Active spheres induce Marangoni flows that drive collective dynamics
Martin Wittmann, Mihail N. Popescu, Alvaro Domínguez, Juliane Simmchen
EEPJ-E manuscript No. (will be inserted by the editor)
Active spheres induce Marangoni flows that drivecollective dynamics
Martin Wittmann · Mihail N. Popescu · Alvaro Dom´ınguez · JulianeSimmchen
Received: Date / Accepted: Date
Abstract
For monolayers of chemically active parti-cles at a fluid interface, collective dynamics are pre-dicted to arise owing to activity-induced Marangoniflow even if the particles are not self-propelled. Here wetest this prediction by employing a monolayer of spheri-cally symmetric active TiO particles located at an oil–water interface with or without addition of a non-ionicsurfactant. Due to the spherical symmetry, an individ-ual particle does not self-propel. However, the gradientsproduced by the photochemical fuel degradation giverise to long-ranged Marangoni flows. For the case inwhich surfactant is added to the system, we indeed ob-serve the emergence of collective motion, with dynam-ics dependent on the particle coverage of the monolayer. Martin Wittmann (contributed equally)
Technical University Dresden, Zellescher Weg 19, DE-01069Dresden, GermanyE-mail: [email protected] N. Popescu (contributed equally)ORCID: 0000-0002-1102-7538
Max Planck Institute for Intelligent Systems, Heisenbergstr.3, D-70569 Stuttgart, GermanyE-mail: [email protected] Dom´ınguez (corresponding author)ORCID: 0000-0002-8529-9667
F´ısica Te´orica, Universidad de Sevilla, Apdo. 1065, 41080Sevilla, SpainInstituto Carlos I de F´ısica Te´orica y Computacional, 18071Granada, SpainE-mail: [email protected] Simmchen (corresponding author)ORCID: 0000-0001-9073-9770
Technical University Dresden, Zellescher Weg 19, DE-01069Dresden, GermanyTel.: +49 351 463-37433Fax: +49 351 463-37164E-mail: [email protected]
The experimental observations are discussed within theframework of a simple theoretical mean field model.
The challenge of endowing micro- and nano-sized par-ticles with motility, without applying external forcesor torques, has received significant interest from bothperspectives of applied and basic science (see, e.g., theinsightful reviews in Refs. [1–3]). As noted in the clas-sic paper of Purcell [4], the issue of motility of suchmicrometer-sized objects in Newtonian liquids of vis-cosity similar to that of water is particularly intriguingand interesting. For such systems the Reynolds numberis very low and thus the hydrodynamics is governed byStokes equations while the motion of the object is in theoverdamped regime. Accordingly, such objects cannotrely on inertia to move steadily (see also the review inRef. [5]). Hence, achieving steady directional motion ofcolloidal particles requires strategies to break the time-reversal symmetry of the Stokes equations [4].One direction which has been intensely pursuedsince the first reports of such particles [6, 7], is makingJanus structures. These have one part of the surface ac-tive (chemically or thermally), while the other is inert(see the left panels in Figure 1). Variations in activ-ity along the surface generate spatial inhomogeneitiesin the chemical composition (or the temperature) ofthe surrounding suspension. Motility then emerges via,e.g., self-phoresis driven by the gradients of these self-generated inhomogeneities [8–15]. The motion of Janusparticles, either in unbounded fluid or in the vicinityof solid-liquid interfaces has been subject of numeroustheoretical and experimental studies. Detailed and in-sightful reviews of such studies can be found, e.g., in a r X i v : . [ c ond - m a t . s o f t ] J a n Martin Wittmann et al.
