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

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 (C_V) 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 (K_t) 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, C_V = 4.19 x 10⁶ J/m³-K and K_t = 0.623 W/m-K,⁷⁶³ hence t_EQ ~ (6.7 x 10⁶) L² (sec) for in vivo medical nanorobot refrigerators. Thus a cold box 1 mm wide requires t_EQ ~ 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 (C_V = 1.8 x 10⁶ J/m³-K) inside a nanorobot has a t_EQ ~ (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 v_fluid ~ L / t_EQ ~ 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