Spatiotemporally controlled emergence of nanoparticle microvortices under electric field

Controlled assembly of nanoparticles (NPs) has garnered much interest over the past two decades. Beyond established techniques, new methods utilizing local short‐range or large‐scale long‐range interactions remain to be explored to achieve diverse micro‐ and nanoscale structures. Here, we report the controlled emergence of vortex‐pair arrays within monodispersed gold nanorods by applying a direct current electric field across a pair of sawtooth electrodes. By employing in situ darkfield microscopy and particle collective analysis, we elucidate the mechanism behind the formation and stabilization of the NP vortices, attributing it to the combined effects of the electrode shape, high NP density, and high solution viscosity. We further explore the controllability of the vortex‐pair arrays and obtain multiple complex vortice patterns. Our findings will facilitate the investigation of efficient and controlled dynamic assembly of NPs under external fields and help manufacture next‐generation optoelectronic functional materials.

the type, effective length, and strength of interparticle interactions.This assembly pathway harnesses transient flows of matter and energy to shape the NPs assemblies dynamically.[22][23][24] Furthermore, vortices hold considerable promise in various real-world applications, such as enhancing mixing in microfluidic devices, manipulating particles, and improving heat transfer.[27][28][29][30] The general paradigm can be summarized as follows.At low particle density, physical stimulation causes faster particle motion with random direction and indiscernible patterns.However, as particle density increases, dynamic assemblies such as homogeneous polar liquid or nonequilibrium steady-state patterns emerge.Further increasing the amplitude or frequency of the external field or ionic concentration of the solution, a single giant vortex or multiple isolated vortices of different sizes spanning the entire device can be obtained.This picture is the outcome of theories, simulations, and experiments mostly performed in devices with straight and smooth edges, which were apparently manufactured using precision tools and methods.The more complex devices with secondary topological microstructure, where the complex interplay of topological constraints and interactions can direct both structural and kinetic variation of the vortices, remain to be explored.
Here, we investigate the controlled emergence of NP vortex-pair arrays in monodisperse gold nanorods (AuNRs) by applying a direct current (dc) electric field to a pair of sawtooth electrodes.Through dark-field microscopy and single particle tracking (SPT) analysis, we capture the entire formation process of the vortex-pair array, which was mainly orchestrated by the sharp teeth-mediated large-scale electrohydrodynamic (EHD) field and the rectification effect of the high-concentration NPs.Multiple factors, including the field amplitude, dispersant viscosity, particle density, NP surface properties, and electrode morphology of the NPs could regulate the formation pathway and fine structure of dynamic vortex-pair assays.By merely adjusting the electric field amplitude within an appropriate viscosity range, we observe four collective states of the NP assembly, including isotropic gas-like, cylindrical uniaxial vortex, vortex-pair array, and chaotic flow.By further exploiting all regulation factors, we obtain diverse complex vortex microstructures, such as doughnut-like and lung-lobe-like vortex-pair arrays, rectangular arrays, and fountain-like complexes.Dynamic living microstructures, such as collision-swallowing-growing vortex and the compressing-spring-like vortex sequence, have also been observed.These controllable and active vortex patterns greatly enrich the high-order structure families obtained from the external field-driven NP assembly.Our studies provide a better understanding of the collective behaviors of colloids in non-equilibrium states, elucidating the interplay between large-scale field regulation and local interactions, and leading to the development of field-driven assembly strategies for complex nanoparticle and microparticle functional structures.

