Mechanisms of Symmetry Breaking in a Multidimensional Flashing

Mechanisms of Symmetry Breaking in a Multidimensional Flashing Particle Ratchet Ofer Kedem,† Bryan Lau,†,‡ and Emily A. Weiss*,†,‡ †

Center for Bio-Inspired Energy Science, Northwestern University, 303 East Superior Street, 11th floor, Chicago, Illinois 60611-3015, United States ‡ Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States S Supporting Information *

ABSTRACT: Ratcheting is a mechanism that produces directional transport of particles by rectifying nondirectional energy using local asymmetries rather than a net bias in the direction of transport. In a flashing ratchet, an oscillating force (here, an AC field) is applied perpendicular to the direction of transport. In an effort to explore the properties of current experimentally realizable ratchet systems, and to design new ones, this paper describes classical simulations of a damped flashing ratchet that transports charged nanoparticles within a transport layer of finite, non-zero thickness. The thickness of the layer, and the decay of the applied field in the z-direction throughout that thickness, provide a mechanism of symmetry breaking in the system that allows the ratchet to produce directional transport using a temporally unbiased oscillation of the AC driving field, a sine wave. Sine waves are conveniently produced experimentally or harvested from natural sources but cannot produce transport in a 1D or pseudo-1D system. The sine wave drive produces transport velocities an order of magnitude higher than those produced by the common on/off drive, but lower than those produced by a temporally biased square wave drive (unequal durations of the positive and negative states). The dependence of the particle velocity on the thickness of the transport layer, and on the homogeneity of the oscillating field within the layer, is presented for all three driving schemes. KEYWORDS: flashing ratchet, particle ratchet, finite element, nonequilibrium, broken symmetry, Brownian motor, simulation

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produce directional transport of charged particles. Understanding the mechanisms by which nondirectional energy is rectified in these complex systems not only poses a fundamentally interesting problem in far-from-equilibrium dynamics, but also allows us to identify and translate the features of highly efficient biological ratchets to artificial devices. For instance, the ratcheting mechanism is a promising strategy for improving the directional transport of electrons within nominally electrically insulating materials for electronic and solar energy conversion applications. Electron ratchets, in many cases, operate in a nearly classical manner,11 so their optimization also benefits from insights gained using classical simulations like those we conduct here. There are three requirements for ratcheting to occur: (i) energy must be supplied to the system to drive it away from equilibrium; (ii) the spatial inversion symmetry along the direction of transport must be broken; and (iii) the temporal symmetry of the system must be broken through irreversible

olecular motors in biological systems use asymmetry in protein structures and in their conformational changes to produce directional motion from nondirectional energy, specifically ATP and thermal noise. Motors perform critical tasks such as transporting cargo,1 pumping ions across membranes,2 and contracting muscle fibers.3 Such motors are ratchets, nonequilibrium devices for transporting mass through the application of nonbiased forces; if that mass is in the form of discrete particles, the systems used to transport them are called particle ratchets.4−7 Synthetic ratchets have been used to transport particles such as plastic microspheres and DNA molecules,8−10 and to separate them based on their mass or charge. They are particularly promising for the transport and separation of particles in highly damped environments environments where particles undergo a high frequency of collisions and achieve no inertial motionand media where strong gradients cannot be applied, for instance, due to electric screening. Rather than using strong gradients across the entire device, ratchets periodically apply local forces to transport particles. This paper describes an investigation, using classical simulations, of a two-dimensional particle ratchet system and, in particular, how the dimensionality of the system influences its ability to rectify a nonbiased oscillating electric field to © XXXX American Chemical Society

Received: May 1, 2017 Accepted: July 12, 2017 Published: July 12, 2017 A

DOI: 10.1021/acsnano.7b02995 ACS Nano XXXX, XXX, XXX−XXX

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Figure 1. (a) Illustration of the working principle of an on/off flashing ratchet−in the on state, particles are trapped in potential wells; upon turning off the potential, particles are free to diffuse isotropically; when the potential is turned on again, particles are preferentially trapped in potential wells to the right, due to the asymmetry of the potential, producing net transport. Reprinted with permission from ref 13. Copyright 2017 Royal Society of Chemistry. (b) Electric potential (colormap) uniform (top) or varying (bottom) in the z-direction. The uniform case is close to the 1D approximation of the ratchet model, whereas experimental realizations typically use asymmetrically spaced pairs of electrodes under the transport layer to apply the potential, producing a z-varying field.