Fig. 1 (A-C): Schematic representations of a Janus (A) and isotropic (B, C) active particles, respectively, in the vicinity of awall (A, B) or at a fluid interface (C). The chemical activity is illustrated via the example of a semiconductor photocatalyistpromoting, upon suitable illumination, the decomposition of hydrogen peroxide (H O ) into water and oxygen. (D-F) Examplesof the calculated concentration c of reaction product (color coded, in arbitrary units) and hydrodynamic flow (streamlines,white). Refs. [2, 16, 17]. The question of the self-phoresis ofactive Janus particles near, or trapped at, interfacesbetween two fluid phases, from now on referred to as fluid interfaces , has been recently tackled both experi-mentally [18–23] and theoretically [20, 24–26].An interesting aspect specific to a fluid interface isthat the interface itself can respond to the activity ofthe particles due to the locally induced changes in sur-face tension, which give rise to Marangoni stresses. Fora Janus particle trapped at the fluid interface, these candirectly drive the motion of the particle along the in-terface [27–30]. Furthermore, the Marangoni flows cancouple back to the motion of a self-phoretic active Janusparticle located near the interface [31]. Concerning col-lective effects, for “carpets” of self-propelled particlesat a fluid interface the interplay between motility andinduced Marangoni flows has been shown to possiblyinduce instabilities and the break up of thin films [32].As can be inferred from the discussion above, mostof the studies of active particles have focused on parti-cles for which self-motility is an intrinsic property. Thecase of chemically or thermally active particles whichlack motility when isolated, but can set in motion whenplaced near confining surfaces or fluid interfaces, hasbeen somewhat less explored. It was predicted that neara fluid interface either self-phoresis [26] or a self-inducedMarangoni flow [33, 34] (as shown schematically in theright column of Fig. 1) can move the particle towardsor away from the planar interface. When one of these particles with uniform activ-ity over its surface is trapped at or near a planar in-terface, there cannot exist any “in-plane” motion, be-cause of the radial symmetry of the in-plane Marangoniflow induced by the particle (see the right column ofFig. 1) (unless there is a mechanism for an in-planesymmetry breaking along the interface, such as, e.g.,the spontaneous autophoresis at large P´eclet numbersdiscussed by Ref. [35]). However, when a collection ofsuch particles forms a monolayer, the radial symmetryof the in-plane hydrodynamic flow is broken by the pres-ence of the other particles . The superposition of thein-plane components of the Marangoni flows inducedby each particle becomes an effective pair interactionwithin the monolayer [34, 37]; this is actually a hydro-dynamic interaction inasmuch as it is due to fluid flow,albeit sourced by the interfacial response, rather thanby the motion of the particles. Consequently, inhomo-geneities in the distribution of active particles within A similar observation is predicted for the case of mono-layers of isotropic active particle at a solid-liquid interface.However, in this case both the hydrodynamic interactionsand the phoretic effective interaction (phoretic response ofone particle to the chemical field produced by another parti-cle) decay at least as r − with the in-plane distance r betweenparticles due to the boundary conditions imposed by the solidwall [36]. In contrast, the in-plane hydrodynamic interactionsdue to induced Marangoni flows have a significantly strongereffect due to their slow decay as r − [34], which is formallyanalogous to gravity or electrostatics in two-dimensions. Itcan have an attractive or repulsive character, depending onhow the surface tension reacts to the activity of the particles.ctivity-induced Marangoni dynamics 3 the monolayer induce motion of the — otherwise im-motile — particles. Therefore, activity–induced collec-tive dynamics within the monolayer may occur in spiteof the absence of single–particle self-motility. As dis-cussed in [37, 38], at least for simple models it can beshown that the state with uniform distribution of par-ticles is unstable against perturbations, and collectivedynamics sets in driven by the response of the interfacein the form of Marangoni stresses. In general, the dynamics within such active mono-layers involve the competition between the Marangonieffective interaction and other interaction, e.g., the di-rect forces that the particles exert on each other. Forexample, the issue of stability of a monolayer under thecompetition between the self–induced Marangoni flowsand the weight–induced capillary attraction, both ofwhich are long-ranged (they decay with the distance r between the particles as r − , for r smaller than thecapillary length) was addressed in Ref. [37]. The ex-istence of stationary states sustained by the competi-tion between the effective Marangoni interaction anda short-ranged repulsion was considered in Ref. [40].It has been shown that in this case stationary statescan emerge, with a radial “onion-like” structure (froma very dense, solid-like center to a very low, gas-likerim) in the particle distribution induced by the pres-ence of the within the monolayer, and hydrodynamicflow within the fluid. The estimates for the length scaleof such spatial structures [40], in agreement with therecently reported experimental set-ups involving mono-layers of active colloids near (or at) fluid interfaces [19–21, 41], suggest that experimental validation of thosetheoretical predictions may be possible in the case of awater-air interface and volatile products of the chem-ical activity . When the Marangoni flows are strong( > µ m/s), which can be the case at an oil–water in-terface, the mean-field analysis in Ref. [40] cannot be apriori justified, and interparticle correlations may playa significant role. This hints to the exciting possibilityof a much richer phenomenology of collective dynamicsemerging in such systems.Motivated by these observations, we designed andcarried-out a study of the emergence of collective dy-namics in a monolayer of chemically active spherical When the system is also laterally confined, as in the caseof a sessile drop containing active particles, the emergent col-lective dynamics driven by the self-induced Marangoni flows,can be very rich; for example, Ref. [39] reported recently theobservation of spontaneous symmetry breaking from a mo-tionless monolayer to states of self-organized motion of par-ticles and polar flow within the drop. Albeit under the drawback of extremely weak (nm/s)Marangoni flows, thus requiring very long experiments, whichis technically challenging. particles sedimented at a oil–water interface in the pres-ence of a non-ionic surfactant. Subsequently, the resultsare discussed and interpreted in the context of previ-ously proposed simple theoretical models of activity-induced Marangoni flows [33, 34, 37, 38, 40].
