**Nanomedicine,
Volume I: Basic Capabilities**

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

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

**6.3.1 Thermal Energy
Conversion Processes**

The second law of thermodynamics says that it is impossible
to to convert heat into useful work if the heat reservoir and the device are
both at the same temperature, as demonstrated by Feynman's classical example
of the Brownian motor using an isothermal ratchet and pawl machine,^{2611}
although nonequilibrium fluctuations, whether generated by macroscale electric
fields or chemical reactions far from equilibrium, can drive a Brownian motor.^{696}
It has also been suggested that reversible-energy-fluctuation converters can
obtain useful electrical work from thermal Nyquist noise, up to power densities
of 10^{15}-10^{16} watts/m^{3} at ~300 K.^{1606,1607}

Of course, a reversible Carnot-cycle heat engine can extract
useful work from even a small temperature differential with a Carnot efficiency
of e% = DT / T. For example, a nanorobot circulating
with the blood between core and peripheral tissues may experience a temperature
variation up to several kelvins during each vascular circuit of duration t_{circ}
~ 60 sec (Section 8.4.1). From this small temperature
differential an ideal biothermal thermomechanical engine may extract a maximum
power:

where nanorobot thermal storage volume is V_{n} =
1 micron^{3}, heat capacity C_{V} = 4.19 x 10^{6} joules/m^{3}-K
for a device filled with water, T_{2} = 310 K at the human body core
and T_{1} = 307 K at the periphery. The thermal store, vacuum-isolated
to prevent heat loss (see below), is equilibrated in the hotter core environment
to T_{2}, which heat is then stored until the device reaches the cooler
peripheral environment at T_{1}. From Eqn.
6.10, this temperature differential yields at most P_{n} ~ 0.002
pW with efficiency e% = (T_{2} - T_{1}) / T_{2} ~ 0.01(1%)
and a peak (accessible) energy density of P_{n} t_{circ} / V_{n}
~ 10^{5} joules/m^{3}. The change in temperature can be made
to cause gas in a three-dimensional coiled piston to slowly expand or contract,
driving a rod back and forth thus providing a cyclical linear mechanical output,
a Stirling engine configuration. The gas expansion is isobaric and reversible
because thermal equilibration time t_{EQ} ~ V_{n} C_{V}
/ h K_{t} = 10^{5} sec for a conduction layer of thickness h
~ 0.5 micron and thermal conductivity K_{t} = 0.623 watt/m-K for water
at 310 K, so t_{EQ} << t_{circ}. (Exploiting the diurnal
variation in mean body temperature, typically ranging from 309.3 K in early
morning to 310.4 K in the evening, produces at most ~2 x 10^{7} pW of
power.) A nanorobot resting on the epidermal surface may exploit the temperature
differential between skin and air, up to 8-13 K (Section
8.4.1.1) giving a maximum Carnot efficiency e% ~ 0.04 (4%); for classical
radiative transfer (see below), the nanorobot develops a net power through an
L^{2} = 10 micron^{2} epidermal contact surface of P_{n}
~ s (T_{2}^{4} - T_{1}^{4})
L^{2} e% = 0.04 pW.

Nakajima^{541} has
built and operated a 50 mm^{3} Stirling engine working at 10 Hz between
273-373 K producing 10^{2} watts (power density 2 x 10^{5} watts/m^{3}),
and has demonstrated the theoretical engineering feasibility of microscale Stirling
engines. In 1993 Jeff Sniedowski of Sandia National Laboratories constructed
a 50-micron steam engine on a silicon chip producing forces ~100 times higher
than those of electrostatic motors of similar size.^{3486
}(The steam was produced electrically.) Computer simulations of a molecular-scale
steam engine have been performed by Donald W. Noid at Oak Ridge National Laboratory.^{3488}
A conservative and practical upper limit to nanorobot Carnot efficiency is probably
~50% (T_{2} = 620 K).

A heat engine may exploit the temperature difference between
the largely isothermal human body acting as a sink and a hot, high-capacity
source of stored heat energy. Because the rate of conductive heat loss is scale-dependent,
such exploitation is not feasible in nanodevices relying on stored heat sources
unless a vacuum isolation suspension is employed (Section
6.3.4.4). As a simple demonstration, consider a vacuum-isolated spherical
thermos bottle of inside radius r, coated with a material of total emissivity
e_{r} and filled with a hot working fluid of heat capacity C_{V}
at initial temperature T_{2}. Conduction and convection are eliminated;
heat loss in vacuo occurs only by radiative transfer. In the classical macro-scopic
formulation, radiated power P_{r} = 4 p r^{2}
e_{r} s (T_{2}^{4} - T_{1}^{4})
(watts), where s = 5.67 x 10^{-8} watts/m^{2}-K^{4}
(Stefan-Boltzmann constant). The thermal energy contained in the hot material
is H_{r} = (4/3) p r^{3} C_{V}
(T_{2}-T_{1}) (joules); hence the time required for half of
the energy to radiate away is:

For r = 1 micron, C_{V} = 4.2 x 10^{6} joules/m^{3}-K
for water, e_{r} = 0.02 for polished silver, T_{1} = 310 K inside
the human body, and T_{2} = 350 K up to 647 K (~critical temperature
of water at 218 atm pressure), t_{1/2} <~
6 sec (at T_{2} __>__ 350 K) starting from an initial (accessible)
energy density of ~10^{8} joules/m^{3}; H_{r} = 1.5
nanojoule for a 1-micron core at 647 K. Almost the entire thermal energy store
(~99%) leaks away in just 40 sec (at T_{2} = 350 K). Smaller thermos
bottles leak even faster, due to the ~r dependence of t_{1/2}.

Radiators lying within <1 micron of a lower-temperature
material surface exhibit near-field anomalous radiative transfer (Section
6.3.4.4 (E)) and thus exhibit different cooling characteristics than Eqn.
6.11 predicts. Taking P_{r} = P_{anomalous} from Eqn.
6.21 for spherical surfaces <200 nm apart, then t_{1/2}
~ 0.01 (h^{2} c^{2} / k^{3}) (r C_{V} / s_{cond}
T_{2}^{2}) (seconds); for T_{2} = 647 K and r = 1 micron,
t_{1/2} ~ 10 sec using a germanium shell
but ~10^{5} sec using a boron shell.

Other thermomechanical transducers include sandwich cantilevers
(Section 4.6.3) made of composite materials with high
coefficients of linear expansion (e.g., heated metal bimorphs^{547}),
Nitinol or other temperature-sensitive shape-memory alloys,^{548}
thermally-driven phase-change microactuators,^{545}
thermally-powered contraction turbines,^{597}
and thermally-driven contractile proteins.^{1261}
Thermochemical transducers that make use of thermal energy stored as a phase
change of a refrigerant can display energy densities of ~10^{8} joules/m^{3},^{1197
}and a thermoacoustic Stirling engine with no moving parts has been demonstrated.^{3267}

A thermoelectric transducer may be constructed from a crystal
with piezoelectric properties. When such a crystal is heated or cooled, charges
are produced on its surfaces (called pyroelectricity^{553}
or heat electricity) setting up mechanical strains in the crystal that produce
the same electrical effect as the application of external forces in piezoelectricity.^{551}
Both bone and tendon exhibit the pyroelectric effect;^{3088}
all pyroelectric materials are piezoelectric, though the converse is not true.^{3089}
Thermocouples are another example of direct thermoelectric energy transduction,
and thermophotovoltaic generators are well-known.^{1983}

A high emissivity blackbody radiator provides thermooptical transduction at temperatures >650 K, increasing in efficiency up to ~6000 K.

Last updated on 18 February 2003