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

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

3.3.1 Simple Nanosieving

Nanometer-scale isoporous molecular sieves (with ovoid, square, or hexagonal holes) are common in almost every taxonomic group of eubacteria and archaeobacteria.⁵²⁵ Other well-known examples of nanoporous structures are the 6-nm pore arrays found in reverse osmosis and kidney dialysis membranes.

Likewise, it is possible for a nanodevice to sort molecules by simple sieving.^987,1177 In this process, a sample containing particles of various sizes suspended in water passes through a graduated series of filters perforated by progressively smaller holes of fixed size and shape. Between each filtration unit, the filtration residue consists almost exclusively of particles having a narrow range of sizes and shapes. For example, a series of n = 100 filtration units could reliably differentiate an input sample containing particles from 0.2-1.2 nm into 100 separate fractions, each fraction consisting predominantly of particles differing in mean diameter by ~0.01 nm. (In a practical system, several passes would be required to achieve complete discrimination; Section 3.2.4.) A ~0.01 nm difference in molecular diameter corresponds to the mean contribution of ~1 additional carbon atom to the size of a small molecule (MW ~ 100 daltons), or to the mean contribution of ~100 additional carbon atoms to the size of a large molecule (MW ~ 100,000 daltons, ~17,000 atoms). A nanomembrane might even permit the (slow, multi-pass) sieving of oxygen from air, since the molecular diameters of N₂ and O₂ differ by ~0.01 nm (Section 3.5.5).

Two opposing forces are at work when moving water and solutes through a membrane. One is the osmotic pressure established by the presence of nonpermeating solutes; the other is the hydraulic or fluid pressure. The velocity of material movement depends on the relative values of the osmotic and hydraulic forces, and on the size of the pores in the filter.

Osmotic pressure p_p is given by the Donnan-van't Hoff formula:⁴⁰³

{Eqn. 3.13}

where R_g = 8.31 joules/mole-K, T is temperature in kelvins (K), c₂ and c₁ are solute concentrations on either side of the membrane in kg/m³ (c₂ > c₁), MW_kg is the molecular weight of the solute in kg/mole, and the Z² term (dependent upon polymer-polymer interactions) is a correction factor for highly concentrated solutions which for some solvents and temperatures may equal zero; Z = net solute charge number and c_s is the concentration in kg/m³ of a second solute, as for example when the first solute is a protein and the second solute is salt, as in human serum. Water at 310 K dissolves a maximum of c₂ = 370 kg/m³ of sodium chloride (a 37.0% solution, by weight), and MW_kg = 0.05844 kg/mole for NaCl, so for salt water solutions the theoretical maximum p_p ~ 1.6 x 10⁷ N/m² ~ 160 atm. Natural bloodstream concentrations of salt produce p_p ~ 3 atm. Since osmotic pressure depends on the number of molecules present, the contribution from large molecules is usually negligible. For instance, total protein concentration in human blood serum is c₂ ~ 73 kg/m³, MW_kg ~ 50 kg/mole (~50,000 daltons), so p_p ~ 0.04 atm.

In theory, extremely large hydraulic counterforces up to 10⁵ atm may be applied in nanomechanical systems, for example by a piston, to overcome osmotic backpressure. As a practical matter, however, rapidly pushing small molecules at high pressure through nanoscale holes is an effective method for generating significant amounts of waste heat; a design compromise is required.

Consider a simple sorting apparatus in which a square piston is used to compress solvent fluid (say, water) trapped in a chamber h_chamber in length and L_chamber² in cross-sectional area, forcing the fluid to filter through pores of radius r_pore (~ target molecule radius) covering a fraction a_H (~50%) of the surface of a square nanoscale sieve of thickness h_sieve and area L_sieve². Solute which is dissolved or suspended in the solvent is sorted based on molecular size; a sequence of sieving runs using sieves having progressively smaller pore radii produces an ordered sequence of molecular size fractions. (In small molecules, adding the mass of one hydrogen atom increases the mean linear dimension of the molecule by 0.11%.)

The first design constraint on this system relates to its maximum operating pressure. If DP is applied pressure (N/m²), then avoiding boiling the solvent water and denaturing proteins requires at least that DP_max < C_V DT_boil, where the heat capacity of water C_V = 4.19 x 10⁶ joules/m³-K and DT_boil = 373 K - 310 K = 63 K give DP_max < 2600 atm. The designs presented below operate at 6% of this maximum (DT ~ 3 K) or less.

