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
3.3.1 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 pp is given by the Donnan-van't Hoff formula:403
where Rg = 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), MWkg 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 cs 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 MWkg = 0.05844 kg/mole for NaCl, so for salt water solutions the theoretical maximum pp ~ 1.6 x 107 N/m2 ~ 160 atm. Natural bloodstream concentrations of salt produce pp ~ 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, MWkg ~ 50 kg/mole (~50,000 daltons), so pp ~ 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 hchamber in length and Lchamber2 in cross-sectional area, forcing the fluid to filter through pores of radius rpore (~ target molecule radius) covering a fraction aH (~50%) of the surface of a square nanoscale sieve of thickness hsieve and area Lsieve2. 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 DPmax < CV DTboil, where the heat capacity of water CV = 4.19 x 106 joules/m3-K and DTboil = 373 K - 310 K = 63 K give DPmax < 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 tp:
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 Npore = aH Lsieve2 / p r2. The volume processing rate (m3/sec) of the sieve 'Vsieve = 'Vchamber = hchamber Lchamber2/tp, hence the molecule processing rate is 'N = 'Vchamber ctarget (molecules/sec) where ctarget is the concentration of target molecules (molecules/m3). At any one time, each pore channel through the sieve can hold at most Nchannel = hsieve / 2 rpore molecules in single file; during each power stroke, at most Nstack = 'N tp / Npore molecules pass through each pore. Hence the time available for molecular rotation trot = tp (Nchannel / Nstack), which assumes the layer of rotating target molecules in the vicinity of the pores approximates sieve thickness hsieve, a reasonable assumption as long as the typical molecular diffusion time (across a distance hsieve) << trot. Taking Nrot ~ 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 trot / 4 p h rpore3)1/2 > 2 p Nrot, where h is solvent viscosity (1.1 x 10-3 kg/m-sec for plasma) at T = 310 K. Solving for minimum tp gives:
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 'Vpore = p rpore4 DPsieve / 8 h hsieve and the volume processing rate through the entire sieve is 'Vsieve = Npore 'Vpore = 'Vchamber; solving for maximum tp gives:
Equating these two bracketing constraints, hsieve > 150 nm for large molecules (rpore ~ 5 nm) but hsieve > 1 nm for small molecules (rpore ~ 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 Dn=109 watts/m3, then:
where Ddevice = device power density (watts/m3), total device power Pdevice = Pchamber + Psieve (watts), chamber fluid flow power Pchamber = p Lsieve4 DPchamber2 / 128 hchamber h, sieve fluid flow power Psieve = aH Lsieve2 rpore2 DPsieve2 / 8 hsieve h, and DPchamber = 16 aH hchamber rpore2 DPsieve / p Lsieve2 hsieve. For small molecules such as NaCl or glucose, DPsieve > 160 atm to overcome maximum osmotic backpressure; for large molecules (e.g., ~50,000 dalton proteins), we assume DPsieve > 1 atm to ensure sieving. The following designs are not optimized but illustrate the tradeoffs involved.
For small molecules (rpore ~ 0.32 nm), an exemplar ~1 micron3 device has hchamber = 1 micron, Lchamber = Lsieve = 0.6 micron, hsieve = 1.5 microns, tp = 0.016 sec, DPsieve = 160 atm, DPchamber = 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 Pdevice = 400 pW and power density Ddevice = 4 x 108 watts/m3.
For large molecules (rpore ~ 5.0 nm), an exemplar ~1 micron3 device has hchamber = 1 micron, Lchamber = Lsieve = 0.9 micron, hsieve = 0.15 microns, tp = 0.010 sec, DPsieve = 1 atm, DPchamber = 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 Pdevice = 80 pW and power density Ddevice = 8 x 107 watts/m3.
Sieve pores can become clogged by particles of radius R ~ rpore if the applied hydraulic pressure DPclog exceeds the thermal energy of the trapped particles, or:
By this criterion, DPclog > 500 atm for small molecules and DPclog > 0.1 atm for large molecules. From the values of DPsieve 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 Wadhesion < DPsieve rpore ~ 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