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Reversible Ratchet Effects for Vortices in Conformal Pinning Arrays

A conformal transformation of a uniform triangular pinning array produces a structure called a conformal crystal which preserves the six-fold ordering of the original lattice but contains a gradient in the pinning density. Here we use numerical simulations to show that vortices in type-II superconductors driven with an ac drive over gradient pinning arrays produce the most pronounced ratchet effect over a wide range of parameters for a conformal array, while square gradient or random gradient arrays with equivalent pinning densities give reduced ratchet effects. In the conformal array, the larger spacing of the pinning sites in the direction transverse to the ac drive permits easy funneling of interstitial vortices for one driving direction, producing the enhanced ratchet effect. In the square array, the transverse spacing between pinning sites is uniform, giving no asymmetry in the funneling of the vortices as the driving direction switches, while in the random array, there are numerous easy-flow channels present for either direction of drive. We find multiple ratchet reversals in the conformal arrays as a function of vortex density and ac amplitude, and correlate the features with a reversal in the vortex ordering, which is greater for motion in the ratchet direction. The enhanced conformal pinning ratchet effect can also be realized for colloidal particles moving over a conformal array, indicating the general usefulness of conformal structures for controlling the motion of particles.


I. INTRODUCTION
When an assembly of particles are placed in an asymmetric potential, a net dc particle flow can arise due to a ratchet effect that occurs when an ac drive is applied or when the substrate is periodically switched on and off in the presence of a thermal bath 1,2 . Ratchet effects on asymmetric substrates have been extensively studied in colloidal systems [3][4][5] , granular matter 6,7 , and polymers 8,9 . Ratchet effects also appear in ac-driven vortices in type-II superconductors in the presence of an asymmetric substrate [10][11][12][13][14][15] , such as a quasi-one-dimensional periodic array produced by asymmetrically modulating the sample thickness 10,[16][17][18][19] , etching funnel-shaped channels for vortex flow 11,[20][21][22][23][24] , introducing asymmetry to the sample edges 25 , or adding periodic pinning arrays in which the individual pinning sites have some form of intrinsic asymmetry 13,14,[26][27][28][29][30][31][32][33][34] . At lower vortex densities when collective interactions between vortices are weak, the ratchet effect produces a dc flow of vortices in the easy flow direction of the asymmetric substrate; however, when collective effects are present it is possible to have reversals of the ratchet effect where for one set of parameters the vortices move in the easy direction while for another set of parameters they move in the hard direction [13][14][15][16]18,[26][27][28][29][30][31][32] . A ratchet effect can also be produced by a pinning array containing symmetric pinning sites arranged with a density gradient. Olson et al. 12 first studied vortex ratchet effects for random gradient array pinning geometries and found that the vortices undergo a net dc flow in the easy direction. Experiments and simulations later showed that for a square array of pinning sites with constant pinning density but with a gradient in pinning site size, a variety of forward and reverse vortex ratchet behaviors occur 35 . Experiments on triangular pinning arrays with a density gradient also revealed a forward ratchet effect at low fields with a reversal at higher fields 36,37 .
Recently a new type of pinning geometry was proposed that is constructed by conformally transforming a triangular pinning lattice to create what is called a conformal pinning array, abbreviated Conf in this work [38][39][40] . As in the original triangular lattice, each pinning site in the transformed array has six neighbors separated by 60 • ; however, the distance to each neighbor is no longer constant, producing a density gradient in the pinning sites 41,42 . In Fig. 1(a) we show a periodic lattice composed of three conformal pinning arrays with the same orientation. Experimental structures with nearly conformal geometries have been observed for magnetically interacting particles subjected to a gravitational force, and due to the arching nature of the conformal array, the magnetic conformal crystals were dubbed gravity's rainbow structures 42 . Similar conformal structures have also been studied in foams 44,45 and in charged particle ordering in confined geometries 45 . In the superconducting system, an arrangement of two Conf arrays placed with their minimum pinning density regions in the center of the sample produces an enhanced critical current or depinning force compared to an equivalent number of pinning sites placed in a uniform periodic, uniform random, or random density gradient array 38 . The enhancement results both from the natural density gradient formed by the vortices as they enter the sample from the edges and form a Bean state, and from the preservation of local sixfold ordering in the Conf array 38,40 . The suppression of easy vortex flow channels which arise in periodic and random arrays by the arching conformal structure also plays a role in the enhancement. Only at the integer matching fields do the periodic pinning arrays produce higher crit- ical currents than the Conf array 40 . Subsequent experimental studies confirmed that the Conf array produces enhanced pinning over a wide range of fields compared to uniform periodic pinning or uniform random pinning arrangements 46,47 . Other experiments have shown enhanced critical currents in systems with periodic arrays when a gradient in the pinning density is introduced 48 . There have also been studies of hyperbolic-tesselation arrays which have a gradient in the pinning density 49 .
