**Kinematic
Self-Replicating Machines**

**©
2004 Robert A. Freitas Jr. and Ralph C. Merkle. All Rights Reserved.**

Robert A. Freitas Jr., Ralph C. Merkle, *Kinematic Self-Replicating Machines*,
Landes Bioscience, Georgetown, TX, 2004.

**B.4.3.1 Bulk Fluid and
Laminar Flows**

As the piston plate is pushed into the molecular assembler
with increasing solvent liquid pressure, fluid flows into the volume vacated
by the piston, approximating fluid flow through a tube. In such situations,
classical continuum models assume, among other things, that the molecular graininess
of the fluid can be ignored. This assumption fails when tube dimensions –
say, tube radius R_{tube} – become comparable to or smaller than the characteristic
molecular length scale (l_{fluid}) of the fluid [208].
In a liquid, l_{fluid} approximates the molecular radius; for n-octane molecules,
l_{fluid} ~ 0.3 nm. Taking pR_{tube}^{2} ~ S_{transducer}, then R_{tube} (~ 29.85 nm) >>
l_{fluid} (~ 0.3 nm), so the classical continuum equations should well approximate
the bulk fluid flows into the piston cavity as the piston is pushed into the
assembler device by rising pressure. It is generally accepted that bulk liquid
behavior exists more than ~5-10 molecular diameters (~2.5-5 nm for octane) from
surfaces [3155, 3187],
and molecular dynamics simulations of liquid-filled pores show that the average
diffusion coefficient in the center of the pore approaches the bulk value at
a distance of more than 10 molecular diameters from the pore’s surface
[3202].

Continuum flow [3156]
is typically governed by the well-known Hagen-Poiseuille Law (or more commonly,
Poiseuille's Law), derived from the Navier-Stokes equations, which states that
a pressure difference of DP_{fluid} between the ends of a rigid tube of radius
R_{tube} and length L_{tube} will drive an aliquot of incompressible fluid of absolute
viscosity h_{fluid} in laminar flow through the entire tube length in a time t_{flow}
= 8 h_{fluid} L_{tube}^{2} / R_{tube}^{2} DP_{fluid}. Taking h_{fluid} = 5.40 x 10^{-4} Pa-sec for n-octane
[3116], R_{tube} ~ 29.85 nm, L_{tube} = 10 nm
(double-band operation, the longest piston throw under normal conditions), and
DP_{fluid} = 2.026 x 10^{5} N/m^{2} (2 atm), then t_{flow} = 2.39 x 10^{-9} sec, an implied
maximum cycling frequency of 418 MHz. However, the actual piston is only operated
at n_{acoustic} = 10 MHz, so the n-octane fluid molecules have plenty of time to
enter the piston cavity, in near-perfect laminar flow (see below) and in equilibrium.
In this regime, the drag power dissipation can be crudely estimated from Stokes’
law which gives the drag power for a sphere of radius R moving through a fluid
of viscosity h at velocity v as p_{Stokes} = 6 p h R v^{2} ~ 12 pW << p_{assembler}
= 56.8 pW during double-band operation (Section B.4.2),
taking R = R_{tube}, h = h_{fluid} for n-octane, and v = v_{piston} ~ (2 L_{tube} n_{acoustic})
= 0.2 m/sec during a double-band cycle of fluid flow. The equivalent drag force
is F_{Stokes} = p_{Stokes} / v ~ 60 pN << DF_{piston} = 568 pN for double-band
operation (Section B.4.2). During single-band operation,
v_{piston} ~ 0.1 m/sec so the Stokes law drag power is p_{Stokes} ~ 3 pW <<
p_{assembler} = 14.2 pW (Section B.4.2). The equivalent
drag force is F_{Stokes} ~ 30 pN << DF_{piston} = 284 pN for single-band operation
(Section B.4.2).

