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 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 va, the speed of the appendage, and by a length La, its distance of travel, which together define a characteristic stirring frequency nstir ~ va/La sec-1. Movement of molecules over a distance La by diffusion alone is scaled by a characteristic time ~La2/D (Section 3.2.2), which defines a characteristic diffusion frequency ndiff ~ D / La2 sec-1. Stirring will be more effective than diffusion only if nstir > ndiff, that is, if va > D / La. For local stirring, La cannot be much larger than the size of the nanodevice itself. Assuming La = 1 micron and D = 10-9 m2/sec for small molecules, then va > 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 m2/sec for large proteins and virus particles, va > 10 microns/sec, well within the normal microbiological range.

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

{Eqn. 3.6}

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

In a classic paper, Berg and Purcell337 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 Rs, 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 m2/sec for small molecules, and using a modest La = 1 micron stirring apparatus giving Rs = 3R, then Pd ~ 3 x 107 watts/m3. This greatly exceeds the 102-106 watts/m3 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 ~109 watts/m3. (Nanomedically safe in vivo power densities are discussed at length in Sections 6.5.2 and 6.5.3.) For D ~ 10-11 m2/sec for large molecules, Pd ~ 3 x 103 watts/m3, which is reasonable even by biological standards. The maximum possible gain from stirring is ~Rs/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 micron3, Pd = 3 x 107 watts/m3, mixing distance Lmix ~ 5 microns, and thermal conductivity Kt = 0.623 watts/m-K for water, then DT ~ (Pd V / Lmix Kt) = 10 microkelvins; taking heat capacity CV = 4.19 x 106 J/m3-K for water, thermal equilibration time tEQ ~ Lmix2 CV / Kt = 0.2 millisec.


Last updated on 7 February 2003