Since conformal pinning arrays have an intrinsic asymmetry, it is natural to ask whether a ratchet effect can occur under application of an ac drive, and if so, whether this ratchet effect would be enhanced compared to other pinning array geometries with density gradients, or whether ratchet reversals could be possible. In Fig. 1 we illustrate three examples of the gradient pinning array geometries we consider in this work: conformal pinning (Conf) [ Fig. 1(a)], random pinning with a gradient (RandG) [ Fig. 1(b)], and a square pinning array containing a gradient along the x direction (SquareG) [ Fig. 1(c)]. In each case, both the gradient and the applied ac drive are along the x direction, while the easy-flow direction for vortex motion is in the negative x direction. In addition to superconducting vortex realizations of these geometries, similar pinning arrangements could also be created in colloidal systems using optical trap arrays.  Fig. 1 at B/B φ = 1.0, Fp = 1.0, and Fac = 0.55. Bottom black curve: Conf array; middle green curve: SquareG array; upper red curve: RandG array. Here the Conf array produces a ratchet that is four times more effective than the SquareG array and 20 times more effective than the RandG array.

II. SIMULATION AND SYSTEM
We consider a two-dimensional system with periodic boundary conditions in the x and y directions. The sample size is L × L with L = 36λ, where distance is measured in units of the London penetration depth λ. The applied magnetic field is perpendicular to the system in theẑ direction. Our results apply to the London limit regime in which the vortices can be treated as rigid objects, when the coherence length ξ is much smaller than λ. The pinning sites are modeled as in previous studies of conformal pinning arrays 38-40 by non-overlapping parabolic circular traps with radius R p and a maximum pinning force of F p . We place the N p pinning sites in a conformal array (Conf), as described in previous work 38 , in a random arrangement with a gradient (RandG), or in a square array with a density gradient along the x direction (SquareG). The width of each pinning array segment is a p = 12λ, and the segments are repeated three times across the sample as shown in Fig. 1. The total density of the pinning sites is n p = N p /L 2 = 1.0. The sample contains N v vortices and we measure the magnetic field in units of B/B φ , where B φ is the matching field at which there is one vortex per pinning site. We obtain the initial vortex configuration by annealing from a high temperature molten state and cooling to T = 0 or to a low but finite fixed temperature. After annealing, we apply an ac driving force to all the vortices.
The dynamics of an individual vortex i is obtained by integrating the following overdamped equation of motion: Here η is the damping constant which is set equal to 1. The repulsive vortex-vortex interaction force is given by F vv , φ 0 is the flux quantum, and µ 0 is the permittivity. The vortex-pinning interaction force is ik , where Θ is the Heaviside step function, r p = 0.25λ is the pinning radius, F p is the pinning strength, R (p) k is the location of pinning site k, R ik . All forces are measured in units of F 0 and lengths in units of λ. Thermal forces are represented by Langevin kicks F T i with the proper- , where k B is the Boltzmann constant. The ac driving force is F ac = F ac sin(ωt)x where F ac is the ac amplitude. To characterize the ratchet effect we measure the average net displacement of all vortices from their starting positions as a function of time, is the x position of vortex i at time t and t 0 is an initial reference time. This measure produces a sinusoidal signal, as shown in Fig. 2; the presence of a net drift indicates that a ratchet effect is occurring. We condense this information into a single number X net , the value of ∆X at t − t 0 = 50 ac drive cycles. Except where otherwise noted, we consider a fixed ac frequency of ω = 0.04 and a time step of δt = 0.02, so that a single drive cycle has a period of 8000 simulation time steps.

