**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

**9.4.2.2 Rotations and
Collisions in sanguo**

From Eqn. 9.62 describing
the axial fluid velocity v_{Pois} as a function of radial distance r
in laminar tube flow, the velocity gradient, or shear rate 'g,
is seen to increase linearly with r, as:

When a suspending liquid is subjected to laminar flow, the
fluid stresses on the surface of suspended rigid bodies cause those bodies to
rotate as they travel down the tube.^{1319}
Rigid free-floating spherical nanorobots will rotate with uniform angular velocity,
or, more specifically, with a rotational frequency of:

For example, a rigid free-floating spherical nanorobot traveling
in an artery with r_{tube} = 500 micron and v_{flow} = 100 mm/sec
(Table
8.2) rotates at n_{nano} ~ 63 Hz at r
= 495 microns (very near the vessel wall), which is much faster than the random
~0.1 Hz Brownian rotation typical for micron-scale nanorobots (Section
3.2.1). Non-spherical rigid particles^{1343}
and red cells^{1344} display variable
angular velocity but spend more time in each orbit aligned with the direction
of flow. Platelets can be seen tumbling in arterioles, especially near the vessel
wall,^{1340} but they tend to align
with the flow, a tendency which is strongest nearest the wall.^{1341}

Gyroscopic effects are present in every rotating body, with
pitch, roll, yaw, and turning producing torques which cause precessions, giving
periodic increases in bearing pressures and stresses. Fortunately, these stresses
remain modest in most nanomechanical designs. For example, consider a rapidly
spinning component inside the tumbling spherical nanorobot described in the
previous paragraph. This internal component is a diamondoid disk of radius r_{disk},
thickness h_{disk}, and density r_{disk},
spinning at an angular velocity w_{disk}
around an axis oriented in the plane of the nanorobot tumble (tumbling is at
frequency n_{nano}), and supported by two
coaxial bearings located a distance x_{bearing} apart. The gyroscopic
reaction force on the bearings is given by:

Taking r_{disk} = 100 nm, h_{disk} = 20 nm,
r_{disk} = 3510 kg/m^{3}, w_{disk}
~ w_{max} ~ 10^{10} rad/sec (Eqn.
4.17), n_{nano} = 63 Hz, and x_{bearing}
~ h_{disk}, then F_{bearing} ~ 2 pN.

In general, if two micron-scale objects are translating at
the same mean speed, more energy is dissipated by the one that produces the
larger amount of vorticity -- that is, an axisymmetric object that rotates as
it swims is less efficient than the nonrotating swimmer.^{1389}
The tangential force required to neutralize the hydrodynamic torque acting on
a rigid sphere of radius R_{nano} and tumble frequency n_{nano}
is torque divided by radius, or:^{1396}

^{ }
{Eqn. 9.69

For R_{nano} = 1 micron, n_{nano}
= 63 Hz, and h** **= 1.1 x 10^{-3} kg/m-sec
for plasma at 310 K, then F_{tumble} = 11 pN. Application of such small
forces may be controlled using measurements of absolute orientation taken from
an onboard nanogyroscope (Section 4.3.4.1). Given
that F_{bearing} and F_{tumble} are of similar magnitude, direct
anti-tumble gyrostabilization and gyroscopic rotational locomotion (gyrorotation)
may be possible in some nanorobot applications.

The velocity gradient often brings suspended particles into
close proximity. Kinetic theories of flowing suspensions have been developed
from experiments on model systems involving two-body collisions between rigid
and deformable spheres^{1345} and
between rigid cylinders.^{1343}
Goldsmith and Mason^{1312} estimate
that the per-object collision frequency K for equal-sized rigid spheres of volume
concentration c flowing at velocity v at a radial distance r from the center
of a tube of radius r_{tube} is:

However, no complete theory yet exists to describe all the interactions of multiple classes of rigid bodies and deformable cells in blood vessels.

