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THE JOURNAL OF CHEMICAL PHYSICS 138, 020901 (2013)
Perspective: Nanomotors without moving parts that propel themselves
in solution
Raymond Kaprala)
Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto,
Ontario M5S 3H6, Canada
(Received 4 November 2012; accepted 11 December 2012; published online 8 January 2013)
Self-propelled nanomotors use chemical energy to produce directed motion. Like many molecular
motors they suffer strong perturbations from the environment in which they move as a result of
thermal fluctuations and do not rely on inertia for their propulsion. Such tiny motors are the subject of
considerable research because of their potential applications, and a variety of synthetic motors have
been made and are being studied for this purpose. Chemically powered self-propelled nanomotors
without moving parts that rely on asymmetric chemical reactions to effect directed motion are the
focus of this article. The mechanisms they use for propulsion, how size and fuel sources influence
their motion, how they cope with strong molecular fluctuations, and how they behave collectively
are described. The practical applications of such nanomotors are largely unrealized and the subject
of speculation. Since molecular motors are ubiquitous in biology and perform a myriad of complex
tasks, the hope is that synthetic motors might be able to perform analogous tasks. They may have
the potential to change our perspective on how chemical dynamics takes place in complex systems.
© 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4773981]
I. INTRODUCTION
The terms “molecular machines” and “molecular motors” often conjure up images from biology. There is a very
good reason for this: nature has fabricated a large number
of machines built from molecular components that are able
to transduce chemical energy and perform a variety of useful
functions. Molecular machines are used in the cell to synthesize specific product molecules and can function as pumps
that generate concentration differences or as motors that convert chemical energy into directed motion. For instance, the
molecular machine ATP (adenosine triphosphate) synthase
utilizes the transport of protons to drive the synthesis of ATP;
the ATPase active ion pump can use some of the free energy
of the hydrolysis of ATP to pump sodium ions across the cell
membrane; the kinesin molecular motor walks on tubulin filaments and is able to actively transport material from one part
of the cell to another. Often such molecular machines do not
act in isolation but cooperate to achieve their goals. In fact,
it is now standard practice to think of the cell, or even structures within the cell, as factories where machines, acting in
concert, are responsible for the organization of much of the
cellular biochemical activity.1–3 For example, oxidative phosphorylation carried out in mitochondria operates through a
chemiosmotic mechanism comprising a sequence of steps involving several molecular machines, including ATP synthase;
even the ATP synthase motor itself is a composite of two cooperating molecular machines.
A few general characteristics of such molecular machines
can be identified. Of course they are small and, as a result,
are subject to strong fluctuations. They must be able to operate effectively in the presence of these strong perturbations
a) Electronic mail: [email protected].
0021-9606/2013/138(2)/020901/10/$30.00
from the environment. Inertia does not play a significant role
in their motion—their Reynolds number, which characterizes
the ratio of inertial to viscous forces, is small. Consequently
they make use of mechanisms for their operation which differ
from those for macroscopic machines.
Within the cell, molecular motors that are responsible for
active transport use chemical energy to drive conformational
changes which lead to directed motion, even under the influence of strong thermal fluctuations.4 Many motors of this
type, such as the kinesins or dyneins, walk along polar filaments so that the random walk that these motors execute is
biased giving rise to the active transport. Similarly, motors
such as RNA polymerase, DNA polymerase, and the helicases walk along DNA strands to carry out their functions. On
larger scales bacteria and other organisms exploit asymmetrical cyclic conformational changes to swim in low Reynolds
number environments.5
A question which naturally arises is, can micron and
nano-scale man-made machines be designed and programmed
in some way to perform tasks which are comparable in their
complexity to those that biological systems now routinely
carry out? One way to attempt to answer this question is
to copy biology and construct artificial molecular machines,
built from proteins or nucleic acids, that use chemical reactions to drive conformational changes and are able to walk
on various materials. There have been several studies where
biased molecular walkers have been synthesized in the laboratory to mimic the motion of motor proteins.6–13 These
molecular motors have been constructed from DNA or composites of protein and nucleic acids. Such machines can be
made with multiple legs (molecular spiders9 ) and often make
use of the fact that DNA can be programmed to identify certain nucleotide sequences. They can be designed to undergo
asymmetric ligation and cleavage, various types of correlated
138, 020901-1
© 2013 American Institute of Physics
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020901-2
Raymond Kapral
attachment or detachment and to walk along programmed
landscapes. Even simpler examples of synthetic molecular
machines include motors that mimic flagellar swimming motion driven by external magnetic fields, coaxial carbon nanotube motors driven by thermal gradients, motors that act as
rotors, switches, brakes, and ratchets, and rotaxanes driven by
electric fields or photochemical stimuli.3, 14–23
Another route can be followed to design synthetic
nanomotors. It is not necessary for a molecular motor to undergo conformational changes to effect directed motion. It is
possible to construct tiny chemically powered motors without
moving parts that rely on asymmetric chemical reactivity instead of asymmetric conformational dynamics, polar tracks,
or programmed landscapes to bias their motion. The most
widely studied examples of this type of motor are bimetallic nanorods,24–29 Janus particle,30, 31 and sphere dimer32, 33
motors.
