The realization of nanoscale machines requires efficient methods by which to rectify unbiased perturbations to perform useful functions in the presence of significant thermal noise. The performance of such Brownian motors often depends sensitively on their operating conditions—in particular, on the relative rates of diffusive and deterministic motions. In this letter, we present a type of Brownian motor that uses contact charge electrophoresis of a colloidal particle within a ratcheted channel to achieve directed transport or perform useful work against an applied load. We analyze the stochastic dynamics of this model ratchet to show that it functions under any operating condition—even in the limit of strong thermal noise and in contrast to existing ratchets. The theoretical results presented here suggest that ratcheted electrophoresis could provide a basis for electrochemically powered, nanoscale machines capable of transport and actuation of nanoscale components.

Nanoscale machines rectify unbiased perturbations to achieve directed transport or perform useful work in the presence of significant thermal noise.1,2 The operation of such Brownian motors requires two critical ingredients: (i) an external stimulus to push the system away from thermodynamic equilibrium and (ii) an asymmetric potential to break the spatial inversion symmetry.1 Importantly, the presence of these necessary ingredients does not ensure that a motor will be functional or in any sense optimal for a given purpose. Effective motors—like those found in biology—are highly optimized and operate only within a narrow range of conditions. This sensitive dependence of ratcheted transport on system parameters presents a significant challenge towards achieving artificial Brownian motors2 that operate reliably over a broad range of conditions.

Simple physical models of ratcheted transport can help to illustrate these challenges and offer insights into the rational design of artificial ratchets. In one common example—the so-called flashing ratchet—a spatially periodic, asymmetric potential is modulated in time to bias the motion of Brownian particles in a preferred direction.3,4 This mechanism has been demonstrated experimentally to bias the motions of colloidal particles5–8 and electrons.9,10 Directed transport depends strongly on the details of the ratchet potential (e.g., the characteristic period L and force F) and is fastest when ratchet parameters are carefully matched to those of the reservoir (e.g., when FL/kBT1). More generally, the performance of Brownian motors depends critically on the relative rates of diffusive and deterministic motions as characterized by the Péclet number, Pe=FL/kBT. While flashing ratchets are most effective when Pe1, many ratchets perform best in the absence of thermal noise when Pe1. These systems often rely on oscillatory forces to drive steady currents of particles,11–14 ions,15 or droplets16 through asymmetrically structured channels. Here, we present a type of ratchet that functions for any value of the Péclet number—even in the limit of strong thermal noise (Pe0).

The ratchet we describe is based on contact charge electrophoresis (CCEP), which uses a constant voltage to drive the oscillatory motion of a conductive particle between two electrodes.17 The particle's motion is rectified by a series of asymmetric ramps positioned along the sides of the channel that direct motion perpendicular to the applied field E (Fig. 1). Notably, particle oscillations are driven not by an oscillating field but by the repeated charging of the particle on contact with the electrodes. The details of the CCEP mechanism have been considered previously;18 here, we use a simpler approximate description. Upon contact with either electrode, a conductive sphere of radius a acquires a net charge, q=±23π3εa2E, where ε is the permittivity of the surrounding fluid.19 The resulting electric force, F=qE, drives the motion of the particle towards the opposite electrode with a velocity, v=F/λ, where the friction coefficient is approximated by the Stokes relation, λ=6πμa, for a sphere in an unbounded fluid of viscosity μ. When the particle contacts a ramp, it moves along the surface until it contacts the opposite electrode, at which point the cycle repeats. In the absence of Brownian motion, this ratchet directs particles down the channel with an average drift velocity, vdriftF/λ, as confirmed in experiment.17 Such motions are easily realized in microfluidic systems and allow for the rapid transport of micron-scale particles17 and droplets20 at velocities of up to 1 cm/s.

FIG. 1.

(a) Schematic illustration of a CCEP ratchet comprised of a conductive sphere between two parallel electrodes. A constant voltage V drives oscillatory particle motion which is rectified by a series of dielectric ramps. The dashed curve denotes the volume accessible to the particle's center; the blue curve shows the deterministic particle trajectory for Pe. (b) and (c) Particle dynamics for high and low Péclet numbers, Pe=100 and Pe=0.01, respectively. The left images show single particle trajectories generated using the Langevin description; the right images show the steady-state particle density p (colormap) and the particle flux (curves) generated by the Smoluchowski description.

