A decade ago, two-dimensional microscopic ﬂow visualization proved the the- oretically predicted existence of electroconvection roles as well as their deci- sive role in destabilizing the concentration polarization layer at ion-selective ﬂuid/membrane interfaces. Electroconvection induces chaotic ﬂow vortices injecting volume having bulk concentration into the ion-depleted di↵usion layer at the interface. Experimental quantiﬁcation of these important ﬂow patterns have so far only been carried out in 2D. Numerical direct simula- tions suggest 3D features, yet experimental proof is lacking. 3D simulations are also limited in covering extended spacial and temporal scales. This study presents a new comprehensive experimental method for the time-resolved recording of the 3D electroconvective velocity ﬁeld near a cation- exchange membrane. For the ﬁrst time, the spatio-temporal velocity ﬁeld can be visualized in 3D at multiples of the overlimiting current density. In con- trast to today’s simulations, these experiments cover length and time scales typical for actual electrodialytic membrane processes. We visualize coherent vortex structures and reveal the changes in the velocity ﬁeld and its statistics during the transition from vortex rolls to vortex ⇤Corresponding author: Email address: manuscripts.cvt@avt.rwth-aachen.de Preprint submitted to Journal of Membrane Science June 29, 2021 rings with increasing current density. The transition is characterized by changes in the rotation direction, mean square velocity, and temporal energy spectrum with only little inﬂuence on the spatial spectrum. These ﬁndings indicate a more signiﬁcant impact of EC’s structural change on the mean square velocities and temporal spectra than on the spatial spectra. This knowledge is a prerequisite for engineering ion-selective surfaces that will enable the operation of electrically driven processes beyond the di↵usion- limited Nernst regime. Keywords: Ion transport, Electroconvection, 3D PTV, Velocity ﬁeld, Electrodialysis 1. Introduction1 Ion-selective membranes are frequently used in industrial processes for 2 water puriﬁcation such as electrodialysis (ED) [1, 2] and ﬂow-electrode ca- 3 pacitive deionization (FCDI) [3, 4, 5], or in systems for energy storage like 4 redox-ﬂow battery systems [6, 7]. In these processes, an electric potential 5 between two electrodes acts as the driving force for ion transport through 6 membranes in contact with an electrolyte. For small currents, the current 7 density increases linearly with the applied potential. However, the transport 8 stagnates at a limiting current density i limwhen concentration polarization 9 (CP) leads to depletion of the Nernst di↵usion layer close to the membrane, 10 see Fig. 1 a) [8]. This limitation is visible as a plateau region in a current 11 density over potential plot in Fig. 1 b). 12 When increasing the current density beyond i lim,o v e r l i m i t i n gc u r r e n t s 13 emerge having their origin in a variety of superimposed phenomena, one of 14 them being electroconvection (EC). Electroconvection can be observed as 15 vortices, which convectively mix the boundary layer supplying the depleted 16 layer with the ion-rich bulk solution, see Fig. 1 a). The increased ion con- 17 centration at the membrane allows for an overlimiting current beyond the 18 di↵usion limit, see Fig. 1 b) [9]. 19 Until now, quantitative studies on the hydrodynamics of EC were lim- 20 ited to computationally demanding 2D and 3D direct numerical simulations 21 (DNSs) which unfortunately only o↵er limited length and time scales [10, 22 11, 12, 13, 14]: experimental quantiﬁcation of velocity proﬁles could only 23 be done using 2D experiments [15, 16, 17]. In fact, understanding and con- 24 trolling the hydrodynamics of EC is expected to facilitate mass transport 25 2 observed as overlimiting currents. Enabling higher current densities results 26 in lower membrane area required to achieve a desired desalination degree: 27 this results in a desired investment costs reduction. It is therefore obvious 28 that understanding the details of the overlimiting transport, in particular 29 EC, is a prerequisite for process optimization of an electrical-ﬁeld driven 30 membrane process. Also, understanding electroconvection’s intricate details 31 comprehensively is important to tailor membrane surfaces such that overlim- 32 iting can be actually utilized in practice. 33 2. Background 34 In a milestone paper of Rubinstein [18], the authors predicted that the 35 onset of electroconvection can be triggered at lower voltages if the surface 36 of the membrane would be ”wavy”. This prediction has spurred research in 37 understanding the phenomena itself, but also into the synthesis of new mem- 38 brane surface topologies to avoid any potential drop in the plateau region 39 prior to the overlimiting currents. With respect to experimental strategies to 40 initiate early electroconvection, membrane modiﬁcation methods developed 41 by the Kuban research community have proven to be e↵ective [19, 20, 21]. 42 As early as 2007, Balster reported that line undulations on the membrane 43 surface normal to the ﬂow direction, having distances in the range of approx- 44 imately 50-200 % of the boundary-layer thickness, lead to an earlier onset of 45 the overlimiting current [22]. Also 2D micropatterns of nanometer-thick lat- 46 eral polyelectrolyte patches induce electroconvection, i.e. macrosopic electro- 47 osmotic chaotic ﬂuid instabilities [23]. Also Roghmans et al. described that 48 inkjet printed microgel patterns induce electroconvection [24]. Yet the work 49 hypothesizes that the surface charge of the patterns may even determine 50 the direction of vortices rendering them either e↵ective or less e↵ective for 51 destabilizing the di↵usion boundary layer and reducing the length of the 52 plateau. Clearly, more experimental visualization methods are highly de- 53 sired to elucidate and quantify the spatio-temporal ﬂuid velocities during 54 electroconvection at the membrane/ﬂuid interface. 