# Graeme A. Bird^{*}

University of Sydney, Sydney N.S.W. 2006, Australia

# Abstract

Direct simulation methods have been used to model the motion of initially stationary microscopic spheres following their release into the atmosphere. The spheres have diameters of the order of 10 nm and acquire energy from the slightly cooled air as they develop Brownian motion. While the individual trajectories vary wildly, an ensemble average over tens of thousands of trajectories yields a classical relaxation curve. The initial energy input from the air to a sphere is determined as a function of its diameter and mass. The energy flux to and from the fluctuations is also calculated along with the inward heat flux at the molecular level. The values of these quantities indicate that, while a sufficiently large number of sufficiently small devices might extract useful energy from air, the requirements with regard to size are beyond current nano-technology.

# Keywords

Air, Energy source, Brownian, Relaxation

# Introduction

Any suggestion that it might be possible harvest energy from air in order to do useful work that eventually returns that energy to the atmosphere would place the author at risk of suffering the fate predicted by Eddington [1] that “…the Second Law of Thermodynamics holds….the supreme position among the Laws of Nature. If…your theory is found to be against the second law of thermodynamics, I can give you no hope; there is nothing for it but to collapse in deepest humiliation”. This is especially the case because the suggestion came to prominence more than a hundred years ago when Smoluchowski [2] proposed the Brownian ratchet and the intervening period has seen many papers that have presented arguments against the proposal. It has become so well known that the term “Brownian ratchet” is now applied [3] to any device that is driven by microscopic fluctuations even when it is unrelated to Brownian motion and does not involve a ratchet.

The haphazard motion of small particles that are suspended in a liquid was first observed by Brown in 1827. This motion was not understood until the early 20th century when the underlying theory and observations [4] not only led to the overdue acceptance of the atomic nature of matter, but provided estimates of the magnitudes of the molecular properties such as Avogadro’s number. The realization that any macroscopic system contains an astronomical number of molecules also led to statistical interpretations of the second law. However, it is only at the start of the 21st century that the validity of second law has been seriously questioned [5, 6], especially for mesoscopic systems that contain only a few thousand molecules.

The conventional theory [7] of Brownian motion assumes thermal equilibrium between the particle and air and applies only to fully developed or equilibrium motion. It predicts that the average displacement of a particle from an arbitrarily chosen initial location is proportional to the square root of the elapsed time from the instant when the molecule was at that location. This means that the “diffusion velocity” is inversely proportional to the square root of the elapsed time. Note that the diffusion velocity relates to the trajectory of the particle and is distinct from the actual velocity which is sometimes called the ballistic velocity. The average values of both the diffusive and ballistic speeds are subject to fluctuation, but the mean value of the ballistic speed does not vary with time and may be regarded as a “steady” quantity. The dependence of the diffusion velocity on time often leads to confusion and it must be emphasised that the starting point for the elapsed time may be at any time after the establishment of fully developed Brownian motion.

A particle in a liquid is simultaneously influenced by a large number of atoms or molecules and the Brownian motion is a result of variations in the net force on the particle. Brownian motion also occurs in a gas but there is then a very short time interval between the successive molecule impacts and the particle executes a variable-step random walk in threedimensional space. The characteristic distance at the molecular scale in a gas is the average distance that a molecule moves between intermolecular collisions. This mean free path in air at the standard temperature and pressure is 49 nm so that the average number of molecules in a cubic mean free path is 3,856. The ratio of the mean free path to the diameter of the particle is called the Knudsen number. A particle with a Knudsen number well above unity together with the adjacent air molecules that maintain its Brownian motion occupy less than a cubic mean free path and constitute a mesoscopic system.

A particle in fully established Brownian motion is in
thermal equilibrium with the air and both the translational
and rotational velocities have a Maxwellian distribution such
that the mean energy in each degree of freedom is ,
where *k* is the Boltzmann constant and *T* is the temperature.
At each collision, either the air molecule does work on the
particle or the particle does work on the molecule. The energy
of the particle fluctuates and there may be a net increase or
decrease over a period of time that is very large in comparison
with the mean collision time. Experiments with micron-sized
latex spheres in water have shown [8] that the intervals over
which there is a net increase in particle energy can be as long
as a few tenths of a second. While this has been claimed to be
a violation of the second law, a similar decrease is equally likely
and the average energy is constant.

