**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

**4.6.2 Piston-Based Temperature
Sensors**

Consider a coiled cylinder of cross-sectional area A filled
with gas at pressure P, with a piston at one end whose movement is resisted
by a constant-force spring (Fig.
4.7). Increasing gas temperature from the coldest temperature at which the
sensor will operate (piston at maximum ingress), T_{0}, to some warmer
temperature T_{1} causes the piston to move from position x_{0}
to position x_{1} while the spring holds pressure constant at P inside
the cylinder during the measurement. If Dx is the
smallest measurable piston displacement and DT =
T_{1} - T_{0}, then the poorest accuracy is

because Dx has a thermal noise
component as well,^{10} and the number
of gas molecules N = n_{d} V_{sensor} where n_{d} ~
10^{28} gas molecules/m^{3} at P = 1000 atm (Table
10.2). Sensitivity is maximized at the largest feasible x_{0}; for
DT / T_{0} = 10^{-6} (~300 microkelvins
at 310 K) and Dx = Dx_{min}
= 1 nm (Section 4.2.1), then x_{0} = 1000 microns,
N = 10^{9} gas molecules at 1000 atm (using the van der Waals equation
of state) and 310 K taking N_{meas} = 1000, giving V_{sensor}
= 10^{-19} m^{3} and thus a cylinder cross-sectional area of
A = V_{sensor} / x_{0} = (10 nm)^{2}. There are ~1000
gas molecules per nanometer of cylinder length, each traveling with mean thermal
velocity v_{t} ~ 500 m/sec (Eqn. 3.3).
Tightly coiled into a cubical volume, the folded sensor size is L_{sensor}
~ V_{sensor}^{1/3} ~ (464 nm)^{3} and measurement time
t_{meas} = N_{meas} Dt_{min}
~ 1 microsec. Sensor mass is ~10^{-16} kg.

The coiled tube design, though not strictly necessary, is
nonetheless convenient because it allows a continuous sensor element to be arbitrarily
distributed through the nanorobot volume, to be concentrated in multiple specific
interior regions, or to be placed inside nanorobot components that will be required
to flex such as metamorphic protuberances. If instead the working gas is placed
in a cubical box of side L_{box} with a sensor piston of area A_{p},
any change in temperature causes the piston to move a distance Dx
= (L_{box}^{3}/A_{p}) (DT
/ T_{0}) ~ (k T_{0} / k_{s})^{1/2}, where k_{s}
= (A_{p}/L_{box}^{2}) (N kT_{0} / L_{box}^{2}),
from Drexler;^{10} hence

which argues for a small piston, a large box, and a high operating
pressure to achieve maximum sensitivity. For A_{p} = 100 nm^{2},
N = 10^{9} molecules, and L_{box} = 464 nm, DT
/ T ~ 10^{-6}.

Last updated on 17 February 2003