© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

4.3.1 Displacement Sensors

Diamondoid parts may contain features, or be positioned, in picometer (1 pm = 10^-12 m) steps, much smaller than the unavoidable RMS thermal vibrations (Section 3.5.6). The nanomanipulator robot arm described in Section 9.3.1.4 may be moved in picometer increments, and the RMS thermal longitudinal displacements at the tip of a 20-nm long, 10-nm wide diamondoid lever is ~1 pm.¹⁰ Displacement sensitivity of 10 pm is routinely achieved in STMs. It is claimed this can be improved down to the ~1 pm level;⁴³³ STM resolution of 2 pm has been demonstrated experimentally.¹²⁶⁰ A C₆₀ molecule used as the active element in an electromechanical amplifier transmits ~100 times more electrical current when physically compressed by 100 pm,⁵⁶¹ suggesting a minimum deformation detection limit of a few picometers. (A strain gauge and a vibration sensor using a (17,0) carbon nanotube (~2000 atoms) has been proposed.²⁹⁰⁸)

Thermally-induced positional uncertainty in the displacement of nanoscale components is approximated¹⁰ by the classical value

{Eqn. 4.8}

where T is temperature (K) and k_s is the Hooke's law spring constant, or restoring force stiffness (N/m). At 310 K, this classical approximation is accurate to within <~10% of the quantum mechanical treatment for harmonic oscillators with RMS displacements Dx >~ 10 pm.¹⁰ Spring stiffness k_s is ~0.1 N/m for nonbonded (noncovalent) interatomic interactions, ~30 N/m for covalent bond angle bending, ~400 N/m for covalent bond stretching, and ~1000 N/m for solid 1 nm³ diamondoid blocks.¹⁰ At 310 K, springs of such stiffness produce minimum displacement uncertainties of 200 pm, 10 pm, 3 pm and 2 pm, respectively.

The log ratio of detection energy to noise energy, the signal/noise ratio (SNR), for a displacement sensor is derived from the harmonic potential (1/2) k_s x² as

{Eqn. 4.9}

Assuming a very stiff k_s = 600 N/m, a minimal SNR = 1 gives a minimum detectable displacement of 6 pm, or 10 pm at a more reasonable SNR = 2 (20 dB). The conservative conclusion is that Dx_min ~ 10 pm displacements should be reliably detectable by medical nanosensors in vivo, in a measurement time t_meas ~ 10^-9 sec (Section 4.3.2). This compares favorably with the stereocilia of the inner ear, which can only detect 100 pm displacements in 10^-5 sec.⁴⁴⁶

Last updated on 17 February 2003