55 Historically, our 2D work on proving the existence of electroconvection 56 through visualization with a particle imaging technique [25] conﬁrmed the 57 predicted vortex-like features at the membrane/liquid interface with grow- 58 ing vortices at increasing current densities. Subsequently, De Valenca et al. 59 quantiﬁed the 2D velocity ﬁeld of EC in the cross-section of the CP layer 60 [15, 16]. They correlated the vortex size and ﬂuid ﬂow velocity with the cur- 61 3 Figure 1: Stages of electroconvection. a) Concentration proﬁles of anions and cations in front of the membrane for three di↵erent regimes (I: i<i lim,I I :i=ilim, and III: i>i lim). b) Sketch of the current versus voltage curve for an electrically driven membrane process. The three regimes in a) appear as the Ohmic, plateau, and overlimiting regions. c) Sketches of the di↵erent vortex structures developing in the overlimiting regime. First, EC appears as vortex rolls that order in a 2D fashion. With increasing i/ilim, the linear structure of vortex rolls breaks up into 3D vortex rings. rent density through the system. This approach was recently advanced by 62 Warren et al. [17] who correlated the e↵ect of the applied potentials to the 63 2D velocity ﬁeld of EC. However, their proposed electrochemical chip had 64 an aspect ratio non-existent in common membrane modules. Other studies 65 also analyzed EC by optically recording tracer substances in 2D but with- 66 out quantiﬁcation of velocities [25, 26, 27, 28, 29, 30, 31]. Recently, Kang 67 et al. visualized the structural transition of EC in the overlimiting current 68 region with and without ﬂow parallel to a membrane surface with confocal 69 microscopy in 3D [32]. This transition from 2D vortex rolls towards 3D vor- 70 tex rings, see Fig. 1 c), was predicted by Demekhin et al. in the ﬁrst 3D 71 DNSs of EC [11]. Although qualitatively accurate, such confocal microscopy 72 methods do not supply information on the 3D EC velocity ﬁeld and its energy 73 transfer. 74 The chaotic 3D velocity ﬁeld of EC is up to now only accessible via in- 75 tensive simulations simultaneously solving the Navier-Stokes, Nernst-Planck, 76 and Poisson-Boltzmann equations [9]. In such a simulation, Druzgalski et al. 77 analyzed and compared the statistics of chaotic EC form 2D to 3D results 78 in an attempt to derive energy spectra needed for the development of a 79 4 statistically averaged reduced-order model [10, 12]. Such a statistical aver- 80 aged model would enable low-cost simulations in the overlimiting regime for 81 process optimization [9]. Experimentally, a simple reduced-order parameter 82 approach has been suggested based on ﬁtting the overlimiting currents uti- 83 lizing a Schmidt number [33]. However, this approach has not been further 84 explored until today. Furthermore, Druzgalski et al. also found qualitative 85 and quantitative di↵erences in their 2D and 3D simulations, emphasizing 86 the need for an new methodology quantifying the 3D hydrodynamics. The 87 required ﬁne resolution in space and time limited their simulations to small 88 geometries and short time spans only. Therefore, the ability to extrapolate 89 their conclusions to the length and time scales of practically relevant devices 90 remains unclear until experimental results of the 3D velocity ﬁeld are avail- 91 able. 92 93 To overcome this lack of experimental data, we developed and report an 94 electrochemical cell allowing the optical recording of the 3D EC velocity ﬁeld 95 close to a cation-exchange membrane using micro particle tracking velocime- 96 try (µPTV). With this setup, one can now make the important step from 2D 97 to 3D velocity ﬁeld quantiﬁcation: at practically relevant length and time 98 scales over a range of overlimiting current densities. We use the velocity 99 data to analyze the structural change of EC from vortex rolls to vortex rings 100 in terms of vortex rotation direction. Additionally, we evaluate the veloc- 101 ity ﬁeld statistics and the ﬂuctuating velocity component. This proposed 102 methodology allows to unravel details of EC hydrodynamics experimentally 103 which in future can be compared to simulated velocity statistics when the 104 reported length and time-scales will be accessible through high performance 105 computing e↵orts. 106 3. Experimental Methods 107 3.1. Electrochemical cell 108 We designed the experimental cell shown in Fig. 2 which is suitable for 109 electrochemical experiments with simultaneous recording of the evolving 3D 110 velocity ﬁeld using microparticle tracking velocimetry ( µPTV). The cell is 111 based on the publication of Linz et al. [34] and consists of a Naﬁon N117 112 (Chemours, Wilmington, Delaware, USA) cation-exchange membrane sealed 113 between two housing parts. The transparency of the membrane allows for 114 imaging of buoyancy stable EC through the membrane [35, 36]. Two copper 115 5 Figure 2: Cell design for electrochemical experiments with simultaneous µPTV recording. The electrochemical cell consists of of two 3D printed housing parts which are sealed with O-rings against two copper electrodes and a Naﬁon N117 membrane. The cathode has a circular hole with a diameter of 9 mm to allow optical access. The hole is sealed with a microscope glass slide, which is not shown in the rendering, on top of the electrode. plates (25 mm ⇥75 mm ⇥0.5m m ) s e a l t h e t o p a n d t h e b o t t o m o f t h e c h i p 116 and act as electrodes in each half cell. The cathode has a circular hole (d = 117 9m m )a n di sg l u e dt oag l a s ss l i d et oe n a b l eo p t i c a la c c e s s . T h i ss e t u pf o r m s 118 electrolyte chambers above and below the membrane, which are ﬁlled with 119 1m M c o p p e r s u l f a t e ( C u S O4)a st h ee l e c t r o l y t e . T h eb o t t o mc h a m b e rh a s 120 ah e i g h to f0 .8m m , m a t c h i n g t h e m a x i m u m f o c a l d e p t h , a n d a d i a m e t e r o f 121 8m m , r e s u l t i n g i n a n a s p e c t r a t i o o f 1 0 . S u c h a l a r g e a s p e c t r a t i o i s d e s i r e d 122 to prevent the conﬁnement of the EC vortices [37, 38]. 123 Ac e l lc o m p r i s i n gt w oc o p p e re l e c t r o d e s ,ac a t i o n - e x c h a n g em e m b r a n e , 124 and CuSO4in aqueous solution as electrolyte is a simple and well studied 125 electrochemical system for evoking EC [15, 39]. In this system, CuSO4dis- 126 sociates to Cu2+and SO2 4and both electrodes favor copper deposition and 127 dissolution as faradaic reactions [40]. 128 During operation, copper is dissolved at the anode, transported through 129 the membrane, and deposited on the cathode. Additionally, this reaction sys- 130 tem limits gas evolution at the electrodes and the membrane to a minimum, 131 which ensures stable reaction conditions and clear optical access without gas 132 bubbles. 133 The copper ions moving towards the cathode result in an ion ﬂux i per 134 membrane area also describes as the current density. The limiting current 135 density of the process, described in Fig. 1, can be calculated by [16]: 136 6 ilim=cB ·F·z tMtE(1) According to de Valenca [16], the bulk concentration c Bis approximated 137 with a linear gradient between electrode and membrane as twice the elec- 138 trolyte starting concentration c 0in the anode chamber. F is Faraday’s con- 139 stant and z the ionic valence. The transport numbers in the membrane and 140 electrolyte t Mand t Edescribing the ion velocity di↵erence are 0.96 and 0.4, 141 respectively. For a 1 mM CuSO4solution, current densities larger than i lim 142 =0.73 A m2will result in the EC vortex formation close to the membrane 143 of the anode chamber [15]. 