III. RATCHET EFFECT
In Fig. 2 we plot the average net displacement ∆X versus time for Conf, RandG, and SquareG arrays with B/B φ = 1.0, F p = 1.0, and F ac = 0.55 during 100 ac drive cycles. Here the overall drift of each curve indicates that all the arrays produce a ratchet effect with the vortices translating in the negative x direction. The Conf array generates the largest ratchet effect, with the vortices translating distances up to 2λ per drive cycle. The ratchet effect for the Conf array is about four times larger than that of the SquareG array and 20 times larger than that of the RandG array.
The relative effectiveness of the different arrays can be more clearly understood by plotting the trajectories of the vortices during the positive and negative portions of a single ac cycle. Figure 3(a,b) shows the trajectories for both halves of the ac cycle in a a RandG array with B/B φ = 1.0, F p = 1.5 and F ac = 1.5. Under both positive and negative drive, the vortices form disordered flow paths with a similar density that is independent of the driving direction. In contrast, in Fig. 3(c) during the positive portion of the ac driving cycle for the Conf array, almost no vortices can cross the densely pinned regions of the sample; instead, the vortices either become trapped at pining sites or remain localized in interstitial cages formed by the pinned vortices. Figure 3(d) shows that occurs only when interstitial vortices manage to squeeze between equally spaced occupied pinning sites. The barrier to this type of vortex motion is the same in each half of the cycle. In contrast, for the Conf array the perpendicular spacing between pinned vortices in the sparse portion of the array is larger than the equivalent spacing between pinned vortices in the dense portion of the array, so the interstitial vortices experience much different effective caging barriers when entering the sparse side of the array than when entering the dense side of the array.
In the RandG arrays, channels of easy vortex flow occur somewhere in the sample with equal probability for both the positive and negative portions of the ac drive cycle. In Fig. 4(a) we plot X net , the average net displacement per vortex after 50 ac drive cycles, versus F p for Conf, RandG, and SquareG samples with B/B φ = 1.0 and F ac = 0.7. For weak pinning F p < 0.7, the vortices move elastically and easily slide over the pinning sites so that there is no ratchet effect in any of the arrays. For F p > 1.5 most of the vortices become increasingly pinned and the ratchet effect is reduced. The optimal ratchet effect occurs for the Conf array at F p = 1.4, where there is a mixture of pinned vortices coexisting with vortices that move temporarily through the interstitial regions as illustrated in Fig. 3(c,d). The SquareG array has a weaker ratchet effect in the range 0.8 < F p < 1.4, with a relatively sharp cutoff at the upper end of this range that occurs when the ability of the pinned vortices to shift inside the pinning sites is reduced, preventing the interstitial vortices from slipping between occupied pinning and causing the motion to become localized, as shown in Fig. 3(e,f). There is a weak ratchet effect for the RandG array with an extremum at F p = 1.6 where the combination of the ac drive and the vortex-vortex interactions causes a portion of the vortices to depin. The maximum magnitude of the ratchet effect for the RandG array is smaller than that for the SquareG array; however, the effect occurs over a wider range of F p . Figure  4(b) shows X net versus F p for B/B φ = 1.5, where there are more interstitial vortices. Here the range of F p over which the ratchet effect occurs for the Conf and RandG arrays extends up to F p = 5.0, with the ratcheting for F p > 2.1 completely dominated by the flow of interstitial vortices. For the SquareG array the ratchet effect is lost for F p > 1.7, the point at which the interstitial vortices can no longer slip through the one-dimensional interstitial channels of the array. In Fig. 4(c) we plot X net versus F p at B/B φ = 0.5, where there are few interstitial vortices. Here most of the motion occurs when vortices jump from one pinning site to another. The ratchet effect for all three arrays vanishes for F p > 1.4 when vortex hopping is suppressed. At this vortex density, the ratchet effect is most pronounced for the SquareG array, where the vortices are able to hop along one-dimensional channels of pinning sites.