FIG. 1.

(a) Schematic illustration of a CCEP ratchet comprised of a conductive sphere between two parallel electrodes. A constant voltage V drives oscillatory particle motion which is rectified by a series of dielectric ramps. The dashed curve denotes the volume accessible to the particle's center; the blue curve shows the deterministic particle trajectory for Pe. (b) and (c) Particle dynamics for high and low Péclet numbers, Pe=100 and Pe=0.01, respectively. The left images show single particle trajectories generated using the Langevin description; the right images show the steady-state particle density p (colormap) and the particle flux (curves) generated by the Smoluchowski description.

Close modal

Here, we consider the operation of this ratchet at smaller scales, at which thermal motions become the dominant force on the particle. The particle dynamics is captured by two complementary descriptions: a Langevin equation for generating stochastic particle trajectories and a Smoluchowski equation for computing the steady particle currents within the channel. Remarkably, we show that the particle velocity down the channel continues to scale as vdriftF/λ regardless of the strength of Brownian motion. This behavior is in sharp contrast to other types of Brownian motors such as flashing ratchets, for which the drift velocity decreases steadily with increasing thermal noise.21 We further investigate the amount of work that can be extracted from such a motor and discuss how thermal motions can be harnessed to improve ratchet performance. Finally, we examine the feasibility of realizing CCEP ratchets at the nanoscale using electrochemical reactions to drive particle motions. Our theoretical results suggest that ratcheted electrophoresis could provide a basis for electrochemically powered, nanoscale machines capable of transporting and actuating nanoscale components.

A simplified model of ratcheted CCEP is illustrated in Figure 1. A conductive sphere is immersed in a dielectric fluid and positioned between two parallel electrodes separated by a distance H=10a. Application of a constant voltage V creates an electric field E=V/H that drives the oscillatory motion of the particle between the electrodes via CCEP.18 This motion is rectified by a periodic array of impermeable dielectric ramps (grey) with height h=4a, width l=4a, and period L=6a. Below, we present two complementary descriptions of the particle's stochastic dynamics.

The geometry of the ratchet is similar to that used in previous experiments17 and is held constant throughout the present study. Several considerations guide the choice of the geometric parameters H, L, h, and l. In experiment, it was observed that directed transport is most reliable when the slope of the ramps is near 45°, such that h = l.17 The length Ll=2a should be large enough to allow the particle to contact the electrode surface; similarly, the length H2h=2a should allow the particle to move freely down the center of the channel in the absence of the field (e.g., due to fluid flow). The ratchet period L should allow directed transport in the deterministic limit (Pe), which implies that L>3.17a given the constraints above. Finally, as the electric driving force scales quadratically with a, the particle size should be comparable to that of the ratchet features.

The Brownian dynamics of a single particle is governed by the following Langevin equation for the particle position r(t)=[x(t),y(t)]:

(1)

The force on the charged particle is the sum of the electrostatic force FE=±qE and the ratchet force FR=UR due to a confining potential UR(r) (Fig. 1). The second term vB(t) is a random velocity process with zero mean vB(t)=0 and delta function correlation vB(t)vB(0)=2Dδ(t)δij, where D=kBT/λ is the diffusion coefficient. We further assume that the particle acquires a charge q on contact with the positively biased electrode at y = a and a charge −q on contact with the opposite electrode at y=Ha. Equation (1) is integrated numerically using a first order scheme22 with initial particle positions drawn from the equilibrium distribution in the absence of the applied field. From multiple realizations of this stochastic process, we compute the drift velocity of the particle down the ratcheted channel as vdrift=[x(t)x(0)]/t. Scaling lengths by the particle radius a, energy by kBT, and time by a2/D, the model is fully characterized by the ratchet potential UR(r)/kBT and the Péclet number Pe=aqE/kBT, which characterizes the relative importance of electrostatic actuation to Brownian motion. Here, we neglect electrostatic and hydrodynamic interactions between the particle and the electrodes; however, a more rigorous description based on Stokesian dynamics19,23 gives similar results.21 

