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

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

4.5.1 Minimum Detectable Pressure

A change in the ambient pressure in a fluid environment (DP) may be determined by measuring the change in volume (DV) of a fixed quantity of gas at constant temperature, relative to a known sensor volume at a known reference pressure. (Thermal sensors elsewhere in the nanorobot provide readings by which corrections can be made for temperature variations.) If the sensor volume consists of a cylinder of fixed cross-sectional area, the change in volume is converted into a linear displacement of a piston located at the interface between gas and fluid environment. The change in Gibbs free energy¹⁰ is DG = DP DV, while thermal noise ~kT. Pressure sensor signal/noise ratio SNR ~ ln (DP DV / kT), so at the minimum DG the minimum detectable pressure DP_min is measured at the maximum possible volume change, or V_sensor (the sensor volume), given approximately by

{Eqn. 4.29}

using the classical ideal gas formulation. At SNR = 2, a (22 nm)³ = 10⁴ nm³ sensor (~5 x 10⁵ atoms or ~10^-20 kg for a cube with 1 nm thick walls) can detect a minimum pressure change DP_min ~ 0.03 atm; a (680 nm)³ = 0.3 micron³ sensor (~3 x 10⁹ atoms, ~6 x 10^-17 kg) detects DP_min ~ 10^-6 atm, probably near the practical limit for mobile in vivo medical nanodevices. Note that DP_min scales approximately as the inverse cube of mean sensor dimension, and immunity to thermal noise scales exponentially with sensor volume. Current silicon micromachined pressure sensors are ~1 mm³ with 10^-3 atm sensitivity,^456,457 a factor of ~10¹² poorer sensitivity than the theoretical minimum for their size. (A field-effect chemical sensor with an active sensing volume of 0.1 mm³ that measures pressure indirectly has detected a 10^-11 atm step in hydrogen gas with t_meas ~ 3 sec,⁴⁵⁸ only a factor of 1000 above the theoretical minimum for a sensor of that size.) The human ear can detect pressure waves as small as 2 x 10^-10 atm.

A less sensitive approach relies on the observation that a change in pressure P alters both density and bulk modulus of a compressed fluid, variations that may be detected by measuring the change in the speed of sound v_s in the working fluid, given by

{Eqn. 4.30}

where B is bulk modulus of the fluid and r is fluid density. (dB/dP)/B may be up to several times larger than (dr/dP)/r for molecular fluids, but there is some (dv_s/dP)/v_s for each fluid. Differentiating Eqn. 4.30 with respect to P and using dr/dP = r/B, then dv_s/dP = (dB/dP - 1) / 2 r^1/2 B^1/2. For water at 1 atm, dB/dP ~ 8.4 atm/atm,⁵⁶⁷ r ~1000 kg/m³ and B = 22,000 atm, giving dv_s/dP ~ 0.25 m/sec-atm. If Dv_s is the measured change in acoustic speed, then:

{Eqn. 4.31}

Velocity can be measured to ~0.1 mm/sec accuracy using a micron-scale velocity sensor and t_meas ~ 1 sec (Section 4.3.2), so a minimum Dv_s ~ 0.1 mm/sec implies DP_min ~ 0.0004 atm.

Last updated on 17 February 2003