Propagating Density Spikes in Light-Powered Motility-Ratchets
Celia Lozano, Benno Liebchen, Borge ten Hagen, Clemens Bechinger, Hartmut Löwen
JJournal Name
Propagating Density Spikes in Light-Powered Motility-Ratchets
Celia Lozano a , ∗ , Benno Liebchen b , c , † , ∗ , Borge ten Hagen b , d , ∗ , Clemens Bechinger a and HartmutLöwen b Combining experiments and computer simulations, we use a spatially periodic and flashing light-field to direct the motion of phototactic active colloids. Here, the colloids self-organize into adensity spike pattern, which resembles a shock wave and propagates over long distances, almostwithout dispersing. The underlying mechanism involves a synchronization of the colloids with thelight-field, so that particles see the same intensity gradient each time the light-pattern is switchedon, but no gradient in between (for example). This creates a pulsating transport whose strengthand direction can be controlled via the flashing protocol and the self-propulsion speed of thecolloids. Our results might be useful for drug delivery applications and can be used to segregateactive colloids by their speed.
Active colloids are autonomously navigating microparticles thatconsume energy while moving. They comprise living microorgan-isms like bacteria, algae and sperm , but also man-made syn-thetic swimmers, which can be produced with desired properties.Such synthetic microswimmers are often based on anisotropic col-loidal Janus particles that are self-propelled by phoretic mecha-nisms either directly induced by catalytic surfaces evoking chem-ical reactions , or initiated by light or other external fields,such as ultrasonic , magnetic or electric ones.While free active colloids show a diffusive random motion onlarge scales , equivalent to the motion of passive colloids athigh temperature, applications to use them e.g. for targeted drugdelivery or nanorobotics require to direct and steer theirmotion on demand.One way to direct the motion of active particles is to exposethem to a periodic but asymmetric potential landscape (ratchet),leading to directed transport , in a way similar as for pas-sive colloids driven out of equilibrium through additional time-dependent fields . Characteristically, such potential ratchets a Fachbereich Physik, Universität Konstanz, Konstanz 78457, Germany. b Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düs-seldorf, 40225 Düsseldorf, Germany. c Institut für Festkörperphysik, Technische Universität Darmstadt, 64289 Darmstadt,Germany. d Physics of Fluids Group and Max Planck Center Twente, Department of Science andTechnology, MESA+ Institute, and J. M. Burgers Centre for Fluid Dynamics, Universityof Twente, 7500 AE Enschede, The Netherlands. † corresponding author: [email protected] ∗ These authors contributed equally. involve forces acting on the center of mass coordinate of the par-ticles, yielding a spatial variation of their potential energy. A ver-satile alternative to create directed transport in active colloids,are so-called motility-ratchets, which specifically exploit the ac-tive nature of the particles and have no direct counterpart forpassive colloids. These ratchets hinge on the spatial modula-tion of the self-propulsion speed (or direction) through an ex-ternal field , not affecting the potential energy of the parti-cles. Here, the required modulation of the self-propulsion speedcan be achieved e.g. for light-sensitive Janus colloids in a suit-able standing light-wave which has been previously discussedin the context of dynamical trapping of active particles in thedark spots of the light field . An interesting extension of suchstatic motility-ratchets, providing an additional handle to controlthe active particle dynamics, is to use a time-dependent motilityfield, as recently considered theoretically , and also experi-mentally for light-sensitive bacteria .In the present work, we combine simulations and experimentsto establish a flashing motility-ratchet for synthetic phototacticcolloids, based on a sawtooth-shaped light pattern with inten-sity I ( x , t ) which we periodically switch on and off (flashing).The gradients of I create an effective torque affecting the col-loids’ self-propulsion direction which can systematically bias theirmotion , yielding a directed transport. Here, the emergenceof transport hinges on a phototactic torque biasing the activeparticles’ direction of motion, as opposed to classical flashingpotential ratchets , usually depending on the interplay oftime-dependent forces and diffusion. This novelty in the work-ing mechanism of the flashing motility-ratchet manifests in a setof remarkable features. In particular, we find that the individ- Journal Name, [year], [vol.] , a r X i v : . [ c ond - m a t . s o f t ] M a y ig. 1 Schematic: Active colloids polarize and self-organize into den-sity spikes propagating resonantly through the flashing light field. Here,the colloids synchronize with the flashing light-field and a macroscopicfraction of them essentially follows the same periodic trajectory, as dic-tated by a limit cycle of the system, so that particles within each densityspike move coherently. This creates a pulsating particle transport withlow dispersion. The shown sawtooth-shaped profile represents the par-ticle’s self-propulsion velocity, which varies between v min and v max whenthe light-pattern is on (“on-phase”), where a , b determine the steepnessof the gradient, and which everywhere equals v off , when the light-field isuniform (middle panel, “off-phase”). ual active particles synchronize with the flashing light-field andself-organize into density-spikes resembling a coherently movingshock-wave. As their most striking feature, these spikes hardlydisperse, opposing the usual situation in potential ratchets, wherethe interplay of time-dependent forces and diffusion leads tostrong dispersion of any localized particle ensemble. Thus, thepresent setup opens a route to use laser-light to create a pulsat-ing transport allowing to ’bombard’ a distant target with short andintense pulses of active particles, as might be interesting, in par-ticular, for drug-delivery applications. Here, the transport velocitycan be systematically controlled via the flashing times, but inter-estingly, it also depends sensitively on the particle speed in theoff-phase. In particular, we find that the transport direction evenreverts when changing the self-propulsion speed of the particlesin the off-phase. This transport reversal can be used, in principle,to segregate ensembles of fast and slow particles and might serveas a useful tool for the preparation of clean ensembles of activeparticle with near identical self-propulsion speed.Before detailing these findings, let us sketch the physical mech-anism underlying the flashing motility-ratchet: If the torque act-ing on the active colloids scales linearly with the gradient ofthe laser field (unsaturated regime) no transport can occur instatic light patterns. (This contrasts