D (Section
3.2.2), which defines a characteristic diffusion frequency n_{diff}
~ D / L_{a}^{2} sec^{-1}. Stirring will be more effective
than diffusion only if n_{stir} > n_{diff},
that is, if v_{a} > D / L_{a}. For local stirring, L_{a}
cannot be much larger than the size of the nanodevice itself. Assuming L_{a}
= 1 micron and D = 10^{-9} m^{2}/sec for small molecules, then
v_{a} > 1000 microns/sec, a faster motion than is exhibited by bacterial
cells but quite modest for nanomechanical devices (Section
9.3.1). With D = 10^{-11} m^{2}/sec for large proteins and
virus particles, v_{a} > 10 microns/sec, well within the normal microbiological
range.

The ratio of stirring time to diffusion time, or Sherwood number, is:

provides a dimensionless measure of the effectiveness of stirring
vs. diffusion. For bacteria absorbing small molecules, N_{Sh} ~ 10^{-2}.
Micron-scale nanodevices with 1-micron appendages capable of 0.01-1 m/sec movement
can achieve N_{Sh} ~ 10-1000 for small to large molecules, hence could
be considerably more effective stirrers.

In a classic paper, Berg and Purcell^{337}
analyzed the viscous frictional energy cost of moving the stirring appendages
so that the fluid surrounding a spherical object (e.g., a nanodevice) of radius
R, out to some maximum stirring radius R_{s}, is maintained approximately
uniform in concentration. The objective is to transfer fluid from a distant
region of relatively high concentration to a place much closer to the nanodevice,
thereby increasing the concentration gradient near the absorbing surface. To
double the passive diffusion current by stirring, the minimum required power
density is:

_{}
{Eqn. 3.7}

If h = 1.1 x 10^{-3} kg/m-sec,
R = 0.5 micron, D = 10^{-9} m^{2}/sec for small molecules, and
using a modest L_{a} = 1 micron stirring apparatus giving R_{s}
= 3R, then P_{d} ~ 3 x 10^{7} watts/m^{3}. This greatly
exceeds the 10^{2}-10^{6} watts/m^{3} power density
commonly available to biological cells (Table
6.8) but lies well within the normal range for nanomechanical systems which
typically operate at up to ~10^{9} watts/m^{3}. (Nanomedically
safe in vivo power densities are discussed at length in Sections 6.5.2
and 6.5.3.) For D ~ 10^{-11} m^{2}/sec
for large molecules, P_{d} ~ 3 x 10^{3} watts/m^{3},
which is reasonable even by biological standards. The maximum possible gain
from stirring is ~R_{s}/R, because the current is ultimately limited
to what can diffuse into the stirred region.

Local heating due to stirring is minor. Given device volume
V ~1 micron^{3}, P_{d} = 3 x 10^{7} watts/m^{3},
mixing distance L_{mix} ~ 5 microns, and thermal conductivity K_{t}
= 0.623 watts/m-K for water, then DT ~ (P_{d}
V / L_{mix} K_{t}) = 10 microkelvins; taking heat capacity C_{V}
= 4.19 x 10^{6} J/m^{3}-K for water, thermal equilibration time
t_{EQ} ~ L_{mix}^{2} C_{V} / K_{t} =
0.2 millisec.

Last updated on 7 February 2003