In order to sharpen the focus of this perspective article,
of the various types of molecular machine that exist or could
be made, the emphasis will be on synthetic motors without
moving parts that use chemical energy to produce directed
motion, although other kinds of motor will be discussed to
place current research in a broader context. Recent work on
the construction and dynamics of chemically powered synthetic micron and nano-scale motors will be described. In addition, some of the important issues and challenges that must
be addressed before these synthetic motors can be used to perform complex tasks will be considered.
II. WHAT MAKES MOTORS WITHOUT MOVING
PARTS MOVE
It is well known that colloidal particles will move when
placed in a fluid with a gradient of some field. Motion of this
type occurs through phoretic mechanisms that are a consequence of the interaction of an inhomogeneous field with the
surface of the particle.34–40 Examples include electrophoresis
driven by an electrical potential, diffusiophoresis due to a concentration gradient of ionic or neutral species, thermophoresis resulting from a temperature gradient and osmophoresis
arising from an osmotic pressure gradient. The description of
such mechanisms is usually cast in terms of the interactions
and dynamics that occur in a thin boundary layer or interfacial
zone around the particle, and the way in which the boundary
layer dynamics couples to the fluid flow in the bulk solvent.
One of the simplest situations to consider is diffusiophoresis involving uncharged solute molecules interacting
with a macroscopic particle. If the concentration of solute
molecules is inhomogeneous on the scale of the characteristic size of the particle, it will experience an unequal force
over its surface within the boundary layer. To compensate for
this force, an inhomogeneous pressure field develops in the
boundary layer which is balanced by forces arising from solvent fluid flow. A feature of such phoretic transport is that
the externally imposed concentration field does not give rise
to a force on the particle plus fluid in the interfacial region.
As a consequence the solvent velocity flow field about the
particle decays quickly with distance from the particle. For a
sphere with radius Rm moving with velocity V by a phoretic
J. Chem. Phys. 138, 020901 (2013)
mechanism the fluid velocity field outside the boundary layer
is given by41 v(r) = (1/2)(Rm /r)3 (3r̂r̂ − 1) · V, where r̂ is a
unit vector normal to the surface. This 1/r3 decay should be
contrasted with the 1/r decay for a particle subject to an external force.42
The gradients in the fields responsible for particle motion by phoretic mechanisms need not be specified by external means but could be generated by the particle itself.
Self-generated asymmetrical electrical fields leading to selfelectrophoretic motion were proposed as a mechanism for
cell propulsion some time ago.43–45 Chemical reactions that
occur on specific parts of the surface of the particle could
lead to particle motion by self-diffusiophoresis. Several of
the synthetic nanomotors46 mentioned above utilize these
mechanisms to achieve propulsion. Janus particle nanomotors
with catalytic and noncatalytic faces have been made from
polystyrene and silica spheres partially coated by platinum.
When placed in a solution of H2 O2 they move with speeds of
several micrometers per second. The catalytic decomposition
of H2 O2 occurs at the Pt face and gives rise to an inhomogeneous distribution of reactants and products in the vicinity
of the Janus particle allowing the diffusiophoretic mechanism
to operate so that the particle is able to propel itself in the
solution.
Theoretical descriptions of nanomotor motion arising
from self-diffusiophoresis, which are based on the theories
for the phoretic motion of colloidal particles in external inhomogeneous fields, have been given.47–51 While the dynamics
of any specific motor will depend on the precise nature of
the chemical reactions that are used to power the motor,52
in order to discuss the factors that control particle motion
by this mechanism consider a general but simple situation.
On a Janus particle with radius Rm , the chemical reaction
A → B occurs on the catalytic (C) face while the noncatalytic
(N) face is chemically inactive. The A and B molecular species
interact with the surface of the Janus particle through potentials of mean force WαS (z), where α = A, B, S = C, N, and z
measures the normal distance of the molecules from the surface (Fig. 1).
In a macroscopic description of the solvent, the solvent
velocity field is taken to vanish at the surface of the Janus
FIG. 1. A Janus motor composed of catalytic (blue) and noncatalytic (red)
spheres. The figure indicates the chemical reaction A → B that converts fuel
A to product B and shows the inhomogeneous distribution of B molecules
around the Janus particle. The boundary layer around the Janus particle
within which the intermolecular forces act is also shown.
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020901-3
Raymond Kapral
J. Chem. Phys. 138, 020901 (2013)
particle as a result of stick boundary conditions. However,
the concentration gradient produced by the chemical reaction
on the catalytic side, in conjunction with the interactions of
the A and B species with the surface of the particle, give rise
to an apparent slip velocity at the outer edge of the boundary layer.53 Its value can be computed using the methods developed for the motion of colloidal particles by the diffusiophoretic mechanism37 and for the A → B reaction is given
by
v(s) (s) =
kB T
(∇ s nA (s))(λ2C (θ ) + λ2N (1 − (θ ))),
η
(1)
where nA is the concentration of species A, s is tangential to
the particle surface, ∇ s is the gradient in this direction, η is
the solvent viscosity, λ2S has the units of length squared, and is
∞
defined through λ2S = 0 dz z(e−βWBS (z) − e−βWAS (z) ), which
contains the effects due to interactions of the reactant and
product molecules with the Janus particle, and (θ ) is a characteristic function that is unity on the catalytic hemisphere
and zero otherwise. The velocity of the Janus particle can
be determined from the slip velocity using the relation V
= −v(s) S , where the angle brackets denote an aver2 −1
)
surface of the particle: · · ·S = (4π Rm
age over the
37, 50
dS
·
·
·.