144 145 3.2. Micro particle image velocimetry 146 Figure 3: Schematic illustration of the setup for stereo micro particle image velocimetry. The setup for stereo micro particle image velocimetry consists of a laser that illuminates tracer particles in a sample volume through a stereo microscope which splits the laser beam in two separate beams. The ﬂuorescence response returns through the microscope and is recorded by two slightly angled high-speed cameras. Returning laser light is blocked by a high-pass ﬁlter. This technique allows the recording of tracer particles in a 3D volume. We use micro particle tracking velocimetry ( µPTV), which is an opti- 147 cal technique for measuring ﬂuid ﬂow in millimeter-scale geometries [41] to 148 record the velocity ﬁeld of EC during our experiments. When using this 149 7 method, tracer particles are illuminated by a high-frequency 532 nm Nd:YAG 150 laser (DM150, Photonics Industries International Inc.) and recorded by two 151 high-speed cameras (Phantom VEO 710L, Vision Research Inc.) through a 152 stereo microscope (SteREO Discovery.V20, Carl Zeiss Microscopy Deutsch- 153 land GmbH, Germany) containing a 1 ⇥objective (Plan-Aprochromat 1 ⇥, 154 Carl Zeiss Microscopy Deutschland GmbH, Germany), see Fig 3. The cam- 155 eras record the resulting particle images inside the microscope’s focal depth 156 from two di↵erent viewing angles. With our setup, the recorded volume 157 at a magniﬁcation of 5 .12⇥and a halfway closed aperture has a size of 158 4.9m m ⇥3.1m m ⇥0.8m mw i t h1 2 8 0p x ⇥800 px resolution in x- and y-direction. 159 The depth of the recorded volume of 0 .8m m m a t c h e s t h e d e p t h o f o u r e l e c - 160 trochemical cell’s anode chamber. Therefore, it is possible to reconstruct the 161 full velocity ﬁeld of EC between membrane and anode. 162 3.3. Combining electrochemical experiments and velocity recording 163 To conduct simultaneous electrochemical and µPTV experiments, the 164 electrolyte in the anode chamber is seeded with 0.001 wt% inert, ﬂuorescent 165 polystyrene microspheres (Thermo Scientiﬁc, Waltham, MA, USA). These 166 tracer particles have a diameter of 3 .2µma n daZ e t ap o t e n t i a lo f 14.9m V 167 measured with a Malvern Zetasizer Nano ZS (Malvern Panalytical Ltd), 168 which are comparable to the study of De Valenca et al. [15]. Di↵erent to 169 De Valenca et al., the microscope is focused below the membrane in the 170 middle of the anode chamber enabling to record the particle tracks over the 171 whole z-dimension of that chamber. The experiments are conducted without 172 forced ﬂow and electrolyte movement only results from EC. 173 174 The chronopotentiometric experiment are conducted at ﬁxed multiples of 175 the limiting current using a potentiostat (Interface 1010E, Gamry, Pennsyl- 176 vania, USA). For each potential, the time until CP is fully built-up and EC 177 starts to develop can be calculated with Sand’s equation [42]: 178 ⌧=⇡D 4·✓c0zF tMtE◆21 i2(2) Here, c 0is the electrolyte’s initial concentration in the anode chamber. 179 The di↵usion coecient of the electrolyte D = 0 .855⇥109m2s1is calcu- 180 lated from both individual di↵usion coecients DCu2+=0.714⇥109m2s1181 and DSO42-=1.065⇥109m2s1with the following equation [43]: 182 8 D=(z++z)D+D z+D++|z|D(3) The + and - signs indicate the values for cation and anion, respectively. 183 However, EC takes longer than the calculated transition time ⌧to reach a 184 steady-state [24]. Therefore, we analyze the ﬂow well after the transition 185 time in the steady sections of the chronopotentiometry to ensure evaluation 186 of the steady-state. 187 188 At 4 ·ilim,t h ef o r m a t i o no ft h eE Cv o r t i c e st a k e sl o n g e rt h a nt h ee l e c - 189 trostatic exclusion of tracer particles from the ESC, e↵ectively preventing 190 recording the EC vortex ﬁeld. In this experiment, the current density is 191 linearly decreased from 8.1 ·ilimto the desired current density at a rate of 192 4⇥103Am2s1. Afterward, a steady-state at 4 ·ilimcould be measured. 193 This approach ensures the presence of particles close to the membrane due 194 to previous EC mixing of the domain. 195 196 When a steady-state in the experiment was reached, the recording of the 197 particle images is started. The ﬂow is recorded for a maximum of 126 s at 198 a frequency of 20 Hz. The time-resolved recording of the velocity ﬁeld also 199 enabled the reconstruction of the velocity ﬁeld during the build-up of EC. 200 201 3.4. Velocity processing 202 The recorded particle images are analyzed with the tools available in 203 the software DaVis (version: 10.0.5.47779, LaVision GmbH, G¨ ottingen, Ger- 204 many). In a preprocessing step, static particle signals are removed by sub- 205 tracting the time-averaged intensity for each pixel. For the reconstruction of 206 the particle tracks, the Shake-the-Box algorithm is used [44, 45]. However, 207 experiments at current densities below 2 ·ilimcannot be processed. Here, 208 the particles move too slow and the velocities are too small to be detected. 209 A ﬁne-scale reconstruction is used for transforming the particle tracks 210 to velocities ( ux,uy,uz)o nar e g u l a rg r i d . T h eﬁ n e - s c a l er e c o n s t r u c t i o ni s 211 performed with the VIC# method implemented in the DaVis Software [46, 212 47].For our experiments, we chose a grid size of ten voxels, which results 213 in reasonable computing times. Therefore, the ﬁnal resolution of the recon- 214 structed velocity ﬁelds is 128 px ⇥80 px ⇥21 px with a voxel size of 38 .4µm. 215 9 To visualize the vortex structure of EC, DaVis o↵ers plotting coherent 216 vortex structures with the 2-method [48]. This method computes vortex 217 core lines by evaluating the eigenvalues of the squared and summed sym- 218 metric and antisymmetric parts of the gradient velocity tensor. Coherent 219 structures are visualized as isosurfaces at a speciﬁc eigenvalue 2. 220 221 After exporting the velocity ﬁeld data to MatLab (version: R2019b, The 222 MathWorks Inc.), the mean square velocities is calculated as a spatial and 223 temporal average in the steady-states of the experiments [10]: 224 u2 i=1 Nt·Lx·LyZ u2 i·dx·dy·dt (4) The equation includes the number of time steps N tand the mesh points 225 in x- and y-direction L xand L y. 226 The ﬂuctuating velocity component is calculated by subtracting the tem- 227 poral and spatial mean velocity from each velocity data point for evaluation 228 of the temporal and spatial energy spectra. To this data, we apply a Han- 229 ning window. The Hanning window or Hann function is a weighting function 230 which is used to reduce the input signal from the middle of the x,y-domain 231 towards its borders [49]: 232 wi(ni)=0 .5✓ 1cos✓ 2⇡ni Ni◆◆ (5) Here, n iand N iare the array of data points and the number of data 233 points in x- or y-direction, respectively. The 2D window is created with 234 wx,y(nx,ny)= wx(nx)·wy(ny). 235 For spectral analysis, a Fourier transformation of the ﬂuctuating velocity 236 components over time or space is performed. The velocities, that were 1D 237 Fourier transformed over time ˜ ui,a r et h e nc o n v e r t e dt ot e m p o r a ls p e c t r a l 238 energies and averaged in the x,y-planes with [10]: 239 E(z,!)=1 2·Nt·Nx·NyZ |˜ux|2+|˜uy|2+|˜uz|2dxdy (6) Here, Nt,Nx,a n d Nyare the number of data points in time, the x-direction, 240 and the y-direction. 