As model active particles we use photocatalytically ac-tive colloids based on spherical, isotropic TiO with anaverage size of 700 nm [42]. Their chemical activity con-sists in promoting the photocatalytic degradation ofhydrogen peroxide, H O , upon UV illumination (seethe schematic in Fig. 1C). The use of photocatalysishas the significant advantage that the chemical activ-ity can be switched on and off despite the fuel (H O )being present in the solution. In previous studies [43]we have shown that very efficient self-motile Janus col-loids (see Fig. 1A) can be made by half-covering suchTiO particles with a thin layer of a metal (e.g., Cu),that alters the reaction rate locally. Without this addi-tional metal layer, only enhanced Brownian motion isobserved (see Appendix B). When the intensity of UVillumination is sufficiently high, a fraction of the parti-cles lift of from their sedimentation location either overa solid wall or over a fluid oil–water interface.In order to test the emergence of activity–inducedMarangoni dynamics with this kind of active particles,we made a specific experimental setup (see Fig. 2A). Itconsists of a cylindrical cell filled with an aqueous so-lution of H O (1 %). This reservoir offers a sufficientquantity of fuel in order to guarantee a stable perox-ide concentration over the duration of an experiment(in the order of several minutes). The bottom wall wasfunctionalized with hexadecyltrimethoxysilane in orderto achieve hydrophobic properties and to enhance thetendency of silicon oil to wet it. We used silicon oil ofviscosity 1000 cSt and mass density 970 kg / m , whichis slightly lower than those of water (1000 kg / m ) andhydrogen peroxide (1100 kg / m ). An oil–water inter-face is created by carefully depositing with a canulaa drop of oil on the hydrophobic bottom wall of thecell. Despite the density mismatch, the wetting forcesprevent the drop from lifting-off by buoyancy and, atthe same time, ensure a pancake-like shape of the drop,which has proven difficult to achieve using microfluidicchambers [41]. These combined effects enable a fluid in-terface that is flat to a good approximation and allowsimaging over an extended area, which is crucial for theoptical microscopy observations. When TiO particlesare added into the cell, they sediment and a monolayeris formed near this quasi-flat fluid interface which will Martin Wittmann et al.
Fig. 2
A) Schematic representation of the experimental setup. B) Schematic representation of the expected location of theparticles (based on the measured in-plane diffusion coefficient) relative to the oil(yellow)–water(blue) fluid interface, withoutor with surfactant added, respectively. The broken lines show typical tracked in-plane trajectories of an inactive particle; thecorresponding values of the in-plane diffusion coefficient D are shown. C) The zeta-potential of particles as a function of addedsurfactant TritonX. The adsorption of the neutral surfactant molecules at the particle’s surface does not alter the chargeconditions, which is in contrast to the case of using anionic or cationic surfactants (see Fig. 9 in Appendix B). be the focus of our analysis. In absence of the drop ofoil the monolayer forms above the substrate solid wall.Once the cell has been prepared, it is placed un-der the microscope and the illumination with UV lightis performed through the objective, from below. Theilluminated area has an almost square shape, and itsdimensions can be varied to a certain extent. For thisstudy, we fixed it to a rectangular area of about 1500 µ m . A consequence of this setup is that only the par-ticles that lie inside this area become chemically active.The rest of the particles are expected to remain inactiveand they serve as passive tracers of the flows arising inthe aqueous phase. per se might cause changes of the fluid interface that wouldinduce flows, we tested the setup with passive SiO par-ticles (see Fig. 6 in Appendix B). We observed indeedno macroscopic or collective motion, apart from the ex-pected Brownian fluctuations. Second, we tested thesetup with TiO particles in the absence of either fuel(H O ) or UV illumination. Only Brownian motion ofthe particles was observed. Accordingly, from these weinfer that particle activity is a prerequisite for collectivemotion.However, when the particles in the cell without sur-factant become active upon exposition to both fuel andUV light, again no collective motion is detected (seeFig. 7 in Appendix B). We only observe, as already re-marked in Sec. 2, that some particles within the irradi-ated area manage to escape the focus plane by driftingin the vertical direction away from the fluid interface. This observation indicates that the particles reside closeto the fluid interface, without actually being trapped byit. Indeed, it is known that pure hydrophilic particlesgenerally do not show much adsorption onto a fluid in-terface [44].Our interpretation of the lack of observable collec-tive motion is that the dynamics may be too weak to bedetected, with velocities that fall under the experimen-tal resolution (well below 0 . µ m / s). Therefore, in orderto enhance the Marangoni flows, we added a surfactantto the aqueous phase with the goal to fix the particlesat the interface, so that a cumulative effect arises dueto the superposition of the Marangoni flows by eachparticle. (The addition of surfactant serve to overcomethe energy barrier that inhibits the attachment of par-ticles to the fluid interface, thus allowing the control ofthe positioning of the particles relative to the interface[45].) The surfactant addidion might also result in anincrease of the responsiveness of the interface by chang-ing the surface