**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.4.3.3.2 Operational
Regimes Defined**

The interaction of the piston plate with the physisorbed n-octane
monolayer during the piston return stroke will be governed by the characteristic
diffusion time t_{diffusion||} of the adsorbed n-octane molecules parallel to the
surface. According to the well-known Einstein-Smoluchowski diffusion equation
[228], t_{diffusion||} ~ (Dx)^{2} / (2 D_{diffusion||})
where Dx is the distance diffused in a fluid having lateral diffusion constant
D_{diffusion||}. For an adsorbed monolayer, Dx may be taken as the mean intermolecular
separation distance within the monolayer, or Dx ~ a_{octane}^{1/2} ~ 0.77 nm for n-octane.
t_{diffusion||} is thus the mean time for a physisorbed molecule to diffuse one
intermolecular separation distance. Although no studies of any kind have been
done on the physisorption of n-octane on diamond surfaces, estimates from molecular
dynamics and other theoretical investigations have found, for example, that
D_{diffusion||} ~ 0.57 x 10^{-9} m^{2}/sec for 1.3 monolayers coverage of liquid n-octane
physisorbed on a-Al_{2}O_{3}(0001) at 300 K [3234],
D_{diffusion||} ~ 7.5 x 10^{-9} m^{2}/sec for an n-octane monolayer on Cu(111) surface
at 200 K [3235], D_{diffusion||} ~ 1.7 x 10^{-9}
m^{2}/sec for butane at 149 K or hexane at 215 K adsorbed on graphite [3236],
D_{diffusion||} ~ 2.1 x 10^{-9} m^{2}/sec for butane at 150 K on Pt(111) at close to
monolayer coverage [3237], D_{diffusion||}
~ 10 x 10^{-9} m^{2}/sec for 1.0 monolayer coverage of liquid benzene on the basal
plane of graphite at 250 K [3238], and
D_{diffusion||} ~ 2.8 x 10^{-9} m^{2}/sec for the translational self-diffusion coefficient
for liquid ethylene adsorbed on the basal plane of graphite at 75 K [3239].
Conservatively taking D_{diffusion||} ~ 1.0 x 10^{-9} m^{2}/sec for liquid octane adsorbed
on hydrogen terminated diamond at 300 K, then t_{diffusion||} ~ 0.3 x 10^{-9} sec.

The characteristic time required for the piston plate to travel
one intermolecular separation distance (Dx) is t_{piston} = Dx / v_{piston}. This
defines three distinct operating regimes for the piston relative to the surface-adsorbed
n-octane monolayer, as discussed further, below:

I. Desorption Regime: t

_{piston}<< t_{diffusion||}(~ 0.3 x 10^{-9}sec)

II. Transitional Regime: t_{piston}~ t_{diffusion||}(~ 0.3 x 10^{-9}sec)

III. Viscous Regime: t_{piston}>> t_{diffusion||}(~ 0.3 x 10^{-9}sec)

Last updated on 13 August 2005