241 For spatial spectral analysis, a 2D Fourier transformation of each ﬂuc- 242 tuating velocity component in each z-plane for each time step is performed. 243 10 The energy is normalized, and averaged over time: 244 E(z,k)= 1 2·Nt·d6 (2·⇡)3·Lx·Ly·LzZ |˜ux|2+|˜uy|2+|˜uz|2dt(7) The result is integrated over annuli with a radius of k=pk2 x+k2 yand a 245 thickness of one in the Fourier-space (see appendix Appendix A). Here, d 246 is the mesh spacing, and L x,Ly,a n dL zare the domain sizes in the x-, y-, 247 and z-directions, respectively. This procedure results in energy graphs over 248 the wavenumber k which is the spatial frequency in x,y-direction. During 249 the integration, the mesh is reﬁned by a factor of ten in both directions to 250 generate smoother graphs. 251 252 3.5. Comparability to simulations 253 To achive camparability to the 3D DNSs of Druzgalski et al. [12]. We 254 chose process parameters that best match their dimensionless numbers sum- 255 marized in Tab. 1. The relevant dimensionless parameters concerning this 256 comparison include: i) the applied voltage divided by the thermal volt- 257 ageV VT=Vz e kBT,i i )t h eS c h m i d tn u m b e r Sc=µ ⇢D,i i i )t h ee l e c t r o h y d r o d y - 258 namic coupling constant =✏ µD·(kBT ze)2, iv) the dimensionless Debye length 259 d=q ✏kBT 2cb(zeL z)2,a n dv )t h ea s p e c tr a t i o a=d Lz. Here, kB,T,z,a n d e 260 are the Boltzmann’s constant, temperature, ionic valence, and the electron 261 charge, respectively. Furthermore, the electrolyte deﬁnes the permittivity 262 ✏,d y n a m i cv i s c o s i t y µ,d i ↵ u s i o nc o e  c i e n tD ,a n db u l kc o n c e n t r a t i o nc b. 263 Lastly, dand L zare the active membrane diameter and the distance between 264 the membrane and electrode, respectively. 265 Out of these parameters, our main targets were a large channel aspect 266 ratio, a large dimensionless Debye length, and the possibility to conduct 267 experiments at large overlimiting currents, which is proportional to large ap- 268 plied voltages. Advantageous to simulation, our setup allows for dimensions 269 and a Debye length found in practically relevant applications. This di↵erence 270 in Debye length of about three orders of magnitude leads to a smaller ex- 271 tended space charge region in the experiments and, thus, velocity generation 272 closer to the membrane [9]. 273 11 Table 1: Comparison of dimensionless numbers of this study and the reference DNSs [12]. *The applied voltage in our experiments includes the voltage drop at the electrodes, in the membrane, and in the second electrolyte chamber. Dimensionless numberDNS [12]This study Applied voltage 120V VT64 - 362 V VT* Schmidt number 1 ⇥1031.17⇥ 103 Electrohydrodyna- mic coupling constant0.5 0.14 Dimensionless De- bye length1⇥1036⇥106 Aspect ratio 6.3 10 i/ilim 10 1.4 - 10.8 4. Results and Discussion 274 Using the electrochemical chip, the movement of tracer particles between 275 the membrane and anode can be followed. Particle tracking velocimetry re- 276 veals the 3D velocity ﬁeld as a consequence of electroconvection at increasing 277 current densities. 278 4.1. Evolution of the velocity ﬁeld with increasing current density 279 First, I-V experiments identify the di↵erent current regimes in a current 280 density over potential graph at multiples of the theoretical limiting current 281 density ranging from 0 ·ilimto 10.8 ·ilim, see Fig. 4 a). The plateau is clearly 282 visible between potentials of about 0 .4V t o 0 .64 V matching the calculated 283 theoretical limiting current density of 0 .73 A m2(Eq. 1). At larger poten- 284 tials, the system enters the overlimiting current region where EC vortices 285 appear. Graphs of the chronopotentiometric experiments are shown in the 286 appendix Appendix B. The data points shown are time-averaged values 287 of the ﬂuctuating signals having their origin in the dynamic nature of the 288 chaotic ﬂow conditions close to the membrane surface. 289 290 12 Figure 4: Development of electroconvection in the overlimiting regime. a)The graph shows mean voltages of chronopotentiometry experiments at increasing multiples of the limiting current. The dotted line is drawn for visual guidance. b) - e) Top views on isosurfaces of coherent vortex structures during experiments at b)4·ilim(at 90 s), c) 6.1·ilim(at 70 s), d)8.1·ilim(at 70 s), and e)10.8 ·ilim(at 81 s) in the steady-state. The velocity is colored in magnitude and direction according to the scale bar from blue to red. We record the velocity ﬁelds with our setup during experiments at di↵er- 291 ent magnitudes of the overlimiting current. Fig. 4 b)-e) show snap-shots of 292 the top view on iso-surfaces of coherent vortex structures at 4 ·ilim,6 . 1 ·ilim, 293 8.1·ilim,a n d1 0 . 8 ·ilim, respectively. With increasing current density, the 294 roll-like structure seen in Fig. 4 b) collapses and vortex rings of oval shape 295 emerge, see Fig. 4 c). With even higher current densities, the vortex rings 296 ﬁrst become more regular, see Fig. 4 d), before chaotic changing of patterns 297 emerges, see Fig. 4 e). While we show the quantiﬁcation of such 3D veloc- 298 ity ﬁeld for the ﬁrst time, this transition from vortex rolls to vortex rings 299 matches the qualitative observations of Demekhin et al. [11] and Kang et 300 al. [32]. 301 302 Additionally to the vortex structure, the measurement of velocity vectors 303 enables the evaluation of rotation directions. At 4 ·ilim,p a i r so fc o u n t e r - 304 rotating vortex rolls are the dominant structure. Fig. 4 b) shows that the 305 rolls of a pair connect in the vicinity of other vortex rolls. The velocity be- 306 tween two rolls of a pair is directed towards the membrane. Contrary, the 307 velocity in the centers of vortex rings (Fig. 4 c)-e)) is directed away from 308 13 the membrane. This change in rotation direction indicates that vortex roll 309 pairs do not simply split up in sections and recombine with themselves when 310 transitioning to rings. Two other mechanisms seem more likely: First, vortex 311 roll pairs could split up in sections and recombine but reverse their rotation 312 direction. This change in direction could be induced by the emergence of 313 as m a l l e rv o r t e xr i n gw i t har e v e r s e dr o t a t i o nd i r e c t i o ni nt h ec e n t e ro fa 314 recombined vortex roll. This smaller ring could increase in size and consume 315 the recombined vortex ring resulting in a ring with a reversed rotation di- 316 rection. Second, vortex roll pairs could split up at their contact points with 317 other pairs and recombine at the contact point. This observation needs fur- 318 ther investigation for a certain explanation. 