**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

**4.5.3 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 / L^{2} for a maximally sensitive spring having an energy of
~kT when stretched by L, and piston mass m, is

where m = r L^{2} h for
a piston of area L^{2}, height h, and density r.
Assuming h/L = 0.1 and r ~ 1000 kg/m^{3},
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 L^{2}
and mass m to which a pressure spike of amplitude P (N/m^{2}) 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_{driven}^{2} w_{res}^{2}))
where w = 2 p n, F_{spike}
= P L^{2} and m is defined above. The driving frequency that produces
stroke length X_{stroke} = L is given by:

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/m^{3}),
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_{sound}^{2}
(N/m^{2}), if r = 3510 kg/m^{3} 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 10^{9} N/m^{2}, r_{water}
= 1000 kg/m^{3}, 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