The resistance to Poiseuille (laminar) flow in a pipe is the
minimum of resistance of all possible flows in a pipe [3157].
If the flow becomes turbulent, the resistance increases. The determinative parameter
is a dimensionless quantity called the Reynolds number [3158-3160],
N_{R}, which is the ratio of the inertial pressure (~ r v^{2}) to the viscous pressure
(~ h v / R) in the flow of a fluid of density r, or, for our piston system,
N_{R} = r_{fluid} v_{piston} L_{char} / h_{fluid} = 0.001, taking r_{fluid} = 702.5 kg/m^{3} for
n-octane [3161] and v_{piston} ~ 0.2 m/sec
during a double-band cycle of fluid flow. The characteristic linear dimension
L_{char} is the ratio of the volume of the fluid to the surface area of the walls
that bound it [3160], giving L_{char} = 4.12
nm for our piston during a double-band cycle. A Reynolds number this low indicates
that bulk-phase fluid flow will be extremely laminar, both into and out of the
piston cavity, and highly uniaxial, with the entire fluid moving parallel to
the local orientation of the walls [3160].

Classical Poiseuille flow is characterized by a parabolic
velocity profile over the cross-section of the channel. However, it is unlikely
that the flow of octane-solvated acetylene into the piston cavity will be able
to develop a parabolic velocity profile across the flow channel (most favored
for cavity aspect ratios [3162] near 1:1,
as here). Brody and Yager [3162] estimate
that small solute molecules of a size similar to the solvent molecules will
have a relatively large diffusion coefficient, hence will move with a plug flow
(~constant velocity) profile whenever N_{R} < 0.001 – very close to our
estimated N_{R} = 0.001. Additionally, when fluid enters a narrow channel from
a wider one, the flow profile is not immediately parabolic [3163].
For low Reynolds number fluid flow, the inlet distance for flow to become 99%
fully developed is ~R_{tube}, but since R_{tube} (= 29.85 nm) > L_{tube} (= 10 nm)
there is insufficient length even in the longest possible piston throw to allow
the flow to become fully developed. These factors argue for plug flow inside
the piston cavity.

In general, a large Reynolds number implies a preponderant
inertial effect and the onset of turbulence; a small Reynolds number implies
a predominant shear effect (e.g. viscosity) and the maintenance of laminar flow,
with no-slip conditions (i.e., no fluid flow at the surface of an object [3164]).
Reynolds [3158] found that the transition
from laminar to turbulent flow typically occurs at N_{R} > 2000-13,000, depending
upon the smoothness of the entry conditions. The lowest transitional value obtainable
experimentally on a rough entrance in macroscale channels was N_{R} ~ 2000, although
when extreme care was taken to establish smooth entry conditions the transition
could be delayed to Reynolds numbers as high as 40,000. Recent studies of Reynolds
numbers in microfluidics flow systems confirm transitional values of N_{R} >100
for the onset of turbulence, and maintenance of laminar flow as long as the
“switching time” t_{switch} on which the pressure driving the flow
varies (e.g., ~10^{-7} sec at 10 MHz) is longer than the characteristic time t_{char}
= r_{fluid} R_{tube}^{2} / h_{fluid} (~10^{-9} sec for our assembler piston) [3162]
– such pressure changes may be considered quasi-static. The Stokes drag
law used earlier should remain applicable as long as flow remains laminar.

Laminar flows in nanoscale channels are well-known in biology.
For example, cellular and nuclear microinjection through fine glass pipette
tips as small as 200 nm in diameter is commonplace in the experimental biological
sciences [3165], and the syringelike T4
bacteriophage ejects ~200,000 nm^{3} of DNA material through a 2.5-nm diameter
hollow core protein nanotube at a flow velocity as high as 0.36 mm/sec, forced
by a ~30 atm head pressure [1758-1762].
Molecular dynamics simulations of fluid flow inside 1.3-1.6 nm diameter single-walled
carbon nanotubes have also been reported [3166,
3167].

Last updated on 13 August 2005