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.5 Wall Stiffness During External Acoustic Forcing, Thermal Noise, or Collision
The proposed molecular assembler in its initial state may be viewed as an evacuated hollow rectangular box surrounded by fluid which is cyclically pressurized between 1-3 atm. This subjects each of the box walls to an oscillating pressure load which may cause each wall to bow out of plane. Each wall of the box is integrally joined to adjacent walls along each edge, hence each wall may be analyzed mechanically  by analogy to a rectangular 1.5:1 clamped plate  or to a >2:1 clamped strip plate . The maximum out-of-plane deflection at the center of a rectangular plate of shortest edge Lplate, thickness hplate, and Young’s modulus Eplate, when clamped on all four sides and loaded by uniform normal pressure DPplate, is given by dplate = kplate (DPplate Lplate4 / Eplate hplate3), where kplate is a constant of order ~0.1 [3266, 3267]. The worst deflection occurs in the longest short wall where Lplate = Yext = 110 nm, hplate = wthick = 10 nm, and Eplate = 1.05 x 1012 N/m2 for diamond. The highest applied pressure pulse is DPplate = 3 atm during a control pulse, giving dplate = 0.004 nm, or DPplate = 2 atm during a power pulse, giving dplate = 0.003 nm, both of which are <0.03 atomic diameter and thus should not affect placement accuracy during mechanosynthesis.
The in-plane stress pattern in a polycrystalline diamond film pressure sensor  constructed of circular radius Rfilm, thickness hfilm, Poisson’s ratio kpois and Young’s modulus Efilm, when clamped along the entire perimeter and loaded by normal pressure DPfilm, is maximized at the center of the disk where radial and tangential strains are equal and the in-plane stretch displacement is given by dstretch = 3 DPfilm Rfilm3 (1 - kpois2) / 8 Efilm hfilm2. Taking DPfilm = 3 atm, Rfilm ~ 0.5 Xext = 110 nm, kpois ~ 0.1, Efilm = 1.05 x 1012 N/m2, and hfilm = 10 nm, then in-plane wall stretch dstretch = 0.002 nm or <10-5 nm per carbon wall atom, which is orders of magnitude too small to affect atomic placement accuracy during mechanosynthesis.
The assembler shell must also be stiff enough to provide a base for positioning the Stewart platform to within a fraction of an atomic diameter near room temperature in the face of thermal noise. The bending stiffness of a clamped square plate of edge Lplate  is given by kbending = Eplate hplate3 / (12 Lplate2 (1 – kpois2)) and the positional uncertainty  of the plate surface due to thermal noise is dbendingkT ~ (kT / kbending)1/2. Taking Eplate = 1.05 x 1012 N/m2, hplate = 10 nm, kpois ~ 0.1, k = 1.381 x 10-23 J/molecule-K, T = 273.15 K, and Lplate ~110 nm, then kbending = 7.3 N/m and dbendingkT ~ 0.02 nm or ~0.1 atomic diameter, thus satisfying our minimal shell stiffness requirement. Stiffness scales as the cube of the wall thickness, so a small increase in wall thickness would be sufficient to compensate either for the uncertainties in this estimate or to increase stiffness and further reduce positional uncertainty if this proves necessary.
The present assembler shell design – essentially comprised of six solid walls forming a rectangular box – was selected for simplicity and was not optimized for stiffness achieved per wall atom. At the cost of increased structural complexity, wall atom count could be reduced without sacrificing stiffness by introducing hollow regions into the wall as, for example, in honeycomb plate , allowing thicker walls and drastically increasing stiffness. Introducing a single floor to ceiling beam in the middle of the floor would greatly increase stiffness for a very modest increase in atom count. Numerous other methods exist for improving stiffness with a similar number of atoms or for maintaining the same stiffness with fewer atoms. We do not explore these approaches here because the present design appears sufficient for our purposes.
Finally, during manufacturing operations the assembler, suspended in liquid n-octane, may collide with other objects of like size (e.g., parent or daughter assemblers, product objects, etc.) or with the walls of the reaction chamber. Both the sedimentation terminal velocity vterminal = 0.037 micron/sec for an assembler falling in liquid n-octane at 1 g (Section B.1) and the Einstein diffusion velocity vdiffuse ~ kT / 3p hsolvent Lassembler2 ~ 58 microns/sec for a diffusion distance of one mean assembler dimension Lassembler ~ 100 nm in solvent with absolute viscosity hsolvent ~ 7.004 x 10-4 Pa-sec (n-octane at 273.15 K)  are dominated by the instantaneous assembler thermal velocity vthermal ~ (3 kT / massembler)1/2 = 64,000 microns/sec . A collisional impact of kinetic energy Eimpact = (1/2) massembler vthermal2 = 5.7 zJ (~2 kT) wherein the colliding assembler is brought to a halt in one atomic diameter (Xhalt ~ 0.15 nm) imparts an impact force of Fimpact = Eimpact / Xhalt = 38 pN; given the high stiffness of diamond and a velocity of onset of the point force far below the speed of sound in diamond (vthermal << vdiamond = 17,300 m/sec ), the applied pressure may be crudely approximated as a uniform normal pressure on a clamped plate, of order ~Fimpact / Xext Yext ~ 0.015 atm << Pmax = 3 atm, yielding a negligible out-of-plane deflection of dplate ~ 0.00002 nm and a negligible in-plane wall stretch of dstretch = 0.00001 nm.
Last updated on 13 August 2005