Letting
û
be
the
vector
from
the
center
of
the
S0
particle sphere to the pole containing the C hemisphere, the
projection of the velocity of the Janus particle in this direction, Vu = û · V, is given by54
Vu =
kB T n0 2
λC + λ2N c1 ,
η 3Rm
species B is less strongly repelled from the Janus C hemisphere than species A.
While the explicit expression for the Janus particle velocity is not simple and depends on the specific value of c1 ,
it is interesting to consider the form of Vu in two limits. In
the diffusion-controlled limit where kf0 /kD 1 with kD
= 4π DRm the Smoluchowski rate constant for a diffusion
controlled reaction, Vu takes the form:
Vu ∼
for the concentration field, nA (r), of the A species, where P
is a Legendre polynomial. The diffusion equation must be
solved subject to a “radiation” boundary condition,55
(4)
that accounts for reaction on the surface of the sphere, and
the boundary condition nA = n0 far from the sphere, in order
to determine the c coefficients. Here kf0 is an intrinsic rate
constant for the reaction taking place in the boundary layer
around the Janus particle.
A number of general features can be gleaned from an
analysis of this expression for the propulsion velocity. The
sign of Vu is determined by the sign of (λ2C + λ2N ), and each
of the two terms in the sum can take either sign depending on
the nature of the interaction potentials of A and B species with
the C and N hemispheres. For example, if WAC = WBC and
WAN = WBN then λ2N = 0 and the sign is determined solely
by λ2C . If the interactions are attractive, λ2C > 0 if species B
is more strongly attracted to the Janus C hemisphere than
species A. Also, if the interactions are repulsive, λ2C > 0 if
kB T n0 2
λC + λ2N c1D ,
η 3Rm
(5)
where c1D is a constant independent of Rm . In the reactioncontrolled limit where kf0 /kD 1 we have
(2)
where D is the (common) A and B molecule diffusion coefficient and n0 = nA + nB is the total concentration of the A
and B species. The coefficient c1 depends on the solution of
the steady-state diffusion equation, D∇ 2 nA = 0, which can be
written in the form:
+1
nA (r, θ ) = n0 1 −
c (Rm /r) P (cos θ )
(3)
2
4π Rm
D∂r nA (r, θ )|Rm = kf0 nA (Rm , θ )(θ ),
FIG. 2. (Left) A sphere dimer nanomotor. The small sphere catalyzes the reaction A → B, while the large sphere is noncatalytic. The inhomogeneous
concentration of product (B) molecules is shown as blue dots. (Right) A
bimetallic nanorod motor. Oxidation reactions take place at the platinum
(grey) end and reduction reactions occur at the gold (yellow) end. The directions of charged particle motions, fluid flow, and the propagation direction
are indicated.
Vu ∼
kf0
1 kB T n0 2
λC + λ2N
.
2
8 η D
4π Rm
(6)
Thus, Vu scales as 1/Rm for a diffusion controlled reaction.
2
2
, kf0 /4π Rm
Since the intrinsic rate constant scales as kf0 ∼ Rm
is independent of Rm and so is Vu in the reaction controlled
limit.
Sphere dimer motors32 are similar to Janus particle motors but the catalytic and noncatalytic portions adopt spherical shapes; the spheres are then linked to make the motor
(see Fig. 2 (left)). The self-propelled motion of these nanomotors has been studied through particle-based mesoscopic
simulations32, 56 using multiparticle collision dynamics57, 58
and, as noted above, they have been made in the laboratory
from silica (noncatalytic) and platinum (catalytic) spheres.33
In the simulations of these motors, the fuel and product
molecules interact with motor spheres with different intermolecular potentials. (Most often the different interactions are
chosen only for the noncatalytic sphere but this is not essential.) The propulsion velocity depends on the sizes of the two
spheres and, of course, on the interaction potentials. In addition to propulsion by self-diffusiophoresis, simulations of
sphere dimer motors that use self-thermophoresis for propulsion have been carried out.59
Perhaps the greatest research effort has been devoted
to experimental investigations of bimetallic nanorod motors. The first such motors comprised platinum and gold
segments24, 25 but subsequently rod-like motors with many
other metallic combinations have been constructed;28, 60–62 in
addition, carbon nanotube metallic motors have been made.29
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020901-4
Raymond Kapral
Evidence suggests that these motors operate through
a self-electrophoretic mechanism.28 In the electrophoretic
mechanism for colloidal motion in an ionic solution due to
an inhomogeneous electrical field,37, 63 the colloidal particle
is taken to be surrounded by an electrical double layer where
the charge on the colloid’s surface due to tightly bound ions
is compensated by a diffuse distribution of counterions. The
motions of the ions within the double layer induced by the
electric field result in a velocity field within this layer and,
as in the diffusiophoretic mechanism, the velocity field at the
outer boundary of the double layer constitutes a slip velocity
that determines the flow field in the region outside the double
layer. The colloidal particle moves in a direction opposite to
the counterion flow in the double layer. The calculation of the
particle velocity is then analogous to that for diffusiophoresis.
A similar mechanism operates for self-electrophoresis. Suppose oxidation and reduction reactions occur on the different
metallic components64, 65 as indicated schematically in Fig. 2
(right). These reactions produce an electron current within the
rod. This electron current induces a concentration gradient of
H+ ions in the electrical double layer surrounding the rod motor, which, in turn, results in a gradient of the electrical field.
