Colloidal particles driven across periodic optical potential energy landscapes
Michael P. N. Juniper, Arthur V. Straube, Dirk G. A. L. Aarts, Roel P. A. Dullens
CColloidal particles driven across periodic optical potential energy landscapes
Michael P. N. Juniper ∗ , Arthur V. Straube, Dirk G. A. L. Aarts, and Roel P. A. Dullens Department of Chemistry, Physical and Theoretical Chemistry Laboratory,University of Oxford, South Parks Road, OX1 3QZ Oxford, United Kingdom Department of Physics, Humboldt-Universit¨at zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
We study the motion of colloidal particles driven by a constant force over a periodic opticalpotential energy landscape. Firstly, the average particle velocity is found as a function of thedriving velocity and the wavelength of the optical potential energy landscape. The relationshipbetween average particle velocity and driving velocity is found to be well described by a theoreticalmodel treating the landscape as sinusoidal, but only at small trap spacings. At larger trap spacings,a non-sinusoidal model for the landscape must be used. Subsequently, the critical velocity requiredfor a particle to move across the landscape is determined as a function of the wavelength of thelandscape. Finally, the velocity of a particle driven at a velocity far exceeding the critical drivingvelocity is examined. Both of these results are again well described by the two theoretical routes,for small and large trap spacings respectively. Brownian motion is found to have a significant effecton the critical driving velocity, but a negligible effect when the driving velocity is high.
PACS numbers: 05.60.-k, 82.70.Dd, 87.80.Cc
I. INTRODUCTION
The phenomenon of a particle travelling over a po-tential energy landscape is important to the behaviourof many physical systems of scientific interest and tech-nological importance. This includes the diverse casesof counter-sliding rough surfaces [1], the movement ofadatoms on atomic surfaces [2], and the motion of mo-bile rings on a poly-rotaxane [3]. Of particular currentrelevance are superconductor effects, such as DC drivenJosephson junctions [4, 5] and charge density waves [6].Such systems are, however, challenging to image [7], mak-ing microscopic scale motion difficult to study at eas-ily accessible temperatures and pressures. Another wellstudied case is vortex motion in type-II superconductors[8–12]. Vortices may be directly imaged by techniques in-cluding Lorentz microscopy [13], Bitter decoration [14],or magneto-optical imaging [15], but direct, controllableaccess to microscopic motion is not available under read-ily accessible experimental conditions. Extensive workusing computer simulation has been conducted [16–18],but there is still a requirement for model systems in whichit is possible to examine behaviour in real space.The experimental model system used in this paper isthat of Brownian particles driven across a periodic opti-cal potential energy landscape. Various techniques havebeen used to drive colloidal systems in optical potentialenergy landscapes [19–24], in order to address numer-ous problems, from tribology [19, 20] to particle sorting[21–24]. Of note is work considering the deflection of par-ticles driven at an oblique angle across two-dimensionaloptical potential energy landscapes, where particle direc-tion is dictated by the competition between the symme- ∗ Present address: The Francis Crick Institute, 44 Lincoln’s InnFields, London, WC2A 3LY, E-mail: [email protected] try of the landscape and the direction of the driving force[22, 24, 25]. Furthermore, motion over both one- and two-dimensional potential energy landscapes has been used inmodels of friction, such as the Prandtl-Tomlinson model[26, 27] and the Frenkel-Kontorova model [27, 28].The non-linearities in systems driven far from equilib-rium by an external force have garnered recent interestin theory and experiment [29–33]. Considerable attentionhas been given to the problem of colloidal particles dif-fusing in a periodic potential [34–36], and diffusing overthreshold potentials [37]. Further to this, the behaviourin a tilted periodic potential has been examined, with thebias leading to transport effects [38–44], giant [45, 46] orsuppressed [47] diffusion.In this article, we particularly focus on the critical driv-ing velocity, upon changing the optical potential energylandscape from sinusoidal to non-sinusoidal by tuning thespacing between the optical traps constituting the land-scape. We also study the average particle velocity wellabove the critical velocity. We compare our experimentalresults to a simple theoretical framework that describesthe potential energy landscape in the limit of small andlarge trap spacing. We show that the Brownian motionof the particles need only be taken into account close tothe critical driving velocity.The paper is organised as follows. In Sec. II, we estab-lish a simple theoretical model to explain the landscapedependent dynamics of the driven particles. The experi-mental methods are outlined in Sec. III, and their resultsare presented and compared to the theory in Sec. IV.Finally, we present our conclusions in Sec. V.