In the limit of small Péclet numbers (Pe1), the particle dynamics is more easily described by the corresponding Smoluchowski equation for the probability density p(r,t) of finding the particle at position r and time t

(2)

where j(r,t) is the particle flux. As the particle can be found in two states of opposite charge, we separate the density into two contributions: p=p++p. At the positively biased electrode, the density of negative particles is zero (p=0 at y = a); likewise, that of positive particles is zero at the opposite electrode (p+=0 at y=Ha). All solid boundaries—namely, the electrodes and the ratchets—are assumed impermeable such that j·n=0, where n is the unit vector normal to the boundary. With these assumptions, we solve Equation (2) numerically to obtain the steady-state densities within a single unit cell of the periodic channel.21 The resulting drift velocity is computed by averaging the steady-state particle flux in the x-direction over the unit cell volume, vdrift=Vex·j(r,)dV.

The complementary descriptions differ only in the hardness of the ratchet walls. In the Langevin description, hard ratchet walls are approximated by a potential UR(z)z8, where z is the distance of the particle center from the wall.21 In practice, each approach has a limited range of applicability depending on the magnitude of the Péclet number. The Lanvegin description is better suited for Pe1, and the Smoluchowski description for Pe1. The two models are in quantitative agreement at intermediate Péclet numbers.

We first examine the steady drift velocity of the particle down the ratcheted channel vdrift as a function of the Péclet number (Fig. 2). There are two limiting regimes corresponding to large and small Péclet numbers. For Pe1, thermal motion is negligible, and the particle follows a nearly deterministic path down the channel in qualitative agreement with previous experimental observations.17 The drift velocity approaches vdrift=311qE/λ in the limit as Pe, where the prefactor depends on the ratchet geometry.

FIG. 2.

Drift velocity vs. Péclet number for the CCEP ratchet (blue) and the analogous rocking ratchet (green). Open and closed markers are obtained using the Langevin and Smoluchowski descriptions, respectively; dashed lines illustrate limiting behaviors. The frequency of the rocking ratchet is near its optimal value: ω=0.458D/a2 for Pe1 and ω=211FE/aλ for Pe1.

FIG. 2.

Drift velocity vs. Péclet number for the CCEP ratchet (blue) and the analogous rocking ratchet (green). Open and closed markers are obtained using the Langevin and Smoluchowski descriptions, respectively; dashed lines illustrate limiting behaviors. The frequency of the rocking ratchet is near its optimal value: ω=0.458D/a2 for Pe1 and ω=211FE/aλ for Pe1.

Close modal

For small Péclet numbers (Pe1), the magnitude of the drift velocity is somewhat reduced; however, it follows the same scaling relation as in the deterministic regime. For a given ratchet geometry, the precise drift velocity can be obtained using perturbation analysis;21 here, vdrift=0.0142qE/λ in the limit as Pe0. This behavior is remarkable in that the qualitative performance of the ratchet is largely unaffected by the presence of Brownian motion. The characteristic displacement of the particle has two contributions: an undirected component due to its Brownian motion, ΔxDt, and directed component due to its steady drift, Δxvdriftt. Owing to differences between these scaling relations, what looks like a random walk at short time scales (tD/vdrift2) conceals a steady drifting motion that emerges largely unperturbed on longer time scales (tD/vdrift2).

This behavior is in contrast to a related “rocking ratchet” that uses the same channel and a prescribed oscillating force directed perpendicular to the channel, FE(t)=FEcos(ωt)ey. At large Péclet numbers, the rocking ratchet performs similarly to the CCEP ratchet with a drift velocity of vdrift=0.174FE/λ at the near optimal frequency of ω=211FE/aλ. At small Péclet numbers, however, the rocking ratchet no longer functions effectively, and the drift velocity decreases with decreasing Péclet number as vdrift=0.0324PeFE/λ at the optimal frequency of ω=0.458D/a2 (Fig. 2).21 