make the field locally inhomogeneous along the z-direction within the layer. Furthermore, we show here that motion along a changing field in the z-direction can in fact be utilized to increase the ratchet current and to enable the use of driving mechanisms that are unavailable in the 1D (or the effectively 1D “z-uniform”) picture. Despite the importance of the transport layer thickness in ratchet performance and its easy tunability in experimental ratchets, the impact of z-direction motion on the ratchet current has not been explored computationally or explicitly studied experimentally. Here, we classically model the motion of nanoparticles within a ratchet where the transport layer (water) has non-zero thickness. The particles experience an oscillating and locally asymmetric field but are subject to no overall bias in the direction of motion (x-direction). The velocity of the particles in the x-direction is therefore due exclusively to a ratcheting mechanism. We find that the particle velocity is largely insensitive to the layer thickness for the common on/off drive of the ratchet potential (Figure 1a), but the velocity obtained using this driving scheme is very low due to the reliance on a slow diffusive step in the “off” state for the majority of transport. Use of a biased square wave drive of the potential (which switches between unequal durations of attractive and repulsive potentials) increases the average velocity by up to 2 orders of magnitude from that obtained with the on/off drive, but this velocity decreases for thicker layers. In contrast, we observe that a non-zero thickness of the transport layer, and a decay of the applied field throughout that layer, enable the use of a temporally unbiased waveform (specifically, a sine wave) to drive the potential. Using an unbiased waveform, like a sine wave, which produces zero current in a 1D ratchet,27 would be highly advantageous for ratchet-based devices, as sine waves are significantly easier to produce than biased waveforms and can be harvested from sources like electromagnetic radiation (e.g., by connecting the electrode array, producing the asymmetric potential, to a suitable antenna).

dissipation of energy from the particle to its environment or through time-asymmetric driving.5,12,13 In general, ratchet systems in the theoretical and experimental literature can be divided into three categories. In a tilting ratchet, a zero-average alternating force is applied across the system in the direction of transport; asymmetries in the system itself (e.g., a semiconductor transport channel with an asymmetric width profile) provide the necessary symmetry-breaking.14−16 A related system, the drif t ratchet, contains a planar substrate with asymmetric inclusions, and the driving is applied perpendicular to the direction of transport.17−19 In a f lashing ratchet, the focus of this work, the particles are subjected to a time-varying, periodic, locally asymmetric field, applied perpendicular to a (typically uniform and symmetric) transport layer, while maintaining a constant overall zero bias across the device in the direction of transport.20−22 The applied field can be stochastically23 or periodically24 switched between “on” and “off” states, as in Figure 1a, or between any combination of two (or more) different states, where at least one state contains asymmetric repeating units. Symmetric but spatially shifted potentials have also been reported to produce ratcheting.25 Almost exclusively in the literature, particles in simulated flashing ratchet systems are limited to motion in a single direction and thus experience a potential that only varies in 1D (Figure 1b, top). While this approximation is reasonable for some cases, such as optically trapped cold-atom ratchets22 or transport across nanowires,21 it is less realistic for most experimental ratchets, such as organic electron ratchets24,26 or ratchets transporting molecules or other particles in water.8,9 In those experimental systems, the transport layer is typically uniform in the y-direction (in the plane of the transport layer, perpendicular to the direction of transport), but not in the zdirection, because the applied field that dictates the ratchet potential, often applied by pairs of electrodes under the transport layer, decays throughout the thickness of the layer (Figure 1b, bottom). The particles thus gain another degree of freedom, and their response in the z-direction to repulsive and attractive forces, their scattering in the z-direction, and their collisions with the top and bottom boundaries of the transport layer become important factors in the operation of the ratchet. This property provides additional means of breaking the symmetry in ratchet devices. While one can, in principle, construct a ratchet device in which the field is uniform in the zdirection (Figure 1b, top) by, say, using two sets of electrodes on top and bottom, it is difficult to overcome small misalignments of these electrodes, or screening effects that

RESULTS AND DISCUSSION We consider an ensemble of noninteracting particles, governed by a Newton−Langevin equation of the form in eq 1. mr(̈ t ) = −q

dV (t,r ) − γr ̇ + FB + ξ(t ) dr

(1)