II. THEORY
The following Langevin equation is used to describethe overdamped motion of a spherical Brownian particledriven by a constant force across a (periodic) potential a r X i v : . [ c ond - m a t . s o f t ] D ec F DC F DC g GLASS CELL WALL ... ... xxy
SOLVENT ...... z U ( x ) ( k B T ) Position, x ( µ m) U T ( x ) ( k B T ) (a)(b) FIG. 1: (a) Schematic of the experimental geometry. Aone dimensional periodic optical potential energy landscape, U T ( x ) , is generated by a line of overlapping optical traps cre-ated from AOD-timeshared focused laser spots separated bya spacing λ . A spherical colloidal particle sedimented to thebottom of the sample cell is driven across the landscape bya constant force F DC . (b) The optical potential energy land-scape, U T ( x ) , corresponding to a trap spacing of λ = . µ mand a laser power per trap of 0.75 mW. The tilted ‘washboard’potential, U ( x ) for F DC / ζ = . µ m s − is shown in the lowerpanel. The symbols are the experimental data and the solidblack line a fit with sine function. The solid orange line in thelower panel illustrates a hypothetical tilted washboard poten-tial corresponding to a subcritical driving force, which leadsto finite barriers in the potential. energy landscape, U T ( x ) , (see Fig. 1) [34, 48]: ζ d x ( t ) d t = F DC + F T ( x ) + ξ ( t ) , (1)where the instantaneous particle velocity, v ( x, t ) = d x / d t ,at position x and time t , depends on the constant (DC)driving force, F DC , the force from the optical potentialenergy landscape, F T ( x ) = − ∂U T / ∂x , the Brownian force, ξ ( t ) , and the friction coefficient, ζ . The impact of ther-mal fluctuations is modeled by Gaussian white noise,such that ⟨ ξ ( t )⟩ = ⟨ ξ ( t ) ξ ( t ′ )⟩ = ζk B T δ ( t − t ′ ) ,where k B T is thermal energy. Thus, in the case U T ( x ) represents a spatially periodic landscape with a wave-length, λ , Eq. (1) describes the motion of a Brownian par- ticle in a tilted ‘washboard’ potential, U ( x ) = − xF DC + U T ( x ) , where F DC determines the tilt (see Fig. 1).The relative importance of the deterministic andstochastic parts of Eq. (1) may be quantified using theP´eclet number. We define this in our context as the ra-tio of the time taken for the particle to diffuse over adistance equivalent to one wavelength of the landscape(the ‘Brownian time’, τ B = λ / D with D = k B T / ζ beingthe diffusion coefficient) and the time taken for the par-ticle to be driven over one wavelength of the landscape, τ D = λ / v , where v is the average particle velocity:Pe = τ B τ D = ζ λ vk B T . (2)The time taken for the particle to be driven over onewavelength of the landscape, τ D , results from the balancebetween the driving force and the force due to the opticalpotential energy landscape, and thus contains the averageparticle velocity, rather than the driving velocity.When Pe ≫
1, the effect of diffusion is negligible rela-tive to the driving force, but as Pe →
1, diffusion becomesmore important. To simplify the analysis of Eq. (1), wewill neglect the stochastic force term, ξ ( t ) , for now. Thisapproximation is instructive and is justified because theP´eclet number is much higher than unity for most driv-ing velocities used here. The (deterministic) equation ofmotion thus becomes: ζ d x ( t ) d t = F DC + F T ( x ) . (3)To define the periodic optical potential energy land-scape, U T ( x ) , we assume that the landscape extends in-finitely, from trap i = −∞ to trap i = ∞ , with traps sep-arated by a spacing λ . Each individual trap i is mod-elled by a Gaussian well V i ( x ) of depth V and stiffness k [22, 24, 49, 50], V i ( x ) = − V exp [− k ( x − λi ) V ] . (4)We stress that although in the vicinity of the trap cen-tre, ∣ x − λi ∣ ≪ λ , Eq. (4) reduces to the conventionallyused harmonic potential, V i ( x ) = k ( x − λi ) /
2, the har-monic approximation generally fails to properly describethe energy landscape; see also Refs. [49, 51], where thenon-harmonic nature of the optical potential is crucial forcapturing equilibrium and non-equilibrium pattern for-mation. As shown in Ref. [50], individual potentials areadditive, so the potential landscape may be expressed as U T ( x ) = ∑ ∞ i =−∞ V i ( x ) , which leads to an optical force F T ( x ) = − k ∞ ∑ i =−∞ ( x − λi ) exp [− k ( x − λi ) V ] . (5)In the experiments, two main observables are consid-ered: the average particle velocity over an integer num-ber of wavelengths of the landscape, v , and the criticaldriving velocity, F C / ζ , required for the particle to move.Firstly we consider the average particle velocity. For theperiodic landscape, the time, ∆ t , in which the particlepasses a single wavelength of the landscape, λ , is (seeEq. (3)): ∆ t = ζ ∫ λ / − λ / [ F DC + F T ( x )] − d x. It thereforefollows that in the deterministic regime: v = λ ∆ t = λ (∫ λ / − λ / ζF DC + F T ( x ) d x ) − . (6)Next, the critical driving force required to cause theparticle to overcome a maximum in the optical force isconsidered. By setting d x / d t = x = x , such that F DC + F T ( x ) =