operating in the saturatedregime.) Intuitively, if noise is negligible, this is because the pho-totactic torque acting on particles crossing a whole spatial period,first turns them into a certain direction and then back to the orig-inal orientation, so that a localized and unbiased initial ensembleremains unbiased for all times. Flashing in turn allows the par-ticles to synchronize with the light field, in a way that they re-peatedly see the same gradient when the light pattern is on and a uniform field in each off-phase, provoking a persistent unidirec-tional motion. This dynamics is based on a limit cycle attractor inthe underlying phase space, which represents a late-time dynam-ics where particles move by exactly one spatial period per flashingcycle, in suitable parameter regimes. Since all particles which areattracted by the same limit cycle show one and the same peri-odic dynamics at late times they move coherently with a speeddictated by the limit-cycle, leading to an almost dispersion-freetransport - a key feature of the present work. For conceptual clarity, we first introduce an idealized flashingmotility ratchet, based on an effective phototactic torque whichscales linearly with the light gradient ( ω ∝ | ∇ I | ). In this case, theemerging transport is flashing-induced and vanishes in static lightpatterns. To see this, consider self-propelled Janus particles, ac-tively moving in 2D with a self-propulsion speed v ( x , t ) , varyingboth in space and time, as controlled by the imposed light-field(see Fig. 1). For simplicity, we specifically consider a quasi 1Dmodulation of the light field, and hence of v . The self-propulsiondirection ˆ u = ( cos φ , sin φ ) changes in response to an effective pho-totactic torque, and also due to rotational diffusion, yielding: ˙ r = v ( x , t ) ˆ u + √ D t ζ r ( t ) , (1) ˙ φ = ω ( x , φ , t ) + √ D r ζ φ ( t ) . (2)Here r = ( x , y ) and D t , D r are translational and rotational diffu-sion coefficients; ζ r ( t ) and ζ φ ( t ) represent Gaussian white noiseof zero mean and unit variance. The key-quantity controlling theparticle dynamics in the light-field is the phototactic alignmentrate ω , which reads ω ( x , φ , t ) = Av ( x , t ) v (cid:48) ( x , t ) sin φ (3)where v (cid:48) ( x , t ) = ∂ v ( x , t ) / ∂ x . Eq. (3) represents a linear relation-ship between alignment rate and intensity gradient, ω ∝ | ∇ I | ,which is realistic for shallow light patterns , but will later begeneralized towards saturation effects. Here, the coefficient A follows from experiments . For the velocity profile v ( x , t ) , wechoose a sawtooth-shape in the on-phase, as sketched in Fig. 1,with segment lengths a , b and minimal and maximal velocities of v min and v max respectively. In the off-phase the velocity is uniform v ( x , t ) = v off .Note that in general, besides creating an effective torque aligningthe particles, light gradients also induce effective forces creatingparticle translations, represented by a term ∝ ∇ I on the r.h.s ofEq. (1). In accordance with , we here neglect such a term forsimplicity, but emphasize its existence for future reference. Let us now explore the dynamics of a representatative particleensemble in the flashing light-field (Fig. 1). We choose randominitial positions and orientations uniformly distributed within oneunit cell of the sawtooth-shaped light pattern ( x ∈ [ , L ) and φ ∈ [ , π ) ) and define the average transport velocity as (cid:104) v (cid:105) = lim t → t end [ x ( t ) − x ( )] / t , where t end is some time, large enough that (cid:104) v (cid:105) Journal Name, [year], [vol.][vol.]
Active colloids are autonomously navigating microparticles thatconsume energy while moving. They comprise living microorgan-isms like bacteria, algae and sperm , but also man-made syn-thetic swimmers, which can be produced with desired properties.Such synthetic microswimmers are often based on anisotropic col-loidal Janus particles that are self-propelled by phoretic mecha-nisms either directly induced by catalytic surfaces evoking chem-ical reactions , or initiated by light or other external fields,such as ultrasonic , magnetic or electric ones.While free active colloids show a diffusive random motion onlarge scales , equivalent to the motion of passive colloids athigh temperature, applications to use them e.g. for targeted drugdelivery or nanorobotics require to direct and steer theirmotion on demand.One way to direct the motion of active particles is to exposethem to a periodic but asymmetric potential landscape (ratchet),leading to directed transport , in a way similar as for pas-sive colloids driven out of equilibrium through additional time-dependent fields . Characteristically, such potential ratchets a Fachbereich Physik, Universität Konstanz, Konstanz 78457, Germany. b Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düs-seldorf, 40225 Düsseldorf, Germany. c Institut für Festkörperphysik, Technische Universität Darmstadt, 64289 Darmstadt,Germany. d Physics of Fluids Group and Max Planck Center Twente, Department of Science andTechnology, MESA+ Institute, and J. M. Burgers Centre for Fluid Dynamics, Universityof Twente, 7500 AE Enschede, The Netherlands. † corresponding author: [email protected] ∗ These authors contributed equally. involve forces acting on the center of mass coordinate of the par-ticles, yielding a spatial variation of their potential energy. A ver-satile alternative to create directed transport in active colloids,are so-called motility-ratchets, which specifically exploit the ac-tive nature of the particles and have no direct counterpart forpassive colloids. These ratchets hinge on the spatial modula-tion of the self-propulsion speed (or direction) through an ex-ternal field , not affecting the potential energy of the parti-cles. Here, the required modulation of the self-propulsion speedcan be achieved e.g. for light-sensitive Janus colloids in a suit-able standing light-wave which has been previously discussedin the context of dynamical trapping of active particles in thedark spots of the light field . An interesting extension of suchstatic motility-ratchets, providing an additional handle to controlthe active particle dynamics, is to use a time-dependent motilityfield, as recently considered theoretically , and also experi-mentally for light-sensitive bacteria .In the present work, we combine simulations and experimentsto establish a flashing motility-ratchet for synthetic phototacticcolloids, based on a sawtooth-shaped light pattern with inten-sity I ( x , t ) which we periodically switch on and off (flashing).The gradients of I create an effective torque affecting the col-loids’ self-propulsion direction which can systematically bias theirmotion , yielding a directed transport. Here, the emergenceof transport hinges on a phototactic torque biasing the activeparticles’ direction of motion, as opposed to classical flashingpotential ratchets , usually depending on the interplay oftime-dependent forces and