© 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 [3265] by analogy to a rectangular 1.5:1 clamped plate [3266] or to a >2:1 clamped strip plate [3267]. The maximum out-of-plane deflection at the center of a rectangular plate of shortest edge L_plate, thickness h_plate, and Young’s modulus E_plate, when clamped on all four sides and loaded by uniform normal pressure DP_plate, is given by d_plate = k_plate (DP_plate L_plate⁴ / E_plate h_plate³), where k_plate is a constant of order ~0.1 [3266, 3267]. The worst deflection occurs in the longest short wall where L_plate = Y_ext = 110 nm, h_plate = w_thick = 10 nm, and E_plate = 1.05 x 10¹² N/m² for diamond. The highest applied pressure pulse is DP_plate = 3 atm during a control pulse, giving d_plate = 0.004 nm, or DP_plate = 2 atm during a power pulse, giving d_plate = 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 [3268] constructed of circular radius R_film, thickness h_film, Poisson’s ratio k_pois and Young’s modulus E_film, when clamped along the entire perimeter and loaded by normal pressure DP_film, is maximized at the center of the disk where radial and tangential strains are equal and the in-plane stretch displacement is given by d_stretch = 3 DP_film R_film³ (1 - k_pois²) / 8 E_film h_film². Taking DP_film = 3 atm, R_film ~ 0.5 X_ext = 110 nm, k_pois ~ 0.1, E_film = 1.05 x 10¹² N/m², and h_film = 10 nm, then in-plane wall stretch d_stretch = 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 L_plate [3267] is given by k_bending = E_plate h_plate³ / (12 L_plate² (1 – k_pois²)) and the positional uncertainty [208] of the plate surface due to thermal noise is d_bendingkT ~ (kT / k_bending)^1/2. Taking E_plate = 1.05 x 10¹² N/m², h_plate = 10 nm, k_pois ~ 0.1, k = 1.381 x 10^-23 J/molecule-K, T = 273.15 K, and L_plate ~110 nm, then k_bending = 7.3 N/m and d_bendingkT ~ 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 [3269], 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 v_terminal = 0.037 micron/sec for an assembler falling in liquid n-octane at 1 g (Section B.1) and the Einstein diffusion velocity v_diffuse ~ kT / 3p h_solvent L_assembler² ~ 58 microns/sec for a diffusion distance of one mean assembler dimension L_assembler ~ 100 nm in solvent with absolute viscosity h_solvent ~ 7.004 x 10^-4 Pa-sec (n-octane at 273.15 K) [3116] are dominated by the instantaneous assembler thermal velocity v_thermal ~ (3 kT / m_assembler)^1/2 = 64,000 microns/sec [228]. A collisional impact of kinetic energy E_impact = (1/2) m_assembler v_thermal² = 5.7 zJ (~2 kT) wherein the colliding assembler is brought to a halt in one atomic diameter (X_halt ~ 0.15 nm) imparts an impact force of F_impact = E_impact / X_halt = 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 (v_thermal << v_diamond = 17,300 m/sec [208]), the applied pressure may be crudely approximated as a uniform normal pressure on a clamped plate, of order ~F_impact / X_ext Y_ext ~ 0.015 atm << P_max = 3 atm, yielding a negligible out-of-plane deflection of d_plate ~ 0.00002 nm and a negligible in-plane wall stretch of d_stretch = 0.00001 nm.

Last updated on 13 August 2005