0. Thissolution describes the locked state because the particle ispinned to the periodic landscape and shifted from one ofits nearest local minima at x min = iλ ( i = , ± , ± , . . . )by δx such that x = x min + δx . The locked state ex-ists only if the constant driving force is small enough, F DC < F C , that there are finite barriers in the full poten-tial (see Fig. 1(b), orange line), with the critical force F C = max x [− F T ( x )] = − F T ( x ∗ ) . (7)Here, x ∗ is the position of the maximum in the opticalforce, defined by F ′ T ( x ∗ ) =
0, where the prime denotes thederivative with respect to x . For F DC > F C there exist nostationary solutions, d x / d t ≠
0. This regime correspondsto the sliding state, meaning that the particle is slidingacross the landscape with a certain averaged speed. Thetransition from the locked to sliding state occurs when F DC = F C . With F T ( x ) given by Eq. (5), the criticalforce may be stated directly: F C = k ∞ ∑ i =−∞ ( x ∗ − λi ) exp [− k ( x ∗ − λi ) V ] . (8)Note that in this regime F DC is greater than, but closeto F C , implying that the P´eclet number is close to oneand diffusion is important, as will be discussed furtherin Sec. II D. Equations (6) and (8) are rigorous, butnot analytically tractable for the full optical landscape(Eq. (5)). We therefore make some approximations inthe description of the optical potential energy landscape. A. Sinusoidal landscape: small trap spacing
Since the optical energy landscape is periodic, it maybe expressed as a Fourier series: U T ( x ) = a + ∞ ∑ m = [ a m cos ( π mxλ ) + b m sin ( π mxλ )] . The calculation of the first Fourier coefficient, a m = ( / λ ) ∫ λ / − λ / U T ( x ) cos ( π mx / λ ) d x , yields: a m = − √ π V / λk / exp (− π m V λ k ) , with m = , , , . . . . The second Fourier coefficient van-ishes, b m ≡
0, because U T ( x ) sin ( π mx / λ ) is an oddfunction integrated within symmetric limits. As a result,the trapping potential U T ( x ) is represented as: U T ( x ) = − √ π V / λk / × [ + ∞ ∑ m = exp (− π m V λ k ) cos ( π mxλ )] . (9)In Ref. [50] it was demonstrated that if the trap spac-ing is sufficiently small, the velocity profile for a particlepassing across the periodic potential is well described by asinusoidal function. It is therefore asserted that for small λ , the potential may be approximated by the leading si-nusoidal term. Indeed, as becomes evident from Eq. (9),at small λ the amplitudes decay exponentially fast with m . For this reason, terms with m > F T ( x ) ≈ − √ ( π V ) / λ k / exp (− π V λ k ) sin ( π xλ ) . (10)The critical driving force, F C , is found from Eq. (7).Accordingly, solving the equation F ′ T ( x ∗ ) = F T ( x ) as in Eq. (10) yields x ∗ = λ / + iλ (where i = , ± , ± , . . . )so that the critical force for small trap spacing λ becomes: F C = − F T ( x ∗ ) = √ ( π V ) / λ k / exp (− π V λ k ) . (11)By taking into account Eqs. (10) and (11), the determin-istic equation of motion, Eq. (3), is reduced to an Adlerequation [52, 53]: ζ d x ( t ) d t = F DC − F C sin ( π xλ ) , (12)which offers a much simpler solution to the average ve-locity, Eq. (6). The time taken for a particle to passone wavelength of the landscape in this case is ∆ t = ζ ∫ λ / − λ / [ F DC − F C sin ( π x / λ )] − d x = ζλ ( F − F ) − / ,leading to an average particle velocity of [54]: v = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ , if F DC ≤ F C ;1 ζ √ F − F , if F DC > F C . (13)These equations will be compared to experimental resultsfor potential energy landscapes with small trap spacings. B. Nearest neighbour landscape: large trap spacing
Now the case of widely spaced traps is considered, bytreating them as almost non-overlapping Gaussian traps(Eq. (4)). An approximation is made that the local opti-cal potential may be described by one central trap, num-bered for simplicity by i =
0, and its two nearest neigh-bours, i = ±
1. Taking this approach, only terms with λ /( V / k ) = 2.03 ( λ = 2 μ m) λ /( V / k ) = 2.53 ( λ = 2.5 μ m) λ /( V / k ) = 4.05 ( λ = 4 μ m) λ /( V / k ) = 5.07 ( λ = 5 μ m) λ /( V / k ) = 10.14 ( λ = 10 μ m) FIG. 2: Comparison of theoretical approximations for theoptical potential energy landscape U T ( x ) . — Periodic land-scape composed of the sum of 11 individual Gaussian opticaltraps, at trap spacing λ given in µ m and units of √ V / k ,using typical trapping parameters k = . × − kg s − and V = k B T . — sinusoidal landscape: small λ , — nearestneighbour landscape: large λ . ∣ i ∣ ≤ U T ( x ) = ∑ i =− V i ( x ) , leadingto the following expression for the optical force: F T ( x ) = − k ∑ i =− ( x − λi ) exp [− k ( x − λi ) V ] . (14)This may be substituted into Eq. (6) to find the averageparticle velocity numerically.In order to calculate the critical driving force for widelyspaced traps, firstly the case is considered where the trapsare infinitely spaced, λ → ∞ , and only a single term ( i =
0) is taken from the sum in Eq. (14). The trapping force isthen F ∞ T ( x ) = − kx exp [− kx / V ] , and maximising thisas in Eq. (7) leads to the position of the maximum opticalforce, x ∞∗ = √ V / k , which in turn provides an expressionfor the critical driving force of a single trap: F ∞ C = − F T ( x ∞∗ ) = √ kV exp [− ] . Next, we take into consideration the optical potential of the nearest neighbouring traps ( i = ±