The electric field acts on the charges in the double layer to
produce a force on the fluid in this region that results in fluid
flow within the diffuse boundary layer. From this point the
calculation of the motor velocity parallels that outlined above.
Nano “batteries” that operate through such mechanisms have
been constructed and their propulsion properties have been
studied.60
In addition to phoretic mechanisms, micron-scale motors
have been built that propel themselves by generating bubbles through chemical reactions. In fact, the early studies of
macroscopic bimetallic platinum-gold motors66 operated by
such a mechanism and this work was a precursor to many
of the early experimental studies of the much smaller micron
and nano-scale motors.27, 67, 68 Motors propelled by polymerization reactions have also been made.69 Self-electrochemical
reactions induced by a magnetic field can drive the motion of
colloidal particles.70 The shapes of motors are not restricted
to the examples of Janus, sphere dimer and rod motors described above and many other geometries are possible, including flexible motors powered by chemical reactions (see
Ref. 71 and references therein), hinged rod motors,72 polymer
motors,73 rectangular motors,74 an active colloidal chucker,75
and platinum-loaded polymer stomatocytes.76 There have also
been studies of liquid drop motors that rely on Marangoni
effects or instabilities for propulsion.77–81 Finally, it is
worth noting that self-propelled motion can occur for systems with symmetric catalysis through a symmetry-breaking
mechanism.82, 83
III. SIZE MATTERS
Fluctuations play important roles in various aspects of the
dynamics of self-propelled motors and molecular machines.
Molecular machines that operate in the cell are very small,
often with characteristic sizes of a few to tens of nanometers. Thermal fluctuations dominate their motions and they
must employ strategies to tame the effects of such fluctua-
J. Chem. Phys. 138, 020901 (2013)
tions. Bacteria and other micron-scale self-propelled swimmers make use of nonreciprocal conformational cycles to
swim in low Reynolds number environments. The effectiveness of swimming by this mechanism is greatly reduced when
the object is small. For example, consider adenylate kinase,
a simple enzymatic protein machine. The protein comprises
214 amino acid residues and has a size of a few nanometers. It has been suggested that this tridomain hinge protein
carries out its catalytic function, the reaction AMP + ADP
2ADP , by executing a specific sequence of hinge opening
and closing events.84 In this mechanism the cyclic conformational changes are nonreciprocal and, in principle, the protein
could “swim” as it performs its catalytic function. Simulations show that, within statistical uncertainty, the motor diffusion coefficient does not differ from that for the protein in the
absence of catalytic activity;85 thus, it is likely that both conformational fluctuations and orientational Brownian motion
mask propulsion.
These same features enter into the effects of size on the
dynamics simple chemically powered motors, which admit a
simpler analysis. While chemically powered motors propel
themselves through solution, they also tumble as a result of
orientational Brownian motion and this influences the form
of the mean square displacement of the motor and its diffusion coefficient.30, 86 For a large spherical particle the reorientation time is given by the Stokes-Einstein-Debye expression,
3
/kB T , where Rm is again the motor radius. Conτr = 4π ηRm
sequently, on the time scale of τ r the particle will reorient,
the direction of the self-propelled motion will change and this
will give rise to diffusive behavior. This effect can be analyzed
approximately as follows: the instantaneous velocity V(t) of
the self-propelled particle can be decomposed into the average
propulsion velocity Vu along the direction û(t), and deviations
from this quantity, V(t) = Vu û(t) + δV(t). The velocity correlation function is then given by
1
1
CV (t) = V(t) · V ≈ Vu2 û(t) · û + δV(t) · δV ,
3
3
(7)
where the average is taken in the steady state and correlations between the instantaneous orientation and velocity fluctuations are neglected. The diffusion coefficient is obtained
from the infinite
∞ time integral of the velocity correlation function, Dm = 0 dt CV (t) . Assuming the orientational correlation function decays exponentially with time constant τ r ,
û(t) · û ≈ exp (−t/τr ), we obtain
Dm =
1 2
V τr + D0 ,
3 u
(8)
∞
where D0 = 0 dtδV(t) · δV/3 is the diffusion coefficient
in the absence of self-propulsion. If the correlations of the
velocity fluctuations are assumed to decay exponentially,
δV(t) · δV ≈ exp(−t/τv ), with time constant τv = M/ζ ,
where ζ ≈ 6π ηRm is the translational friction coefficient and
M is the motor mass,87 D0 takes the Stokes-Einstein form D0
= kB T/ζ .
The mean square displacement (MSD) of the motor can
also be computed from the velocity
t correlation function using the expression, L(t)2 = 6 0 dt Dm (t ), where Dm (t)
t
= 0 dt CV (t ) is the time dependent motor diffusion coeffi-
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020901-5
Raymond Kapral
cient. Assuming the exponentially decaying correlation functions given above, for times t τv the MSD takes the form:
1
L(t)2 ≈ 6 D0 + Vu2 τr t − 2Vu2 τr2 (1 − e−t/τr ). (9)
3
For short times, τv t τr , the behavior of the mean
square displacement is dominated by ballistic motion, L(t)2
≈ Vu2 t 2 , while for long times it exhibits diffusive behavior,
L(t)2 ≈ 6(D0 + 13 Vu2 τr )t, with an effective motor diffusion
coefficient Dm given above.