diffusion. This novelty in the work-ing mechanism of the flashing motility-ratchet manifests in a setof remarkable features. In particular, we find that the individ- Journal Name, [year], [vol.] , a r X i v : . [ c ond - m a t . s o f t ] M a y ig. 1 Schematic: Active colloids polarize and self-organize into den-sity spikes propagating resonantly through the flashing light field. Here,the colloids synchronize with the flashing light-field and a macroscopicfraction of them essentially follows the same periodic trajectory, as dic-tated by a limit cycle of the system, so that particles within each densityspike move coherently. This creates a pulsating particle transport withlow dispersion. The shown sawtooth-shaped profile represents the par-ticle’s self-propulsion velocity, which varies between v min and v max whenthe light-pattern is on (“on-phase”), where a , b determine the steepnessof the gradient, and which everywhere equals v off , when the light-field isuniform (middle panel, “off-phase”). ual active particles synchronize with the flashing light-field andself-organize into density-spikes resembling a coherently movingshock-wave. As their most striking feature, these spikes hardlydisperse, opposing the usual situation in potential ratchets, wherethe interplay of time-dependent forces and diffusion leads tostrong dispersion of any localized particle ensemble. Thus, thepresent setup opens a route to use laser-light to create a pulsat-ing transport allowing to ’bombard’ a distant target with short andintense pulses of active particles, as might be interesting, in par-ticular, for drug-delivery applications. Here, the transport velocitycan be systematically controlled via the flashing times, but inter-estingly, it also depends sensitively on the particle speed in theoff-phase. In particular, we find that the transport direction evenreverts when changing the self-propulsion speed of the particlesin the off-phase. This transport reversal can be used, in principle,to segregate ensembles of fast and slow particles and might serveas a useful tool for the preparation of clean ensembles of activeparticle with near identical self-propulsion speed.Before detailing these findings, let us sketch the physical mech-anism underlying the flashing motility-ratchet: If the torque act-ing on the active colloids scales linearly with the gradient ofthe laser field (unsaturated regime) no transport can occur instatic light patterns. (This contrasts operating in the saturatedregime.) Intuitively, if noise is negligible, this is because the pho-totactic torque acting on particles crossing a whole spatial period,first turns them into a certain direction and then back to the orig-inal orientation, so that a localized and unbiased initial ensembleremains unbiased for all times. Flashing in turn allows the par-ticles to synchronize with the light field, in a way that they re-peatedly see the same gradient when the light pattern is on and a uniform field in each off-phase, provoking a persistent unidirec-tional motion. This dynamics is based on a limit cycle attractor inthe underlying phase space, which represents a late-time dynam-ics where particles move by exactly one spatial period per flashingcycle, in suitable parameter regimes. Since all particles which areattracted by the same limit cycle show one and the same peri-odic dynamics at late times they move coherently with a speeddictated by the limit-cycle, leading to an almost dispersion-freetransport - a key feature of the present work. For conceptual clarity, we first introduce an idealized flashingmotility ratchet, based on an effective phototactic torque whichscales linearly with the light gradient ( ω ∝ | ∇ I | ). In this case, theemerging transport is flashing-induced and vanishes in static lightpatterns. To see this, consider self-propelled Janus particles, ac-tively moving in 2D with a self-propulsion speed v ( x , t ) , varyingboth in space and time, as controlled by the imposed light-field(see Fig. 1). For simplicity, we specifically consider a quasi 1Dmodulation of the light field, and hence of v . The self-propulsiondirection ˆ u = ( cos φ , sin φ ) changes in response to an effective pho-totactic torque, and also due to rotational diffusion, yielding: ˙ r = v ( x , t ) ˆ u + √ D t ζ r ( t ) , (1) ˙ φ = ω ( x , φ , t ) + √ D r ζ φ ( t ) . (2)Here r = ( x , y ) and D t , D r are translational and rotational diffu-sion coefficients; ζ r ( t ) and ζ φ ( t ) represent Gaussian white noiseof zero mean and unit variance. The key-quantity controlling theparticle dynamics in the light-field is the phototactic alignmentrate ω , which reads ω ( x , φ , t ) = Av ( x , t ) v (cid:48) ( x , t ) sin φ (3)where v (cid:48) ( x , t ) = ∂ v ( x , t ) / ∂ x . Eq. (3) represents a linear relation-ship between alignment rate and intensity gradient, ω ∝ | ∇ I | ,which is realistic for shallow light patterns , but will later begeneralized towards saturation effects. Here, the coefficient A follows from experiments . For the velocity profile v ( x , t ) , wechoose a sawtooth-shape in the on-phase, as sketched in Fig. 1,with segment lengths a , b and minimal and maximal velocities of v min and v max respectively. In the off-phase the velocity is uniform v ( x , t ) = v off .Note that in general, besides creating an effective torque aligningthe particles, light gradients also induce effective forces creatingparticle translations, represented by a term ∝ ∇ I on the r.h.s ofEq. (1). In accordance with , we here neglect such a term forsimplicity, but emphasize its existence for future reference. Let us now explore the dynamics of a representatative particleensemble in the flashing light-field (Fig. 1). We choose randominitial positions and orientations uniformly distributed within oneunit cell of the sawtooth-shaped light pattern ( x ∈ [ , L ) and φ ∈ [ , π ) ) and define the average transport velocity as (cid:104) v (cid:105) = lim t → t end [ x ( t ) − x ( )] / t , where t end is some time, large enough that (cid:104) v (cid:105) Journal Name, [year], [vol.][vol.] , s basically stationary. Now performing Brownian dynamics sim-ulations (see ∗ for details) until t = t end for various t on but fixedflashing period T = t on + t off , we generically find a directed trans-port for all values of t on (Fig. 2, inset). In contrast, however, forthe static cases t on = and t off = T (uniform light field) and t on = T and t off = (static sawtooth-shaped light pattern) the transportvanishes. That is the directed transport is flashing-induced.Before exploring the origin and properties of the flashing-induced transport, let us first understand its absence in uniformlight fields. To see this, note that particles crossing a whole spa-tial period of a static light pattern ( t on = s ), do not experienceany net alignment, which can be seen as follows for the noise-free case: from Eqs. (1,2,3) and ˙ φ = ˙ x ( ∂ φ / ∂ x ) , it follows that forvanishing noise we have ∂ φ / ∂ x = Av (cid:48) ( x ) tan φ ( x ) , which yields, af-ter integration over a spatial period φ ( x + L ) = φ ( x ) . Thus, in theabsence of noise, if all particles are initialized with uniformely dis-tributed orientations in an intensity-maximum of the light field,exactly half of them permanently move to the left and to the rightrespectively, i.e. there is no transport. (For spatially distributedinitial ensembles, we provide a slightly more general argumentfor the absence of transport in footnote .) Our argument forthe absence of transport breaks down in the presence of flashing.Intuitively, flashing allows the particles to synchronize with thelight field in a way that they “see” the same gradient each timethe light-field is switched on and no gradient when the light-fieldis off. This repeatedly aligns their self-propulsion in the same di-rection, as we further detail in section 4.The synchronization of the particle motion with the light-fieldhas a number of remarkable consequencies. One of them is shownin Fig. 2 (inset), where we can see that the standard deviation σ = (cid:112) (cid:104) v (cid:105) − (cid:104) v (cid:105) (cid:28) (cid:104) v (cid:105) , for most values of t on . That is, the trans-ported particle ensemble hardly disperses, but moves rather co-herently through the flashing light field. In particular, σ has aminimum at t on ≈ . s where the transport velocity is maximal. Tofurther illuminate the particle dynamics in the flashing motility-ratchet, let us consider the spatial particle distribution in the light-pattern after, a snapshot of which after 20 driving cycles is shownin Fig. 2. As we can see in this figure, for t on = s , where we haveno transport, most particles are localized around their initial po-sitions; they collectively move back and forth in the light-pattern.Conversely, for t on = s the particle distribution is strongly asym-metric and shows a train of pronounced peaks resembling a shockwave. This train persistently moves to the right, with a speed ofprecisely one spatial periode per flashign cycle ( v = L / T ).Apart from exploring the dependence of the transport velocityon parameters of the environment, such as the flashing duration t on , it is instructive to also explore the dependence of the trans-port velocity on particle properties: In Fig. 3, we show the trans-port velocity as a function of v off (self-propulsion speed in the offstate), finding a remarkable structure, comprising distinct peaks ∗ Brownian dynamics simulations have been performed using a standard forward Eu-ler algorithm with a time step of dt = . T . Each point in curves showing averagesis based on N p = trajectories (initial conditions), integrated for T and distri-butions are based on N p = . initial conditions. The system size is infinite, i.e.the simulations do not require any boundary conditions. Fig. 2
Propagating density spikes: Spatial particle distribution after t = s for N p = . and different t on shown in the key. Inset: Mean trans-port velocity (cid:104) v (cid:105) and standard deviation σ = (cid:112) (cid:104) ( v − (cid:104) v (cid:105) ) (cid:105) as a functionof t on , calculated for N p = and t end = T per data point. The dashedline shows the zero line to guide the eye. Parameters: t on + t off = s ; a = . µ m ; b = . µ m ; v min = µ m / s ; v max = µ m / s and v off = µ m / s ; A ≈ . ( A = C C / R for particles with radius R = . µ m and C = . , C = . s being experimentally determined parameters, see ). in the transport velocity, and in particular a transport-reversal for µ m / s (cid:46) v off (cid:46) µ m / s . (Note here, that the structure of thepeaks is statistically well converged, as indicated by the fact thatthe standard deviation σ is much smaller than (cid:104) v (cid:105) for most datapoints.) Following the observed transport reversal, when consid-ering a mixture of particles with different velocities, say µ m / s and µ m / s , the flashing light-field will transport them in oppo-site directions and segregate the mixture. This could be used inprinciple as a tool to prepare clean ensembles of active particleswith a uniform self-propulsion speed. Notice that the presenceof a current reversal alone, which is naturally present in manypotential ratchets , is not sufficient to achieve a clean and near-complete segregation of two initially distributed particle species.Instead, such a segregation requires |(cid:104) v (cid:105)| (cid:38) σ for each species,a condition which is naturally fulfilled for the flashing motility-ratchet (Fig. 3). This shows once more that the intrinsically lowdispersion of the emerging transport might be useful for practicalapplications. Notice however, that σ increases roughly linearlywith v off (Fig. 3) i.e. with the distance particles travel within eachoff-phase. That is, the particle distribution gets broader as v off in-creases allowing for a clean segregation ’only’ for particles whichare not too fast. To understand the mechanism leading to a coherently travellingdensity spike pattern in detail, it is instructive to first explore typ-ical trajectories in the noise-free case ( D r = D t = ), as visualizedin Fig. 4a for the case t on = s . Here, within one or several drivingcycles, the orientation of the particles, which align antiparallel tothe light-gradient, converges to a steady value of either φ = (or φ = π n , n being integer) or to φ = ( n + ) π (particles typically Journal Name, [year], [vol.] , ig. 3 Transport reversal: Mean velocity (cid:104) v (cid:105) and standard deviation σ = (cid:112) (cid:104) ( v − (cid:104) v (cid:105) ) (cid:105) for t on = s as a function of v off . Other parameters as inFig. 2. need about − s to fully turn ) corresponding to perfect align-ment with the axis along which the light-pattern is modulated(upper panel). Once they are aligned, particles no longer expe-rience a torque from the light-field ( p × ∇ I ( x , t ) = for all x , t )and persistently self-propel in one and the same direction. Con-versely to their constant orientation, the speed of the particlesstill evolves rather irregularly at this stage: it simply reads v = v off in each off-phase, but in the on-phase v ∈ ( v min , v max ) it dependson the particle’s position at the instance the light-field is switchedon. At late times, the dynamics converges to a limit cycle at-tractor, whose existence is expected from the Poincaré-Bendixontheorem, resulting in a regular, periodic dynamics (Fig. 4a, mid-dle and lower panel). That is particles fully synchronize with theflashing light-field with a temporally periodic speed, whose av-erage is exactly v = L / T (middle panel), i.e. particles propagateby exactly one spatial period L per T = s . As illustrated by theblue and the orange curve in Fig. 4a (middle panel), dependingon the initial conditions, particles follow a corresponding orbit ei-ther to the left or to the right. In this minimal case where onlytwo limit cycles are relevant, each particle propagates the sameone-flashing-cycle-averaged velocity of v = L / T either to the leftor to the right (Fig. 4a, middle and lower panel). The resultinglate-time transport velocity is then fully determined by the ratioof initial conditions reaching the limit cycle corresponding to mo-tion to the left and to the right, respectively, i.e. to the ’basins ofattraction’ of the two limit cycles. Specifically for the consideredcase of t on = s , most of the particles reach an attractor point-ing to the right, which generates the transport. (In general, de-pending on t on besides limit cycles representing a 1:1 resonance,also limit cycles allowing particles to traverse n spatial periodswithin m driving cycles are allowed, leading to a particle speed of v = nL / ( mT ) .)In the presence of noise, φ does of course not fully converge to π or π but fluctuates even at late times (upper panel in Fig. 4b).However, most of the time (for t on = s ), particles still follow anear-resonant dynamics and move with v ≈ L / T (middle panel inFig. 