In Fig. 5(a) we plot X net versus the ac drive amplitude F ac for Conf, RandG, and SquareG samples with F p = 1.0 and B/B φ = 1.0. For F ac < 0.35, the vortices are mostly pinned and the ratchet effect is absent for all three of the pinning geometries. At intermediate F ac the Conf array has the strongest ratchet effect, with an extremum in X net at F ac = 0.55. The SquareG array has the next most effective ratchet effect, with an optimal magnitude at F ac = 0.5. For higher values of F ac , the vortices are all in motion during some portion of the driving cycle and the ratchet effect gradually decreases to zero with increasing F ac . We observe a similar trend at B/B φ = 1.5 as shown in Fig. 5(b). Here the ratchet effect for the Conf array extends up to much larger values of F ac ; however, the maximum value of |X net | is slightly smaller than for the B/B φ = 1.0 case. For B/B φ = 0.5 in Fig. 5(c), the dominant motion is hopping of vortices from pinning site to pinning site. Here the ratchet effect is strongest for the SquareG array, similar to what is shown in Fig. 4(c).

IV. RATCHET REVERSAL
In Fig. 6(a) we plot X net versus B/B φ for Conf, RandG, and SquareG arrays with F p = 1.0 and F ac = In the SquareG array, due to the periodic ordering along the y direction, some commensuration effects occur, such as enhanced pinning near B/B φ = 1.0 which locally suppresses the ratchet effect. Figure 6(b) shows X net versus B/B φ for F p = 1.5. In this case, the ratchet effect for the SquareG array is lost for B/B φ < 1.2 when the vortices become strongly pinned at the pinning sites. In general, the ratchet effect for the Conf array is stronger than that for the SquareG and RandG arrays, with a local extremum for the ratchet effect in the negative or normal direction occurring at B/B φ = 1.05. The SquareG array has a local extremum in X net in the negative direction at B/B φ = 1.5, followed by a sharp drop in X net for B/B φ > 2.25. We find a ratchet reversal in the Conf array, where X net switches from negative to positive over the range 2.1 < B/B φ < 3.375. There is a local maximum in the positive ratchet effect at B/B φ = 2.4. In Fig. 7(a) we show a highlight of X net versus B/B φ from Fig. 6(b) for the Conf array indicating that two reversals in the ratchet effect occur. Figure 7(b) illustrates ∆X vs time in ac drive cycles for the system in Fig. 7(a) at B/B φ = 1.9, where the motion is in the negative x direction, and at B/B φ = 2.4, where the motion is in the positive x direction, showing more clearly the change in the net direction of vortex motion.
In Fig. 8(a) we plot X net versus F ac for a Conf array with F p = 1. there is no ratchet effect since the vortices move only small distances. A negative ratchet effect occurs for 0.1 < F ac < 0.45, while for 0.45 ≤ F ac < 1.3 there is a positive ratchet effect with a maximum amplitude at F ac = 0.7. There is another transition to a weaker negative ratchet effect for F ac > 0.13, and X net gradually approaches zero for high values of F ac . Figure 8(b) shows that at B/B φ = 2.1, the ratchet effect is always negative; however, there are still local features in the response such as at 0.3 < F ac < 1.0 where the negative ratchet effect is strongly reduced. In Fig. 8(c) at B/B φ = 1.7, the ratchet effect is strongly negative with an extremum in X net near F ac = 0.5. The ratchet effect goes to zero for increasing F ac . In Fig. 9(a) we show X net versus F p for a Conf array at B/B φ = 2.4 and F ac = 0.7. There is a negative ratchet effect for 0 < F p < 1.25, a positive ratchet effect for 1.25 ≤ F p < 2.1, and a much larger negative ratchet effect for F p > 3.0. At intermediate F p when there is a positive ratchet effect, vortices can be temporarily trapped by pinning sites. The negative ratchet effect for large F p arises from the interstitial flow of vortices, and X net saturates at large F p since the caging barrier experienced by interstitial vortices from the neighboring pinned vortices does not increase with increasing F p . In Fig. 9(b), at F ac = 1.6 there is a negative ratchet effect for 0 < F p < 1.625, a positive ratchet effect for 1.625 ≤ F p < 4.25, and another negative ratchet regime for F p > 4. 