It is instructive to consider the values of these quantities
for motors with various sizes. From the discussion above, the
diffusion coefficient of a spherical particle (in the absence of
self-propulsion) is given by D0 = kB T/6π ηRm . At room temperature kB T ≈ 4 × 10−21 m2 kg/s. Taking η to be the viscosity of water, η ≈ 10−3 kg/ms we have D0 ≈ 0.2 μm2 /s
for a spherical motor with Rm = 1 μm. The velocity relaxation time is τv ≈ 1 μs, while the orientational relaxation
time for particles of this size is τr ≈ 3 s, so that τv τr .
Taking this micron-size motor to move with a velocity of
Vu ≈ 10 μm/s, we find 13 Vu2 τr = 100 μm2 /s. Consequently,
translational Brownian motion plays only a small role and Dm
is dominated by the contribution from self-propulsion.
Contrast these results with those for motor with Rm
= 10 nm, the size range that is typical for small motor proteins. In this case the velocity relaxation time is τv ≈ 0.1 ns
and τr ≈ 3 μs so that τv τr again. Assuming the motor
is operating in the diffusion controlled limit88 we have Vu
≈ 1 mm/s. These values yield D0 ≈ 20 μm2 /s and 13 Vu2 τr
≈ 1 μm2 /s. Therefore, Dm is dominated by the translational
Brownian motion contribution: directed motion is highly
compromised for a small motor with this size. This behavior is evident in the plots of L(t)2 in Fig. 3. A 1 μm particle shows well defined ballistic and diffusive regimes with a
cross-over at tc equal to a few seconds. By comparison, the
cross-over time for a 0.01 μm particle is in the μs regime and
is not visible in the figure.89
Orientational Brownian motion is not the only stochastic element that can alter the dynamics of chemically pow-
FIG. 3. Log-log plot of the mean square displacement (MSD) in Eq. (9),
L(t)2 (μm2 ) versus time (s) for two self-propelled particle sizes, Rm
= 1 μm (thick blue dashed line) and Rm = 0.01 μm (thick black line). The
plot for Rm = 1 μm shows ballistic and diffusive regimes and the straight
lines (thin black lines) are plots of Vu2 t 2 at short times and 6Dm t at long times.
The cross-over between the ballistic and diffusion regimes occurs where these
two straight lines cross. The plot for Rm = 0.01 μm shows only diffusive behavior on the time scale of the figure.
J. Chem. Phys. 138, 020901 (2013)
ered motors and molecular machines. Since they operate by
chemical reactions, concentration fluctuations associated with
reactive events can also influence the dynamics and lead to
changes in the character of the mean square displacement and
diffusion coefficients.86 Recently it was shown experimentally that enzymes exhibit enhanced diffusion when carrying
out their catalytic functions, compared to the same enzymes
in the absence of such chemical activity.90 In principle, asymmetric chemical reactivity leading to self propulsion or reactive concentration fluctuations could lead to such effects and
the investigation of the mechanisms underlying enhanced diffusion in these systems is an interesting research topic.
The size of the motor that is needed will very likely be determined by the application for which it is intended. If a very
small motor with a characteristic size of the order of a few
nanometers or even smaller Angstrom size is required, then
strategies similar to those used by molecular machines in the
cell could be used to ameliorate the effects of fluctuations;
for example, confinement of some sort, prescribed tracks for
motion,91, 92 or even the exploitation of chemical patterns as
described below. Feynman made the now often-quoted observation that “There’s Plenty of Room at the Bottom.”93 This
is certainly true for the chemically powered motors considered here, but new strategies will have to be employed to harness their motion for the execution of specific tasks when their
sizes are very small.
IV. WHAT KIND OF FUEL IS USED AND WHERE IS IT?
In most cases the fuel used by the synthetic chemically
powered motors that have been studied is supplied by the
medium in which they move. Hydrogen peroxide is often the
fuel of choice; however, other fuels have been used to power
motor motion, including glucose,94 hydrazine,29 and polymerization reactions.69 Motors could be designed to carry their
own fuel. They can also be propelled by external means, such
as magnetic fields or light, but here we consider situations
where the chemical fuel in the environment provides the energy for propulsion. Motor motion will cease when the supply
of fuel is exhausted and, typically, observations of motion are
carried out under transient nonequilibrium conditions.
In the biological realm, molecular machines consume
fuel in the environment that is produced by complex networks
of chemical reactions, which themselves often involve other
molecular machines for this purpose. For instance, as discussed earlier, the kinesin and dynein motors use ATP produced in the cell as fuel to execute biased random walks
on filaments to carry out their active transport functions. On
larger scales bacteria such as Escherichia coli use internal
fuel sources to power flagellar motors that are responsible for
propulsion. (Of course, the environment ultimately supplies
the nutrients needed for the synthesis of fuel within the cellular organism.) The system exists in a nonequilibrium state
which is necessary for machine operation. Given this motivation, it is interesting to consider how reactions in the environment can influence motor motion. Such considerations are
important since potential medical and other applications may,
of necessity, have to make use of fuel sources supplied by a
sequence of often complex chemical reactions.