A stationary particle has zero kinetic energy and, if it
has temperature *T* and is suddenly immersed in air also at a
temperature *T*, a finite time is required for particle to develop
Brownian motion. As noted above, it gains an average of
per degree of freedom and this energy comes from the nearby
molecules that are cooled to a temperature slightly below *T*.
This contradicts the Kelvin statement [9] of the second law
that: “It is impossible, by means of inanimate material agency, to
derive mechanical effort from any portion of matter by cooling
it below the temperature of the coldest of the surrounding
objects”. There is a net transfer of energy from the thermal
motion of air to a particle only while it is not in equilibrium
with the air. Should there be a process to transfer energy from
the particle, it would not come to equilibrium and the energy
transfer could be continuous. There are no such processes for a
free and inert particle, but if piezoelectric nanowires [10] could
be produced with diameters of a few nanometers rather than
the microns of existing wires, continuous energy could almost
certainly be produced. This statement is based on quantitative
studies [11, 12] of the initiation of the Brownian motion of a
spherical particle. These studies are described in Section II and
permit the informed speculation in Section III about whether
or not it will ever be possible to harvest useful energy from air.

# Initiation of the Brownian Motion of a Spherical Particle

The development of Brownian motion by an initially
stationary sphere is not amenable to mathematical analysis
but, because it involves the interaction of the particle with
a limited number of molecules, it is an ideal subject for a
probabilistic simulation of the physics. A direct Monte Carlo
simulation is particularly straightforward when the Knudsen
number based on the sphere diameter is large in comparison
with unity. There is then little likelihood that the molecules
that are incident on the sphere have been affected by the
reflected molecules and it is a called a “free-molecule” or
“collisionless” flow. The trajectory of the sphere is calculated
as it interacts with a representative set of incident molecules.
The random impact points are generated with a probability
that depends on the velocity of the sphere and the properties
of the incident molecules are sampled from their known
distributions in an equilibrium gas at temperature *T*. The
initial study [11] assumed elastic collisions between the air
molecules and the particle, while the second employed the
more realistic diffuse model for the gas-surface interaction.
Diffuse reflection is inelastic in that the sum of the molecule
and particle energies is not conserved. There is, therefore,
heat transfer to or from the particle at each collision and its
temperature fluctuates. Also, diffuse reflection exerts a shear
stress as well as a normal force, or pressure, on the particle so
that it develops a fluctuating rotational as well as translational
velocity. There are three rotational degrees of freedom in
addition to the three translational degrees of freedom so that
the average translational and rotational energies are equal.

A stationary sphere of given mass and diameter was
exposed at zero time to air at standard temperature and
pressure and was then subjected to representative molecular
collisions. Most calculations were made for a sphere with a
Knudsen number *Kn*=10 and a specific weight equal to that of
water. The sphere in this “reference case” has a diameter of 4.9
nm and a mass of 6.16 x 10^{-23} kg, while an average air molecule
has an effective diameter of 0.415 nm and a mass of 4.81x10^{-26}
kg. The mass ratio . is defined as the ratio of the particle
mass to the average molecular mass and is equal to 1281. A
large number of independent trajectories were calculated and
there were very large variations in the history of the particle
energies. The particle is subject to an average of 264 molecular
impacts per nanosecond and the colored lines in Figure 1
shows the translational energy of a 10 typical particles over
the first 20 ns. The particle energy is divided by (3/2) k so that
it is expressed as degrees Kelvin. It should be emphasised that
this is the “temperature” that corresponds to the kinetic energy
and is unrelated to the actual temperature of the particle. This
is assumed to be uniform and equal to the air temperature of
273 K at zero time and, because energy is not conserved in
an inelastic collision, it fluctuates by a few degrees during the
subsequent motion.

