**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 N_{2} and O_{2} 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}

where R_{g} = 8.31 joules/mole-K, T is temperature
in kelvins (K), c_{2} and c_{1} are solute concentrations on
either side of the membrane in kg/m^{3} (c_{2} > c_{1}),
MW_{kg} is the molecular weight of the solute in kg/mole, and the Z^{2}
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/m^{3} 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 c_{2} = 370 kg/m^{3} 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 10^{7} N/m^{2} ~ 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 c_{2} ~ 73 kg/m^{3}, MW_{kg} ~
50 kg/mole (~50,000 daltons), so p_{p} ~
0.04 atm.

In theory, extremely large hydraulic counterforces up to 10^{5}
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_{chamber}^{2} 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_{sieve}^{2}. 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/m^{2}), 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 10^{6} joules/m^{3}-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_{sieve}^{2} / p r^{2}.
The volume processing rate (m^{3}/sec) of the sieve 'V_{sieve}
= 'V_{chamber} = h_{chamber} L_{chamber}^{2}/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/m^{3}). 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_{pore}^{3})^{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:

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_{pore}^{4} 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:

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}=10^{9} watts/m^{3}, then:

where D_{device} = device power density (watts/m^{3}),
total device power P_{device} = P_{chamber} + P_{sieve}
(watts), chamber fluid flow power P_{chamber} = p
L_{sieve}^{4} DP_{chamber}^{2}
/ 128 h_{chamber} h, sieve fluid flow power
P_{sieve} = a_{H} L_{sieve}^{2}
r_{pore}^{2} DP_{sieve}^{2}
/ 8 h_{sieve} h, and DP_{chamber}
= 16 a_{H} h_{chamber} r_{pore}^{2}
DP_{sieve} / p
L_{sieve}^{2} 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 micron^{3} 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}
m^{3}/sec for a 0.1 M solution of target molecules, yielding a processing
rate of 1.5 x 10^{9} 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 10^{8} watts/m^{3}.

For large molecules (r_{pore} ~ 5.0 nm), an exemplar
~1 micron^{3} 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}
m^{3}/sec for a 0.001 M solution of target molecules, yielding a processing
rate of 5.9 x 10^{7} 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 10^{7} watts/m^{3}.

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:

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/m^{2} for small molecules and ~ 0.5 x 10^{-3}
J/m^{2} 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