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020901-6
Raymond Kapral
J. Chem. Phys. 138, 020901 (2013)
As a simple illustration of how reactions in the environment can influence nanomotor motion consider the dynamics
of a sphere dimer motor in a chemically active medium where
a cubic autocatalytic reaction occurs.95 The reactions involved
in motor propulsion are
k1
A+C B + C, on the motor,
k−1
(10)
k2
B + 2A 3A, in the environment.
k−2
Of course, the maintenance of far-from-equilibrium conditions in the bulk fluid is achieved by flows of reagents but
these are assumed to be accounted for in pool species whose
effects are incorporated in the reaction rate constants. Figure 4
shows the system for irreversible versions of these reaction
k1
steps: the motor is powered by the reaction A → B and the
k2
autocatalytic reaction B + 2A → 3A occurs in the bulk solution. This latter reaction regenerates the fuel A needed for
motor motion and destroys the product B. Thus, bulk phase
reactions can influence the concentrations of product and fuel
and this, in turn, will affect motor motion. For the general
reversible reaction case the diffusion equation can be solved
subject to suitable boundary conditions far from and on the
motor. Far from the motor the concentration fields are given
by the steady state values of the cubic autocatalytic reaction
in the bulk phase, n̄B and n̄A , while a radiation boundary condition (Eq. (4)) is applied at the catalytic sphere to account
for the reactions occurring there. This yields the following
expression for the product concentration field as a function of
distance from the center of the catalytic sphere:
0
0
k1 n̄A − k−1
n̄B kD
e−κ(r−Rm )
. (11)
nB (r) = n̄B + 0
0
k1 + k−1
+ kD (1 + κRm ) 4π Dr
A
The A concentration field is determined from the conservation
condition, n0 = nA (r) + nB (r). An inverse length κ
= (k2 + k−2 )n̄2A /D appears in this formula. The inverse
length κ depends on the (common) diffusion coefficient D
of the A and B species and the rate constants k2 and k−2 for
0
are
the autocatalytic reactions. The rate constants k10 and k−1
intrinsic rate constants for the forward and reverse reactions
within the boundary layer around the motor catalytic sphere.
This result should be contrasted with the situation when there
are no reactions in the bulk. In this case the B concentration
field is given by
nB (r) =
k10
k10 kD n0
1
.
0
+ k−1
+ kD 4π Dr
(12)
We see that the slow 1/r decay of the concentration field is
“screened” by reactions in the environment leading to more
a rapid exponentially damped decay. Since the gradient of B
across the motor is essential for propulsion, the more rapid
decay of the concentration field can change the propulsion
characteristics of the motor. If B is destroyed too quickly the
B field gradient will be reduced leading to lower nanomotor
velocities.
Self-propulsion is a nonequilibrium phenomenon and
cannot occur in a system at equilibrium. It is instructive to see
how this comes about for this system. At equilibrium detailed
balance applies and the rates of both the motor and bulk reactions are zero. The rate constants of the reactions then must
satisfy the relation,
eq
eq
k1 /k−1 = k−2 /k2 = nB /nA ,
eq
nA
(13)
eq
nB
where
and
are the equilibrium densities of these
species. In this case Eq. (11) predicts that nB (r) = n̄B outside the catalytic sphere; hence, the propulsion velocity will
be zero.
The theoretical results described above are consistent
with simulations of motor dynamics in an active medium.95
The B concentration fields are presented in Fig. 5 for irreversible reactions on the motor and in the environment (red
triangles) and reversible reactions that satisfy detailed balance
(black circles). The rapid decay of nB (r) in the former case is
evident in the figure. In addition, simulations show that the
A
C
N
A
B
B +2A
B
B
3A
FIG. 4. A schematic representation of a sphere dimer motor that catalyzes
the irreversible reaction A → B on the C sphere. The product B is consumed
in the environment by the autocatalytic reaction B + 2A → 3A, which regenerates the fuel A for use by the motor.
FIG. 5. The product B concentration field as a function of distance r from the
center of the catalytic sphere. The curves correspond to two nonequilibrium
situations: irreversible reactions (red triangles) and reversible reactions that
satisfy detailed balance (black circles) (see Ref. 95).
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020901-7
Raymond Kapral
J. Chem. Phys. 138, 020901 (2013)
motor velocity decreases with increasing magnitude of the k2
rate constant. Compared with this nonequilibrium case that
violates detailed balance, when detailed balance is satisfied
one sees that the density field is approximately constant in the
regions outside the sphere dimer (the small variations seen in
the graph are due to spherical averaging that includes the noncatalytic sphere) in accord with theoretical predictions. Thus,
no propulsion occurs in the absence of nonequilibrium B gradients and simulations verify that the average propulsion velocity of the motor is zero. All of these effects point to the
important role that reactions in the environment can play in
nanometer dynamics.
Far-from-equilibrium chemically active media need not
be homogeneous or in simple steady states but can exist in
various more complicated spatiotemporal states. These include states with chemical oscillations or chaotic dynamics as
well as patterned states whose characteristic lengths can vary
from macroscopic to nanometer scales.96 In such cases the effects of the environment on motor dynamics may no longer be
a simple alteration of the velocity. For example, if the medium
exists in an inhomogeneous steady state comprising fuel-rich
and fuel-poor domains with a specific geometry, motor motion will be confined to the fuel-rich regions, similar to the
confinement that one finds in micro-channel arrays. In this
way nano-scale motors could overcome some of the effects of
fast orientational motion by moving on nano-scale patterns.
