© 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), T₀, to some warmer temperature T₁ causes the piston to move from position x₀ to position x₁ while the spring holds pressure constant at P inside the cylinder during the measurement. If Dx is the smallest measurable piston displacement and DT = T₁ - T₀, then the poorest accuracy is

{Eqn. 4.35}

because Dx has a thermal noise component as well,¹⁰ and the number of gas molecules N = n_d V_sensor where n_d ~ 10²⁸ gas molecules/m³ at P = 1000 atm (Table 10.2). Sensitivity is maximized at the largest feasible x₀; for DT / T₀ = 10^-6 (~300 microkelvins at 310 K) and Dx = Dx_min = 1 nm (Section 4.2.1), then x₀ = 1000 microns, N = 10⁹ 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 m³ and thus a cylinder cross-sectional area of A = V_sensor / x₀ = (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_box³/A_p) (DT / T₀) ~ (k T₀ / k_s)^1/2, where k_s = (A_p/L_box²) (N kT₀ / L_box²), from Drexler;¹⁰ hence

{Eqn. 4.36}

which argues for a small piston, a large box, and a high operating pressure to achieve maximum sensitivity. For A_p = 100 nm², N = 10⁹ molecules, and L_box = 464 nm, DT / T ~ 10^-6.

Last updated on 17 February 2003