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

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

10.5.2 Viscosity and Locomotion in Ice

The viscosity of liquids generally declines with rising temperature following Andrade's formula (Section 9.4.1.1); water is ~6 times more viscous at 273 K (near freezing) than at 373 K (near boiling) (Table 9.4). Viscosity affects internal fluid transfers but also locomotion. Natation in liquid nitrogen should require relatively less power than in water, since liquid air (at 81 K) is only one-quarter as viscous as liquid water at 310 K (Table 9.4).

An equally relevant but far more serious challenge is locomotion through solid water ice. Just below freezing, crystalline ice viscosity is ~10¹⁰ kg/m-sec, requiring a 1-micron nanorobot to expend on the order of ~200,000 pW to creep forward at 1 micron/sec (Eqn. 9.73) by viscoplastic flow in which ice crystals are deformed without breaking. Just halfway from freezing to liquid nitrogen temperature, at 164 K, viscosity has already risen to ~10²¹ kg/m-sec, roughly equivalent to solid mantle rock, and the power requirement has increased 100-billionfold, clearly prohibitive. (Microdroplets of pure water may be supercooled in the liquid state to 235 K at 1 atm, or 181 K at ~2000 atm.²⁹⁶⁵).

One solution that avoids this problem at temperatures near the melting point is baronatation, which depends upon the fact that water, almost uniquely, is less dense as a solid than a liquid (i.e., ice floats), suggesting a freezing point depression effect with increasing pressure that is visible as the short downleg from 0°C to ­16°C in the phase diagram for Ice Ih (Figure 10.11). This is confirmed experimentally by suspending two heavy weights from a wire stretched across a block of ice. The wire passes slowly through the block, the wire exerting a pressure that melts a thin layer of water ahead of it, allowing the wire to progress; as the water passes behind the wire to the lower pressure region, it refreezes, a process known as regelation.¹⁶⁹⁷ The melting ice ahead of the wire absorbs the heat of fusion while the refreezing water gives up the heat of fusion, with heat steadily transferred by the wire, hence a good conductor cuts better than a poor one.¹⁶⁹⁷ The barostatic freezing point depression constant for ice is 134 atm/°C^390,2050 up to ~2100 atm. By exerting a higher pressure (force per unit area) ahead of it than behind it, a baronatating nanorobot of roughly conical geometry can progress slowly through ice that is no colder than ­16°C.

Taking the heat of fusion for water ice as DH_fus = 306 pJ/micron³ (334 J/gm at 0°C) and assuming at least ~3 micron³ of ice must be melted to allow 1 micron of forward progress, then baronatation power requirement is very conservatively estimated as P_baro ~ 3 DH_fus v_nano ~ 900 pW for v_nano = 1 micron/sec. (This is energy flow, which is not necessarily energy dissipation, since most of the heat loss will come from water refreezing at the rear of the nanorobot and only the losses from finite thermal conductivity must be made up.) Below ­16°C the ice would have to be heated to that temperature before melting can occur, requiring a probably prohibitive additional power dissipation P_heat ~ L_nano K_t DT ~22,000,000 pW to produce a DT = 10°C warming of size L_nano = 1 micron in ice assuming thermal conductivity of K_t ~ ~2,200,000 pW/micron-°C at -24°C. Bejan and Tyvand²⁹⁶¹ have analyzed gravity-induced pressure-melting of ice due to the passage of solid bodies having square, disklike, or cylindrical contact surfaces. The details of compression-driven phase transitions in ice are also being studied computationally using the tools of molecular dynamics.^2966,2967

The freezing point depression effect (Section 10.5.3) in which solute molecules are released ahead and recovered behind might serve as the basis for a similar drive system concept at temperatures just below the melting point.

Burrowing by progressive voids is yet another alternative that will work over a wider range of cold temperatures. The binding energy per hydrogen bond in the ice ahead is E_HBond = 33 zJ/bond (4.6 Kcal/mole,²⁰³⁶), there are two hydrogen bonds per water molecule, and n_water = 3 x 10¹⁰ water molecules/micron³ in ice at 273 K, so H-bond-breaking power is at most P_HBond ~ 2 E_HBond n_water L_nano² v_nano = 2000 pW for a nanorobot of dimension L = 1 micron and velocity v_nano = 1 micron/sec. (Again, this is a conservative estimate because it may be possible to recapture some of the energy that is liberated as the water molecules are returned to an ice lattice at the rear of the nanorobot.)* Additionally, a molecule handling device of an efficiency comparable to the telescoping manipulator arm (~10 zJ/nm per molecule; Section 3.4.3) that moves ice molecules a total distance ~10 nm to and from a conveyor device of efficiency ~10^-6 zJ/nm per molecule (Section 3.4.3) running a ~1 micron course consumes E_transport = 100.001 zJ/molecule, so molecule-moving power is at most P_move ~ E_transport n_water L_nano² v_nano = 3000 pW, hence total motive power for a 1-micron ice-burrowing nanorobot moving at 1 micron/sec is at most ~5000 pW. Moving ice in small chunks may be another energy-saving alternative.

* An additional complication is that the uppermost molecular layers of ice may not be fully frozen. Experiments by Van Hove and Somorjai²⁶⁹⁹ show that even as cold as 90 K, the amplitude of vibrational motions in the topmost surface water monolayer is several times that of the water molecules buried deeper in the bulk ice. The second molecular layer also has enhanced vibrational motion, but far less than the top monolayer. This excess motion is attributed to a lack of water molecules above the monolayer, hence the monolayer molecules have fewer motion-restricting hydrogen bonds to other water molecules than those of the layers beneath the surface. At 200 K, the amorphous film becomes thicker; above 230 K, the film becomes a quasi-liquid layer measured as ~12 nm thick at 249 K and ~30 nm thick at 268 K, rising to ~70 nm thick at 272.5 K.²⁷⁰¹

Last updated on 24 February 2003