The cubic autocatalytic medium discussed above supports
propagating chemical fronts and simulations have shown that
sphere dimer motors are reflected by such chemical waves,
which supports this suggestion.97
The last issue we consider in this section is how well motors use their fuel. The answer is, not very well. The thermodynamic efficiency of the power transduction of the motor
is defined as the ratio between the power associated with the
work done by the motor against an external conservative force
Fex and the power input due to chemical reaction98, 99 and can
be determined from
ηT D = −
Vz Fex
,
μR
(14)
where R is the net chemical reaction rate and μ is the
change in the chemical potential in the reaction. For sphere
dimer motors ηTD ranges from a few hundreds to a few tenths
of a percent.100 Low efficiencies are found for nanorod64 and
Janus particle51 nanomotors. While low efficiency may not be
a concern for many applications, the efficiency could be improved by new motor designs that incorporate some features
that biological motors use to operate efficiently.101
V. MOTORS CAN ACT TOGETHER
In the biological realm molecular motors often cooperate to perform tasks. For example, the kinesin-1, kinesin-3,
and dynein motors cooperate in the long range movement
of early endosomes in the cell, and the bi-directional motion of chitin synthase involves the cooperation of myosin-5
and kinesin-1 motors.102 One can imagine scenarios in which
synthetic motors could be made to cooperate to carry out specific functions. While very little work has been done on com-
plex scenarios involving cooperation among many synthetic
motors, there have been some studies of interactions among
chemically powered nanomotors and the collective dynamics of many motors. Suspensions of bimetallic nanorod motors have been shown to exhibit collective behavior which is
similar to that of E. coli bacterial suspensions as monitored
by enhanced tracer diffusion.103 Micrometer-scale silver chloride particles move under ultra-violet illumination and display
schooling behavior where these particles move into regions of
higher concentration. When photo-inactive silica particles are
added more complex collective behavior is observed.104 Selforganized composite motors have been constructed as a first
step in the fabrication of complex nanomachine systems.105
Direct attractive interactions between motors can cause
motors to aggregate and propagate or rotate as a unit. This
is the case for Silica-Pt sphere dimer motors where aligned
dimer pairs are formed, which then propagate as a unit in solution. Other configurations where triplets of dimers with less
simple structure form and execute chemically powered rotational motion.33 Simulations of the dynamics of two sphere
dimers also show the formation of dimer pairs which are
held together by solvent-exclusion forces and undergo chemically powered translational and rotational motion as well
as Brownian motion.106 The dynamics of pairs of Silica-Pt
Janus particle motors has been studied experimentally. The
Janus particles form bound species which can execute different types of motion depending on the orientation of the
Janus particle faces in the Janus pair.107 Furthermore, interactions between metal-composite rod motors that rotate when
isolated have been shown to lead to cooperative rotational
motion.108 Experiments on large collections of chemically
powered Au-Pt Janus motors demonstrated that the motors
undergo chemotactic aggregation to form clusters with distinctive dynamics.109
Cooperative motion among nanomotors can arise from
different kinds of interaction, including direct intermolecular interactions between motor components, solvent depletion
forces, hydrodynamic interactions, and concentration gradients. These interactions can lead to new types of motion that
are absent for isolated motors; for instance, swarming behavior where groups of motors move as a dynamical unit and
active self-assembly of motors into specific structures.
An example of the kinds of cooperative motion that can
occur is provided by the dynamics of many sphere dimer motors. The motors are again powered by the chemical reaction A
→ B and propagate in a direction with catalytic sphere in front
where the chemical concentration of product B is highest.110
In an ensemble of such motors an individual motor will sense
the concentration fields of nearby motors and propagate in
the direction of those motors, very much like the chemotactic response of a microorganism sensing a chemical gradient
and moving in response to it. This mechanism will lead to
active self-assembly of motors, even in the absence of direct
attractive intermolecular forces between them. Once the motors are in close proximity, they will orient themselves to accommodate the local collective concentration field. Motor geometry will then also play a role in determining the form of
the self-assembled aggregate. If the density of motors in the
system is not too high, aggregates with different shapes can
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020901-8
Raymond Kapral
FIG. 6. A self-assembled aggregate of six sphere dimer motors that propagates as a unit, along with a dimer pair and two individual dimer motors.
The aggregates persist for long periods of time but breakup as a result of
collisions with other dimers or aggregates and reform due to concentration
gradient-driven motion (see Ref. 110).
exist, and their shapes will determine the nature of their dynamics. An example is shown in Fig. 6 where an aggregate
composed of six approximately aligned sphere dimers propagate as unit in the direction indicated in the figure, i.e., it
forms a dynamic “swarm” that propagates coherently through
the system. Fluctuations within the swarm or collisions with
other motors or swarms can lead to swarm breakup and the
formation of new swarms. For higher motor densities the motors self-assemble into nearly ordered arrays whose structure
is determined by individual motor geometry. The collective
behavior of these diffusiophoretic chemically powered motors
is governed largely by chemical gradients but hydrodynamic
interactions are present and influence some aspects of the dynamics. For other active particles,111 including swimmers that
move by carrying out non-reciprocal conformational changes,
hydrodynamic coupling is the primary mechanism for coupling motor motions.112, 113
A large research effort is being devoted to the exploration of the collective dynamics of interacting active objects.