4b), temporarily resembling trajectories of the underlying de- terministic system. Such a dynamics can prolong for many drivingcycles. From time to time, however, noise relocates a particle fromone limit cycle of the underlying deterministic system to anotherone, leading, at least temporarily to a different dynamics. Thedynamics shown in Fig. 4b (middle panel) illustrates this: In thetime interval between t ∼ s and t ∼ s , the orange trajectorycrosses from the “ φ = -attractor” of the underlying noise-free sys-tem over to the “ φ ≈ − π attractor”, yielding the same dynamics,but in between, it follows the “ φ = − π -attractor” for a few flash-ing cycles and moves into the opposite direction. In the presenceof noise, the resulting transport velocity is therefore mainly deter-mined by the time particles spend in the basin of attraction of thelimit cycles of the underlying noise-free system. When this basinof attraction is small, as e.g. for t on = s for the “ φ = ( n + ) π -attractor” which leads to motion to the left, particles all move tothe right in the long-time average.The spike pattern observed above now follows quite naturallyfrom the described particle-light-field synchronization. Here,each spike represents a package of particles which have followedthe same limit cycle dynamics for the same amount of time. Inparticular, particles which have moved for s = T with avelocity of v ≈ L / T = . µ m / s , have traversed a distance of x ≈ L ≈ µ m , which corresponds to the first large peak inFig. 2 for t on = s ; the following peaks correspond to particleshaving traversed a distance of x ≈ L , L , L .. , correspondingto particles which have either initially ’lost’ time by aligning witha certain delay, or by moving temporarily into the opposite direc-tion. Finally, there is a small peak of particles having traverseda distance x > L ; this corresponds to particles which have beenpushed forward during the on-phases before synchronizing withthe light-field. (The spike pattern is less clean for t on = s and t on = s , where the relevant attractors do not correspond to 1:1resonances.) In experiments the phototactic alignment rate of a Janus colloidwith the intensity gradient does not generally increase linearly, aswe have assumed so far, but saturates for steep intensity gradi-ents due to thermal coupling . Thus, to compare the flashingmotility ratchet with experiments in the next section, we wish tounderstand the impact of saturation effects first. To this end, weuse the following expression for the phototactic alignment rate ω ( x , φ , t ) = v ( x , t ) sin φ sgn (cid:0) v (cid:48) ( x , t ) (cid:1) C R (cid:0) − exp (cid:0) − C (cid:12)(cid:12) v (cid:48) ( x , t ) (cid:12)(cid:12)(cid:1)(cid:1) (4)which approximately reduces to Eq. (3) for | v (cid:48) ( x , t ) | (cid:28) / C and A = C C / R (and exactly for | v (cid:48) ( x , t ) | → ). See Fig. 5 for a com-parison of Eqs. (3) and (4), both in the saturated and in the un-saturated regime. Here, C controls the crossover from the linearto the saturated region. To see the impact of saturation effects, letus now explore the transport velocity as a function of t on (Fig. 6,inset) for C = . s as previously determined in . Just as for theunsaturated case (Fig. 2), the transport increases about linearlyfor small t on , reaches a maximum at t on ≈ . s of (cid:104) v (cid:105) (cid:38) µ m / s and then decays again, suggesting that saturation effects have lit-tle bearing on the flashing motility ratchet. However, at larger Journal Name, [year], [vol.][vol.]
Propagating density spikes: Spatial particle distribution after t = s for N p = . and different t on shown in the key. Inset: Mean trans-port velocity (cid:104) v (cid:105) and standard deviation σ = (cid:112) (cid:104) ( v − (cid:104) v (cid:105) ) (cid:105) as a functionof t on , calculated for N p = and t end = T per data point. The dashedline shows the zero line to guide the eye. Parameters: t on + t off = s ; a = . µ m ; b = . µ m ; v min = µ m / s ; v max = µ m / s and v off = µ m / s ; A ≈ . ( A = C C / R for particles with radius R = . µ m and C = . , C = . s being experimentally determined parameters, see ). in the transport velocity, and in particular a transport-reversal for µ m / s (cid:46) v off (cid:46) µ m / s . (Note here, that the structure of thepeaks is statistically well converged, as indicated by the fact thatthe standard deviation σ is much smaller than (cid:104) v (cid:105) for most datapoints.) Following the observed transport reversal, when consid-ering a mixture of particles with different velocities, say µ m / s and µ m / s , the flashing light-field will transport them in oppo-site directions and segregate the mixture. This could be used inprinciple as a tool to prepare clean ensembles of active particleswith a uniform self-propulsion speed. Notice that the presenceof a current reversal alone, which is naturally present in manypotential ratchets , is not sufficient to achieve a clean and near-complete segregation of two initially distributed particle species.Instead, such a segregation requires |(cid:104) v (cid:105)| (cid:38) σ for each species,a condition which is naturally fulfilled for the flashing motility-ratchet (Fig. 3). This shows once more that the intrinsically lowdispersion of the emerging transport might be useful for practicalapplications. Notice however, that σ increases roughly linearlywith v off (Fig. 3) i.e. with the distance particles travel within eachoff-phase. That is, the particle distribution gets broader as v off in-creases allowing for a clean segregation ’only’ for particles whichare not too fast. To understand the mechanism leading to a coherently travellingdensity spike pattern in detail, it is instructive to first explore typ-ical trajectories in the noise-free case ( D r = D t = ), as visualizedin Fig. 4a for the case t on = s . Here, within one or several drivingcycles, the orientation of the particles, which align antiparallel tothe light-gradient, converges to a steady value of either φ = (or φ = π n , n being integer) or to φ = ( n + ) π (particles typically Journal Name, [year], [vol.] , ig. 3 Transport reversal: Mean velocity (cid:104) v (cid:105) and standard deviation σ = (cid:112) (cid:104) ( v − (cid:104) v (cid:105) ) (cid:105) for t on = s as a function of v off . Other parameters as inFig. 