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

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

4.5.3 Temporal Pressure Gradients

Driving the sensor at a frequency near n_res may cause large amplitude excursions unless the motion is heavily damped. The lowest possible undamped resonance frequency n_res for piston-type acoustic sensors of size ~ L with minimum spring constant k_s ~ 2 kT / L² for a maximally sensitive spring having an energy of ~kT when stretched by L, and piston mass m, is

{Eqn. 4.32}

where m = r L² h for a piston of area L², height h, and density r. Assuming h/L = 0.1 and r ~ 1000 kg/m³, n_res = 21 MHz for an L = 22 nm sensor (k_s = 1.8 x 10^-5 N/m) and n_res = 4 KHz for an L = 680 nm sensor (k_s = 1.9 x 10^-8 N/m). Resonance frequency may be adjusted by selecting appropriate k_s and m. In practical devices resonance will occur at a slightly lower frequency than n_res and with some broadening of the peak, because m should also include the masses of the shank of the piston plus a time-varying portion of the fluid mass occupying the piston's chamber (depending upon the time-varying position of the piston).

Consider the surface of a piston of height h, area L² and mass m to which a pressure spike of amplitude P (N/m²) and frequency n_driven is applied, producing a stroke length X_stroke. The solution for the equation of motion of an undamped forced harmonic oscillator has a maximum amplitude of X_stroke = F_spike / abs ( m (w_driven² w_res²)) where w = 2 p n, F_spike = P L² and m is defined above. The driving frequency that produces stroke length X_stroke = L is given by:

{Eqn. 4.33}

For an L = 22 nm sensor driven to full throw by a minimum detectable DP_min = 0.03 atm pressure spike (again taking h/L = 0.1 and r ~ 1000 kg/m³), n_driven = 34 MHz; n_driven = 45 GHz when the sensor is driven to full throw by the maximum possible pressure spike P_max = 39,000 atm consistent with water remaining in the liquid state (see below). The piston may be driven at these frequencies, or slower, using triangular-wave pulse trains of pressure spikes of these magnitudes.

For an L = 680 nm sensor driven to full throw by a minimum detectable DP_min = 10^-6 atm pressure spike, n_driven = 6 KHz; n_driven = 1.5 GHz when the sensor is driven to full throw by the maximum pressure spike P_max = 39,000 atm.

What is the maximum tolerable pressure spike in aqueous media? The requirement of subsonic piston motion in water gives the highest maximum: P_max < (1/2) r v_sound² (N/m²), if r = 3510 kg/m³ for a solid diamondoid piston and the speed of sound v_s ~ 1450 m/sec in water (Eqn. 4.30) for bulk modulus B_water = 2.2 x 10⁹ N/m², r_water = 1000 kg/m³, giving P_max < 39,000 atm. However, rapid dissipative pressure spikes P_max > 26,000 atm add sufficient energy to boil 310 K water, and isothermal static compression of pure water at 310 K causes crystallization into Ice VI near ~11,500 atm. Additional restrictions on (and features of) high-frequency pressure transducers are described in Sections 6.3.3, 6.4.1, and 7.2.2.

Last updated on 17 February 2003