Observations and modeling of phenomena such as the flocking of birds, the schooling of fish, and swarming, clustering,
and giant number fluctuations in suspensions of active colloidal particles and microorganisms are topics that are under
study.114–123 The investigation of such phenomena in large ensembles of micron- and nano-scale synthetic chemically powered motors is an open area of research. If collections of motors are to be used in applications an understanding of the full
scope of their collective dynamical behavior is required.
VI. PIE IN THE SKY OR JUST AROUND
THE CORNER?
Speculations about potential applications of chemically
powered nanomotors have appeared a number of times in the
literature.27, 124–130 Many of these applications center around
issues concerning cargo transport and navigation in channels and experiments have been carried out that demonstrate
J. Chem. Phys. 138, 020901 (2013)
that such applications are possible. The motions of striped
nanorod motors containing Ni could be controlled by external
magnetic fields and targeted in microchannels towards cargo,
which could then be transported to other locations.125, 131, 132
Simulations of sphere dimer motors moving upstream in a
fluid flowing in a channel have shown how a motor interacts
with a confined fluid flow of the type encountered in microfluidic applications.133 Similarly, the delivery of cargo by Pt-Au
nanomotors could be achieved using photochemical means
by employing special linkages between the motor and cargo
that can respond to light.134 It has been observed that Pt-Au
nanorod motors are sensitive to the presence of metal cations
in their environment and this fact has led to the suggestion that
such motors could be used to detect trace amounts of some
metals in solution.135 These exploratory studies could eventually lead to uses such as drug delivery to specific parts of the
body and the detection and destruction of specific pollutants.
For applications that involve the introduction of motors
into the body to perform tasks such as the targeting of specific pathogens or malignant cells, motors must be able to
sense the target and find it by moving through a flowing confined medium that contains a variety of other material, some
of which may attack the motor. For this purpose motors will
have to contain biological sensing elements129 to attain their
target and utilize fuel that is readily available in the body
or carry their own fuel. Furthermore, once the motors have
achieved their goals they should be readily degraded or eliminated. These are challenging criteria that could be met by constructing the motors from biomaterials and using biochemical
reactions to power motor motion.
We are quite far from practical applications and to
achieve such aims it is likely that nanomotors will have to
be much more sophisticated than those that have been made
thus far and will need to use new fuel sources. Research into
the construction of more complex motors has begun where
the motor components self-assemble to form a compound
motor.105 We have seen that chemically powered motors can
use chemical gradients to actively self-assemble into complex
structures with interesting dynamical properties.
Self-assembly is, of course, a wide-spread phenomenon
that occurs in many different contexts.96 Complex selfassembled chemical structures have been built using the principles of supramolecular chemistry.136 In active media containing many chemically powered motors with specific functional groups, one could speculate that active self-assembly
driven by chemical gradients could bring the building blocks
for supramolecular assembly into proximity where specific
forces could come into play to form desired structures.
In a much broader context, in most cases solution phase
chemical dynamics relies on diffusion to bring chemical
species into contact where reaction can take place. The random motions that this diffusive search entails may not be the
most effective way for molecules to find each other or for a
molecule to find a suitable catalytic site. Taking a cue from
biology, it could be much more effective if active motion using synthetic motors could be used to direct chemical species
into configurations that lead to reactions. For instance, motors could be used to target specific reaction sites and use
their directed motion to move to a reaction zone quickly. In
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020901-9
Raymond Kapral
addition to sensing chemical gradients chemically powered
motors could follow specific tracks on surfaces like molecular spiders or use chemical patterns in nonequilibrium states
to guide the motor motion. In this way they could overcome
some of the effects of orientational Brownian dynamics which
can mask directed motion.
While most of the applications currently being considered will likely use motors with sizes of the order of microns
or hundreds of nanometers, one can speculate about uses for
very tiny motors with molecular-scale dimensions. Such motors would be small enough to operate at the level of the interior of a single cell, much like the protein motors that are
involved in many cellular biological functions. Even at this
small size chemically powered directed motion can play an
important role. Perhaps such motors could be designed to act
in concert with protein molecular machines to achieve specific
functions at the cellular level.
Many of the potential applications mentioned above are
highly speculative and will take decades to be realized, if at
all. However, chemically powered motors are interesting objects, both because of the fundamental challenges they present
(they are small nonequilibrium devices and their full theoretical description is at the forefront of nonequilibrium statistical
mechanics), and because they have the potential to change the
way we think about dynamics in complex systems. Although
these goals are very far from being realized, the very recognition that they might be attained, coupled with the experimental and theoretical work currently being carried out make this
a fruitful and interesting area of research.
ACKNOWLEDGMENTS
I would like to express my thanks to my colleague
Geoff Ozin whose work stimulated my interest in selfpropelled nanomotors. I have had the pleasure of working on
the problems discussed in this paper with my co-workers G.
Rückner, Y.-G. Tao, S. Thakur, P. de Buyl, and P. Colberg.
This work was supported in part by a grant from the Natural
Sciences and Engineering Research Council of Canada. Computations were performed on the GPC supercomputer at the
SciNet HPC Consortium.137 SciNet is funded by the Canada
Foundation for Innovation under the auspices of Compute
Canada, the Government of Ontario, Ontario Research FundResearch Excellence, and the University of Toronto.
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Raymond Kapral
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