1) to account for thedependence of F C on the large but finite λ . Accordingly,by substituting an ansatz x ∗ = x ∞∗ + δx ∗ into Eq. (14)and then solving F ′ T ( x ∗ ) =
0, retaining only the leadingterm, we obtain the exponentially small correction δx ∗ =−√ k / V λ exp [− kλ / V ] . The critical force for large λ then follows from Eq. (7): F C = k ∑ i =− ( x ∗ − λi ) exp [− k ( x ∗ − λi ) V ] , (15)with x ∗ = √ V k − √ kV λ exp [− kλ V ] . (16)These expressions for the average velocity and the criticalforce will be compared to experimental measurements forperiodic optical potential energy landscapes with largetrap spacings. C. Comparing the approximations
Figure 2 shows periodic optical potential energy land-scapes comprising eleven single Gaussian traps, withspacings varying from 2 µ m to 10 µ m, and typical trap-ping parameters of k = . × − kg s − and V = k B T .Note that we also give the trap spacings in units of √ V / k , which is a measure for the width of a single trap.For comparison we additionally show the landscapes cor-responding to the sinusoidal (small λ , as in Eq. (9) with asingle term m = λ , as in U T ( x ) = ∑ i =− V i ( x ) ) approximations. The fivelarge panels show the full landscape for each trap spacing,and the small panels show details of these, to highlightthe comparisons between the different models.Several features should be noted from the comparisons.Firstly, the height of the barrier in the landscape in-creases with trap spacing, until it is equal to the depthof a single trap, when the trap separation is very large( λ = µ m). At very small trap spacing ( λ = µ m),the barrier is on the order of 4 k B T , easily accessible bydiffusion alone. However already at λ = . µ m, the bar-rier is 20 k B T , making diffusion from one minimum toanother unlikely. Secondly, it is observed that the sinu-soidal (small λ ) approximation describes the full land-scape very well for 2 µ m ≤ λ ≤ . µ m. Above this, how-ever, both the form and depth of the potential landscapeare poorly described. Conversely, the nearest neighbour(large λ ) approximation does not describe the landscapewell below λ = µ m, but is a very good description when λ ≥ µ m. The small and large λ approximations there-fore cover the whole interval of required values of λ andhave a small overlapping interval, 4 µ m ≤ λ ≤ . µ m. D. The effect of Brownian noise
So far, a deterministic Langevin equation has beenused, which is only valid at high P´eclet number, Pe. Justabove the critical driving velocity, the particle velocity isvery low and Pe ∼
1, so that the stochastic force term, ξ ( t ) , is of the same magnitude as the driving force, F DC .In this regime, it is therefore necessary to consider theeffect of Brownian motion on the particle velocity.The average velocity of an overdamped Brownian par-ticle in a tilted periodic potential, U ( x ) = − xF DC + U T ( x ) ,can be expressed as [34, 54, 55] v = λk B Tζ J ( − exp [− F DC λk B T ]) (17)with J = ∫ λ / − λ / exp [− U ( x ) k B T ] d x ∫ x + λx exp [ U ( x ′ ) k B T ] d x ′ . (18)Generally, for an arbitrary periodic potential U T ( x ) , inte-gral (18) admits no analytic representation and Eq. (17)has to be computed numerically. For the case of small λ , however, the optical potential is sinusoidal (see Sec.II A) and Eq. (18) can be explicitly integrated to yield J = λ exp (− π F)∣ I i F (F C )∣ [55]. Here, I i F ( x ) is themodified Bessel function of the first kind, and F = F DC λ π k B T ; F C = F C λ π k B T , with F C taken from Eq. (11). Thus, the average velocityof a particle driven over a landscape with small λ is: v = ( k B Tλζ ) sinh ( π F)∣ I i F (F C )∣ . (19)Critical driving forces are determined numerically fromthese expressions, as described in Sec. III D below. III. EXPERIMENTAL METHODSA. Colloidal model system
The colloidal system used is composed of DynabeadsM-270 (diameter 3 µ m), in 20% EtOH aq , held in a quartzglass cell (Hellma) with internal dimensions of 9 mm ×
20 mm × µ m. After filling we wait for sufficient timefor residual flows to be absent. The particles are muchmore dense than the solvent, so they sediment into a sin-gle layer near the lower wall of the glass sample cell. Thefriction coefficient, ζ , in the absence of any optical land-scape is found from diffusion to be ζ = . × − kg s − ,which is slightly higher than would be expected fromStokes friction alone ( ζ Stokes = π ηa ), due to the prox-imity of the wall. Particle concentration is low, so thatonly a single particle is visible in the field of view. B. Experimental setup and parameters
The experimental setup consists of an infra-red(1064 nm) laser, controlled using a pair of perpendic-ular acousto-optical-deflectors (AODs), and focused us-ing a 50 × , NA = .
55 microscope objective [50]. One-dimensional periodic optical landscapes, with trap spac-ings 2 . µ m ≤ λ ≤ µ m, are generated in Aresis Tweezsoftware, controlled using a LabView interface. The trapsare time-shared at 5 kHz, such that on the timescale ofthe particles (with a Brownian time of ∼
50 s, and a driventime of at least ∼ / k and V ) are consistent. A laser power of 350 mW isset, and 46 traps are used, corresponding to ∼ .