25. The vortices at the pinning sites remain permanently pinned for F p > 4.25. The positive ratchet effect is larger and extends out to higher values of F p for the F ac = 1.6 system compared to the F ac = 0.7 system. Figure 9(c) shows that at F ac = 0.225, there is a weak negative ratchet effect for all values of F p . Although we focus here on the Conf array, we also found that some weak ratchet reversals are possible in the SquareG array; however, we did not observe a vortex ratchet reversal for the RandG array. In order to better understand the vortex dynamics and ordering during an individual ac cycle, in Fig. 10 we plot the time series of the fraction of sixfold-coordinated vortices, P 6 = N −1 v Nv i=1 δ(z i − 6), versus time. Here z i , the coordination number of vortex i, is obtained from a Voronoi construction. Superimposed over the plot is a curve showing the phase of the ac drive, and the points at which the drive reaches its maximum positive value are marked MP while those at which the drive reaches its maximum negative value are marked MN. Figure 10(a) shows P 6 versus time for the system from Fig. 9(a) with B/B φ = 2.4 and F ac = 0.7 at F p = 0.875 where there is a negative ratchet effect. Here P 6 = 0.65 at the start of each drive cycle when the drive magnitude is zero, deceases slightly when the drive becomes positive and the system disorders, then reaches its highest values of P 6 ≈ 0.8 in the MP portions of the drive cycle and P 6 ≈ 0.83 in the MN portions of the drive cycle. When the magnitude of the ac drive is maximum, all the vortices move elastically, and since they are slightly more ordered during the negative cycle of the drive, they can The system is most ordered whenever the magnitude of the ac drive is maximum; however, the ordering peaks for the negative portions of the drive cycle are slightly higher than those for the positive portions of the drive cycle. (b) At Fp = 1.5 the ratchet effect is positive. The system is most ordered whenever the magnitude of the ac drive is close to zero, but the net motion is determined by the relatively larger ordering at MP points compared to MN points. slide slightly further in the negative x direction than in the positive x direction, giving a negative ratchet effect. At F p = 1.5 in Fig. 10(b), the ratchet effect is positive and the vortex ordering is reversed. The vortices are now the most ordered when the magnitude of the ac drive is close to zero, and they are disordered when the ac drive magnitude reaches a maximum. During the MP portion of the drive cycle, P 6 = 0.5, while in the MN portion of the drive cycle the system is more disordered with P 6 = 0.47. The more ordered vortices are able to slide slightly further in the positive x direction, resulting in a net positive ratchet effect. There is also an asymmetry in the ordering at the zero force portions of the drive cycle. The value of P 6 at cycle times of 0.0 and 1.0 is smaller than that at times of 0.5 and 1.5; however, since the vortices are not moving during this portion of the cycle, this asymmetry does not produce a preferred direction of motion. In general we find that the ordering of the vortices at the MP and MN points of the drive determines the direction of the ratchet motion, with the net ratchet effect occurring in whichever drive direction generates the most ordered vortex arrangement.

V. THERMAL EFFECTS
We next consider thermal effects on the ratchet response. For weak pinning, the addition of thermal fluctuations monotonically decreases the ratchet effect for all three geometries, as shown in Fig. 11(a) for F p = 1.0, F ac = 0.7, and B/B φ = 1.0. As before, the ratchet effect is most pronounced for the conformal array. As F p increases, the vortices become more strongly pinned, and the addition of thermal fluctuations can increase the ratchet effect by permitting vortices to escape from pinning sites or interstitial caging sites via thermal activation. In Fig. 11(b) we plot X net versus F T in samples with F p increased to F p = 3.0, showing a strong ratchet effect in the Conf array. Here, the ratchet effect is lost at small F T since the vortices are strongly pinned, and the ratchet effect also disappears for high values of F T when the thermal fluctuations become so strong that the vortices enter a molten state that interacts too weakly with the substrate for an asymmetry in the response to positive and negative drives to be noticeable. The largest ratchet signatures appear for intermediate F T . In general, when F p increases, the point at which the magnitude of the ratchet effect is largest shifts to higher values of F T .
We find that X net increases linearly as the period of the ac driving cycle increases, as illustrated in Fig. 12 for Conf, RandG, and SquareG arrays with B/B φ = 1.0, F p = 1.0, and F ac = 0.7. As the other parameters are varied, we find a robust increase in the magnitude of the ratchet effect with decreasing ac frequency.

