© 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_a²/D (Section 3.2.2), which defines a characteristic diffusion frequency n_diff ~ D / L_a² 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 m²/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 m²/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:

{Eqn. 3.6}

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 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 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 m²/sec for small molecules, and using a modest L_a = 1 micron stirring apparatus giving R_s = 3R, then P_d ~ 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, P_d ~ 3 x 10³ watts/m³, 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 micron³, P_d = 3 x 10⁷ watts/m³, 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 10⁶ J/m³-K for water, thermal equilibration time t_EQ ~ L_mix² C_V / K_t = 0.2 millisec.

Last updated on 7 February 2003