75 mWper trap at the sample position. This gives typical trap-ping parameters for trap stiffness, k = . × − kg s − ,and trap strength, V = k B T . The particle is forcedcloser to the wall by the optical potential energy land-scape, thus increasing the friction over the value quotedabove. In order to account for this difference, we mea-sured the friction coefficient felt by a particle in a singletrap with the parameters described here, and found avalue of ζ = . × − kg s − , a difference of around 10%relative to the case where the optical landscape is ab-sent. As this variation is much less than the variation inthe velocity due to the optical landscape, this effect doesnot significantly affect our measurements. The numberof traps which fit in the field of view changes with trapspacing, so excess traps are positioned at the edges of thefield of view, in lines parallel to the experimental land-scape. These extra traps are sufficiently far away as tonot influence the experiment, and have the added ad-vantage of catching extra particles which diffuse into thefield of view.The driving force is provided by a PI-542.2CD piezo-stage, controlled using the LabView interface, moving at0 . µ m s − ≤ F DC / ζ ≤ µ m s − . C. Average velocity experiments
Images are focused onto a Ximea CMOS camera usinga 40 × , NA = .
50 microscope objective. Experimental pa-rameters are set automatically in the LabView interface,and six repeats are made at each driving velocity for eachtrap spacing. Particle position is recorded live at 40 Hzfrom the camera image. Average velocity is found by lin-early fitting to a graph of x ( t ) , over an integer numberof wavelengths of the landscape. Instantaneous veloc-ity is found as the numerical derivative of the x ( t ) data.The measured average properties have typically been av-eraged over six repeats, and the error bars correspond tothe standard deviation of the repeats. (a) (b) (c) FIG. 3: Trajectories of Brownian particles driven over a periodic potential. (a) Individual particle trajectories for cases withdiffering trap spacings and driving velocities, but the same average velocity. Lines are spaced in t for ease of comparison.(b) Particle velocity as a function of time for the four trajectories in (a). (c) Particle velocity as a function of position, for thefour trajectories in (a). The dashed line in the right panel is at v = . µ m s − D. Critical velocity experiments
The driving velocity is iterated to find the critical driv-ing velocity at which the particle starts to slide acrossthe optical potential energy landscape, with a maximumresolution of 0 . µ m s − , which constitutes the exper-imental uncertainty. A particle is said to be pinned ifit does not move after the stage has moved 100 µ m, orthree minutes has elapsed, whichever happens first.To determine the critical driving velocity fromEqs. (17) (large λ ) and (19) (small λ ), a numeri-cal search is conducted to find the driving velocity atwhich the particle average velocity first exceeds a cut-off, set to the maximum resolution of the experiments(0 . µ m s − ). Thus, at each trap spacing (with incre-ments of ∆ λ = . µ m), the driving velocity is increaseduntil v is found to exceed 0 . µ m s − , and that drivingvelocity is then defined as the critical driving velocity.The results of these numerical experiments are presentedand compared to experimental results in Sec. IV. IV. RESULTS AND DISCUSSION
Results are presented from experiments in which a col-loidal particle was driven across a periodic optical po-tential energy landscape. Figure 3(a) shows four parti-cle trajectories with the same average velocity, but eachdriven with different driving velocities, F DC / ζ , across aperiodic optical potential energy landscape with a differ-ent wavelength, λ . The plateaus are spaced by the trapspacing as they are caused by the particle sitting in atrap for a period of time. The distribution of the waitingtimes between ‘hops’ from one trap to the next is a resultof the combined effect of the external drive and thermalfluctuations, and does not reflect the uniformity of theunderlying optical potential energy landscape.Figure 3(b) shows the particle velocities, v ( t ) , obtainedfrom the four trajectories in Fig. 3(a). When the particleis delayed in the vicinity of a trap centre, its velocity is close to zero. When a particle hops to the next minimumin the optical landscape, there is a spike in the velocity.The irregularity with which the particle hops to anothertrap illustrates the importance of thermal fluctuations atrelatively low driving velocities.Combining the y -axes of Figs. 3(a) and 3(b) leads toa measure of velocity as a function of position. Figure3(c) shows v ( x ) for the four cases shown in the previ-ous two panels. The regions of minimal velocity are nowevenly spaced by λ , as expected from the periodic poten-tial energy landscape, and the regions where the particleis almost stationary are roughly the same size. A numer-ical integration of v ( x ) , which is directly proportionalto F T ( x ) , indeed yields a uniform and periodic U T ( x ) ,as shown for λ = . µ m in Fig. 1(b). In the case of λ = µ m, when the traps are widely spaced, the par-ticles regain the driving velocity (dashed line) after theyhave escaped the influence of each individual trap. A. Average particle velocity
