© 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:

{Eqn. 9.66}

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:

{Eqn. 9.67}

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 particles¹³⁴³ and red cells¹³⁴⁴ 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:

{Eqn. 9.68}

Taking r_disk = 100 nm, h_disk = 20 nm, r_disk = 3510 kg/m³, w_disk ~ w_max ~ 10¹⁰ 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 spheres¹³⁴⁵ and between rigid cylinders.¹³⁴³ Goldsmith and Mason¹³¹² 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:

{Eqn. 9.70}

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/m³), number density n_i (m^-3), and volumetric concentration c_i = (4 p / 3) n_i R_i³ 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:

{Eqn. 9.71}

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:

{Eqn. 9.72}

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_nano²; taking r_nano ~ 1000 kg/m³ and v_nano ~ 1 cm/sec, then p_shear ~ 0.1 N/m². By comparison, the time-averaged shear stress for blood circulation in normal vessels^{362,1346,1347,1352} is 1-2 N/m² (range 0.5-5.6 N/m²)³⁸⁶, reaching up to 10-40 N/m² 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/m²,^{1346,1349-1351} the shearing stress of 5-100 N/m² 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/m² known to initiate major changes in the endothelial cells in the arteries.³⁶⁵

Last updated on 21 February 2003