In eq 1, m and q are the mass and charge of the particle, respectively; V(t,r) = g(t)·U(r) is the spatially and temporally B


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dimensionless position r̂ = r/L; time t ̂ = t/τ0 where τ0 = γL2/ Aq and A is the amplitude of the applied potential (discussed below); and potential V̂ (t ̂, r )̂ = V(t,r)/A. The rescaled thermal noise term is ⟨ξ (̂ t )̂ ξ (̂ s )̂ ⟩ = 2D̂ δ(t ̂ − s )̂ , where D̂ = kBT/Aq. The dimensionless Langevin equation and wall definitions can now be written as eq 4 (dropping the hat notation for convenience). In eq 4,

varying potential; γ is the viscous drag coefficient; FB describes the force acting on the particles upon collisions with the top and bottom boundaries of the simulation area; and ξ(t) is a δcorrelated Gaussian white noise term (serving as the Brownian force), such that ⟨ξ(t)⟩ = 0 and ⟨ξ(t)ξ(s)⟩ = 2γkBTδ(t − s), in accordance with the dissipation−fluctuation theorem, where kB is the Boltzmann constant and T is the temperature of the system.28 The friction coefficient is γ = 6πμrp, where μ is the dynamic viscosity of the medium and rp is the radius of a particle. The diffusion coefficient of the particles is given by the Stokes−Einstein relation, D = kBT/γ.28 The simulation area is a 2D rectangle of height d and length l. The term FB in eq 1 describes diffuse scattering from the boundaries, which are defined by eq 2. In eq 2, wb(x) and wt(x) define the bottom and top boundaries of the wb(x) = −d /2;

wt(x) = d /2

κr(̈ t ) = −

wb(x) = −d /2L ;

wt(x) = d /2L

(4)

κ = m/τ0γ is a dimensionless particle mass and λ = L/Aq is a characteristic reciprocal force. The dimensionless version implies that if the physical parameters are varied such that the dimensionless parameters remain unchanged, the performance of the ratchet would be the same. For example, a change in particle mass m can be compensated for by a corresponding change of τ0 (which might result in further changes to other terms). Ratchets with an On/Off (Attractive/Diffusive) Drive Are Largely Insensitive to Layer Thickness, but Perform Better with z-Uniform Potentials. We first briefly explore the effect of transport layer thickness in a system with on/off driving, eq 5 and Figure 2a,b, as it is commonly used in

(2)

transport layer, respectively. The postscattering velocity v(t + dt) is defined by eq 3. In eq 3, ν(t + dt ) = |v(t )| sin θt ̂ + |v(t )| cos θn ̂

dV (t,r ) − r ̇ + λFB + ξ(t ); dr

(3)

π , 2

θ = acos Γ − and Γ is a uniformly distributed random number between −1 and 1, and t ̂ and n̂ are unit vectors tangential and normal to the boundary, respectively. The ratchet potential U(r) is periodic in x with period L, such that U(x) = U(x + L), and is defined explicitly below. We define the temporal component, g(t) = A·waveform(2πft), where A is the amplitude and f is the frequency of oscillation. We use three temporal waveformsan on/off drive, a square wave with different duty ratios (ratios of times in the positive and negative states), and a sine wave. Though an on/off drive is equivalent to a square wave drive with a constant offset, it is a special case, as the off state allows for free diffusion of the particles, and so we treat it separately from the square wave case. For each driving scheme, we compare the transport produced by the ratchet for two types of applied potentials. In the first “zuniform” type, the potential is uniform in the z-direction and therefore is defined solely by the x-coordinate (Figure 1b, top). This scenario is similar to most published models in that the potential only varies spatially in a single dimension. In the second “z-varying” type, the potential is applied to the bottom edge of the system and produces, in the transport layer, a field that decays in the z-direction with the distance from the bottom to the top of the layer (Figure 1b, bottom). The ratchet potential thus varies in two dimensions. The latter scenario is similar to experimental realizations of flashing ratchets, where the potential is applied by electrodes beneath the transport layer. As experimental ratchet systems are typically uniform in the y-direction, we only model motion in the x−z plane. The primary observable is the mean velocity of the particles in the xdirection, taken over the last eight oscillations of the potential, during which the particles have achieved a dynamic steadystatea constant rate of change in the x-position per oscillation (Figure S1). The Methods section contains additional details regarding the simulation. To ease the translation to experimental systems, we present results for realistic parameters; however, our findings are valid for a wide range of values. The Langevin equation proscribing the motion of the particles can be written in dimensionless form by normalizing the parameters (e.g., position) by the physical characteristics of the system.30,31 We define the