As a crude approximation, consider the laminar flow of a
fluid suspension of two types of spherical objects (i = 1, 2) of radius R_{i},
density r_{i} (kg/m^{3}), number
density n_{i} (m^{-3}), and volumetric concentration c_{i}
= (4 p / 3) n_{i} R_{i}^{3}
expressed as a fraction. Each object occupies a cubic fluid volume of dimension
L_{i} ~ n_{i}^{-1/3} = (4 p
R_{i} /3 c_{i})^{1/3}, and in laminar flow, two radially-adjacent
boxes move past each other (a "collision") at a mean relative velocity of v_{ij}
~ 'g (R_{i} + R_{j}) when R_{i},
R_{j} << r_{tube}, and interact for a time t_{ij}
~ L_{i} / v_{ij} (j = 1, 2). From the viewpoint of object i,
the probability that after N_{coll ij} collisions at least one object
j has been encountered is p_{j} = 1 (1 c_{j})N_{coll
ij}. Taking p_{j} ~ 0.9 (the final result is relatively insensitive
to the exact threshold selected), and multiplying by 2 because there is a concentric
layer on either side of object i, then the mean collision rate experienced by
each object i with an object j also present in the suspension is K_{ij}
~ 2 (N_{coll ij} t_{ij})^{-1}, which, at a radial distance
r in the tube, may be written as:

and the mean free path (I_{ij}) between an object
i and an object j in the suspension (neglecting all velocity profile and margination
effects) is given approximately by:

For example, in a suspension consisting of R_{i}
= 1 micron nanorobots with volume concentration c_{i} = 0.10 (Nct =
10%), and R_{j} ~ 1.5 micron platelets with c_{j} = 0.0035 (mean
physiological plateletocrit), flowing through a small artery with r_{tube}
= 500 microns at mean velocity v_{flow} = 100 mm/sec, and at a radial
distance r = 495 microns (near the tube wall), then for nanorobot/platelet interactions,
K_{ij} ~ 2 collisions/sec, mean collision velocity v_{ij} ~
2 mm/sec, and I_{ij} = 9.7 microns. For nanorobot/nanorobot interactions,
K_{ij} ~ 40 collisions/sec (cf. K = 200 collisions/sec using Eqn.
9.70), v_{ij} ~ 1.6 mm/sec, and I_{ij} = 3.5 microns. These
figures are crude estimates at best, since they ignore many possible complicating
factors such as aggregation and margination, collision inelasticities and n-body
collisions (n > 2), nonspherical object shapes, the presence or absence of
specific macromolecules, receptor interactions, pulsatile flow, ionic strength
of the medium, and cell surface electrostatics.^{1325,3545}

Free-floating nanorobots that collide with blood vessel walls
produce negligible shear forces, given the no-slip condition at the wall. Powered
nanorobots of radius R_{nano} that impact a vessel wall at velocity
v_{nano} may apply a maximum shear stress of p_{shear} ~ r_{nano}
v_{nano}^{2}; taking r_{nano}
~ 1000 kg/m^{3} and v_{nano} ~ 1 cm/sec, then p_{shear}
~ 0.1 N/m^{2}. By comparison, the time-averaged shear stress for blood
circulation in normal vessels^{362,1346,1347,1352}
is 1-2 N/m^{2} ^{}(range 0.5-5.6 N/m^{2})^{386},
reaching up to 10-40 N/m^{2} when small arteries and arterioles are
partially occluded as by atherosclerosis or vascular spasm.^{1348,1349}
This may also be compared to the threshold limit for shear stress-induced platelet
aggregation of 6-9 N/m^{2},^{1346,1349-1351}
the shearing stress of 5-100 N/m^{2} acting at the interface between
a leukocyte and an endothelium when the leukocyte is adhering to or rolling
on the endothelium of a venule,^{366}
and the critical shear stress of 42 N/m^{2} known to initiate major
changes in the endothelial cells in the arteries.^{365}

Last updated on 21 February 2003