**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

**4.6.3 Thermal-Expansion
Temperature Sensors**

Consider a thin rod of length x and width w, constructed as
a sandwich of two dissimilar materials each of thickness h, having coefficients
of linear expansion a_{1} and a_{2},
respectively (Fig.
4.8). When heated, the two materials expand differentially to lengths of
x (1 + a_{i} DT)
(neglecting the quadratic and higher-order terms in a_{i}), producing
a cantilever deflection of q_{t} = (x/h)
(a_{1} - a_{2})
(radians) from simple geometry. The minimum detectable temperature change DT_{min}
= 2 p Dx_{min} / x q_{t}
or

where volume of the sandwich rod V_{rod} = 2 h w x.
Taking Dx_{min} = 10 pm, a_{1}
= 2.38 x 10^{-5} /K for aluminum,^{460}
a_{2} = 8 x 10^{-7} /K for diamond
at 310 K,^{460} h = w = 5 nm, and
assuming sensor volume V_{sensor} ~ 2 V_{rod} in a tightly packed
three-dimensional coil configuration, DT_{min}
= 1 microkelvin for an L^{3} ~ (227 nm)^{3} sensor ~ 1% of the
volume of a 1-micron nanorobot. Use of a suitable negative-coefficient material
for a_{2}, such as the ceramics zirconium
tungstate^{1025} and cordierite,
can further reduce L by 5%-10%.

Measuring the volume of the sensor is essentially like taking
an inventory of total stored thermal energy, which has a kT N^{1/2}
fluctuation term from adding the N uncorrelated kT fluctuations; hence

Hence, to achieve DT ~ 1 microkelvin
at T_{0} = 310 K with this sensor may require N_{meas} ~ 47
million independent measurements of Dt_{min}
= 1 nanosec each, or t_{meas} ~ 47 millisec.

Micromechanical silicon cantilever heat sensors (microcalorimeters)
of ~20,000 micron^{3} volume have already achieved DT_{min}
= 10 microkelvins at room temperature (DT_{min}
/ T ~ 3 x 10^{-8}), responding to ~1 pJ of heat in a measurement time
t_{meas} ~ 1 millisec.^{461}
Thermal-expansion nanosensors might also exploit the temperature sensitivity
of viscoelastic materials (e.g., modulus of relaxation).

Last updated on 17 February 2003