© 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_tube² ~ 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_tube² / R_tube² 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 10⁵ N/m² (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 v² ~ 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 v²) 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/m³ 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_tube² / 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 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