2. need about − s to fully turn ) corresponding to perfect align-ment with the axis along which the light-pattern is modulated(upper panel). Once they are aligned, particles no longer expe-rience a torque from the light-field ( p × ∇ I ( x , t ) = for all x , t )and persistently self-propel in one and the same direction. Con-versely to their constant orientation, the speed of the particlesstill evolves rather irregularly at this stage: it simply reads v = v off in each off-phase, but in the on-phase v ∈ ( v min , v max ) it dependson the particle’s position at the instance the light-field is switchedon. At late times, the dynamics converges to a limit cycle at-tractor, whose existence is expected from the Poincaré-Bendixontheorem, resulting in a regular, periodic dynamics (Fig. 4a, mid-dle and lower panel). That is particles fully synchronize with theflashing light-field with a temporally periodic speed, whose av-erage is exactly v = L / T (middle panel), i.e. particles propagateby exactly one spatial period L per T = s . As illustrated by theblue and the orange curve in Fig. 4a (middle panel), dependingon the initial conditions, particles follow a corresponding orbit ei-ther to the left or to the right. In this minimal case where onlytwo limit cycles are relevant, each particle propagates the sameone-flashing-cycle-averaged velocity of v = L / T either to the leftor to the right (Fig. 4a, middle and lower panel). The resultinglate-time transport velocity is then fully determined by the ratioof initial conditions reaching the limit cycle corresponding to mo-tion to the left and to the right, respectively, i.e. to the ’basins ofattraction’ of the two limit cycles. Specifically for the consideredcase of t on = s , most of the particles reach an attractor point-ing to the right, which generates the transport. (In general, de-pending on t on besides limit cycles representing a 1:1 resonance,also limit cycles allowing particles to traverse n spatial periodswithin m driving cycles are allowed, leading to a particle speed of v = nL / ( mT ) .)In the presence of noise, φ does of course not fully converge to π or π but fluctuates even at late times (upper panel in Fig. 4b).However, most of the time (for t on = s ), particles still follow anear-resonant dynamics and move with v ≈ L / T (middle panel inFig. 4b), temporarily resembling trajectories of the underlying de- terministic system. Such a dynamics can prolong for many drivingcycles. From time to time, however, noise relocates a particle fromone limit cycle of the underlying deterministic system to anotherone, leading, at least temporarily to a different dynamics. Thedynamics shown in Fig. 4b (middle panel) illustrates this: In thetime interval between t ∼ s and t ∼ s , the orange trajectorycrosses from the “ φ = -attractor” of the underlying noise-free sys-tem over to the “ φ ≈ − π attractor”, yielding the same dynamics,but in between, it follows the “ φ = − π -attractor” for a few flash-ing cycles and moves into the opposite direction. In the presenceof noise, the resulting transport velocity is therefore mainly deter-mined by the time particles spend in the basin of attraction of thelimit cycles of the underlying noise-free system. When this basinof attraction is small, as e.g. for t on = s for the “ φ = ( n + ) π -attractor” which leads to motion to the left, particles all move tothe right in the long-time average.The spike pattern observed above now follows quite naturallyfrom the described particle-light-field synchronization. Here,each spike represents a package of particles which have followedthe same limit cycle dynamics for the same amount of time. Inparticular, particles which have moved for s = T with avelocity of v ≈ L / T = . µ m / s , have traversed a distance of x ≈ L ≈ µ m , which corresponds to the first large peak inFig. 2 for t on = s ; the following peaks correspond to particleshaving traversed a distance of x ≈ L , L , L .. , correspondingto particles which have either initially ’lost’ time by aligning witha certain delay, or by moving temporarily into the opposite direc-tion. Finally, there is a small peak of particles having traverseda distance x > L ; this corresponds to particles which have beenpushed forward during the on-phases before synchronizing withthe light-field. (The spike pattern is less clean for t on = s and t on = s , where the relevant attractors do not correspond to 1:1resonances.) In experiments the phototactic alignment rate of a Janus colloidwith the intensity gradient does not generally increase linearly, aswe have assumed so far, but saturates for steep intensity gradi-ents due to thermal coupling . Thus, to compare the flashingmotility ratchet with experiments in the next section, we wish tounderstand the impact of saturation effects first. To this end, weuse the following expression for the phototactic alignment rate ω ( x , φ , t ) = v ( x , t ) sin φ sgn (cid:0) v (cid:48) ( x , t ) (cid:1) C R (cid:0) − exp (cid:0) − C (cid:12)(cid:12) v (cid:48) ( x , t ) (cid:12)(cid:12)(cid:1)(cid:1) (4)which approximately reduces to Eq. (3) for | v (cid:48) ( x , t ) | (cid:28) / C and A = C C / R (and exactly for | v (cid:48) ( x , t ) | → ). See Fig. 5 for a com-parison of Eqs. (3) and (4), both in the saturated and in the un-saturated regime. Here, C controls the crossover from the linearto the saturated region. To see the impact of saturation effects, letus now explore the transport velocity as a function of t on (Fig. 6,inset) for C = . s as previously determined in . Just as for theunsaturated case (Fig. 2), the transport increases about linearlyfor small t on , reaches a maximum at t on ≈ . s of (cid:104) v (cid:105) (cid:38) µ m / s and then decays again, suggesting that saturation effects have lit-tle bearing on the flashing motility ratchet. However, at larger Journal Name, [year], [vol.][vol.] , ig. 4 Transport mechanism in the flashing motility ratchet without (a)and with (b) noise for t on = s and two representative trajectories (blueand orange) with initial orientations φ = . and φ = . . Without noise,particles fully align with the symmetry axis of the light-field ( p × ∇ I = ),yielding φ = π + π n or φ = π n ( n ∈ Z ), representing motion to the leftand to the right, respectively. At late times (middle and lower panel),particles reach a limit cycle attractor and move periodically, with an av-erage speed of exactly v = L / T (middle panel) either to the left or to theright. In the presence of noise, the particle orientations do not fully con-verge but fluctuate. Here particles show a dynamics which is similar tothe noise-free dynamics most of the time (middle panel), but they cancross-over from one attractor to another one, which can, for example,lead to temporary motion into the opposite direction (middle panel). Theshown time intervals have been chosen to representatively illustrate thedescribed dynamics (periodic motion and attractor hopping). Parametersas in Fig. 2. Fig. 5