319 320 4.1.1. Particle tracks of vortex rolls 321 Figure 5: Snapshot of particle tracks for 4 ·ilimat 90 s. Top view of the particle tracks in the reconstructed 3D volume and a cross-section (A-A) with a track length of 49 time steps (2.45 s). A-A shows a 2D cross-section of the 3D particle ﬁeld between the electrode and membrane. The velocity is colored in magnitude and direction according to the scale bar from blue to red. The axes are scaled by the channel height L z= 800 µm. Black domains indicate regions where no tracer particles are present due to space charge exclusion as proven by [38] . Fig. 5 shows the 3D reconstructed particle tracks at 4 ·ilimwhich is in 322 the region of vortex rolls. The vortex roll pairs, described in the previous 323 section, are visible in the top view as pairs of particle tracks with opposing 324 rotation directions. De Valenca et al. and Davidson et al. already reported 325 that the slight Zeta potential of the tracer particles leads to their exclusion 326 14 from locations with high anion concentration [15, 38]. These locations appear 327 as black areas which are void of particles. Another distinct observation is 328 particles moving on separate orbits along each vortex roll, forming hollow 329 cylindrical structures (Fig. 5). The cross-section A-A shows that the particles 330 are moving on exclusive trajectories. A possible explanation could be the 331 formation of secondary vortices between these separate circles comparable 332 to Taylor ﬂow [50]. These vortices could exclude the particles from the rest 333 of the roll. Another explanation could be the formation of regions of high 334 ion concentration inside the vortex rolls. Such regions would also exclude 335 tracer particles. To identify the validity of these hypotheses, quantitative 336 information on the concentration ﬁeld would be necessary. 337 4.1.2. Veloctity ﬁeld of chaotic vortex rings 338 Figure 6: Snapshot of particle tracks and the corresponding velocity correlation for 10.8 ·ilimat 81 s. a) Top view of the particle tracks in the reconstructed 3D volume with a track length of 1 s . A-A shows a 2D cross-section of the 3D particle ﬁeld between the electrode and membrane. b) Top view of the isosurface of coherent vortex structures in the reconstructed 3D volume showing a chaotic pattern of vortex rings at 81 s. c) Velocity maps of the cross-section B-B, C-C, and D-D. The white symbols mark the positions of the highest positive (plus) and negative (cross) velocities in each cross-section. The cross- sections A-A and D-D show velocity ﬁelds calculated from the particle tracks. Videos of the time-resolved particle tracks and vortex structure can be found as supplementary material. The velocity is colored in magnitude and direction according to the scale bar from blue to red. Fig. 6 shows a snapshot of the steady-state of EC in the regime of chaotic 339 vortex rings at 10.8 ·ilim.T h e t o p v i e w o f r e c o n s t r u c t e d p a r t i c l e t r a c k s 340 in Fig. 6 a)reveals the appearance of vortex rings with varying diameters. 341 In the cross-section A-A, the particle tracks show circular movement at the 342 15 borders of vortex rolls, for example, at x/L z=4 . 5 . T h e s ec i r c u l a rt r a c k s 343 strongly resemble structures recorded in the 2D experiments of Valenca et 344 al. [15]. 345 346 For further investigation, the particle tracks are converted to a 3D veloc- 347 ity ﬁeld. From this velocity ﬁeld, coherent vortex structures are extracted 348 and displayed as isosurfaces as shown in Fig. 6 b).T h e t o p v i e w o f t h e 349 vortex ﬁeld structure shows coherent vortex rings as reported by Demekhin 350 et al. [11]. The largest displayed velocities seem to appear in the centers of 351 small vortex rings and at contact points of multiple rings. To conﬁrm this 352 observation, cross-sections of the vortex ﬁeld through such structures are dis- 353 played in Fig. 6 c)B-B, C-C, and D-D. The analysis of the cross-sections 354 prove that the largest absolute velocities in the respective planes appear close 355 to the membrane and between coherent vortex rings. These velocities are up 356 to 4.8 times larger than the average absolute velocities in their respective 357 plane. This velocity distribution can be explained by continuity. The area 358 with velocities towards the electrode is proportionally larger than the area 359 with velocities towards the membrane (Fig. 6 b))w h i c hr e s u l t si nh i g h e r 360 velocities for the latter. 361 362 These local velocity hot spots might be connected to local current den- 363 sity hot spots at the membrane. At the positions of high velocities towards 364 the membrane, a larger quantity of high-concentrated electrolyte solution is 365 transported from the bulk to the cation-depleted membrane surface. The re- 366 sulting local jump in cation concentration will, in turn, lead to a local jump 367 in ionic current through the membrane. Thus, the numerically-predicted for- 368 mation of current density hot spots [12] could be connected to the velocity 369 hot-spot and veriﬁed by our experiments. 370 371 4.2. Velocity statistics of electroconvection 372 The spatio-temporal 3D velocity information allows to quantify the ve- 373 locity statistics with increasing current density. This enables to compare 374 the experimental statistics for chaotic vortex rolls to the velocity statistics 375 simulated by Druzgalski et al. from 3D DNSs [12]. 376 16 Figure 7: Comparison of mean square velocity, temporal- and spatial energy spectra be- tween 3D µPTV (upper row) and 3D DNSs (lower row) [12]. The graphs present a direct comparison of our 3D µPTV results at 10.8 ·ilim(1000 time steps, 74 .5 s to 124 .5 s) a) - c) to the 3D DNSs results of Druzgalski et al. d) - f) redrawn from [12] with permission from Physical Review Letters . a) and d) show the mean square of one in-plane and the out-of-plane velocity components over the dimensionless distance to the membrane z/L z with Lz=0.8mm. b) and e) show temporal energy spectra of the ﬂuctuating velocity components. c) and f) show spatial energy spectra of the ﬂuctuating velocity components. The spectra of 3D µPTV results b) and c) are plotted for dimensionless distance to the membrane of 0.05, 0.10, 0.25, and 0.80. The linear zoom of the spatial spectrum empha- sises the maxima at low wavenumbers. e) and f) are plotted for the normalized distances to the membrane of 0.037, 0.104, 0.250, and 0.820. 17 4.2.1. Comparison of velocity statistics to simulation 377 Fig. 7 shows the velocity statistics calculated from our experiment with 378 the statistics extracted from the simulations by Druzgalski et al. [12] for 3D 379 DNSs. The statistical analysis quantiﬁes (a) the mean square velocity and 380 (b) the energy distribution across frequencies. 381 The mean square velocity for the 3D µPTV measurements and 3D sim- 382 ulation [12] are plotted over the normalized distance to the membrane z/L z 383 in Fig. 7 a)and b).T h e m e a n s q u a r e s o f t h e i n - p l a n e a n d o u t - o f - p l a n e 384 velocity components show local maxima at about the same distances to the 385 membrane of z/L z=0 ,0 . 2 5 ,a n d0 . 