**Nanomedicine,
Volume I: Basic Capabilities**

**©
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
energy^{10 }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

using the classical ideal gas formulation. At SNR = 2, a (22
nm)^{3} = 10^{4} nm^{3} sensor (~5 x 10^{5}
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 micron^{3} sensor (~3 x 10^{9}
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 mm^{3} with 10^{-3} atm sensitivity,^{456,457}
a factor of ~10^{12} poorer sensitivity than the theoretical minimum
for their size. (A field-effect chemical sensor with an active sensing volume
of 0.1 mm^{3} 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

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/m^{3} 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:

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