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 Rtube – become comparable to or smaller than the characteristic molecular length scale (lfluid) of the fluid . In a liquid, lfluid approximates the molecular radius; for n-octane molecules, lfluid ~ 0.3 nm. Taking pRtube2 ~ Stransducer, then Rtube (~ 29.85 nm) >> lfluid (~ 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 .
Continuum flow  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 DPfluid between the ends of a rigid tube of radius Rtube and length Ltube will drive an aliquot of incompressible fluid of absolute viscosity hfluid in laminar flow through the entire tube length in a time tflow = 8 hfluid Ltube2 / Rtube2 DPfluid. Taking hfluid = 5.40 x 10-4 Pa-sec for n-octane , Rtube ~ 29.85 nm, Ltube = 10 nm (double-band operation, the longest piston throw under normal conditions), and DPfluid = 2.026 x 105 N/m2 (2 atm), then tflow = 2.39 x 10-9 sec, an implied maximum cycling frequency of 418 MHz. However, the actual piston is only operated at nacoustic = 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 pStokes = 6 p h R v2 ~ 12 pW << passembler = 56.8 pW during double-band operation (Section B.4.2), taking R = Rtube, h = hfluid for n-octane, and v = vpiston ~ (2 Ltube nacoustic) = 0.2 m/sec during a double-band cycle of fluid flow. The equivalent drag force is FStokes = pStokes / v ~ 60 pN << DFpiston = 568 pN for double-band operation (Section B.4.2). During single-band operation, vpiston ~ 0.1 m/sec so the Stokes law drag power is pStokes ~ 3 pW << passembler = 14.2 pW (Section B.4.2). The equivalent drag force is FStokes ~ 30 pN << DFpiston = 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 . If the flow becomes turbulent, the resistance increases. The determinative parameter is a dimensionless quantity called the Reynolds number [3158-3160], NR, which is the ratio of the inertial pressure (~ r v2) to the viscous pressure (~ h v / R) in the flow of a fluid of density r, or, for our piston system, NR = rfluid vpiston Lchar / hfluid = 0.001, taking rfluid = 702.5 kg/m3 for n-octane  and vpiston ~ 0.2 m/sec during a double-band cycle of fluid flow. The characteristic linear dimension Lchar is the ratio of the volume of the fluid to the surface area of the walls that bound it , giving Lchar = 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 .
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  near 1:1, as here). Brody and Yager  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 NR < 0.001 – very close to our estimated NR = 0.001. Additionally, when fluid enters a narrow channel from a wider one, the flow profile is not immediately parabolic . For low Reynolds number fluid flow, the inlet distance for flow to become 99% fully developed is ~Rtube, but since Rtube (= 29.85 nm) > Ltube (= 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 ). Reynolds  found that the transition from laminar to turbulent flow typically occurs at NR > 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 NR ~ 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 NR >100 for the onset of turbulence, and maintenance of laminar flow as long as the “switching time” tswitch on which the pressure driving the flow varies (e.g., ~10-7 sec at 10 MHz) is longer than the characteristic time tchar = rfluid Rtube2 / hfluid (~10-9 sec for our assembler piston)  – 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 , and the syringelike T4 bacteriophage ejects ~200,000 nm3 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