tension, however, there is no direct wayof measuring this effect.After testing different types of surfactants, we optedfor TritonX, a non-ionic surfactant that does not signifi-cantly influence the zeta-potential of the particles in or-der to keep the charge conditions constant (see Fig. 9 inAppendix B). In the absence of surfactant and irradia-tion, the diffusion coefficient of the particles sedimentedon the fluid interface is about 0 . µ m / s, which is com-parable to the value when the particles reside on a solidsubstrate (i.e., in the absence of the drop of oil). Afteraddition of TritonX, the diffusion coefficient decreasedsignificantly to a value 0 . µ m / s. This decrease is dueto the influence of the higher viscosity in the silicon oilphase [46]. Corroborated by the absence of lift-off whenthe particles are active, this indicates that the particlesare effectively trapped at the interface. ctivity-induced Marangoni dynamics 5 Therefore, we repeated the experiment with activeTiO particles in the presence of a low (0.05 wt%) con-centration of TritonX. Now we observe the emergenceof collective flows of particles in the plane of the in-terface (see Figs. 3 and 4). In order to support thatthis is due to a Marangoni flow, we tested that, whenthe particles reside on a solid substrate, the addition ofsurfactant does not induce any in-plane collective mo-tion (see Figs. 10 and 11 in the Appendix B). Whenthe fluid-fluid interface is replaced by a fluid-solid in-terface we do not observe any collective dynamics (seeAppendix B.5), which convincingly rules out the chem-ical (“phoretic”) interactions as the source of the ob-served behavior. To dismiss the assumption of ther-mally driven Marangoni flows (induced by a hypothet-ical heating of the titania particles due to UV absorp-tion), we compared the behavior of UV-illuminated ac-tive particles at the oil-(water plus peroxide) interfacein presence and absence of surfactant and found col-lective dynamics only in the first case. Moreover, thesesame observations allow us to also rule out mechanismsbased on Marangoni stresses due to some trace amountsof contaminants at the interface, a scenario invokedby Ref. [30] in the context of the motion of heatedAu@SiO Janus particles at an oil-water interface.In summary, we have identified the minimal ingredi-ents required for the emergence of significant activity–induced collective dynamics with spherically symmetricparticles in our experimental setup: besides the photo-catalytic particles fuelled by hydrogen peroxide, and afluid interface in our specific conditions also a surfactantis necessary in order to generate observable Marangoniflows.3.2 Dependence of the collective dynamics on theareal density of particlesThe photocatalytic activity of the TiO particles scaleswith light intensity and fuel concentration, both lead-ing to an increase in product concentration. We alsoconfirmed that the behavior of the system depends onthese two factors, resembling strongly to what is pre-sented here. However, it turns out that the impact ofthe areal density of particles in the monolayer on theemerging dynamics is much more significant. For thisreason, we decided to keep the light intensity and thefuel concentration fixed and, for the rest of this study,to investigate the dynamics at different particle densi-ties.At low particle areal density (12 % surface cover-age), we observe the effect displayed in Fig. 3. Beforeirradiation with UV light, the particles are uniformlydistributed over the fluid interface (Fig. 3A). Once the Fig. 3
Behaviour at low areal fraction: A) Before onset ofillumination the particles are uniformly distributed over theinterface. B) When UV irradiation is on (the violet squareindicates the illuminated area, but the camera is not sensi-tive to UV light) an inward flow sets in, bringing particlestowards (and into) the irradiated area. C) The flow reversesdirection after approximately 5 s; the outwards flowing in-duces a depletion of particles from the illuminated area. D)The distance to the center of the illuminated area for sometracked particles (the trajectories in B and C) is plotted as afunction of the time after UV irradiation is switched on. (Thescale bars correspond to 50 µ m.) UV irradiation in the marked area is switched on, a col-lective flow towards the center of the illuminated regionis observed (Fig. 3B). After a few seconds, and withoutany changes in the experimental parameters, the direc-tion of the flow reverses (Fig. 3C). A steady state isnot reached within the experimental times. When theillumination is turned off, the flows stop and the systemrelaxes to its original, equilibrium state.At higher particle concentrations (32 % surface cov-erage), the experiment starts analogously to the lowconcentration setup: before irradiation with UV light,the particles are uniformly distributed over the fluid in-terface, albeit with a higher surface coverage (Fig. 4A).Switching on the UV irradiation in the area indicatedin Fig. 4 also causes a collective flow towards the irra-diated area. But, differently to the previous case, wedo not observe any velocity reversal here. The flowdrags the particles towards the center of the illumi-nated area, where they form a cluster (see Fig. 4D).For the trajectories shown in Fig. 4D, corresponding toparticles passing through the same location at differ-ent times after the UV was turned on, the velocity foreach tracked trajectory is approximately constant (seeFig. 4E). Comparing these velocities, it can be inferredthat the magnitude increases with the time passed sincethe illumination was switched on (compare the slopesof the lines in Fig. 4E, from left to right). Alternatively,
Martin Wittmann et al.