The first main observable from the experiments is theaverage velocity of a particle travelling over many wave-lengths of the periodic optical potential energy land-scape. Figure 4 shows the dependence of the averageparticle velocity, v , on the driving velocity, F DC / ζ , at atrap spacing of λ = . µ m. At low driving velocities, theparticle does not move across the landscape. The drivingforce then reaches a critical value, F DC = F C , after whichthe average particle velocity rises sharply from zero, be-fore the gradient decreases, and ends roughly collinear tothe line for the case of no traps. The average velocity willalways be lower than the case of no traps, due to the timethe particle is delayed by each trap (see Fig. 3). The ex-perimental data are fitted with the deterministic expres-sion for the average velocity for a particle driven over asinusoidal potential energy landscape, ζv = √ F − F ,see Eq. (13), which describes the data well, and givesa critical driving velocity of 1 . µ m s − , which is very FIG. 4: Average particle velocity against driving velocity for λ = . µ m, compared to the case of no traps. △ experimentaldata (error bars: standard deviation of repeats). — fit corre-sponding to a sinusoidal optical potential energy landscape, ζv = √ F − F with fitting parameter F C . – × – no traps. similar to its direct measurement using the approach de-scribed in Sec. III D.Next, the dependence of the average particle velocityon the driving velocity for different trap spacing is con-sidered. Figure 5 shows the mean particle velocity as afunction of driving velocity for six trap spacings, from λ = . µ m to λ = µ m. All of the experimental linesshow the same shape, but remain below the no trap line.The critical driving velocity increases with λ , which isdiscussed further in Sec. IV B. Each data set is fittedwith Eq. (13), which describes the average velocity of aparticle driven over a sinusoidal optical potential energylandscape in the absence of noise. This fit is accurateup to λ = µ m, but fits less well at λ = µ m. This isexpected, as the sinusoidal nature of the potential energylandscape only holds for small trap spacing (see Fig. 2).At higher driving velocity, there is a decrease in averagevelocity as trap spacing increases. This is due to the timethe particle is delayed by each trap, which increases withtrap spacing due to the increase in the critical force, untila plateau at around λ = µ m, when the traps no longeroverlap (see Fig. 7, described in Sec. IV C).The behaviour at low driving velocity is less well de-scribed by the deterministic equation. The inset in Fig. 5shows the lower parts of the curves for λ = . µ m to λ = µ m, with the previously described fits and nu-merical solutions to the stochastic Langevin equation,Eq. (19), for a sinusoidal optical potential energy land-scape. The effect of Brownian noise is only noticeableat very low average velocity, when Pe ∼
1. This mani-fests as a small deviation from the deterministic velocityWhile the deterministic velocity abruptly goes to zero atthe critical driving velocity, the fits to Eq. (19) show aclear ‘smoothing’ of the average velocity upon approach-ing the critical driving velocity, in agreement with the
FIG. 5: Average particle velocity against driving velocityfor varying trap spacing. ⊖ , ⊟ , ▽ , △ , ◻ , ◯ experimental data(error bars: standard deviation of repeats). — fit corre-sponding to a sinusoidal optical potential energy landscape, ζv = √ F − F . – × – no traps. Inset: the effect of Browniannoise on particle velocity close to the critical driving velocity. — numerical solutions for a Brownian particle driven over asinusoidal optical potential energy landscape, Eq. (19). experimental data. B. Critical driving velocity
Now the dependence of the critical driving velocity onthe trap spacing is considered in more detail, and com-pared to theoretical predictions excluding and includingBrownian noise. The experimental data shown in Fig-ure 6 show the critical driving velocity, obtained as de-scribed in Sec. III D, as a function of the trap spacing.It is observed that at small trap spacings there is es-sentially no critical driving velocity, as the height of thebarriers between the minima in the periodic optical po-tential energy landscape are on the order of a few k B T ,and the particle is able to diffuse across the landscape(see Fig. 2). At λ ≈ µ m, the critical driving velocityincreases sharply, until it plateaus at λ ≈ µ m, with avalue of F C / ζ = . µ m s − , when the critical drivingforce becomes the force required to escape a single iso-lated Gaussian trap.Four theoretical predictions are plotted on Fig. 6.Firstly, the dotted line and the dashed line show the de-terministic solutions for the small λ case, Eq. (11), andthe large λ case, Eq. (15), respectively. The determin-istic expression for a sinusoidal landscape (small λ ) iseffective from λ = λ ≈ µ m, while the deterministicexpression for the nearest neighbour landscape (large λ )works better for large trap spacings. These deterministictheoretical predictions generally follow the same trend asthe experimental data, though the critical driving veloc-ity increases from zero at too low a trap spacing, and theplateau at large λ lies at too high a value of F C / ζ . Thisis because around the critical driving velocity the P´ecletnumber is on the order of unity, and the experimental FIG. 6: Critical driving velocity as a function of trap spacing. ◯ experimental data (error bars: experimental uncertainty).Lines: theoretical predictions. Sinusoidal potential energylandscape: ⋯⋯ deterministic force, Eq. (11); — stochasticforce, according to Eq. (19). Nearest neighbour landscape: - - - deterministic force, Eq. (15), — stochastic force, ac-cording to Eq. (17). F C / ζ will be lowered