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

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

3.2.2 Passive Diffusive Intake

Medical nanodevices will frequently be called upon to absorb some particular material from the external aqueous operating environment. Molecular diffusion presents a fundamental limit to the speed at which this absorption can occur. (Once a block of solution has passed into the interior of a nanodevice, it may be divergently subdivided and transported at ~0.01-1 m/sec along internal pathways of characteristic dimension ~1 micron far faster than the <1 mm/sec bulk diffusion velocity across 1 micron distances; Section 9.2.7.5.)

For a spherical nanodevice of radius R, the maximum diffusive intake current is:

{Eqn. 3.4}

where J is the number of molecules/sec presented to the entire surface of the device, assumed to be 100% absorbed (but see Section 4.2.5), D (m²/sec) is the translational Brownian diffusion coefficient for the molecule to be absorbed, and C (molecules/m³) is the steady-state concentration of the molecule far from the device.³³⁷ (Blood concentrations in gm/cm³ from Appendix B are converted to molecules/m³ by multiplying Appendix B figures by (10⁶ x N_A/MW), where N_A = 6.023 x 10²³ molecules/mole (Avogadro's number) and MW = molecular weight in gm/mole or daltons.) For rigid spherical particles of radius R, where R >> R_H2O, the EinsteinStokes equation³⁶³ gives:

{Eqn. 3.5}

though this is only an approximation because D varies slightly with concentration, with departure from molecular sphericalness, and other factors.

Measured diffusion coefficients in water for various molecules of physiological interest, converted to 310 K, are in Table 3.3. Diffusion coefficient data for ionic salts such as NaCl and KCl, which dissociate in water and diffuse as independent ions, are for solvated electrolytes. A 1-micron (diameter) spherical nanodevice suspended in arterial blood plasma at 310 K, with C = 7.3 x 10²² molecules/m³ of oxygen and D = 2.0 x 10^-9 m²/sec, encounters a flow rate of J = 9.2 x 10⁸ molecules/sec of O₂ impinging upon its surface. (The same calculation applied to serum glucose yields J = 1.3 x 10¹⁰ molecules/sec.) The characteristic time for change mediated by diffusion in a region of size L scales as ~L²/D (Eqn. 3.9, below). Across the diameter of an L = 1 micron nanodevice, small molecules such as glucose diffuse in ~0.001 sec, small proteins like hemoglobin in ~0.01 sec, and virus particles diffuse in ~0.1 sec. Diffusion coefficients of the same molecules in air at room temperature are a factor of ~60 higher, because h_air ~ 183 micropoise at 20°C.

In blood, the diffusivity of larger particles is significantly elevated because local fluid motions generated by individual red cell rotation lead to greater random excursions of the particles.³⁸⁸ The effective diffusivity D_e = D + D_r, where the rotation-induced increase in diffusivity D_r ~ 0.25 R_rbc² 'g, with red cell radius R_rbc ~ 2.8 microns (taken for convenience as a spherical volume equivalent) and a typical blood shear rate (Section 9.4.1.1) 'g ~ 500 sec^-1, giving D_r ~10^-9 m²/sec in normal whole blood. The elevation of diffusivity caused by red cell stirring is just 50% for O₂ molecules. However, for large proteins and viruses the effective diffusivity increases 10-100 times, and the effective diffusivity of particles the size of platelets is a factor of 10,000 higher than for Brownian molecular diffusion.

The diffusion current to the surface of a nanodevice can also be estimated for various nonspherical configurations.³³⁷ For instance, the diffusion current to both sides of an isolated thin disk of radius R is given by J = 8 R D C. The two-sided current to a square thin plate of area L² is J = (8/p^1/2) L D C. The steady-state diffusion current to an isolated cylinder of length L_c and radius R is approximated by J = 2 p L_c DC / (ln(2L_c/R) - 1 ), for L_c >> R. The diffusion current through a circular hole of radius R in an impermeable wall separating regions of concentration c₁ and c₂ is J = 4 R D (c₁ - c₂).

Last updated on 7 February 2003