⎧ A , nτ ≤ t < (n + 1/2)τ g (t ) = ⎨ ⎩ 0, (n + 1/2)τ ≤ t < (n + 1)τ ⎪ ⎪

n∈ (5)

Figure 2. (a) Temporal on/off drive; (b) the shape of the potential for the “on” and “off” states; (c) mean particle velocity in the xdirection, vx as a function of transport layer thickness for a zuniform (shades of red) and z-varying (shades of blue) potential at different oscillation frequencies (indicated), A = 10 V. For panel c, a more negative value of vx corresponds to higher current. C


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Figure 3. (a) Temporal square wave drive, 0.4 duty ratio; (b) shape of the potential, at the bottom boundary, for the positive and negative states; (c) scheme of the transport mechanism for a z-uniform potential−previously trapped particles travel in both direction in the short phase, but only the ones moving to the right travel far enough to reach the next potential well and transport right-ward when the potential reverses polarity in the following long phase; (d) mean particle velocity in the x-direction, vx as a function of transport layer thickness for a zuniform (red) and z-varying potential (shades of blue) at different oscillation frequencies (indicated), for a square wave drive, 0.4 duty ratio, A = 10 V.

wells, here they are actively repelled. Oscillating the potential between two states of equal magnitude and opposite polarity, for equal durations, however, will produce no transport, even in the case of a z-varying potential. To understand why, let us assume that the asymmetric ratchet potential U(x) is driven by a simple square wave g(t), oscillating between +1 and −1, for equal durations. The particles will experience the potentials U(x) and −U(x). If the asymmetry of U(x) tends to transport particles to the left, then that of −U(x) will transport them to the right. As the potentials are applied for equal durations, the currents they produce will be exactly equal and opposite and cancel. The symmetry of the driving function, therefore, nullifies the current. One way to address this limitation is by applying the two states for unequal durations (i.e., a temporally biased waveform), so that their time-averaged magnitudes differ.27 We thus oscillate the potential, defined spatially by the biharmonic expression in eq 6, as a temporally biased square wave, with states of equal magnitude but unequal durations, as defined in eq 7, and plotted in Figure 3a,b. In eq 7, dutyratio is the fraction of the

the literature on particle and electron ratchets and is mechanistically the simplest flashing ratchet. In eq 5, τ = 1/f is the period of oscillation. The ratchet potential shape U(x) is biharmonic, as defined by eq 6, and plotted in Figure 2b. In the “off” state, particles diffuse isotropically. When ⎛ 2πx ⎞ ⎛ 4πx ⎞ ⎟ + a sin⎜ ⎟ ; a = 1, a = 0.25 U (x) = a1 sin⎜ 2 2 ⎝ L ⎠ ⎝ L ⎠ 1

(6)

the potential is turned on, particles are accelerated toward potential wells, and ratcheting occurs because the asymmetric potential causes some particles, depending on their xcoordinate after diffusion, to relax asymmetrically to adjacent potential wells during the “on” stage (Figure 1a, and movies M1 and M2). For the studied range, the amplitude of the potential only weakly affects the current (Figure S2a), as the velocitylimiting step is the diffusion of the particles, rather than their trapping. With the on/off driving scheme, the velocity is generally lower for the z-varying (Figure 1b, bottom) case than for the z-uniform case (Figure 1b, top), as the asymmetric shape of the field decays to a more symmetric shape on going from the bottom boundary to the top boundary, so particles traveling near the top boundary experience a more symmetric field. For the z-uniform potential, the thickness of the layer has no impact on the velocity, as expected; for the z-varying case, thinner layers produce higher velocities (Figure 2c). If the polarity of the potential in the “on” state is reversed, so is the direction of transport, Figure S2b. Eliminating the Diffusive Stage by Using a Temporally Biased Square Wave Drive Produces 100 Times Faster Transport than an On/Off Drive but Loses Effectiveness for Thicker Layers. As noted previously by Tarlie and Astumian for a 1D system,27 the transport velocity can be increased drastically by eliminating the slow diffusive stage of the on/off drive and instead continuously driving the particles by alternating the polarity of the potential (Figure 3a), such that the positions of the potential wells and barriers alternate in time (Figure 3b). Rather than waiting for slow diffusion to move the particles from their former potential

