{Eqn. 4.10}
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
4.3.2 Velocity and Flow Rate Sensors
Typical thermal velocities of nanoscale diamondoid components at 310 K, from Eqn. 3.3, are ~60 m/sec for 1 nm3 objects, ~2 m/sec for (10 nm)3 objects, 0.06 m/sec for (100 nm)3 objects, and 0.02 m/sec for 1 micron3 objects. These thermal motions are in degrees of freedom whose range of motion is largely confined, by design, to sub-nanometer distances (Section 4.3.1) within nanoscale measuring devices. However, objects of nanomedical interest may have velocities ranging from ~10-9 m/sec (Section 9.4.4.2) to ~103 m/sec (e.g., the speed of sound in water), and may include biological elements in the environment or internal machine components. Thus it is useful to explore the fundamental limitations involved in measuring the full range of this physical variable.
There are at least two simple ways to determine a velocity. First, velocity may be measured by transferring the kinetic energy of a moving object or fluid volume to a sensor element. The movement of the sensor element is then clocked as it traverses a known distance. Second, and more efficiently, the passage of the moving object may sequentially trip two latches located a known distance apart, again revealing the moving object's velocity; alternatively, detecting the presence or absence of a protruding feature on a passing body using probe rods (e.g., Fig. 4.5) can determine velocity without any requirement for large kinetic energy transfers. The need for physical gating movements by sensor components and computational register-shift operations using diamondoid rods moving at ~GHz clocking speeds10 suggests a conservative minimum stable sensor cycle time of ~10-9 sec (Section 10.1).
The two principal restrictions on the maximum detectable velocity
vmax are the maximum linear dimension of the sensor element, or Lsensor,
and the minimum event time that may be accurately discriminated, or Dtmin.
For a sensor element with thermal noise energy DE
~ 10 kT = 43 zJ at 310 K, the quantum mechanical limit is Dtmin
>~h / (2 DE) ~ 10-15
sec. Dtmin is also limited by the thermal
variation in length of the individual latches. If the minimum detectable latch
displacement is Dx ~ 10 pm (Section
4.3.1), and the maximum speed of latch displacement is approximately the
speed of sound in the latch material (e.g., vsound ~ 17,300 m/sec
for diamond), then Dt ~ Dx
/ vsound ~ 10-15 sec, near the quantum mechanical limit.
Dt is further restricted by the minimum detectable
phase variance between latches located at either end of the sensor. An acoustically-transmitted
clock pulse emitted from a centrally-located source that is Dx
closer to one of the two latches can discriminate a phase variance of >~Dx
/ Lsensor ~ 10-5 taking Lsensor ~ 1 micron,
so the minimum discriminable time is again Dtmin
~ 10-5 Lsensor / vsound ~ 10-15
sec; hence vmax ~ Lsensor / Dtmin
~ 109 m/sec ~ c (= 3 x 108 m/sec, the speed of light).
However, unlike photons, physical objects moving in an aqueous medium are normally limited to the speed of sound in that medium,* e.g., vsound ~ 1500 m/sec in water at 310 K (Table 6.7). If vmax is taken as ~103 m/sec, then for a sensor of size Lsensor ~ 1 micron the required time discrimination is only Dtmin ~ Lsensor / vmax ~ 10-9 sec.
* Crystal dislocations,3042 high-speed bullets, detonation waves and other nanomedically-relevant phenomena can exceed the local speed of sound, but supersonic operations are hardly conservative and a turbulent medium adds additional sources of error to the measurement.
The minimum detectable velocity vmin is limited by the minimum measurable linear displacement Dx ~ 10 pm and the limits of our patience in making the measurement. For a maximum measurement time of tmeas ~ 1 sec, vmin ~ Dx / tmeas ~ 0.01 nm/sec in vacuo. In a nanomedically-relevant aqueous medium, the systematic motion of a free-floating (untethered) object of radius R is masked by added Brownian noise as defined in Eqn 3.1, giving the limiting vbrownian = DX / t >~ 260 nm/sec for R = 10 micron and vbrownian >~ 2.6 micron/sec for R = 100 nm, where t = tmeas = 1 sec. The thermal motions of tethered objects moving along surfaces (e.g., vesicles transported along microtubules by kinesin motors; Section 9.4.6) are more restricted, hence Brownian velocity masking is reduced.
Modest accuracy may be achieved using acoustic reflection in a Doppler velocimeter. Employing acoustic waves of frequency n, the minimum measurable velocity change Dv = vsound / (n tmeas) = 150 microns/sec in an aqueous medium at 310 K with vsound = 1500 m/sec, n ~ 10 MHz, and tmeas = 1 sec, corresponding to the shifted frequency n (1 + (Dv / vsound)) having enough time to make one extra cycle during tmeas. Even at such long wavelengths (~150 microns), significant backscattering of ultrasound from ~7.82-micron red cells is observed experimentally.3044
Velocity sensors may also be used to estimate volumetric flow rates in fluidic channels442 and blood vessels443 by measuring the speed at which fluid passes a nanodevice resting on the vessel wall. The parabolic velocity profile of v is a function of radial distance r from the center of a cylindrical vessel of radius R for Hagen-Poiseuille flow361 (Section 9.2.5):
where absolute viscosity h = 1.1 centipoise (1.1 x 10-3 kg/m-sec) for plasma at 310 K and DP is the pressure change along the length of a vessel of length Lv. A 1-micron nanodevice obtains measurements of v at various distances from the vessel wall, then fits these data to Eqn. 4.10 to find the choice of R that minimizes the variation in estimated DP / Lv, which should be roughly constant under nonaccelerative conditions. Once R is determined, DP / Lv is known as well as the cross-sectional area (pR2) of the vessel lumen, and a simple integration of the radial velocity distribution provides the flow rate in m3/sec. Unfortunately, the usefulness of this technique is limited by boundary layer effects (Section 9.4.2.6).
Last updated on 17 February 2003