The enhanced performance of the CCEP ratchet derives from the fact that the electric driving force does not change direction until the particle contacts the opposite electrode. This behavior increases the likelihood that the ratchet teeth will be used to translate forces perpendicular to the channel into motion parallel to the channel. For the rocking ratchet, it is quite often the case that the particle has not yet crossed the channel (or rather has crossed and returned) when the direction of the oscillating force changes sign. As a result, the performance of this and related Brownian ratchets deteriorates as the Péclet number decreases.21 

We now consider the ability of the CCEP ratchet to do work against a constant load force directed parallel to the channel in opposition to the particle's motion, FL=FLex (Fig. 3). In the deterministic regime (Pe1), there exists a critical load force FL* beyond which the ratchet will no longer function (FL*=0.283qE for the present geometry). Just above this stall force, the particle repeatedly fails to “catch” the next tooth of the ratchet as it moves backwards down the channel. For subcritical loads (FL<FL*), the ratchet operates reliably but with a slightly reduced drift velocity. In the opposite limit of small Péclet numbers (Pe1), the CCEP motor continues to function; however, the drift velocity and the stall force are diminished by an order of magnitude. In particular, the stall force is reduced to FL*=0.0314qE in the limit as Pe0 as computed via perturbation analysis.21 We emphasize that the numeric prefactor is less important than the scaling behavior, which is unaffected by the presence of strong thermal noise. The corresponding power output and mechanical efficiency of the CCEP ratchet are detailed in the supplementary material.21 

FIG. 3.

Drift velocity vs. load force for different Péclet numbers for the CCEP ratchet in Fig. 1. Open and closed markers show results obtained using the Langevin and Smoluchowski descriptions, respectively.

FIG. 3.

Drift velocity vs. load force for different Péclet numbers for the CCEP ratchet in Fig. 1. Open and closed markers show results obtained using the Langevin and Smoluchowski descriptions, respectively.

Close modal

Interestingly, the presence of some thermal noise can actually enable the ratchet to do work against larger loads than is possible in the deterministic regime (see Fig. 3, Pe=10). Under these conditions, random diffusive motions of the particle can sometimes help it to reach the next tooth of the ratchet, which is otherwise impossible for FL>FL* when Pe. This effect can be further exaggerated by increasing the period of the ratchet L such that deterministic operation is no longer possible.21 Such a ratchet functions—if at all—due to the presence of thermal noise in close analogy to the flashing ratchet.1,2 Nevertheless, it appears more desirable to use CCEP ratchets that function effectively (though not necessarily optimally) for all Pe.

Finally, we consider the performance of a CCEP ratchet under realistic experimental conditions (Fig. 4). We start from an actual implementation of this ratchet within a microfluidic channel17 and extrapolate to smaller scales. Here, the charge acquired by the particle is approximated as q=±23π3εa2E; the friction coefficient as λ=6πμa. The insulating carrier fluid (e.g., an aliphatic hydrocarbon such as decane) is assumed to have a relative permittivity of εr=2, a dynamic viscosity of μ=103Pas, and a dielectric strength of E=106V/m, which determines the maximum field. With these assumptions, the cross-over Péclet number, Pe=1, corresponds to a particle radius of a22nm and a speed of vdrift20μm/s. The applied voltage is only V200mV, which could potentially be generated by suitable electrochemical reactions, such as those used to power catalytic nanomotors.24,25

FIG. 4.

Drift velocity vs. particle size for different electric fields (solid curves); other parameters are held constant—namely, the relative permittivity (εr=2), the viscosity (μ=103Pas), and the ratchet geometry (H/a=10,L/a=6,l/a=h/a=4). The deterministic regime (Pe1, darker) is separated from the Brownian regime (Pe1, lighter) by the condition Pe=1 (dashed curve). The dashed-dotted curve shows when the particle charge is equal to the fundamental charge, q = e; the dotted curve shows when the applied voltage is equal to the thermal potential, V=kBT/e.

FIG. 4.

Drift velocity vs. particle size for different electric fields (solid curves); other parameters are held constant—namely, the relative permittivity (εr=2), the viscosity (μ=103Pas), and the ratchet geometry (H/a=10,L/a=6,l/a=h/a=4). The deterministic regime (Pe1, darker) is separated from the Brownian regime (Pe1, lighter) by the condition Pe=1 (dashed curve). The dashed-dotted curve shows when the particle charge is equal to the fundamental charge, q = e; the dotted curve shows when the applied voltage is equal to the thermal potential, V=kBT/e.