Fig. 4
Behaviour at high areal fraction: A) Before onset of illumination the particles are uniformly distributed over the surface.B) When UV irradiation is on (the violet square indicates the illuminated area) an inwards flow sets in and brings particlestowards (and into) the illuminated area. C) Formation of a cluster within the illuminated region. D) As long as the UV lightremains on, the flow is maintained and the cluster grows. E) The distance to the center of the illuminated area for the trackedparticles (trajectories shown in panel D) as a function of the time after the UV irradiation is switched on. The straight linesrepresent a fit to a motion with constant velocity V i . F) Plot of the velocity magnitude | V i | of a particle which at time t i (afterillumination is turned on) is at a given distance R i to the center of the illuminated area (in this example, R i ≈ µ m). Thedashed line is the fit by Eq. (4). (The scale bars correspond to 50 µ m.) by considering a fixed time, e.g., t = 4 s, it can also beinferred that the velocity of a particle is larger whenits distance from the illuminated area is larger. Thisinflow leads to a steady growth of the cluster; we donot observe particles escaping the cluster as long as theUV light is on. After turning off the UV light, the flowsalso stop; however, the dense cluster remains compactand merely breaks apart into larger pieces as the sys-tem relaxes. This fact can be interpreted as indicativeof Marangoni flows that are sufficiently strong to pushthe particles so close as to foster van der Waals forcescausing an essentially irreversible aggregation.In summary, we can state that the observed collec-tive flows depend significantly on the coverage by ac-tive particles. We now attempt to frame these findingsinto a simple theoretical model for activity–inducedMarangoni flows.3.3 Theoretical model and analysisBased on the experimental analysis, which pinpointsthe activity-induced Marangoni flows as the most plau-sible source of the collective dynamics within the mono-layer, the interpretation of the results is attemptedwithin the theoretical framework proposed in Refs. [34, 38]. This is the simplest model for the collective mo-tion of active particles by the self-induced Marangoniflow. Succinctly, it treats the monolayer as a contin-uum dragged by the ambient flow. This flow is de-scribed in the Stokes approximation, and it is drivenby the chemical gradients determined according to thediffusion equation (see Fig. 1F). The evolution withinthe monolayer is thus due to the hydrodynamic inter-actions between the particles (sourced by the interfa-cial response); even though the particles are lackingself-motility, the hydrodynamic interactions are longranged ( ∼ /r , see, c.f., Eq. (3)) because of the inducedMarangoni flows.Thus, by adopting a coarse–grained approach to de-scribe the large scale dynamics, we employ continuumfields that are assumed to vary slowly over the micro-scopic length scales (size of the particles, mean interpar-ticle separation, etc. . . ). The monolayer plane is iden-tified with z = 0; the vector (cid:126)r = ( x, y ) denotes the in-plane position and ∇ := ( ∂ x , ∂ y ) the two–dimensional(2D) nabla operator in the monolayer plane. The arealnumber density of particles in the monolayer is given bythe field (cid:37) ( (cid:126)r, t ), which is the only relevant field becausethe particles are not intrinsically motile (i.e., there areno fields associated to a polar or a nematic order). Weassume that there is no particle flux in or out of the ctivity-induced Marangoni dynamics 7 monolayer; accordingly, (cid:37) ( (cid:126)r, t ) is a conserved quantityand obeys the continuity equation ∂(cid:37)∂t = −∇ · ( (cid:37)(cid:126)u ) , (1)as the particles are dragged by the ambient Marangoniflow (cid:126)u ( (cid:126)r ) induced by the activity of the particles, i.e., weassume that the only relevant cause of particle motionis this flow . Notice that, although the Marangoni flowexists in the bulk of the fluid phases, the only relevantcontribution to the dynamics is the 2D flow evaluatedat the monolayer plane. A particularly important con-sequence is that, although the three-dimensional (3D)flow is incompressible, the projection of the flow ontothe monolayer plane is compressible (see, c.f., Eq. (2)),and thus Eq. (1) does describe the emergence of an in-homogeneous distribution within the monolayer.The activity of the particles is modeled as a sourceterm in the concentration of a chemical involved in thereaction (e.g., O as product, or H O as reactant). Onecan assume that the molecular diffusion is much fasterthan the time scales associated with the collective mo-tion, so that the distribution of chemical is adapted tothe instantaneous configuration of the particles. Conse-quently, the concentration field can be found from the3D Fick’s law with sources due to the active particles,i.e., by neglecting the time dependence as well as thedrag by the ambient flows (low P´eclet number), andsolving a Poisson equation — see the color-coded fieldin Fig. 1F.Since the surface tension of the interface dependson the local chemical composition, spatial variations ofthe surface tension develop due to the inhomogeneousdistribution of chemicals in the fluid media. These gra-dients in surface tension translate to tangential