VI. RATCHET EFFECTS FOR COLLOIDAL PARTICLES
Ratchet effects can be generated in systems of colloidal particles interacting with various types of periodic arrays of traps that are created using optical means [50][51][52][53] . The ability to make structures similar to conformal lattices has been demonstrated by Xiao et al. 54 , who examined a colloidal ratchet effect on optical traps forming Fibonacci spirals. In that case the ratchet effect is induced by rotating the potential through a three-step cycle. Here we consider the ac-driven motion of colloidal particles Here the maximum ratchet effect occurs when P6 = 0.6, indicating that although colloid-colloid interactions remain important, the system is in a disordered state. When the system forms a crystalline state with P6 ≈ 1.0, the ratchet effect disappears.
over a Conf array. The equation of motion for colloids is similar to that given in Eqn. 1 for vortices, except that the pairwise repulsive colloid-colloid interaction potential has the form V (R ij ) = A c exp(−κR ij )/R ij , where E 0 = q 2 Z * 2 /4πǫǫ 0 a 0 , q is the dimensionless interaction strength, Z * is the effective charge of the colloidal particles, ǫ is the solvent dielectric constant, and 1/κ is the screening length which we set equal to 1.0. The number of colloids in the sample is N c . For our parameters, the interactions between colloids for R ij < 1 is much larger than the interactions between vortices separated by the same distance, while for R ij > 1 the colloidal interaction strength falls off much more rapidly than the vortex-vortex interaction strength, so that nearest neighbor interactions are dominant in the colloidal system. We have conduced a series of simulations for colloidal particles moving through Conf, RandG, and SquareG pinning landscapes under an ac driving force, and find results very similar to those obtained in the vortex system. For example, in Fig. 13 we plot ∆X versus time for colloids interacting with Conf, SquareG, and RandG arrays for N c /N p = 1.0, F p = 1.0, F ac = 0.55, and A c = 0.01, where we observe that just as in the vortex case, the Conf array produces the most pronounced ratchet effect, the SquareG array shows a weak ratchet effect, and the RandG array does not exhibit a ratchet effect. Since the effective charge on the colloids can be changed ex-perimentally, it is possible to hold the substrate strength fixed and modify how strongly the colloids interact with one another. In Fig. 14(a) we plot X net versus A c for the Conf array from Fig. 13, and in Fig. 14(b) we show the corresponding fraction of sixfold coordinated colloids P 6 versus A c . For A c = 0 the colloids all become pinned in the pinning sites since F ac < F p . As A c increases, the colloid-colloid interactions become important and a ratchet effect arises with a maximum amplitude near A c = 1.0. The largest ratchet effect is associated with a sixfold ordering fraction of P 6 = 0.6, indicating that the colloids are still disordered, with some colloids trapped in pinning sites and others occupying interstitial regions between pins. For A c > 0.01 the ratchet effect begins to diminish with increasing A c while simultaneously P 6 increases, indicating an increase in the ordering of the colloids. For A c > 0.04, the colloids form a rigid triangular lattice as indicated by the fact that P 6 ≈ 1.0, and the ratchet effect disappears. These results show that in order for a ratchet effect to appear in the gradient pinning arrangements, it is generally necessary for plasticity or defects in the colloid or vortex lattice to appear. Our results with the colloidal system indicate that pronounced ratchet effects should be realizable in a variety of systems where assemblies of interacting particles are driven with an ac drive over conformal array substrates.

VII. SUMMARY
We examine ratchet effects for ac driven vortices interacting with a conformal pinning array and with square and random pinning arrays containing a gradient along one direction. In general, the conformal pinning array produces the most pronounced ratchet effect, particu-larly for fields greater than the first matching field. The enhanced effectiveness of the conformal ratchet results in part from the fact that the pinning sites in the low density portion of the array are widely spaced not only parallel to but also perpendicular to the net pinning gradient direction, permitting the easy flow of interstitial vortices through the sparse portion of the array. In contrast, for the square pinning array with a gradient, the perpendicular distance between pinning sites is constant throughout the array, producing the same barrier for interstitial motion in both the sparse and dense portions of the array and reducing the relative magnitude of the ratchet effect for fields at which interstitial vortices are present. For the random pinning array with a gradient, channels of easy vortex flow form for driving in either direction, significantly reducing the effective asymmetry of the array. We find that the conformal array exhibits a series of vortex ratchet reversals as a function of vortex density, ac drive amplitude, and pinning strength, and we show that the direction of the ratchet is determined by the amount of order present in the vortex lattice at different phases of the ac driving cycle. Finally, we demonstrate that the conformal array also produces a larger ratchet effect compared to square and random pinning arrays with a gradient in systems of colloidal particles, suggesting that pronounced ratchet effects should be a general feature of particles moving over conformal arrays.