Close modal

At the nanoscale, additional factors neglected by the present analysis may become increasingly important. First, the capacitive charge q acquired by the particle becomes comparable to the elementary charge, and the effects of charge quantization are expected to influence the mechanistic details of particle charging (Fig. 4; dashed curve q = e). Provided the particle's charge changes only upon contact with the bounding electrodes, these details are unlikely to alter the qualitative performance of the ratchet. Second, surface forces such as van der Waals interactions may become increasingly large compared to the electric force causing the particle to “stick” to the electrodes. Fortunately, there exist strategies to mitigate surface adhesion,26 though one must also avoid inhibiting the charge transfer process. Third, the voltage between the electrodes can become comparable to the thermal potential (Fig. 4; dotted curve V=kBT/e), such that charge transfer is increasingly subject to thermal fluctuations. In light of these complicating factors, some assumptions of the present model become questionable in the Brownian regime—most notably, the use of classical electrostatics to approximate the charge q. Nevertheless, the operation of the CCEP ratchet is insensitive to these details provided the particle charges repeatedly on contact with the bounding electrodes. In this way, ratcheted electrophoresis can enable directed transport in the face of seemingly overwhelming thermal noise.

We examined the stochastic dynamics of particles moving via CCEP within a ratcheted channel and quantified their directed motion as a function of the thermal noise. We showed that CCEP ratchets are relatively insensitive to noise and can operate over a wide range of conditions in contrast to existing ratchets. This robust performance derives from the unique characteristics of CCEP motion—namely, that changes in the electric force are directly coupled to the location of the particle and its progress down the channel. We argue that rectified CCEP motions may be well-suited to directing the transport and/or actuation of nanoscale components powered by electrochemical potential gradients. The present results should be useful in guiding the rational design of such nanoscale machines where thermal noise plays a significant or even dominant role.

This work was supported by the National Science Foundation under Grant No. CBET-1351704.