⎧ A , nτ ≤ t < (n + dutyratio)τ ⎪ g (t ) = ⎨ ⎪ ⎩−A , (n + dutyratio)τ ≤ t < (n + 1)τ

n∈ (7)

oscillation spent in the positive (g(t) = A) state. The different durations of the positive and negative phases of the temporal waveform, quantified as the duty ratio, together with the asymmetry of the potential, result in particles traveling far enough to reach the adjacent well on one side, but not the other, and thus produce overall directional transport (Figure 3c and movies M3 and M4). The particle velocity for the square wave drive is two orders of magnitude greater than that for the on/off drive for a potential of the same amplitude, as it allows the use of significantly higher oscillation frequencies. In the z-uniform case, the thickness of the transport layer has essentially no impact on the velocity for this driving scheme. In the z-varying case, however, the velocity approaches zero as the D


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Figure 4. (a) Temporally unbiased sine wave drive; (b) electric potential along the x-axis, at the bottom boundary of the transport layer, throughout a cycle of the sine wave; (c) scheme of the transport mechanism; the particles trapped at the bottom boundary are repelled toward the top, where they experience a weak repulsive potential; though both attractive and repulsive states are applied for equal durations, the weaker magnitude of the repulsive state transports particles more slowly, and so particles moving to the left at the top boundary cannot reach the next potential well before the potential turns attractive again, and they are trapped at the bottom boundary; (d) mean particle velocity in the x-direction as a function of transport layer thickness for sine wave driving at different flashing frequency (indicated), T = 293 K; (e) velocity for different temperatures (indicated), varying TBrownian; f = 10 kHz. A = 10 V in all cases.

thickness increases from 10 to 800 nm (Figure 3d). As the field decays in the z-direction, the particles are driven to travel in arcs in the x-z plane between potential wells on the bottom boundary, rather than only along the x-direction, as for the zuniform case (Figure S2c and movies M3 and M4). For a thin transport layer, the top boundary quickly arrests the zcomponent of the trajectory, and transport progresses much like in the z-uniform case (Figure 3c). For a thick layer, however, the particles travel farther from the bottom boundary, and experience weaker and more symmetric potentials. The particles are therefore subject to weaker x-acceleration, requiring longer times to reach an adjacent well, and thereby reducing the overall velocity. Thicker transport layers yield lower velocities for a wide range of duty ratios and frequencies, Figure S3. At a duty ratio of 0.5 (equal positive and negative durations), the particles experience mirror images of the asymmetric potential equal duration, which average to zero, and so no transport is produced; the velocity reverses direction between duty ratios above and below 0.5 (compare Figure 3d with Figure S2d). The velocity, and its dependence on the thickness of the transport layer, is sensitive to the oscillation frequency. For a high frequency ( f = 160 kHz) the transport even reverses direction for thick layers. Current reversals in response to the variation of an operating parameter (e.g., frequency) are a commonly observed in ratchets.5 Overall, a square wave drive can produce velocities two orders of magnitude larger than those produced by an on/off

drive but comes with two complications: (i) at frequencies that produce its peak performance, the velocity is highly sensitive to the thickness of the layer for the more experimentally accessible z-varying potential, and (ii) it requires a temporally biased waveform (unequal positive and negative durations), which is more complex to produce experimentally than an on/off drive. The Thickness of the Transport Layer Can Be Used To Break the Symmetry, Allowing Use of a Temporally Unbiased Drive−A Sine Wave. We have previously observed transport for temporally unbiased drives (e.g., sine and triangle waveforms) in an experimental electron ratchet system with a thick transport layer, which led us to suspect that motion of particles in the z-direction influences the current in such systems.26 We show here that transport within a multidimensional ratchet system driven by an unbiased sine wave can indeed occur, by utilizing motion in the z-direction to break the symmetry of the drive. In the previous section, we broke the symmetry between the two states of the potential by applying them for unequal durations (Figure 3c); here, instead, we use equal durations, but take advantage of motion in the z-direction to effect unequal magnitudes (Figure 4c). This mechanism has two necessary conditions: (i) the transport layer must have a non-zero thickness, with a ratchet potential that decays strongly in the z-direction, and (ii) the potential should oscillate in time between a fully attractive and a fully repulsive state (as in Figure 4b), as opposed to being positive at some x-coordinates and simultaneously negative at other x-coordinates (as would be the case for a simple biharmonic, Figure 3b). In ratchets that satisfy these conditions, the particles are alternately trapped at the E