7 5i nt h ee x p e r i m e n ta n dt h es i m u l a - 386 tion. A deviation is identiﬁed at the z/L z=0a n d1p o s i t i o n st h a tc a nb e 387 attributed to the ideal boundary conditions of the simulations compared to 388 the experiments. The maxima positions originate from the vortex generation 389 close to the membrane leading to the ﬁrst maximum. The second maximum 390 in the out-of-plane component marks half the vortex height with mostly ve- 391 locity towards or away from the membrane. The last local maximum in the 392 in-plane velocity is located at the average vortex height where the ﬂuid turns 393 its direction. However, the ratio between the amplitudes of both components 394 di↵ers between our measurements and the DNS. 395 An explanation for the di↵erences between our measurements and the 396 DNSs of Druzgalski et al. can be the discrepancy in the electric double layer 397 thickness by three orders of magnitude, see Tab. 1. The consequence of this 398 di↵erence is a decreased ESC region and, thus, the appearance of the maxi- 399 mum in-plane velocities closer to the membrane in the experiments [12]. The 400 velocities in z-direction are averaged in segments of 38 .4µm(⇠0.05 ·z/L z) 401 length. This averaging always results in values above zero in the planes 402 next to the membrane and electrode. Compared to DNSs, our experiments 403 also include electrodes, the membrane, and a second electrolyte chamber, 404 which inﬂuences the electrochemical cell’s resistance and chemistry. This 405 comparison proves that the velocity ﬁelds and its statistical mean features 406 originating in the EC can be successfully extracted at practically relevant 407 length and time scales. 408 The µPTV velocity ﬁeld data is used to calculate the temporal spectra 409 in planes parallel to the membrane shown in Fig. 7 b).T h e e n e r g y g r a p h s 410 show similar trends in energy levels compared to the numerical results of 411 Druzgalski et al. [12] in Fig 7 e). 412 Close to the membrane (z/L z=0 . 0 5a n d0 . 1 0 ) ,t h eg r a p h s ’s l o p ec o n - 413 18 tinuously decreases from small to large frequencies. At z/L z=0 . 2 5 ,as i m - 414 ilar energy level is reached in the beginning but shows a more prominent 415 decrease in slope afterwards. The temporal spectrum closest to the elec- 416 trode notably start at similar energy levels. Di↵erent to the DNSs refer- 417 ence, the energy follows a linear decrease that can be ﬁtted by a power-law 418 of E x+y+z, temporal ⇠!1.44between angular frequencies of !=2 . 8 ·102and 419 !=1 . 3 ·104.B e t w e e n f r e q u e n c i e s o f !=1 . 3 ·104and!=2 . 1 ·104,ﬂ u c t u a t i o n s 420 appear followed by a decrease in energy. All graphs show a change in slope 421 above an angular frequency of !=3 . 5 ·104. 422 The ﬂuctuations and changes in slope most probably originate from small- 423 scale noise in the experiments and the measurement error of PTV at small 424 scales [51]. 425 The spectra for planes close to the membrane indicate the dissipation of 426 kinetic energy across all frequencies without power-law dependencies, as also 427 reported by Druzgalski et al. for all x,y-planes (Fig. 7 e))[ 1 2 ] . 428 The power-law relation close to the electrode can be explained by the 429 forces that evoke EC. The body force that generates EC vortices is located 430 close to the membrane [9]. Therefore, vortices will propagate from the mem- 431 brane towards the electrode. The shape of the spectra also indicates the 432 generation of ﬂuctuations of all time scales at the membrane. The ﬂuctu- 433 ations with small frequencies, e.g. large vortices, could propagate towards 434 the electrode and dissipate their energy, whereas ﬂuctuations with larger fre- 435 quencies might not reach the regions close to the electrode. 436 437 The spatial energy spectra calculated from our experiment’s velocity ﬁelds 438 are displayed in Fig 7 c). All graphs follow the same trend at di↵erent energy 439 levels. They show a local maximum at a wavenumber of k ⇠4.40. The slope 440 of the graphs decreases at k = 22 with two local maxima above k = 44. 441 These local maxima at high frequencies in the spatial µPTV energy spectra 442 can also be linked to small-scale noise and the measurement error of µPTV 443 for small scales [51]. The spatial energy spectra reported by Druzgalski et 444 al. [12] are reproduced in Fig 7 f).C o m p a r a b l e t o o u r e x p e r i m e n t , a l l g r a p h s 445 show an increase in energy with a maximum at a wavenumber of k ⇠4 [12]. 446 The peak at a wavenumber k ⇠4.40 in the experiment (Fig. 7 c))c o r - 447 responds to a length of 1.43 times the electrode-to-membrane distance (L z) 448 carrying the largest energy. This length of 1.43 ·Lzmatches well with liter- 449 ature and the diameter of the vortex rings in Fig. 6 b). 450 Due to the time-resolved recording of the velocity ﬁeld, the evolution of 451 19 the EC statistics over the whole experimental time are additionally analyzed 452 in the appendix Appendix C. 453 454 4.3. Development of velocity statistics with increasing current density 455 Figure 8: Development of the mean square velocity, temporal and spatial energy spectra in the overlimiting regime for 4 ·ilim, 6.1 ·ilim, 8.1 ·ilim, and 10.8 ·ilim. The graphs represent an average over 1000 time steps (74 .5 s to 124 .5s ) . T h e ﬁ r s t r o w s h o w s t h e development of the mean square of one in-plane and the out-of-plane velocity components over the plane-to-membrane distance from 3D experiments. The second and third row show the development of the temporal and spatial energy spectra, respectively. Note the changing axes for the graphs showing the mean square velocities and the linear zoom on the maxima of the spatial energy spectra. Fig. 8 shows the mean square velocity, temporal and spatial energy spec- 456 tra for the experiments at 4 ·ilim,6 . 1 ·ilim,8 . 1 ·ilim,a n d1 0 . 8 ·ilim. All 457 graphs have shapes typical for EC as also seen in Fig. 7. 458 459 20 The overall mean square velocities increase with an increasing current 460 density, as expected due to the higher energy input. The biggest di↵erence 461 appears between 8.1 ·ilimand 10.8 ·ilim. Here, the sharp maximum in the 462 out-of-plane velocity and the local maximum in the in-plane velocity seen 463 at 8.1 ·ilimbroaden at 10.8 ·ilim. Additionally, the local maximum in the 464 in-plane velocity shifts it’s position from ⇠0.5 L zto⇠0.75 L z.T h i s b e h a v i o r 465 can be explained by the change from the distinct vortex ring pattern into a 466 more chaotic pattern, described in Fig. 4, with vortices that span the whole 467 distance between membrane and electrode as seen in Fig. 6. 468 469 The energy spectra in Fig. 8 also reﬂect the overall higher velocities with 470 increased energies for temporal and spatial ﬂuctuations with increasing cur- 471 rent densities. Again, the quality of the results at high frequencies in the 472 temporal as well as the spatial energy spectra su↵er from small-scale noise 473 and the measurement error of PTV at small scales [51]. 