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

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

9.4.2.5.1 Surface Deformation

The use of flexible or metamorphic surfaces (Section 5.3) allows natative structures to deform asymmetrically. For example, it is impossible to "row a boat" that is fully submerged in a viscous fluid, because the stiff oars are simply reciprocating. But if the structures are flexible, then the oar can bend one way during the first half of the stroke and the other way during the second half, providing an asymmetry that permits the object to advance (Fig. 9.20). If the oar is metamorphic and can also vary its total surface area throughout the cycle, then this asymmetry can be enhanced.

Another example is the metamorphic doughnut-shaped nanorobot (Fig. 9.21A). An invaginating torus (e.g., a "smoke ring" motion) can swim along a vector normal to the torus plane because its outer surface is larger and moving faster than its inner surface (in the doughnut's "hole"), giving rise to a differential viscous force in the same direction as the inner surface is moving -- the opposite travel direction from what might be expected in an inertia-dominated environment. Differential rotation of circumaxial surface segments allows steering.

Yet another example proposed by Purcell is a single device constructed as two reversibly-adhered continuously-rotating spheroids (Fig. 9.21B). The spheroids stick together but are free to roll all over each other's surfaces, by selectively binding and releasing mateable surface elements. A differential viscous force arises because the outermost surface of the pair is always fully bathed in viscous fluid but the inner contact surface is partially shielded from the fluid. By altering speeds and rotational axes, the pair can establish a velocity vector in any direction in three-dimensional space. A similar locomotive method is provided by Solem's "viscous-lift helicopter" design (Fig. 9.22). Viscous drag on each of the four rotating wheels is F_wheel ~ (16 p / 3) h n_wheel R_wheel² for wheels of radius R_wheel turning at frequency n_wheel (Hz) in a medium of viscosity h, and device mass is m_nano ~ 2 p r_nano R_wheel³ for a device of density r_nano.¹⁹⁸² Taking n_wheel = 10 KHz, R_wheel = 1 micron, r_nano = 2000 kg/m³, and h ~ 10^-3 for water, then F_wheel ~ 170 pN per wheel, or ~0.7 nN for the whole device. There is a vast range of possibilities, according to Purcell:³⁸⁹ "Turn anything -- if it isn't perfectly symmetrical, you'll swim."

While incompressible tangential surface deformations alone are not propulsive, even single-sphere swimmers can translate or rotate using specific cyclic, nonreciprocal, compressible surface distortions (i.e., traveling waves). Such waves have been suggested as the mode of locomotion employed by appendageless spheroidal cyanobacteria,¹³⁸⁸ and surface undulations passing backward from the advancing edge of the cell have been observed in mammalian fibroblasts.¹⁴⁶⁷ Swimming speed for an object of radius R_nano is ~10 R_nano per second using a radial deformation of e ~ 0.05 (5%); with n = 10 surface ripples, energy efficiency is approximated by e% ~ (3 p² n / 128) e² = 0.006 (0.6%) for e << 1.¹³⁸⁸ (If the waves are not shallow, and (n e) > 1, then the peak-to-peak spacing becomes less than the peak-to-valley depth, whereupon friction from the vertical movement of the waves' walls should dominate energy consumption.)

A more familiar example of a deformable natation structure is the simple cilium (Section 9.3.1.1). In ciliate protozoa, cilia are typically much shorter than body length and are arranged in large numbers in rows on the body surface. Individual cilia have a regular beat pattern, typically including a rapid forward power stroke at full extension, followed by a slower return or recovery stroke with the cilium bent close to the body surface. Ciliary arrays (Section 9.3.4) may also display metachronism, or time-synchronized wave patterns, that can be used for precision steering and speed control, as, for example, by a 60 micron x 220 micron (~600,000 micron³) Paramecium caudatum, which rotates as it progresses forward in its path (Fig. 9.23) using the ~2500 cilia on its outer surface.⁵²⁶ Metachronal waves traveling in the same direction as the power stroke are symplectic; such waves are antiplectic if they travel against the power stroke, and are dexioplectic or laeoplectic (or, more generally, diaplectic) when the power stroke is to either side of the line of wave propagation.^1405,3555 The swimming paramecium may adjust its wave patterns while negotiating a liquid medium of varying viscosity.^1406,3558 These and similar results should be generalized and extended to the case of nanorobotic ciliary propulsion.

For a medical nanorobot coated with propulsive cilia with wave velocity v_cilium, then v_nano/v_cilium = 0.125-0.25 and e% = 0.125-0.30 (12.5%-30%).^1379,1380 If v_nano = 1 cm/sec, then v_cilium ~ 4-8 cm/sec giving p_shear ~ 2-6 N/m² (Section 9.4.2.2), well within the normal range for human blood and probably nonthrombogenic to platelets.

Ciliary propulsion even by large protozoa costs relatively little energy. For example, a 220-micron long paramecium swimming at ~1 mm/sec in water experiences ~1 nN of drag force and consumes P_drag ~ (1 pW / e%) of power to locomote, but the animal can be very inefficient because ~10,000 pW are available to a paramecium with power density ~10⁴ watts/m³ (est. from Table 6.8). Observed propulsion velocities of paramecia range from 0.2-2.5 mm/sec.^1460,3586

Last updated on 21 February 2003