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temperature value into two parameters, Tviscosity and TBrownian. We vary TBrownian, while maintaining a constant Tviscosity. The transport velocity and optimal layer thickness both increase with increasing temperature from 77 to 293 K, Figure 4e, but do not change when the temperature is further raised to 373 K. This finding indicates that some degree of Brownian motion is beneficial to the transport mechanism, even though the particles in this ratchet are continuously driven and have no explicit diffusion step. The additional kinetic energy allows the particles to spread further along the upper boundary, where the potential is weak, and reach adjacent wells when the potential turns fully attractive again. This type of unbiased driving scheme, which utilizes a modified potential to harness z-direction motion, produces superior velocities to the on/off drive, while allowing the use of a simple sine wave, rather than a temporally biased square wave. Its performance can be optimized by choosing a thickness for the transport layer appropriate for the desired driving frequency. We also compared the performance of the ratchet transport mechanism to a simple static bias across the length of the transport layer (in the x-direction), without any ratchet potential. The ratchet, using the on/off, square and sine ratchet drives, achieved velocities of approximately 0.1, 20, and 1 mm/ s, respectively, under the optimal conditions specific to each drive. These velocities correspond to static fields of approximately 320, 64300, and 3200 V/cm, respectively (see the Supporting Information).

bottom and top boundaries (Figure 4c, Figure S5c, and movie M5), since the potential oscillates between fully attractive and fully repulsive states. The alternating positions of the particles at the bottom and top boundaries mean that the attractive potential that they experience is stronger than the repulsive potential, since they are at the bottom boundary when the attractive potential is applied. Therefore, for the particles, the opposite asymmetries of the two states are of unequal magnitude and do not nullify one another. After reaching a boundary, the particles travel parallel to it (at a small distance from it, due to the particles’ thermal fluctuations and collisions with the boundaries), in a direction determined by the local potential gradient. When the potential reverses polarity, the x-position of the particle, to the left or right of a potential barrier, determines to which direction it travels (Figure 4c and movie M5). For a simple biharmonic spatial potential with an unbiased drive, no current arises because the particles jump between potential wells on the lower boundary, never localizing on the top boundary, and so experience equal and opposite potentials. We set the temporal waveform to be a sine wave, eq 8 (Figure 4a), and use a modified biharmonic spatial potential, which switches between purely positive and negative states (Figure 4b), eq 9, to satisfy condition (ii) listed above. For a zuniform potential (Figure 1b, top), no

g (t ) = A sin(2πft ) U (x ) =

(8)

⎛ 2πx ⎞⎞ 1 ⎛ ⎛ 4πx ⎞⎞ 1 ⎛ ⎟⎟ + ⎟⎟ ; a1⎜1 + sin⎜ a 2⎜1 + sin⎜ ⎝ L ⎠⎠ ⎝ L ⎠⎠ 2 ⎝ 2 ⎝

a1 = 1,

a 2 = 0.25

CONCLUSIONS In this work, we have explored a simulated electrostatic flashing particle ratchet where the particles are free to move in two dimensions, compared to previous work in the field, which limited particles to motion in a single dimension. The addition of this second degree of freedom, coupled with an applied electric potential that decays throughout the thickness of the layer, results in a more realistic representation of most experimental ratchet devices. We identify a ratchet transport mechanism where we use the variation in the applied field through the thickness of the transport layer to break the symmetry of particle motion when driven by an unbiased temporal waveform; this strategy allows the use of a simple sine wave to produce ratcheting, a result not possible in a 1D (or effectively 1D) system. A sine wave is easier to produce than biased waveforms, and its use in a ratchet system simplifies its design and operation, and could enable the use of energy harvested from the environment, like electromagnetic waves, to power a flashing ratchet. The sine wave also produces transport velocities an order of magnitude faster than produced by the commonly studied on/off drive, and can be optimized by careful choice of the transport layer thickness. We find that the thickness of the transport layer is of critical importance for all cases where a z-varying potential is used, which is the case in most experimental implementations of flashing ratchets.