474 All temporal energy spectra in Fig. 8 can be partly ﬁtted by a power- 475 law relation which was also seen in Fig. 7 c).T h e s l o p e o f t h e ﬁ t t i n g ﬁ r s t 476 increases from !0.68at 4 ·ilimto!0.36at 8.1 ·ilimand then decreases to 477 !1.39at 10.8 ·ilim. 478 The graphs at 4 ·ilimadditionally show a region of constant energy be- 479 tween !=1 ⇥103and!=1 ⇥104which is less visible at higher current 480 densities. The reason for this steady section could be the absence of informa- 481 tion on velocities between the vortex rolls, as seen in Fig. 5. The velocities 482 in these regions potentially ﬂuctuate in small length scales and at high fre- 483 quencies. 484 The di↵erence in the temporal energy graphs between 4 ·ilimand 6.1 ·ilim 485 coincides with the structural transition from vortex rolls to vortex rings al- 486 ready seen in Fig. 4. Moreover, the changes between the graphs at 8.1 ·ilim 487 and 10.8 ·ilimmatch the transition from regular vortex rings to chaotic vor- 488 tex rings shown in Fig. 4. 489 490 In contrast, the shapes of the spatial spectra do not change signiﬁcantly 491 between current densities. However, the overall energy increases and the 492 maxima emphasised in the linearly scaled inserts change positions from 5.02 k 493 and 4.48 k to 8.95 k and ﬁnally to 4.40 k. These maxima positions reﬂect the 494 changes in structure that were seen in Fig. 4. The correlated diameter of the 495 vortex structures ﬁrst reduces from 6.1 ·ilimto 8.1 ·ilimand then increases 496 again at 10.8 ·ilim.T h e l i n e a r l y s c a l e d i n s e r t a t 1 0 . 8 ·ilimshows a broader 497 21 energy distribution over multiple wavenumbers compared to the graphs at 498 lower current densities. The absence of a sharp maximum indicates that 499 no single dominant length scale exists but a composition of di↵erent length 500 scales. This composition is caused by the chaotic change in vortex structure 501 at this large overlimiting current density with the ﬂuctuating size of vortex 502 rings. 503 Overall, our experiments conﬁrm that large vortices are the dominant 504 actors during EC [12]. The results additionally indicate that an increasing 505 current density seems to have a more considerable impact on the mean square 506 velocities and temporal spectra than on the spatial spectra. 507 508 5. Conclusion 509 We established the ﬁrst experimental setup for the time-resolved record- 510 ing of the 3D velocity ﬁeld during electroconvection (EC) close to a cation- 511 exchange membrane. With this setup, Whe EC velocity ﬁeld was recorded 512 in the steady-state at multiples of the overlimiting current density covering 513 length and time scales found in industrial applications inaccessible by current 514 3D direct numerical simulations (DNSs). 515 We visualized coherent vortex structures and revealed the change in vor- 516 tex structure from vortex rolls to vortex rings along the iV-graph. This 517 change in structure is coherent with the ﬁndings reported by Demekhin et 518 al. and Kang et al. [11, 32]. Additionally, our method enables evaluating the 519 rotation direction of vortex structures. We found reversed rotation directions 520 between vortex roll pairs and vortex rings, raising the question of the ﬂuid 521 mechanics during the transition from rolls to rings. 522 The particle tracks in the regime of vortex rings showed the movement of 523 particles on separate orbits. However, possible explanations for this behavior 524 need further physical evidence. 525 The velocity measurement in the regime of chaotic vortex rings revealed 526 local velocity hot spots between the vortex rings directed towards the mem- 527 brane. These possibly enable the transport of the cation-rich solution at 528 the electrode towards the membrane. This phenomenon could explain the 529 predicted current density hot spots by 3D DNSs [12]. 530 Further statistical analysis of the velocity ﬁeld disclosed the nature of 531 energy transfer with a partial power-law behavior in the temporal spectrum. 532 The spatial energy spectrum showed that the length scale carrying the largest 533 22 energy is 1.43 ·Lz.T h e s e ﬁ n d i n g s s h o w e d g o o d a g r e e m e n t o f g e n e r a l t r e n d s 534 with the reported statistics of Druzgalski et al. for 3D DNSs [12]. Di↵erences 535 were found especially for high frequencies. 536 Lastly, the evolution of the vortex statistics for increasing limiting cur- 537 rent densities was analyzed. The results reveal changes in the mean square 538 velocities and the temporal energy spectra during the transition of EC from 539 vortex rolls over vortex rings to chaotic behavior. The spatial energy spectra 540 showed similar trends for all analyzed current densities. These ﬁndings in- 541 dicated a more signiﬁcant impact of EC’s structural change on the temporal 542 spectra than the spatial spectra. 543 All in all, the presented results provide a ﬁrst insight of the fundamental 544 physics of EC at practically relevant length and time scales. The experi- 545 mental technique also builds a foundation to solve the remaining challenges 546 in future detailed 3D studies. Especially the combination of experimental 547 studies, that cover larger scales, and simulations, that more precisely resolve 548 small scales, could lead to signiﬁcant progress. Furthermore, the methodol- 549 ogy permits studying the impact of membrane surface modiﬁcations on the 550 hydrodynamics of EC. These modiﬁcations aim to manipulate the 3D vor- 551 tex build-up and reduce the limiting current plateau with increasing suc- 552 cess [52, 38, 53, 54] Additionally, knowledge of the velocity statistics enables 553 the development of a RANS-like statistically averaged reduced-order model. 554 Such a model will allow for feasibility analysis and optimization of electrically 555 driven membrane processes in the overlimiting regime at a low computational 556 cost. Full comprehension of EC will push the feasibility of desalination pro- 557 cesses beyond the limiting current. 558 559 Appendix A. Integration and reﬁnement 560 Fig. A.9 displays a graphical representation of the integration and reﬁne- 561 ment scheme for the spatial spectra also described by Druzgalski et al. [12]. 562 On the left side (Fig. A.9 a)), the two yellow circles with a di↵erence in 563 radius of one form the annuls for integration. The wavenumber k increases 564 from the center of the k-space in steps of one. The grey highlighted data 565 points show the selection for the non-reﬁned integration. In Fig. A.9 b), 566 ai n t e g r a t i o ns c h e m ew i t har e ﬁ n e m e n tf a c t o ro ft e ni nb o t hd i r e c t i o n si s 567 displayed. The reﬁned integration results in a smoother graph especially for 568 small wavenumbers by distributing the information in one cell between two 569 23 Figure A.9: Graphical representation of the spatial spectra integration and reﬁnement scheme. a)Example of the integration for k = 1. b)Example of the reﬁned integration for k = 1. The detail on the right shows the selected fraction of the data point in one original pixel adjacent annuli. The detail on the right shows the selected fraction of the 570 data point. 