due to the effect of Brownian noise,hence the deterministic predictions will overestimate theexperimental critical velocity.The numerical solutions including Brownian noise forthe small λ case (Eq. (19)) and the large λ case (Eq. (17))are also shown in Fig. 6. These predictions show thesame trends as the deterministic lines, but have gener-ally lower values. The numerical results including noisecompare very favourably with the experimental results,with two caveats resulting from the fact that the mean-ing of the experimental and numerical critical velocitiesis not identical (see Sec. III D). Firstly, whereas the ex-perimentally measured critical driving velocity is that re-quired for a particle to escape a single minimum in thelandscape within a reasonable period of time, the numer-ical method gives a critical driving velocity equal to theDC driving velocity required to achieve a certain thresh-old average velocity (set to 0 . µ m s − , see Sec. III D).Therefore, at very small trap spacings, where the barriersof the landscape are on the order of k B T , the presenceof noise negates these barriers and the average velocityis approximately equal to the applied driving velocity.The second caveat applies at large λ , where a small de-crease in the numerically determined critical velocity isobserved with increasing λ as the impact of the traps onthe average velocity becomes less pronounced comparedto the space between them. C. Driving far above the critical driving velocity
When the particle is driven by a velocity well in ex-cess of the critical driving velocity, the effect of the trap
FIG. 7: Effect of the trap spacing on the average velocity of aparticle at a driving velocity of 5 . µ m s − . ◯ experimentaldata (error bars: standard deviation of repeats). Sinusoidalpotential energy landscape: ⋯ deterministic force, Eq. (13), — stochastic force, Eq. (19). Nearest neighbour landscape:- - - deterministic force, Eq. (6), — stochastic force, Eq. (17). spacing on the average particle velocity is a result of theamount of time the particle is delayed by each trap (seeFig. 5). Figure 7 shows this effect more clearly by plottingthe average particle velocity against the trap spacing, for F DC ≫ F C . The experimental data show that velocitymarkedly decreases between λ = µ m and λ = µ m,before gradually increasing again.Four theoretical predictions are plotted on Fig. 7. Thedeterministic solutions for the small λ case, Eq. (13) andthe large λ case, Eq. (6), are shown along with the noisysolutions for small λ , Eq. (19), and large λ , Eq. (17),with F T ( x ) from Eq. (14). Here there is no differencebetween the predictions from the deterministic and noisycases, which is consistent with the driving velocities inthis regime being far beyond the critical velocity, sothat the P´eclet number is large and Brownian noise isthus unimportant. Again the small λ approximationworks up to λ ≈ µ m, and the large λ approximationworks above λ ≈ . µ m, consistent with Fig. 6. Theinitial rapid decrease in v as λ increases is due to theincreased time the particle is delayed by each trap,due to the increased optical potential barrier. As thetraps no longer overlap for larger λ , v then increases,and the time spent between traps takes over, bringingthe average velocity back up towards the driving velocity. V. CONCLUSIONS
We have studied the behaviour of Brownian particlesdriven by a constant force through quasi-one-dimensionalperiodic optical potential energy landscapes. Firstly, wehave seen a critical driving velocity, below which the par-ticle is pinned to the potential energy landscape. We haveconsidered potential energy landscapes with a range oftrap spacings, and the critical driving velocity has beenfound to depend on the wavelength. At small trap spac-ings, the critical driving velocity is very low, and it thenincreases, reaching a plateau when the individual trapsare fully separated. Above the critical driving velocitythe particle slides, with an average velocity that increasesnon-linearly with the driving velocity. The particle veloc-ity is also found to depend on the landscape wavelengthwhen the driving velocity is far in excess of the criti-cal driving velocity. At small trap spacings, increasingthe landscape wavelength reduces the average particlevelocity, as each barrier in the optical potential becomeshigher and delays the particle for longer. At larger trapspacings, average particle velocity increases again, as theparticle velocity is determined by the time spent betweenfully separated traps.We have made theoretical predictions correspondingto two different approximations for the optical potentialenergy landscape. When trap spacing is small the land-scape is treated as sinusoidal, and when trap spacing islarge it is treated as a sum of three individual Gaussiantraps. These approximations have been broadened to include the effect of Brownian noise. The trend for theaverage particle velocity as a function of driving velocityand trap spacing has been shown to very accuratelymatch the theoretical prediction from the small trapspacing approximation, up to a limiting landscapewavelength. Including the effect of Brownian noiseallows more realistic prediction of the average particlevelocity close to the critical driving velocity. Criticaldriving velocities themselves are also found to agreewell with theoretical predictions, especially once theeffect of Brownian noise has been taken into account.At higher driving forces, however, it is shown thatthe deterministic solutions alone provide an adequatedescription.