Red and blue curves represent Eqs. (3) and (4), respectively,deep in the saturated (left) and in the unsaturated (right) regime. Dashedlines show averages of full curves. Parameters as in Fig. 2
Fig. 6
Impact of torque saturation on particle distribution and transportvelocity. Figures and parameters as Fig. 2 but in the presence of torquesaturation, Eq. (4), ( C = . R = . µ m ; C = . s , obtained from ex-periments). values of t on , the transport plateaus (Fig. 6), rather than decay-ing towards zero and reaches a finite value of (cid:104) v (cid:105) (cid:38) . µ m / s for t on → s . That is, saturation effects create a directed transporteven for a stationary (non-flashing) light field. This special casehas been previously explored in . Why do we obtain a directedparticle transport here even in the absence of flashing? Note first,that the argument for the absence of transport in a static light pat-tern as given in section 3 breaks down in the presence of torquesaturation, as ω ( x , φ ) is no longer spatially periodic and a particlecrossing a whole spatial periode can therefore experience a netchange of its orientation (even in the absence of noise).More constructively, on average, particles need significantlymore time to cross the long and shallow b -segments where ∇ I is small and the alignment rate is essentially unsaturated (i.e. ω ∝ | ∇ I | ) than for crossing the short and steep a -segments, where ω saturates and is only slightly larger than in the b -segments.Hence, particles leaving a steep a -segment are aligned onlyweakly as compared to those leaving a b -segment. Thus, parti-cles leaving the a -segement to the left are commonly reflected inthe adjacent b -segment, since phototaxis opposes their swimmingdirection. Conversely, particles leaving an a -segment to the rightalmost certainly manage to cross the adjacent b -segement wherephototaxis is supportive and speeds them up. This breaks the left-right symmetry, initiating a transport to the right. (In the extremecase of an almost vertical a -segement and an extremely flat andlong b -segment, basically all particles would get reflected, whenleaving an a -segement to the left, and the ratchet would serve asan “active particle diode”.)To further illuminate the impact of saturation effects, let us ex-plore the particle distribution in the lattice, say after 20 drivingcycles. From Fig. 6, we can see that the distribution for t on = s is essentially the same as in the absence of saturation effects(Fig. 2), preserving the shock-wave like profile. As for the trans-port itself, however, the distribution significantly deviates fromthe unsaturated case when t on ∼ T . Following these observations,for our upcoming experiments we expect that saturation effectsdo not play a strong role if t on (cid:28) T . However, if t on ∼ T , satura- Journal Name, [year], [vol.] , ion effects are expected to significantly impact the transport. To test the flashing motility-ratchet, we now compare our resultswith experiments. We use light-activated Janus colloids, whichare composed of optically transparent silica spheres ( R = . µ m )being capped on one side with nm of carbon. The active colloidsare suspended in a critical mixture of water and 2,6-lutidine (luti-dine mass fraction 0.286), whose lower critical point is at a tem-perature of T c = . ◦ C . When the solution is kept at a tempera-ture of ◦ C < T c , the colloids perform diffusive Brownian motion.Upon laser illumination (at wavelength λ = nm ), where lightis absorbed at the caps only, the solvent locally demixes. Thiscreates a concentration gradient accross each colloid’s surface,leading to self-propulsion. The resulting self-propulsion speedscales linearly with the laser intensity . The light pattern, andhence also the motility-pattern, is created by a laser line focus( λ = nm ) being scanned across our sample cell by means of agalvanostatically driven mirror with a frequency of Hz . Syn-chronization of the scanning motion with the input voltage ofan electro-optical modulator leads to a quasi-static illuminationlandscape. The alternation between an instensity ladscape andan homogenous one were fully automated using a customizedsoftware written in LABVIEW. Since the remixing timescale of thebinary mixture is on the order of ms , the periodic mirrormotion is fast enough to produce stable particle self-propulsion.Particle positions and orientations were obtained by digital videomicroscopy with a frame rate of 13 fps.We use a very dilute suspension of microswimmers to avoidparticle interactions and subject it to the described flashingratchet. Here we track trajectories of particles initialized in oneand the same a -segment with uncontrolled initial orientations andlet them evolve for 13 T . We repeat the experiment several timesand average over N p > trajectories. In Fig. 7a we comparethe resulting transport velocity in experiments with our modelfor various flashing times, finding close quantitative agreement.For the considered parameters, the transport velocity rapidly in-creases from (cid:104) v (cid:105) = at t on = s and approaches a maximal speed ofabout (cid:104) v (cid:105) (cid:38) µ m / s for t on ∼ − s , which is about 2-3 times largerthan for the static case. For larger switching times, the transportvelocity decreases monotonously with t on . To also compare thedispersion in the model and in the experiments, we show the stan-dard deviation σ also in Fig. 7a, finding near quantitative agree-ment. It is instructive to visualize this also in a different way. Todo this, we define the coefficient of variation as CV = (cid:104) v (cid:105) / σ (qual-ity factor), which is the ratio of the mean transport velocity overthe standard deviation, shown in Fig. 7b. While there are notabledeviations between experiments and simulations for small and forlarge t on (Fig. 7b), we find a rather good agreement in the regime s ≤ t on ≤ s , where the transport is dominated by flashing, ratherthan by torque-saturation effects. This reflects that also in experi-ments the synchronization between particles and light-field leadsto a transport with little dispersion. It is instructive to also resolvethe time-evolution of the average particle speed in the N -th flash-ing periode, defined as (cid:104) ˙ x (cid:105) ( N ) , where the average is taken bothover the particle ensemble and the N -th flashing period. Here, both in experiments and in simulations, the transport convergesto its steady state value within a few driving cycles in most cases,but takes significantly longer in those cases, where dispersion islarge ( t on = s and t on = s ). Journal Name, [year], [vol.][vol.]