forcesat the interface (Marangoni stresses), which are trans-mitted to the fluids and generate the Marangoni flow.Assuming again that the flow adapts instantaneouslyto the particle configuration, the associated 3D veloc-ity field is described by the Stokes equations for incom-pressible flow (low Reynolds number) — see the whitestreamlines in Fig. 1F.Finally, one assumes that the surface tensionchanges linearly with the local concentration, which isa reasonable hypothesis if the range of variations is nottoo large. When this assumption is combined with thesolutions of Fick’s law for the concentration of chemicaland of Stokes’ equation for the flow, one ends up with asimple relationship between the particle density of the The model can be easily extended to account for otherdriving forces, e.g., the thermal (Brownian) forces and thedirect interactions between particles (e.g., steric repulsion),see Ref. [40]. monolayer and the Marangoni flow at the monolayerplane (see Ref. [40] for the detailed derivation): (cid:126)u = −∇ Φ, ∇ · (cid:126)u = G(cid:37) ( (cid:126)r ) , (cid:126)r ∈ illuminated area,0 , (cid:126)r (cid:54)∈ illuminated area, (2)where the constant G (which can be either positive ornegative) is proportional to the activity of the parti-cles .Equations (1) and (2) form a closed system that al-lows one to obtain the monolayer density (cid:37) ( (cid:126)r, t ). Noticethat Eq. (2) means that the 2D flow at the monolayer isactually a Newtonian field; e.g., for G < (cid:37) ( (cid:126)r,
0) = (cid:37) (see Appendix C). In particu-lar, for G < u r ( r, t ) = − A e t/T πT r , (3)where T = ( | G | (cid:37) ) − is a characteristic time scale setby the activity, and A is the area of the illuminatedregion.One can compare this result with the quantitativeobservations presented in Fig. 4 as follows. During theobservation time, the velocity of a particle does notchange significantly (see Fig. 4E), so that it would begiven by its initial value: a particle located at a distance R i in the non-illuminated region at the time t i (mea-sured from the onset of the illumination at t = 0) willhave a velocity V i = − C i e t i /T , C i = A πT R i . (4)This expression is used to fit the experimental datapoints, as shown in Fig. 4F; it can be seen that it pro-vides a good approximation with the two fitting param-eters T ≈
11 s, C i ≈ . µ m / s. With these numbers,Eq. (3) also predicts that the velocity of a particle isindeed approximately constant during the observationtime (see Eq. (8) in Appendix C). More precisely, if the activity of a particle is quantified bythe time rate Q at which the chemical reaction proceeds, then G = Qb/ [2 π ( η + η )( D + λD )]. Here b is the proportionalitycoefficient between the changes in the surface tension and thechanges in the chemical concentration (in the linear approxi-mation), η , η are the viscosities of the fluid phases, D , D are the diffusivities of the chemical in the fluids, and λ is theratio of solvabilities. Martin Wittmann et al. In spite of these results, there are discrepancies in-dicating that the experimental system is too rich to befully captured by the simple theoretical model we haveemployed. First, the combination
T C i ≈ . µ m ofthe fitting parameters differs significantly from the pre-diction A/ πR i ≈ . µ m for the value R i ≈ µ mof the trajectories depicted in Fig. 4E and the area A ≈ µ m of the illuminated region. Second, andmore importantly, at given time t the velocity field ac-cording to Eq. (3) decays with the distance r , while thetrajectories in Fig. 4E, which are probing this veloc-ity field, actually show a flow that at fixed t increasesin magnitude with r . This means, in particular, thatthis flow is compressible even in the non-illuminatedregions, in disagreement with Eq. (2). Additionally, thetheoretical model cannot explain the trajectories withvelocity reversal depicted in Fig. 3D, because the signof the velocity is fixed by the sign of the constant G (see Eq. (5) in Appendix C).These arguments suggest that, contrary to themodel assumptions, the activity that drives the flowis not located only in the illuminated region. The ob-servation that, in the absence of surfactants, no collec-tive motion is detected, allows one to further conjecturethat the surfactant is sensitive to the chemical reac-tion at the active particles, e.g., by desorbing from theTiO particles when they are active or by reacting withproduced intermediate reactive oxygen species. Accord-ingly, one may attempt to extend the theoretical modelby allowing for additional sources of Marangoni flow,so that ∇ · (cid:126)u (cid:54) = 0 also in the non-illuminated region.In any case, these findings raise new questions, whoseanswer would require further experimental studies andcomplementary theoretical modeling. In conclusion, we have set up an experimental studyto test the prediction of emergent collective dynam-ics driven by activity induced Marangoni flows, ratherthan self-propulsion, within a monolayer of chemicallyactive particle at a fluid interface. The setup involvesTiO particles, which under UV illumination promotethe photocatalytic decomposition of hydrogen perox-ide in aqueous solutions. The particles are spherical byconstruction, so that they lack self-motility. We studiedthe configuration of particles sedimented at a