The wide spread in the particle energies at any instant is evident in Figure 1. An ensemble average must be made over tens of thousands of particles before the fluctuations cease to be noticeable. The black line in Figure 1 is the average of 100,000 particle trajectories and exactly fits a classical exponential relaxation curve leading to the uniform 273 K that is expected from equipartition. The relaxation time τ based on the initial slope of this curve is 2.1 ns and the mean initial power input from the gas to the particle is given by the equilibrium energy divided by this time, or 0.0086 nW. Other cases were calculated and the results fit the empirical equation

where is the ratio of the particle diameter to the effective molecular diameter.

The moment of inertia of the particle assumes a solid sphere and the relaxation time for rotational energy was slightly faster than that for translational energy. The moment of inertia for a hollow sphere is larger by a factor of 5/3 and, for the same mass, the relaxation time increases by the same factor.

The particle energy *E* increases when the air does work on
the particle and decreases when the particle does work on the
air. The energy gain or loss may be sampled at each collision
and the power imparted to the particle by the fluctuations is
given by summing over the collisions that involve an energy
increase and dividing by the time over which the summation
is made. The average power imparted to the particle is

and a similar sum over the negative energy changes determines
average power imparted by the particle to the air. Note that
the sum of the energy transfers is proportional to time so that
the result is independent of Δ*t* . While the choice of time
step is arbitrary, it affects only the sampling interval between
the data points. Figure 2 shows the value from an average of
100,000 particle trajectories for the reference case and this
demonstrates that the air does more work on the particle than
the particle does work on the air during the establishment of
the motion.

For the reference case, the mean rate of energy transfer
to and from the fluctuations is about seven times the mean
initial power imparted to the particle. The empirical result for
the energy flux in nanoWatts from calculations with a range of
particle sizes and masses is

# Molecular Fluxes at the Nanoscale

The number density of molecules in air at standard
temperature and pressure is n = 2.68678×10^{25} /m^{3}.
This is a very large number but the average molecular mass
m is similarly small at 0.481×10^{−24} kg so the density
*ρ* = *nm* =1.29 kg/m^{3} is of order unity. Standard kinetic
theory results for the fluxes of number, momentum and energy
that are incident to a surface element are, respectively,

where

is the average molecular spe ed, a molecule in equilibrium air at
standard temperature moves with an average speed of 447 m/s
between collisions with other molecule that occur with a
frequency such that the mean free path λ is approximately 49
nm. Note that this motion is only at the nanoscale and the
macroscopic velocities that appear in the continuum
formulations such as the Navier-Stokes equations are zero.
The macroscopic density has already been defined in terms of
the molecular properties and the combination of equation 4
into equation 5 shows that the momentum flux due to the
incident molecules is , or half the macroscopic
pressure. The molecules that are reflected from the surface
move in the opposite direction and supply the other half of the
macroscopic pressure which is 101,278 N/m^{2} at STP. This is a
large number because the expression for contains the
square of the near sonic average molecular speed. The incident
heat flux is proportional to this velocity cubed and has the
extremely large value of 22.61 MW/m^{2}. However, if the
surface is at the same temperature as the gas, the energy flux of
reflected molecules the same and the net heat flux is zero. Any
net heat flux to or from a surface is entirely due to a temperature
gradient normal to the surface. The coefficient of heat
conduction of air is such that the macroscopic heat flux is only
0.0243 W/m^{2} for a temperature gradient of one degree K per
meter. The macroscopic and molecular fluxes are similar and
this macroscopic gradient requires a temperature slip at the
surface of 273 x 0.0243/22,610,000 or only 2.93 x 10^{-7} K.

The only consequence at the macroscopic level of the
extremely high one-way heat flux at the molecular level is that
it justifies the conventional assumption of zero temperature
slip at a surface. However, it is the reason for the interest in air
as a potential energy source at the nanoscale. The heat flux of
22.61 MW/m^{2} is about twenty thousand times the maximum
solar heat flux at the surface of the earth and there is an even
larger factor with regard to the maximum usable energy flux
associated with wind. Figure 2 shows that, during the time
that an ensemble-averaged Brownian particle extracts energy
from air, the average energy flux of the molecules incident on
its surface is larger than the average energy flux carried away
by the reflected molecules.

