Nanomedicine, Volume I: Basic Capabilities

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

10.5.4 Heat Conductivity and Capacity, and Refrigeration

Thermophysical characteristics of various materials at 310 K are given in Table 8.12, but what is the temperature dependency? Heat capacity (CV) generally rises with temperature. For example, the heat capacity of ice rises from 0.63 x 106 J/m3-K at -196°C (liquid N2) to 1.7 x 106 J/m3-K at -30°C.763 Thermal conductivity (Kt) may have a more complicated relationship with temperature. For example, the thermal conductivity of liquid water rises with temperature, from 0.561 W/m-K at 0°C to 0.681 W/m-K at 100°C.763 The thermal conductivity of sapphire normal to the optical or c-axis also rises from 2.3 W/m-K at 310 K to 6.0 W/m-K at 900 K, according to one source;2153 however, in the direction parallel to the optical axis, the thermal conductivity of sapphire rises from 0.3 W/m-K at 3 K to a peak of ~200 W/m-K near 70 K, then falls to ~30 W/m-K at 310 K and ~6 W/m-K at 1000 K;2154 conductivity for ruby falls from ~20 W/m-K at 310 K to ~6 W/m-K at 1000 K.2154 In the case of diamond, thermal conductivity rises from ~10 W/m-K at 3 K to a peak of 12,500 W/m-K at 69 K, then falls to ~2000 W/m-K at 300 K.2154

Is it possible to build micron-scale refrigerators? Refrigerators serve to maintain a temperature differential between an enclosed volume and the external environment. Leaving aside vacuum isolation levitation techniques (Section 6.3.4.4), thermal equilibration of a volume of size L by conduction* requires an equilibration time of approximately:

{Eqn. 10.24}

For water at 310 K, CV = 4.19 x 106 J/m3-K and Kt = 0.623 W/m-K,763 hence tEQ ~ (6.7 x 106) L2 (sec) for in vivo medical nanorobot refrigerators. Thus a cold box 1 mm wide requires tEQ ~ 7 sec, but a 1 micron3 box equilibrates in only ~7 microsec. Diamond is far more conductive; a 1 micron3 cold box embedded in a surplus of diamondoid structure (CV = 1.8 x 106 J/m3-K) inside a nanorobot has a tEQ ~ (900) L2 = 0.9 nanosec equilibration time (giving a ~1011 kelvin/sec cool rate, assuming DT ~ 100 K). Thus in order to avoid thermal re-equilibration with the surroundings, a ~1 micron3 refrigeration mechanism must circulate working coolant fluid at a velocity vfluid ~ L / tEQ ~ 1000 m/sec, very near the speed of sound in most fluids, hence sub-ambient refrigerators smaller than ~1 micron3 are not feasible using this method.

* Also, the nature of radiative transfer (Section 6.3.4.4 (E)) changes drastically as the size of the box gets close to the wavelength of the peak of the distribution, e.g., 3-30 microns (100-1000 K; Eqn. 6.20) with heat transfer rates enhanced by up to several orders of magnitude,652,653 making sub-ambient cooling much more difficult.

Drexler10 proposes a working fluid consisting of encapsulated submicron water ice particles with surface structures preventing aggregation and flexible enough to allow repeated expand/contract cycles as the contained ice alternately freezes and thaws, combined with a low-viscosity, low-melting point carrier such as a light hydrocarbon. The heat absorbed by a substance that melts at constant temperature (the melting point) is the heat (or enthalpy) of fusion; the much larger heat required to boil a liquid is the heat (or enthalpy) of vaporization (Table 10.8); the heat of sublimation is simply the sum of the two.390,2050 Phase changes provide the most efficient cooling -- ice absorbs 306 pJ/micron3 of heat when it melts at 0°C, and water absorbs 2170 pJ/micron3 when it boils at 100°C, but water at 37°C absorbs only 42 pJ/micron3 when it warms by 10°C. The exact temperature of a reversible phase transition in a refrigerant working fluid can be precisely controlled by judicious selection of operating pressures and fluid materials. Thermal-driven phase-change microactuators have been tested.545

Many other possible refrigeration technologies are known but have yet to be investigated in the context of nanorobot refrigeration, including "magnetic cooling" by adiabatic demagnetization1031 or magnetocaloric refrigerators,2159 Seebeck effect or Peltier effect electronic cooling,1034,1035,2160,2933 thermoacoustic refrigeration (Section 6.3.3), optical refrigerators,549 chemomechanical turbines operated in reverse,597 heat of solvation cooling mediated by molecular sorting rotors (e.g., KMnO4 cools solvent water by 44,000 J/mole, which is ~750 pJ/micron3 or ~73 zJ/molecule, as it dissolves,763) heat of allotropic-transition cooling (e.g., transition from red a-HgI2 to yellow b-HgI2 absorbs ~13,000 J/mole or ~180 pJ/micron3,2036) and acoustic, polymeric, or mechanical prevention of ice crystallization during supercooling of working fluid.

Last updated on 24 February 2003