571 Appendix B. Chronopotentiometry Graphs 572 Figure B.10: Chronopotentiometry graphs of the experiments at a) 4 ·ilim, b) 6.1 ·ilim, c) 8.1 ·ilim, and d) 10.8 ·ilim. The red area displays the recorded sections. Fig. B.10 shows the chronopotentiometry graphs during the experiments 573 at 4 ·ilim,6 . 1 ·ilim,8 . 1 ·ilim,a n d1 0 . 8 ·ilim. The graph in Fig. B.10 a) 574 directly starts at the steady state potential due to the linearly decreased 575 24 current density prior to the experiment as described in the main document’s 576 method section. 577 At current densities of 6.1 ·ilimand 8.1 ·ilim, the graphs (Fig. B.10 b) 578 and d)) show a similar trend. After the calculated transition times of 3 .94 s 579 and 2 .24 s, the systems resistance increases until EC reaches a steady state. 580 In the experiment at the highest current density of 10.8 ·ilim(Fig. B.10 d)), 581 the potential increases over the course of the measurement. This behavior 582 indicates increasing resistance of the system due to chemical changes or gas 583 bubble formation. Both phenomena could originate from possible water dis- 584 sociation at the membrane and the electrodes at this large potentials. The 585 resulting OHions possibly form solid CuO or Cu(OH)2scaling at or in- 586 side the membrane [55]. In our experiments, we did not observe any visual 587 changes of neither the membrane nor the electrodes. Excess H+and OH 588 would form H2and O2gas bubbles at the electrodes. However, gas bubble 589 formation was only observed for experimental times larger than 500 s. 590 Appendix C. Temporal Development of Chaotic Electroconvec- 591 tion 592 Our set up also allows to analyze the development of the EC velocities, 593 vortex structure, and statistics over time. Fig. C.11 and C.12 show this 594 development for the case of 10.8 ·ilimin steps of 20 seconds for the ﬁrst 120 s 595 of the experiment. 596 The development of the coherent vortex structures is depicted in Fig. C.11. 597 The ﬁrst coherent structures already build up at 20 s of the experiment ap- 598 pearing as mostly regular vortex rings. The structures grow in diameter over 599 the course of 80 s while loosing their regular size and shape. Up to this stage, 600 the velocity in most centers of coherent structures is directed towards the 601 electrode. The velocity ﬁeld becomes more chaotic between 100 s and 120 s. 602 In this stage, coherent structures appear in a wide range of shapes and sizes 603 with velocities directed both ways. 604 Fig. C.12 shows the change in one in-plane and the out-of-plane mean 605 square velocities. The distinct velocity proﬁle of EC is not completely build 606 up in the ﬁrst 20 s. Similar to later stages of EC, the maxima of both velocity 607 components are close to the membrane. However, the out-of-plane velocity 608 is higher than the in-plane component. Both velocity maxima ﬁrst increase 609 between 20 s to 40 s and decrease to a steady state value thereafter. At this 610 point, the proﬁles are fully developed. 611 25 Figure C.11: Temporal development of the EC vortex structure for 10.8 ·ilim. The top views of the structure of coherent vortices is depicted at a)20 s, b)40 s, c)60 s, d)80 s, e)100 s, and f)120 s of the experiment. 26 Figure C.12: Temporal development of the mean square velocity, temporal and spatial energy spectra for 10.8 ·ilim. The ﬁrst row shows the development of the mean square of one in-plane and the out-of-plane velocity components over the plane-to-membrane distance from 3D experiments between 0 s and 120 s. The second and third row show the development of the temporal and spatial energy spectra between 0 s and 120 s, respectively. Note the changing axes for the graphs showing the linear zoom on the maxima of the spatial energy spectra. 27 The above perceptions reﬂect in the development of the temporal and 612 spatial spectra (Fig. C.12). As explained in the previous section, the local 613 maxima at high frequencies in both the temporal as well as the spatial energy 614 spectra and the following change in slope originate from small-scale noise and 615 the measurement error of PTV for small scales [51]. Only slight changes are 616 visible in the overall shapes of the spectra for both temporal and spatial 617 analysis. However, some distinct observations can be made. First, the slope 618 of the temporal spectrum for the plane closest to the electrode (0 .8·Lz) 619 changes between 0 s and 40 s from !0.79to!1.35indicating that the energy 620 reaching the region closer to the electrode dissipates faster when EC is fulls 621 developed. Second, the energy levels in both spectra increase from 0 s to 20 s 622 and decrease again after 40 s as already seen in the mean square velocities. 623 Third, the position of the maxima of the 2D spatial spectra change to smaller 624 wavenumbers for all planes over time. The maxima positions shift from 625 11.6 kto 5.8 kbetween 0s and 100s as can be seen in the linear zoom of each 626 spectrum. The shift of the maxima correlates to increasing contributions to 627 the energy of vortex structures with larger length scales in the respective 628 planes. The structures with the highest energy share grow from 0 .61·Lzto 629 1.43·Lzover the course of 80 s. The negligible changes in all graphs conﬁrm 630 that the experiment was in a statistical steady state after about 40 s. 631 Acknowledgments 632 This work received funding from the European Research Council (ERC) 633 under the European Union’s Horizon 2020 research and innovation program 634 (grant agreement no. 694946). The DFG also supported this work through 635 the project SFB 985 Functional Microgels and Microgel Systems in project 636 B6. The measurements were conducted with a high-speed stereomicroscope 637 PIV funded by the Major Research Instrumentation Program according to 638 Art. 91b GG in the Research Building NW1481006 “NGP2 – Center for 639 Next Generation Processes and Products”. M.W. acknowledges DFG fund- 640 ing through the Gottfried Wilhelm Leibniz Award 2019 (grant ID = WE 641 4678/12-1) and the support from the Alexander-von-Humboldt foundation. 642 The authors acknowledge F. Roghmans, G. Linz, A. Kalde, J. L¨ olsberg, A. 643 Cinar, K. Baitalow, and D. Wypysek for their input and the fruitful discus- 644 sions. 645 28 Author contributions 646 Felix Stockmeier: Conceptualization, Methodology, Formal analysis, In- 647 vestigation, Data Curation, Writing - Original Draft, Visualization, Project 648 administration. Michael Schatz: Formal analysis, Investigation, Visualiza- 649 tion. Malte Habermann: Formal analysis, Investigation, Visualization. John 650 Linkhorst: Writing - Review & Editing, Project administration, Supervi- 651 sion. Ali Mani: Interpretation of the data; Writing - Review & Editing . 652 Matthias Wessling: Conceptualization, Resources, Project administration, 653 Supervision, Writing - Review & Editing, Funding acquisition. 654 Competing interests 655 The authors declare that there are no competing interests. 656