quasi-flatsilicon oil–water interface, in the presence of the neu-tral surfactant TritonX in the aqueous phase. Upon UVillumination of a small central region, we have indeedobserved the emergence of radial flows dragging the par-ticles in the monolayer towards or away from the illu- minated region, depending on the areal particle densitywithin the monolayer.Through a set of complementary experiments, wecould identify activity induced Marangoni flows as themost plausible cause of this dynamics and rule out var-ious other a priori possible mechanisms (such as, e.g.,phoretic interactions or UV-response of the surfactant).Surprisingly, the dynamics exhibits a significant, quali-tative change upon increasing the average areal densityin the monolayer. At low densities, an initial transientinflow towards the illuminated region is replaced, withinfew seconds, by a monotonic outwards flow, so that agrowing area around the illuminated region emerges de-pleted of particles. In contrast, at large areal densitiesthe inflow persists, leading to the steady growth of aparticle cluster in the illuminated region.The trajectories tracked for individual particles wereanalyzed within the framework of the simplest mean–field model for the dynamics driven by activity inducedMarangoni flow [40]. It turns out that the experimen-tal setup is very rich and cannot be captured fully bythe model. The most relevant conclusion is the findingthat, contrary to expectations, the interfacial stressesthat drive the Marangoni flow are not localized in theilluminated area, but rather they also exist outside, al-though the photochemical activity in the illuminatedarea remains the driving factor. This led to the con-jecture that the role played by the surfactant is morethan just facilitating the entrapment of the particles atthe interface or providing a more responsive interface.For instance, one can conceivably argue that the surfac-tant, which is not restricted to stay in the illuminatedarea, may be sensitive to the photochemical reactionat the TiO particles. Future studies are required tounderstand and rationalize these findings. Acknowledgements
JS and MNP thank the SPP 1726 forproviding a platform for scientific exchange and the FreigeistFellowship grant number 91619 from Volkswagen founda-tion. AD acknowledges support from the Spanish Govern-ment through Grant FIS2017-87117-P, partially financed byFEDER funds.
A Experimental Details
A.1 Materials
Chemicals: All chemicals and solvents were used in analyt-ical grade without any additional treatment: titanium (IV)isopropoxide (Alfa Aesar Co. Ltd.); dodecylamine (Fluka);silicone oil (Sigma-Aldrich).Particle synthesis: TiO particles were synthesized via an im-proved version of a previously published method [43]. In brief,0.18 mL of water was added to mixture solution of 105 mLmethanol and 45 mL acetonitrile. Then 0.28 g of DDA wasctivity-induced Marangoni dynamics 9dissolved in the mixture and stirred for about 10 min. 1 mL ofTTIP was added dropwisely and stirred for 12 hours, precip-itates were removed and stirring was continued for 24 hours,when the process was repeated until no further precipitateswere observed. Then the particles were washed with methanolthree times and were calcined in a tubular furnace under ni-trogen flow for 2 hours at 600 ◦ C.Surface functionalization: 24x24 mm glass slides were washedby sonication in aceton and ethanol and subsequently plasmacleaned. The surface functionalization was performed in thegas phase in a desiccator at 5 mbar for two days using 30 µ Lhexadecyltrimethoxysilane per glass slide.
A.2 Methods
XRD: The phases of the TiO particles were identified us-ing XRD. Measurements were carried out with a Bruker 2Dphaser in a 2 Θ range of 20 ◦ –80 ◦ . The XRD of TiO particles(5) only includes reflexes corresponding to the anatase andrutile phases, confirming the presence of a mixture of bothpolymorphic forms of TiO . Fig. 5
XRD of TiO particles.Zeta-potential measurements: The zeta-potential was mea-sured with a Malvern Zetasizer Nano ZSP in autotitrationmode.Video recording: An inverted optical microscope (Carl ZeissMicroscope GmbH) equipped with a Zeiss Colibri LED lampand a “N-Achroplan” 63x/0.95 M27 objective were used forobservation. The wavelength of the UV light was 385 nm,the UV lamp power was fixed at 100% lamp intensity, cor-responding to 315 mW. The particle behavior was recordedwith a Zeiss camera (Axiocam 702 Mono) and a frame rateof 40 fps.Experiments on liquid interfaces: The cylindrical cell wasfilled with 800 µ L of a solution containing 1 % H O and0.05 wt% TritonX. A drop of silicon oil was placed on thehydrophobic bottom using a canula. 10 - 50 µ L of particlesdispersed in water (10 mg/mL) were added to the cell andgiven time to settle down before the observation with themicroscope was started. Reference experiments on solid substrate: The experimentson solid substrate were carried out in the same cylindricalcell without the oil drop.Video analysis: Videos were analyzed using ImageJ 1.52p.Tracking was done with the Manual Tracking plugin.
B Reference experiments
B.1 Reference experiments on a fluid interface usingpassive SiO particles In the experiments with passive silica particles on the solidsubstrate and on the fluid interface, no motion induced bythe UV light was observed (see Fig. 6).