1.
P.
Reimann
, “
Brownian motors: Noisy transport far from equilibrium
,”
Phys. Rep.
361
,
57
265
(
2002
).
2.
P.
Hänggi
and
F.
Marchesoni
, “
Artificial Brownian motors: Controlling transport on the nanoscale
,”
Rev. Mod. Phys.
81
,
387
442
(
2009
).
3.
R. D.
Astumian
and
M.
Bier
, “
Fluctuation driven ratchets: Molecular motors
,”
Phys. Rev. Lett.
72
,
1766
1769
(
1994
).
4.
J.
Prost
,
J. F.
Chauwin
,
L.
Peliti
, and
A.
Ajdari
, “
Asymmetric pumping of particles
,”
Phys. Rev. Lett.
72
,
2652
2655
(
1994
).
5.
J.
Rousselet
,
L.
Salome
,
A.
Ajdari
, and
J.
Prost
, “
Directional motion of Brownian particles induced by a periodic asymmetric potential
,”
Nature
370
,
446
448
(
1994
).
6.
L. P.
Faucheux
,
L. S.
Bourdieu
,
P. D.
Kaplan
, and
A. J.
Libchaber
, “
Optical thermal ratchet
,”
Phys. Rev. Lett.
74
,
1504
1507
(
1995
).
7.
S.
Verleger
,
A.
Grimm
,
C.
Kreuter
,
H. M.
Tan
,
J. A.
van Kan
,
A.
Erbe
,
E.
Scheer
, and
J. R. C.
van der Maarel
, “
A single-channel microparticle sieve based on Brownian ratchets
,”
Lab Chip
12
,
1238
1241
(
2012
).
8.
W. C.
Germs
,
E. M.
Roeling
,
L. J.
Van Ijzendoorn
,
B.
Smalbrugge
,
T.
De Vries
,
E. J.
Geluk
,
R. A. J.
Janssen
, and
M.
Kemerink
, “
High-efficiency dielectrophoretic ratchet
,”
Phys. Rev. E
86
,
041106
(
2012
).
9.
E. M.
Roeling
,
W. C.
Germs
,
B.
Smalbrugge
,
E. J.
Geluk
,
T.
de Vries
,
R. A. J.
Janssen
, and
M.
Kemerink
, “
Organic electronic ratchets doing work
,”
Nat. Mater.
10
,
51
55
(
2011
).
10.
P.
Hänggi
, “
Organic electronics: Harvesting randomness
,”
Nat. Mater.
10
,
6
7
(
2011
).
11.
C.
Kettner
,
P.
Reimann
,
P.
Hänggi
, and
F.
Müller
, “
Drift ratchet
,”
Phys. Rev. E
61
,
312
323
(
2000
).
12.
C.
Marquet
,
A.
Buguin
,
L.
Talini
, and
P.
Silberzan
, “
Rectified motion of colloids in asymmetrically structured channels
,”
Phys. Rev. Lett.
88
,
168301
(
2002
).
13.
S.
Matthias
and
F.
Müller
, “
Asymmetric pores in a silicon membrane acting as massively parallel Brownian ratchets
,”
Nature
424
,
53
57
(
2003
).
14.
K.
Loutherback
,
J.
Puchalla
,
R. H.
Austin
, and
J. C.
Sturm
, “
Deterministic microfluidic ratchet
,”
Phys. Rev. Lett.
102
,
045301
(
2009
).
15.
M.
Tagliazucchi
and
I.
Szleifer
, “
Salt pumping by voltage-gated nanochannels
,”
J. Phys. Chem. Lett.
6
,
3534
3539
(
2015
).
16.
L.
Gorre
,
E.
Ioannidis
, and
P.
Silberzan
, “
Rectified motion of a mercury drop in an asymmetric structure
,”
Europhys. Lett.
33
,
267
272
(
1996
).
17.
A. M.
Drews
,
H.-Y.
Lee
, and
K. J. M.
Bishop
, “
Ratcheted electrophoresis for rapid particle transport
,”
Lab Chip
13
,
4295
4298
(
2013
).
18.
A. M.
Drews
,
C. A.
Cartier
, and
K. J. M.
Bishop
, “
Contact charge electrophoresis: Experiment and theory
,”
Langmuir
31
,
3808
3814
(
2015
).
19.
A. M.
Drews
,
M.
Kowalik
, and
K. J. M.
Bishop
, “
Charge and force on a conductive sphere between two parallel electrodes: A Stokesian dynamics approach
,”
J. Appl. Phys.
116
,
074903
(
2014
).
20.
D. J.
Im
, “
Next generation digital microfluidic technology: Electrophoresis of charged droplets
,”
Korean J. Chem. Eng.
32
,
1001
1008
(
2015
).
21.
See supplementary material at http://dx.doi.org/10.1063/1.4950801 for details.
22.
D. L.
Ermak
, “
A computer simulation of charged particles in solution. I. Technique and equilibrium properties
,”
J. Chem. Phys.
62
,
4189
4196
(
1975
).
23.
J. W.
Swan
and
J. F.
Brady
, “
Particle motion between parallel walls: Hydrodynamics and simulation
,”
Phys. Fluids
22
,
103301
(
2010
).
24.
Y.
Wang
,
R. M.
Hernandez
,
D. J.
Bartlett
,
J. M.
Bingham
,
T. R.
Kline
,
A.
Sen
, and
T. E.
Mallouk
, “
Bipolar electrochemical mechanism for the propulsion of catalytic nanomotors in hydrogen peroxide solutions
,”
Langmuir
22
,
10451
10456
(
2006
).
25.
W.
Wang
,
W.
Duan
,
S.
Ahmed
,
T. E.
Mallouk
, and
A.
Sen
, “
Small power: Autonomous nano- and micromotors propelled by self-generated gradients
,”
Nano Today
8
,
531
554
(
2013
).
26.
K. J. M.
Bishop
,
C. E.
Wilmer
,
S.
Soh
, and
B. A.
Grzybowski
, “
Nanoscale forces and their uses in self-assembly
,”
Small
5
,
1600
1630
(
2009
).

Supplementary Material