**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
nm^{3} 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 micron^{3} 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 ~10^{3} 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 speeds^{10}
suggests a conservative minimum stable sensor cycle time of ~10^{-9}
sec (Section 10.1).

The two principal restrictions on the maximum detectable velocity
v_{max} are the maximum linear dimension of the sensor element, or L_{sensor},
and the minimum event time that may be accurately discriminated, or Dt_{min}.
For a sensor element with thermal noise energy DE
~ 10 kT = 43 zJ at 310 K, the quantum mechanical limit is Dt_{min}
>~ / (2 DE) ~ 10*h*^{-15}
sec. Dt_{min} 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., v_{sound} ~ 17,300 m/sec
for diamond), then Dt ~ Dx
/ v_{sound} ~ 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
/ L_{sensor} ~ 10^{-5} taking L_{sensor} ~ 1 micron,
so the minimum discriminable time is again Dt_{min}
~ 10^{-5} L_{sensor} / v_{sound} ~ 10^{-15}
sec; hence v_{max} ~ L_{sensor} / Dt_{min}
~ 10^{9} m/sec ~ c (= 3 x 10^{8} 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., v_{sound}
~ 1500 m/sec in water at 310 K (Table
6.7). If v_{max} is taken as ~10^{3} m/sec, then for a sensor
of size L_{sensor} ~ 1 micron the required time discrimination is only
Dt_{min} ~ L_{sensor} / v_{max}
~ 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 v_{min} 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 t_{meas} ~ 1 sec, v_{min} ~ Dx
/ t_{meas} ~ 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 v_{brownian} = DX
/ t >~ 260 nm/sec for R = 10 micron and v_{brownian}
>~** **2.6 micron/sec for R = 100 nm, where t
= t_{meas} = 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 = v_{sound}
/ (n t_{meas}) = 150 microns/sec in an aqueous
medium at 310 K with v_{sound} = 1500 m/sec, n
~ 10 MHz, and t_{meas} = 1 sec, corresponding to the shifted frequency
n (1 + (Dv / v_{sound}))
having enough time to make one extra cycle during t_{meas}. 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 channels^{442} and
blood vessels^{443} 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 flow^{361}
(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 L_{v}.
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
/ L_{v}, which should be roughly constant under nonaccelerative conditions.
Once R is determined, DP / L_{v} is known
as well as the cross-sectional area (pR^{2})
of the vessel lumen, and a simple integration of the radial velocity distribution
provides the flow rate in m^{3}/sec. Unfortunately, the usefulness of
this technique is limited by boundary layer effects (Section
9.4.2.6).

Last updated on 17 February 2003