Impact of torque saturation on particle distribution and transportvelocity. Figures and parameters as Fig. 2 but in the presence of torquesaturation, Eq. (4), ( C = . R = . µ m ; C = . s , obtained from ex-periments). values of t on , the transport plateaus (Fig. 6), rather than decay-ing towards zero and reaches a finite value of (cid:104) v (cid:105) (cid:38) . µ m / s for t on → s . That is, saturation effects create a directed transporteven for a stationary (non-flashing) light field. This special casehas been previously explored in . Why do we obtain a directedparticle transport here even in the absence of flashing? Note first,that the argument for the absence of transport in a static light pat-tern as given in section 3 breaks down in the presence of torquesaturation, as ω ( x , φ ) is no longer spatially periodic and a particlecrossing a whole spatial periode can therefore experience a netchange of its orientation (even in the absence of noise).More constructively, on average, particles need significantlymore time to cross the long and shallow b -segments where ∇ I is small and the alignment rate is essentially unsaturated (i.e. ω ∝ | ∇ I | ) than for crossing the short and steep a -segments, where ω saturates and is only slightly larger than in the b -segments.Hence, particles leaving a steep a -segment are aligned onlyweakly as compared to those leaving a b -segment. Thus, parti-cles leaving the a -segement to the left are commonly reflected inthe adjacent b -segment, since phototaxis opposes their swimmingdirection. Conversely, particles leaving an a -segment to the rightalmost certainly manage to cross the adjacent b -segement wherephototaxis is supportive and speeds them up. This breaks the left-right symmetry, initiating a transport to the right. (In the extremecase of an almost vertical a -segement and an extremely flat andlong b -segment, basically all particles would get reflected, whenleaving an a -segement to the left, and the ratchet would serve asan “active particle diode”.)To further illuminate the impact of saturation effects, let us ex-plore the particle distribution in the lattice, say after 20 drivingcycles. From Fig. 6, we can see that the distribution for t on = s is essentially the same as in the absence of saturation effects(Fig. 2), preserving the shock-wave like profile. As for the trans-port itself, however, the distribution significantly deviates fromthe unsaturated case when t on ∼ T . Following these observations,for our upcoming experiments we expect that saturation effectsdo not play a strong role if t on (cid:28) T . However, if t on ∼ T , satura- Journal Name, [year], [vol.] , ion effects are expected to significantly impact the transport. To test the flashing motility-ratchet, we now compare our resultswith experiments. We use light-activated Janus colloids, whichare composed of optically transparent silica spheres ( R = . µ m )being capped on one side with nm of carbon. The active colloidsare suspended in a critical mixture of water and 2,6-lutidine (luti-dine mass fraction 0.286), whose lower critical point is at a tem-perature of T c = . ◦ C . When the solution is kept at a tempera-ture of ◦ C < T c , the colloids perform diffusive Brownian motion.Upon laser illumination (at wavelength λ = nm ), where lightis absorbed at the caps only, the solvent locally demixes. Thiscreates a concentration gradient accross each colloid’s surface,leading to self-propulsion. The resulting self-propulsion speedscales linearly with the laser intensity . The light pattern, andhence also the motility-pattern, is created by a laser line focus( λ = nm ) being scanned across our sample cell by means of agalvanostatically driven mirror with a frequency of Hz . Syn-chronization of the scanning motion with the input voltage ofan electro-optical modulator leads to a quasi-static illuminationlandscape. The alternation between an instensity ladscape andan homogenous one were fully automated using a customizedsoftware written in LABVIEW. Since the remixing timescale of thebinary mixture is on the order of ms , the periodic mirrormotion is fast enough to produce stable particle self-propulsion.Particle positions and orientations were obtained by digital videomicroscopy with a frame rate of 13 fps.We use a very dilute suspension of microswimmers to avoidparticle interactions and subject it to the described flashingratchet. Here we track trajectories of particles initialized in oneand the same a -segment with uncontrolled initial orientations andlet them evolve for 13 T . We repeat the experiment several timesand average over N p > trajectories. In Fig. 7a we comparethe resulting transport velocity in experiments with our modelfor various flashing times, finding close quantitative agreement.For the considered parameters, the transport velocity rapidly in-creases from (cid:104) v (cid:105) = at t on = s and approaches a maximal speed ofabout (cid:104) v (cid:105) (cid:38) µ m / s for t on ∼ − s , which is about 2-3 times largerthan for the static case. For larger switching times, the transportvelocity decreases monotonously with t on . To also compare thedispersion in the model and in the experiments, we show the stan-dard deviation σ also in Fig. 7a, finding near quantitative agree-ment. It is instructive to visualize this also in a different way. Todo this, we define the coefficient of variation as CV = (cid:104) v (cid:105) / σ (qual-ity factor), which is the ratio of the mean transport velocity overthe standard deviation, shown in Fig. 7b. While there are notabledeviations between experiments and simulations for small and forlarge t on (Fig. 7b), we find a rather good agreement in the regime s ≤ t on ≤ s , where the transport is dominated by flashing, ratherthan by torque-saturation effects. This reflects that also in experi-ments the synchronization between particles and light-field leadsto a transport with little dispersion. It is instructive to also resolvethe time-evolution of the average particle speed in the N -th flash-ing periode, defined as (cid:104) ˙ x (cid:105) ( N ) , where the average is taken bothover the particle ensemble and the N -th flashing period. Here, both in experiments and in simulations, the transport convergesto its steady state value within a few driving cycles in most cases,but takes significantly longer in those cases, where dispersion islarge ( t on = s and t on = s ). Journal Name, [year], [vol.][vol.] , ig. 7 Comparison of model and experiments: (a) Average velocity (cid:104) v (cid:105) and standard deviation σ , (b) coefficient of variation ( CV = (cid:104) v (cid:105) / σ ), averagedover the entire simulation/experiment (see text), as a function of t on . Panel (c) shows the average particle velocity in the N -th flashing cycle (cid:104) ˙ x (cid:105) fordifferent t on shown in the key. Parameters, both in experiments and simulations T f = t on + t off = s , v max = µ m / s , v off = µ m / s ; v min = . µ m / s and C = . R = . µ m ; C = . s from fits to experiments in . Active colloids can synchronize with a sawtooth-shaped flashing-light field and self-organize into a pattern of coherently propa-gating density spikes. This pattern hardly disperses and yields apulsating particle transport, which might be useful e.g. for tar-geted drug delivery applications. The transport velocity can betailored by the parameters of the flashing motility field, and re-markably, it reverts when the particle’s self-propulsion velocity ex-ceeds a certain threshold. Thus, the present setup can be used asa device for segregating particle ensembles. To observe the latteraspect also in experiments, it would be interesting to do experi-ments with mixtures of active colloids which are faster than theones used in the present study, in the future. Our results can be straightforwardly generalized to more complex spatio-temporalmotility patterns including random landscapes in space and time,and can be used as a platform to transfer ideas from potentialratchets to active systems.
Acknowledgments
H.L. and C.B. acknowledge financial support through the pri-ority programme SPP 1726 of the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation). C.B. acknowledgesfinancial support by the ERC Advanced Grant ASCIR (GrantNo.693683). B.t.H. gratefully acknowledges received fundingthrough a Postdoctoral Research Fellowship from the DeutscheForschungsgemeinschaft – HA 8020/1-1.
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