(9)

transport is produced (not shown) because condition (i) is not satisfied. For a z-varying potential (Figure 1b, bottom), where we ground the top boundary of the transport layer to facilitate a rapid change in the potential as a function of the z-position, we observe transport in the x-direction, with a magnitude that depends on the thickness of the transport layer and the frequency of oscillation (Figure 4d). The maximal velocity we observe is an order-of-magnitude higher than that produced by the commonly used on/off drive, but an order of magnitude lower than for a biased square wave. We note that, under these conditions of a modified potential and grounded top boundary, an unbiased square wave drive (0.5 duty ratio) also produces transport of similar magnitude and thickness dependence to the sine wave drive (Figure S4a). The velocity of the particles displays a peaked dependence on the thickness of the transport layer, Figure 4d, with the sine wave drive. The optimal thickness decreases with increasing flashing frequency, as the particles must travel between the two boundaries twice per oscillation, so a thinner layer allows for shorter travel times. The particles are not perfectly localized at the boundaries−thermal fluctuations (Brownian motion) allow the particles to inhabit a finite thickness near the boundaries. If the thickness of the transport layer approaches this finite thickness, the particles no longer travel between two welldefined potentials, and the ratcheting diminishes (below ∼50 nm in Figure 4d). Increasing temperature decreases the viscosity of the medium and increases the intensity of Brownian motion in the system (see the Supporting Information), which results in higher transport velocity (Figure S5a,b). To independently examine the effect of Brownian motion on the performance of the devices, we artificially separate the

METHODS In every run, we simulate 1000 independent particles, each for 10 oscillations of the potential; the actual duration varies with the oscillation frequency used. The Supporting Information contains movies of particle trajectories under the influence of the ratchet potential. At time = 0, the particles are released from the exact center of the simulation area (0,0) and spread radially, with an initial velocity of 10 mm/sec, though this velocity is almost immediately eliminated F


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ACS Nano by friction, and the specific value has no impact on the observable. The medium, for the purposes of dielectric constant (εr = 80) and dynamic viscosity (μ(293 K) = 0.001009 Pa·s), is water. The particles have the density of Au (19.3 g/cm3), radius r = 2.5 nm, and a charge of −e. Unless otherwise noted, we simulate the system at T = 293.15 K (the temperature controls the viscosity of the medium and the intensity of the Brownian force). The spatial periodicity of the potential, L, is 1 μm for all simulations shown here. Upon hitting the left or right boundaries, particles exit the simulation area, but the length, 16−24 spatial periods L, is selected to be long enough so that no (or a negligible fraction of) particles reach the right or left edges of the simulation area during the simulation period. Here, we simulate an extended system (multiple periods of the potential) to more easily visualize the transport mechanism; we confirmed that the results we obtain are however equivalent to simulating a single period with periodic boundary conditions in the x-direction, where particles are randomly distributed throughout the layer at t = 0, with zero initial velocity (Figure S4b). The model includes no particle−particle interactions or screening of the electric field by the charged particles, so each simulation is equivalent to a collection of individual single-particle simulations; the trajectories of the various particles differ from one another due to the presence of random forces, which are the Brownian force and the diffuse reflections from the top and bottom boundaries. The values of the parameters used in the simulation are in Table S1. All of the simulations presented in this work are performed using the finiteelement software COMSOL Multiphysics 5.2−5.3. The time steps taken by the solver are chosen by the generalized-α method.29

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ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b02995. Movie captions; simulation parameters; mean particle xposition vs time for several cases; additional velocity vs transport layer thickness plots for different drives and temperature; plots showing the trajectories of particles under biased square and unbiased sine wave drives; and the results of simulations using a static bias to transport particles in the x-direction (PDF) Movies tracking the motion of particles for the different cases studied (ZIP)

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Ofer Kedem: 0000-0001-7757-8335 Emily A. Weiss: 0000-0001-5834-463X Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This material is based upon work supported as part of the Center for Bio-Inspired Energy Science, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DESC0000989. REFERENCES (1) Wang, W.; Cao, L.; Wang, C.; Gigant, B.; Knossow, M. Kinesin, 30 Years Later: Recent Insights from Structural Studies. Protein Sci. 2015, 24, 1047−1056. G


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