Fig. 6
Passive SiO particles in a solution of TritonX andH O on oil–water interface. Left: before UV irradiation.Middle: after 5 s of UV irradiation. Right: after 10 s of UVirradiation. B.2 Reference experiments on a fluid interface withoutsurfactant addition
In these experiments we have observed that irradiation withUV light causes enhanced Brownian diffusion of the TiO particles, but does not lead to collective dynamics. Some par-ticles lift off and move upwards out of focus (see Fig. 7) if theUV intensity is sufficiently high. Fig. 7
TiO particles without addition of surfactant on oil–water interface. Left: before UV irradiation. Middle: after 5 sof UV irradiation. Right: after 10 s of UV irradiation. B.3 Reference experiments on a fluid interface withoutH O addition If H O is not added to the solution, the UV light does notinduce any particle motion (see Fig. 8). B.4 Choosing a surfactant — Zeta-potentialmeasurements
Zeta-potential titrations were performed using Malvern Ze-tasizer Nano ZSP in autotitration mode using the attached0 Martin Wittmann et al.
Fig. 8
TiO particles on oil–water interface in a solution ofTritonX without H O . Left: before UV irradiation. Middle:after 5 s of UV irradiation. Right: after 10 s of UV irradiation.Multi Purpose Titrator. The titration was carried out addingthe respective surfactant (SDS, CTAB and TritonX) and zeta-potential values were recorded after every ml of the solutionof surfactant was added. Fig. 9
Zeta-potential titrations of TiO particles with differ-ent surfactants (SDS, CTAB and TritonX). B.5 Reference experiments on solid substrates
Similarly to the observations in the case of TiO particleson the oil–water interface without surfactant addition, whenthe interface is replaced by a wall the irradiation with UVlight causes solely an enhanced Brownian diffusion and anupwards motion of the particles, irrespective of the absence(see Fig. 10) or presence (see Fig. 11) of surfactant. Fig. 10
TiO particles without surfactant on a solid sub-strate. Left: before UV irradiation. Middle: after 5 s of UVirradiation. Right: after 10 s of UV irradiation. C Theoretical model and analysis
Here we solve Eqs. (1) and (2), under the assumption of radialsymmetry, in order to compare with the observations. We take
Fig. 11
TiO particles in a solution of TritonX on a solidsubstrate. Left: before UV irradiation. Middle: after 5 s ofUV irradiation. Right: after 10 s of UV irradiation.the initial distribution of particles in the monolayer to be ho-mogeneous, with density (cid:37) , and we approximate the illumi-nated area as a disk of fixed radius L . Therefore, the evolveddensity field will have radial symmetry about the center ofthe illuminated area. The velocity field of the Marangoni flow,which will only have a radial component u r ( r, t ), can be ob-tained easily by applying Gauss theorem to Eq. (2): thus, theflow outside of the illuminated area is given by u r ( r, t ) = GN ( t )2 πr , r > L, (5)where N ( t ) is the number of particles inside the illuminatedarea. By applying Gauss theorem over the illuminated areato Eq. (1) and combining with Eq. (5), one obtains ∂N∂t = − πLu r ( L, t ) (cid:37) ( L, t ) = − GN(cid:37) ( L, t ) . (6)Assuming G <
0, the velocity field (5) describes a “fallingflow” into the illuminated region, so that all the particlesapproaching the rim of this region, and which determinethe value (cid:37) ( L, t ) of the density there, come from the non-illuminated region. But the 2D flow is incompressible in thisregion (see Eq. (2)), and consequently, the density at the rimdoes not change in time and it is equal to the initial one , (cid:37) ( L, t ) = (cid:37) . Equation (6) then renders N ( t ) = N e t/T , (7)where T = 1 / ( | G | (cid:37) ) is a characteristic time scale and N = πL (cid:37) is the initial number of particles inside the il-luminated area. Combining Eqs. (7) and (5), one arrives atthe prediction in Eq. (3) with A = πL .The distance R ( t ) of a particle to the center can be ob-tained by solving the equation of motion dR/dt = u r ( R, t )with the initial condition R ( t i ) = R i . The initial velocity isgiven by Eq. (4), and the correction thereof can be estimatedby Taylor–expanding the velocity V ( t ) = u r ( R ( t ) , t ) aroundthe initial time: V − V i V i ≈ V i dVdt (cid:12)(cid:12)(cid:12)(cid:12) t = t i ( t − t i ) = (cid:18) − V i TR i (cid:19) t − t i T . (8)With the value T ≈
11 s obtained from the fit in Fig. 4F, thisexpression predicts that the relative error of approximating V ≈ V i is at most ≈
10 % for the trajectories depicted inFig. 4E. More specifically, in the non-illuminated region the ma-terial derivative of the monolayer density vanishes, d(cid:37)/dt = ∂(cid:37)/∂t + (cid:126)u · ∇ (cid:37) = 0, according to Eq. (1) with ∇ · (cid:126)u = 0.ctivity-induced Marangoni dynamics 11 References
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