Acknowledgements
We thank Alice Thorneywork, Samantha Ivell and UrsZimmermann for useful discussions. EPSRC is acknowl-edged for financial support. [1] F. Heslot, T. Baumberger, B. Perrin, B. Caroli, andC. Caroli, Physical Review E , 4973 (1994).[2] P. Talkner, E. Hershkovitz, E. Pollak, and P. H¨anggi,Surface Science , 198 (1999).[3] M. B. Pinson, E. M. Sevick, and D. R. M. Williams,Macromolecules , 4191 (2013).[4] B. D. Josephson, Physics Letters , 251 (1962).[5] P. W. Anderson and J. M. Rowell, Physical Review Let-ters , 230 (1963).[6] G. Gr¨uner, Reviews of Modern Physics , 1129 (1988).[7] J. M. Carpinelli, H. H. Weitering, E. W. Plummer, andR. Stumpf, Nature , 398 (1996).[8] A. A. Abrikosov, Reviews of Modern Physics , 975(2004).[9] R. Besseling, P. H. Kes, T. Drose, and V. M. Vinokur,New Journal of Physics , 71 (2005).[10] R. Besseling, Ph.D. thesis, Universiteit Leiden (2001).[11] P. Martinoli, Physical Review B , 1175 (1978).[12] V. Gotcheva and S. Teitel, Physical Review Letters ,2126 (2001).[13] K. Harada, T. Matsuda, J. Bonevich, M. Igarashi,S. Kondo, G. Pozzi, U. Kawabe, and A. Tonomura, Na-ture , 51 (1992).[14] L. Y. Vinnikov, J. Karpinski, S. M. Kazakov, J. Jun,J. Anderegg, S. L. Budko, and P. C. Canfield, PhysicalReview B , 092512 (2003).[15] V. K. Vlasko-Vlasov, A. E. Koshelev, U. Welp, W. Kwok,A. Rydh, G. W. Crabtree, and K. Kadowaki, Magneto-optical imaging of Josephson vortices in layered supercon-ductors (Springer, 2004), chap. Magneto-Optical Imag-ing, pp. 39–46.[16] A. E. Koshelev and V. M. Vinokur, Physical Review Let-ters , 3580 (1994).[17] C. Reichhardt, C. J. Olson, and F. Nori, Physical Review Letters , 2648 (1997).[18] C. J. Olson, C. Reichhardt, and F. Nori, Physical ReviewLetters , 3757 (1998).[19] T. Bohlein, J. Mikhael, and C. Bechinger, Nature Mate-rials , 126 (2012).[20] T. Bohlein and C. Bechinger, Physical Review Letters , 058301 (2012).[21] M. P. MacDonald, G. C. Spalding, and K. Dholakia, Na-ture , 421 (2003).[22] K. Ladavac, K. Kasza, and D. G. Grier, Physical ReviewE , 010901 (2004).[23] A. Jonas and P. Zemanek, Electrophoresis , 4813(2008).[24] K. Xiao and D. G. Grier, Physical Review E , 051407(2010).[25] M. Pelton, K. Ladavac, and D. G. Grier, Physical ReviewE , 031108 (2004).[26] V. L. Popov and J. A. T. Gray, Z. Angew. Math. Mech. , 683 (2012).[27] A. Vanossi and O. M. Braun, J. Phys.: Condens. Matter , 305017 (2007).[28] O. M. Braun and Y. S. Kivshar, The Frenkel-KontorovaModel; Concepts, Methods, and Applications (Springer,2010).[29] C. Reichhardt and C. J. Olson Reichhardt, Physical Re-view Letters , 028301 (2006).[30] V. Blickle, T. Speck, C. Lutz, U. Seifert, andC. Bechinger, Physical Review Letters , 210601 (2007).[31] A. I. Shushin, Journal of Physical Chemistry A , 9065(2009).[32] J. M. Sancho and A. M. Lacasta, The European PhysicalJournal Special Topics , 49 (2010).[33] M. P. N. Juniper, A. V. Straube, R. Besseling, D. G. A. L.Aarts, and R. P. A. Dullens, Nature Communications , , 031104 (2002).[35] M. ˇSiler and P. Zem´anek, New Journal of Physics ,083001 (2010).[36] C. Dalle-Ferrier, M. Kruger, R. D. L. Hanes, S. Walta,M. C. Jenkins, and S. U. Egelhaaf, Soft Matter , 2064(2011).[37] G. Costantini and F. Marchesoni, Europhysics Letters , 491 (1999).[38] K. Lindenberg, A. M. Lacasta, J. M. Sancho, and A. H.Romero, New Journal of Physics , 29 (2005).[39] M. Evstigneev, O. Zvyagolskaya, S. Bleil, R. Eichhorn,C. Bechinger, and P. Reimann, Physical Review E ,041107 (2008).[40] Y. Roichman, V. Wong, and D. G. Grier, Physical Re-view E , 011407 (2007).[41] A. V. Straube and P. Tierno, Europhysics Letters ,28001 (2013).[42] X. G. Ma, P. Y. Lai, and P. G. Tong, Soft Matter , 8826(2013).[43] X. G. Ma, P. Y. Lai, B. J. Ackerson, and P. G. Tong,Soft Matter , 1182 (2015).[44] X. G. Ma, P. Y. Lai, B. J. Ackerson, and P. G. Tong,Physical Review E , 042306 (2015). [45] S. H. Lee and D. G. Grier, Physical Review Letters ,190601 (2006).[46] P. Reimann, C. Van den Broeck, H. Linke, P. H¨anggi,J. M. Rubi, and A. Perez-Madrid, Physical Review Let-ters , 010602 (2001).[47] S. Martens, A. V. Straube, G. Schmid, L. Schimansky-Geier, and P. H¨anggi, Phys. Rev. Lett. , 010601(2013).[48] P. H¨anggi and F. Marchesoni, Rev. Mod. Phys. , 387(2009).[49] A. V. Straube, A. A. Louis, J. Baumgartl, C. Bechinger,and R. P. A. Dullens, Europhysics Letters , 48008(2011).[50] M. P. N. Juniper, R. Besseling, D. G. A. L. Aarts, andR. P. A. Dullens, Optics Express , 28707 (2012).[51] A. V. Straube, R. P. A. Dullens, L. Schimansky-Geier,and A. A. Louis, The Journal of Chemical Physics ,134908 (2013).[52] R. Adler, Proceedings of the IRE , 351 (1946).[53] R. E. Goldstein, M. Polin, and I. Tuval, Physical ReviewLetters , 168103 (2009).[54] M. Gitterman, The Noisy Pendulum (World ScientificPublishing, Singapore, 2008).[55] R. L. Stratonovich,