The average energy flux to and from the fully established
or equilibrium fluctuations may be compared with the incident
energy flux to the surface. The surface area of a sphere with
4.9 nm diameter is 7.543 x 10^{-17} m^{2} and the energy flux to the
surface 1.705 nW. Figure 2 shows that the flux to and from
the equilibrium Brownian motion of the sphere is only 3%
of the incident energy flux to its surface, the effect of sphere
diameter on this ratio is probably the best indicator of the
required size of any object or device that could harvest energy
from air. For a particle of given density, the diameter enters
both the mass ratio and Knudsen number so that, ignoring the
factor based on the radius ratio, the energy flux in equation 3 is
proportional to the square root of the diameter. However, the
surface area is proportional to the square of the diameter so
that the energy flux decreases as the power of the diameter.

Also, if it was possible to harvest energy from a device of the size of the sphere, the energy would be minute and useful energy could only come from a vast number of devices fabricated similarly to a semiconductor chip. The number of devices would be inversely proportional to the diameter so that the energy output per unit area would decline as the power of the diameter. equation 3 relates to the equilibrium or fully established rate of energy transfer, while equation 1 for the initial power to a stationary sphere relates to the nonequilibrium rate. This equation has the mass rather than the square root of the mass in the denominator so that, instead if the energy flux increasing as the square root of the diameter, it is inversely proportional to the diameter. The energy output per unit area in the nonequilibrium case decreases as the fifth power of the diameter.

It is unlikely than any energy harvesting device could be
smaller than the 5 nm diameter of the sphere in the reference
case. This is about ten times larger than an average air molecule
and the device would be comprised of about a thousand
atoms. Should the device be a flexible piezoelectric nanowire,
the length could be much larger than the diameter, but the
power output would be unlikely to exceed 0.01 nw. Useful
power would require an enormously large number of such
devices with sufficient spacing to prevent interference between
adjacent wires. For example, if the 5 nm diameter wires were
spaced at 100 nm, there would be 10^{14} wires per square meter
and the power output would be one KW/m^{2}.

# Conclusions

The initiation of the Brownian motion of microscopic particles demonstrates that, despite some interpretations of the Second Law, energy can be extracted from air at the nanoscale. Direct simulation studies have determined the magnitude of the rates of energy transfer that are involved in this process for a sphere and show that they decrease strongly with the diameter of the sphere. The energy imparted to the motion an inert object is not a useful source of energy, but energy could similarly be transferred to a flexible piezoelectric nanowire if it was possible to fabricate a sufficiently small wire. Speculative calculations indicate that useful energy could be harvested if a surface was covered by a sufficiently large number of nanowires. This is beyond current technology and may never be feasible. At the same time, the future might see nano-devices that are unrelated to nanowires but can more effectively take advantage of the enormous one-way molecular energy flux of more than 22 Mega Watts per square meter in equilibrium air at standard temperature and pressure. For power to be generated by a nano-device, the energy imparted by the incident molecules must be larger than the energy carried away by the reflected molecules. There must be very few intermolecular collisions in the vicinity of the device in order to preserve the consequent nonequilibrium velocity distribution function. This is most easily achieved if the device is smaller than the mean free path and the spacing between devices is larger than the mean free path. Nanofabrication is a burgeoning field and it is important that there should be an awareness of the potential to harvest energy from air.

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^{*}Correspondence to:

Graeme A. Bird

University of Sydney

Sydney, N.S.W. 2006, Australia

E-mail: gab@gab.com.au

**Received:** November 10, 2016

**Accepted:** February 03, 2017

**Published:** February 07, 2017

**Citation:** Bird GA. 2017. The Possibility of Harvesting Useful Energy from the Thermal Motion of Air Molecules. *NanoWorld J* 3(1): 18-22.

**Copyright:** © 2017 Bird. This is an Open Access article distributed under the terms of the Creative Commons Attribution 4.0 International License (CC-BY) (http://creativecommons.org/licenses/by/4.0/) which permits commercial use, including reproduction, adaptation, and distribution of the article provided the original author and source are credited.