There are two major design constraints on the duration of the power stroke, or t_p:

1. Molecular Rotation Constraint -- Flow through the sieve must be slow enough to allow molecules to align with the holes. Assuming round pores, the total number of pores in the sieve is N_pore = a_H L_sieve² / p r². The volume processing rate (m³/sec) of the sieve 'V_sieve = 'V_chamber = h_chamber L_chamber²/t_p, hence the molecule processing rate is 'N = 'V_chamber c_target (molecules/sec) where c_target is the concentration of target molecules (molecules/m³). At any one time, each pore channel through the sieve can hold at most N_channel = h_sieve / 2 r_pore molecules in single file; during each power stroke, at most N_stack = 'N t_p / N_pore molecules pass through each pore. Hence the time available for molecular rotation t_rot = t_p (N_channel / N_stack), which assumes the layer of rotating target molecules in the vicinity of the pores approximates sieve thickness h_sieve, a reasonable assumption as long as the typical molecular diffusion time (across a distance h_sieve) << t_rot. Taking N_rot ~ 10 as the mean number of molecular revolutions needed to ensure proper pore alignment with noncircular sieve holes (the most difficult case), then, from Eqn. 3.2, Da = (kT t_rot / 4 p h r_pore³)^1/2 > 2 p N_rot, where h is solvent viscosity (1.1 x 10^-3 kg/m-sec for plasma) at T = 310 K. Solving for minimum t_p gives:

{Eqn. 3.14}

2. Pressure/Flow Constraint -- Flow through the sieve must be fast enough to establish a sufficient pressure to oppose osmotic backflows. From the Hagen-Poiseuille law (Section 9.2.5), the volume processing rate through each pore is 'V_pore = p r_pore⁴ DP_sieve / 8 h h_sieve and the volume processing rate through the entire sieve is 'V_sieve = N_pore 'V_pore = 'V_chamber; solving for maximum t_p gives:

{Eqn. 3.15}

Equating these two bracketing constraints, h_sieve > 150 nm for large molecules (r_pore ~ 5 nm) but h_sieve > 1 nm for small molecules (r_pore ~ 0.32 nm).

As a final constraint, power released by fluid flow through the chamber and sieve must not exceed safe thermogenic limits. Given the maximum safe power density for in vivo nanomachines given in Section 6.5.3 as D_n=10⁹ watts/m³, then:

{Eqn. 3.16}

where D_device = device power density (watts/m³), total device power P_device = P_chamber + P_sieve (watts), chamber fluid flow power P_chamber = p L_sieve⁴ DP_chamber² / 128 h_chamber h, sieve fluid flow power P_sieve = a_H L_sieve² r_pore² DP_sieve² / 8 h_sieve h, and DP_chamber = 16 a_H h_chamber r_pore² DP_sieve / p L_sieve² h_sieve. For small molecules such as NaCl or glucose, DP_sieve > 160 atm to overcome maximum osmotic backpressure; for large molecules (e.g., ~50,000 dalton proteins), we assume DP_sieve > 1 atm to ensure sieving. The following designs are not optimized but illustrate the tradeoffs involved.

For small molecules (r_pore ~ 0.32 nm), an exemplar ~1 micron³ device has h_chamber = 1 micron, L_chamber = L_sieve = 0.6 micron, h_sieve = 1.5 microns, t_p = 0.016 sec, DP_sieve = 160 atm, DP_chamber = 0.0001 atm. Piston velocity ~ 60 micron/sec and 'V = 2.5 x 10^-17 m³/sec for a 0.1 M solution of target molecules, yielding a processing rate of 1.5 x 10⁹ molecules/sec (1.5 x 10^-16 kg/sec); the device processes its own mass every ~7 sec or every ~430 power strokes. Device power P_device = 400 pW and power density D_device = 4 x 10⁸ watts/m³.

For large molecules (r_pore ~ 5.0 nm), an exemplar ~1 micron³ device has h_chamber = 1 micron, L_chamber = L_sieve = 0.9 micron, h_sieve = 0.15 microns, t_p = 0.010 sec, DP_sieve = 1 atm, DP_chamber = 0.0004 atm. Piston velocity ~ 100 micron/sec and 'V = 9.8 x 10^-17 m³/sec for a 0.001 M solution of target molecules, yielding a processing rate of 5.9 x 10⁷ molecules/sec (4.9 x 10^-15 kg/sec); the device processes its own mass every ~0.2 sec or every ~20 power strokes. Device power P_device = 80 pW and power density D_device = 8 x 10⁷ watts/m³.

Sieve pores can become clogged by particles of radius R ~ r_pore if the applied hydraulic pressure DP_clog exceeds the thermal energy of the trapped particles, or:

{Eqn. 3.17}

By this criterion, DP_clog > 500 atm for small molecules and DP_clog > 0.1 atm for large molecules. From the values of DP_sieve given above, clogging is unlikely for small molecules but is possible at the highest concentrations of large molecules. In 310 K water, large molecules diffuse ~3 nm and small molecules diffuse ~17 nm in ~10^-7 sec, just far enough to clear the hole, so a ~10 MHz sawtooth pressure profile imposed on the power stroke may ensure sufficient backflushing action to avoid serious blockages. To reduce the possibility of clogging due to surface force adhesion (Section 9.2.3), as a design criterion the work of adhesion should be reduced to W_adhesion < DP_sieve r_pore ~ 5 x 10^-3 J/m² for small molecules and ~ 0.5 x 10^-3 J/m² for large molecules likely to come into contact with sieve pore surfaces. Clogging due to long-term random polymerizations can be minimized by periodically exchanging the entire contents of the input chamber with fresh solution, by operating the device at reduced power density, or by periodically replacing the sieve.

Last updated on 7 February 2003