© 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.4.3.3.4 Operation in the Viscous Regime

Again assuming double-band piston operation, a piston frequency of n_piston = 10 MHz << n_transitional = 128 MHz. Specifically, at 10 MHz, t_piston = 3.85 x 10^-9 sec >> t_diffusion|| (~ 0.3 x 10^-9 sec) and so the piston is found to be operating well into the viscous regime. When the piston is operated in the viscous regime, there is sufficient time for adsorbed molecules to laterally diffuse and escape from the approaching piston plate during the return stroke. As a result, the adsorbed molecules need not be mechanically desorbed from the surface in order to allow the piston plate to pass. Instead, the piston moves through the monolayer much like a solid object passing through a viscous fluid, and the drag power required for this motion can be crudely estimated from Stokes’ law (Section B.4.3.1), as follows.

First, according to the Einstein-Stokes equation [228], viscosity is inversely proportional to the diffusion coefficient. Since the self-diffusion constant D_bulk ~ 2.9 x 10^-9 m²/sec for bulk n-octane at 300 K [3234], then the effective lateral viscosity of the adsorbed octane monolayer is h_monolayer ~ h_bulk (D_bulk / D_diffusion||) ~ 1.57 x 10^-3 Pa-sec, taking h_bulk = 5.40 x 10^-4 Pa-sec for bulk liquid n-octane [3116] and D_diffusion|| ~ 1.0 x 10^-9 m²/sec for liquid octane adsorbed on hydrogen terminated diamond at 300 K (Section B.4.3.3.2). Second, the Stokes’ law drag force and drag power are both proportional to area^1/2. Taking monolayer thickness d_monolayer ~ 0.5 nm (Section B.4.3.3.1), then the effective Stokes radius for the portion of the piston surface in contact with the monolayer is R_monolayer ~ ((2Y_int + 2Z_int) d_monolayer / p)^1/2 = 6.2 nm. Hence the additional Stokes’ law drag power for the piston plate edge moving through the physisorbed n-octane monolayer at a viscous regime velocity v_piston = 0.2 m/sec is p_monolayer ~ 6 p h_monolayer R_monolayer v_piston² ~ 7.3 pW << p_assembler (= 56.8 pW; Section B.4.2), and so the piston moves easily through the monolayer (at low speeds) with only modest energy losses.

Note that while no-slip conditions at the walls are customarily assumed in analyses of low Reynolds number fluid flow regimes and Poiseuille flows [3164], there are cases where this assumption does not hold [3253]. Most notably, one molecular dynamics study [3254] of simulated liquid-phase constant-temperature linear alkane chains confined between atomistic pure titanium walls under shear flow conditions (walls translating in opposite directions) found that the alkane chains exhibited a tendency for slip at the walls, as evidenced by a bifurcated density profile, by small discontinuities in the velocity and temperature profiles at the wall-fluid interface, and by higher mean-square end-to-end distance along the shear flow direction.

To summarize: During double-band operation, the gross acoustic power input to the piston is p_assembler = 56.8 pW (Section B.4.2), which, when reduced by the bulk drag loss of p_Stokes ~ 12 pW (Section B.4.3.1) and the monolayer drag loss of p_monolayer ~ 7.3 pW, gives a net acoustic power input delivered to the piston of p_piston ~ 37.5 pW and an available force DF_piston = 2 p_piston / v_piston = 375 pN during the power stroke. During single-band operation, the gross acoustic power input to the piston of p_assembler = 14.2 pW (Section B.4.2) is reduced by the bulk drag loss of p_Stokes ~ 3 pW (Section B.4.3.1) and the monolayer drag loss of p_monolayer ~ 1.8 pW, giving a net acoustic power input delivered to the piston of p_piston ~ 9.4 pW and an available force DF_piston = 2 p_piston / v_piston = 188 pN during the power stroke.

Last updated on 13 August 2005