© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

9.4.2.5.2 Inclined Plane

Another class of mechanisms for natation makes use of the inclined plane, a basic mechanical device that can convert viscous forces into forward motion. The simplest example is a threaded screw. In standard propeller theory at high Reynolds number, forward thrust is proportional to the rate at which a mass of fluid can be ejected out the rear (e.g., inertial forces). However, at low Reynolds number, the fluid that is pushed backwards by the rotating tilted planes does not provide thrust primarily by its inertial movement, but rather serves as a resistive medium against which the device can push itself forward. In the world of the nanorobot, the environment is very thick and viscous. The motive effect is not unlike the forward motion achieved by a threaded screw as it is screwed into a piece of wood using a screwdriver.³⁵⁸⁰

The motive force and power consumption of a microscale screw drive (Fig. 9.24) with pitch angle j and mean radius R_screw may be very crudely approximated as follows. Consider a helical ribbon of width w_thread and total length l_thread that is wrapped around an axially-translating cylindrical body, making a pitch angle j as measured from normal to the direction of travel of the screw body. From Stokes law (Eqn. 9.73), a square element of that ribbon with area w_thread² experiences a maximum drag force of ~6 p h w_thread v_thread. There are n_element = l_thread/w_thread square elements in the entire ribbon; neglecting flow field interactions of the elements and of the solid center for this approximation, the maximum laminar drag force on the entire ribbon is F_max ~ 6 p h l_thread v_thread. Viscous drag is lowest at j = 0° (edge on) and highest at j = 90° (face on); a factor of (3 - cos(2j))/4 captures the experimental behavior of needle-shaped bodies which fall in viscous media about half as fast sideways as they do end-on (Section 9.4.2.4), with periodicity of p. The number of threads around the screw is N_thread = l_thread cos(j)) / (2 p R_screw) and the screw rotates at a frequency n_screw = v_thread / (2 p R_screw), hence the total force required to turn the screw is:

{Eqn. 9.76}

with total drag power P_screw ~ F_screw v_thread = 12 p⁴ h n_screw² N_thread R_screw³ (3 - cos(2j)) / cos(j). To further simplify the calculation, we assume a "no slip" condition such that each complete revolution of the screw carries the nanorobot forward by a distance ~ 2 p R_screw tan(j), although more slip may occur as the thread becomes looser (e.g., at high j and low N_thread). To avoid turbulence in fluid passing through the threads, from Eqns. 9.29 and 9.65, we must require v_thread << 2000 h / r L, a condition easily met for L ~ 1 micron devices. Under "no slip" conditions, the velocity of forward translation is approximated by v_nano ~ 2 p R_screw n_screw tan(j), giving from Stokes law a net forward towing force of F_nano ~ 6 p h R_screw v_nano and a net mechanical efficiency e% ~ 2 cos(j) tan²(j) /[p N_thread (3 - cos(2j))].

Taking R_screw = 1 micron, w_thread = 0.1 micron, N_thread = 1 turn, j = 60°, h = 1.1 x 10^-3 kg/m-sec for plasma at 310 K, and n_screw = 920 Hz, then v_nano = 1 cm/sec, F_nano ~ 200 pN, v_thread = 0.6 cm/sec, total power requirement is P_screw ~ 7.6 pW, efficiency is e% ~ 0.27 (27%), and the pressure at the screw thread surface is p_thread = F_screw / (l_thread w_thread) ~ 10³ N/m² << ~5 x 10⁵ N/m² (required to induce transient cavitation in water at this frequency; Section 6.4.1). The outside edge of the screw thread is blunted to minimize energy transfer to impacted biological blood elements. A second counterrotating reverse-threaded screw mounted coaxially doubles the motive force while reducing net viscous torque on the natator to zero. If each screw of the screw-pair is mounted on gimbals, the nanorobot can achieve controlled translation and rotation in any direction in three-dimensional space; changing leading screw rotation from clockwise (CW) to counterclockwise (CCW) enables the nanodevice to reverse direction or to undertake more complex motions.

Another well-known instance of the inclined plane in locomotion is the corkscrew drive (Fig. 9.25), of which the bacterial flagellum is the most familiar biological example.^{216,581,1395,1397} The flagellum works because of the differential viscous forces felt by thin cylinders passing through fluid at various angles of attack (e.g., because F_nanoN # F_nanoP; Section 9.4.2.4). The typical bacterial flagellum is a closely-packed rigid helix ~20 nm in diameter (with a ~3 nm flagellin protein core), and its length is almost always more than 100 times its thickness,³³⁸ up to 10 microns long. The bacterial flagellum is turned by a ~0.0001 pW motor that rotates up to 300 Hz at 310 K (~15 Hz under load) and can reverse its direction of rotation in ~1 millisec (Section 6.3.4.2). The bacterium typically uses about 0.1% of its available metabolic energy (under growth conditions) to run the flagellum.⁵⁸¹ Forward motion may be achieved using either planar waves or (more efficient) spiral/helical waves. The highest swimming speed attainable by a flagellate (body length up to 50 microns with flagella >100 microns long in eukaryotes) is ~50% of the front-to-back wave speed of its flagellum,¹³⁸⁰ although ~20% is more typical.¹⁴⁰¹ Measured swimming speeds are up to 100-200 microns/sec in various sperm species.¹⁴⁴⁹ The flexural rigidity of the bull sperm flagellar tail is 30 x 10^-21 N-m² in the rigor state, and 2 x 10^-21 N-m² in the more flexible state in the presence of ATP;¹⁴⁵¹ see Eqn. 8.6. Energy efficiency may vary widely (Section 9.4.2.4).

Last updated on 21 February 2003