Electric control platform and characterization methods
Figure 1A shows the schematic diagram of our electrical device and the imaging setup.Rather than precision microfabrication, the original electrode pairs were obtained by dividing an entire indium tin oxide (ITO) glass slide into separated conductive areas with a high-power laser direct-writing machine.By burning round holes on the ∼185 nm thick ITO layer sequentially while translating the glass slide from one side to the other in a straight line in step size larger than the laser beam diameter, a pair of electrodes with wave-like or sawtooth edges were obtained.Unless mentioned otherwise, the channel between the two electrodes is ∼300 μm wide; the height, width, and period of each "tooth" are ∼200 nm, ∼12 μm, and ∼75 μm, respectively.To produce the vortex pair patterns, 5 μL AuNR water/glycerol solution containing ∼ 4.2 × 10 7 NPs, is deposited between an ITO glass side and a cover glass, resulting in a solution layer of ∼200 μm thick.
The dc electric signal is provided using a waveform generator.Using a darkfield microscope, we could in situ monitor the dynamic assembly of AuNRs energized by the electric field.
In a typical case, a 3D hollow oval-like vortex array (Figure 1B and Movie S1) would emerge in ∼10 s after applying 3 V dc to the AuNR solution containing 80% v/v glycerol/H 2 O, in which the spatial positions (in the x-y plane) of the vortices in the array are largely matched with that of the sharp teeth one by one.The lengths of the long and short axes of the vortices are ∼120 μm and ∼40 μm, respectively.The 3D NP vortices with a height of ∼30 μm are floating at ∼65 μm above the electrodes.To analyze the trajectories of individual NPs or NP aggregates in the vortex array, the particle collective analysis (PECAN) method is used. [31]In PECAN, a series of smaller sub-trajectories are extracted from the longer mother trajectories using a moving window operation with a specific temporal window.These sub-trajectories intrinsically are the fragments of the mother trajectories but contain the ordered temporal information with a defined time interval.To further obtain the specific spatiotemporal distribution, these sub-trajectories are assigned to local regions divided by spatial grids.Subsequently, critical kinetic parameters, including the normalized net displacement along the y-axis d N y i , translational diffusion coefficient D T and exponential factor α of each particle, are computed for each sub-trajectory (see more details in "Kinetic parameters calculation" of Experimental Section/Methods).To facilitate data readability, a pseudo-color mapping technique is employed, which visually represents the spatiotemporal variations of these parameters across the spatial grid based on their relative values.
Using PECAN, more details on the hollow oval-like vortex array were revealed (Figure 1B and S1).According to the distribution of d N y i (Figure S1A,B), the two adjacent vortices have opposite chirality, and their rotational directions are almost mirror symmetrical.The movement directions of the vortices in the array are alternately arranged in a "clockwisecounterclockwise" manner, therefore, two adjacent vortices can be regarded as a "vortex pair".The information revealed by the heat map of the D T is also interesting (Figure S1C,D).The gap with a fast diffusional rate (red region) seems to be the "metabolism" zone where new particles are received and old particles are released.The gap with the moderate diffusional rate (bluish-yellow region) stabilizes the vortex structure, and so does the blue region with the extremely low diffusional rate in each vortex.There may be a local laminar flow structure that provides an ideal buffer space for maintaining stability.Taken together, this 3D dynamic vortex-pair array not only has a uniform vortex size but also has a regular distribution of diffusional rate and movement direction.

Emergence dynamics of the vortex-pair array
To explore the emergence processes of the vortex-pair array (Movie S2), we tracked the trajectories of individual AuNRs or small AuNR aggregates and calculated the spatiotemporal distribution characteristics of d N y , scattering intensity, and D T (Figure 2).According to the time-dependent variation of the three parameters, the entire formation processes of the vortex-pair array can be divided into four stages (S1-S4).Initially, all AuNRs exhibit stochastic gas-like motion (S1) within a small region with D T of ∼0.29 μm 2 /s and scattering intensity of ∼4.2 × 10 5 (arbitrary unit).After applying the dc signal, the AuNRs near the insulating region move directionally to one electrode with an almost constant D T and a sharply increased d N y (S2 and Figure 2D(i)).The scattering intensity is gradually increasing, indicating the enrichment of the AuNRs near the electrode.After the relaxation time of ∼13 s, the particle intensity reaches a new plateau of ∼7.0 × 10 5 where the particle density is ∼1.7 times the initial value in S1 and the d N y decreases to 0. A small vortex-pair array emerges (S3 and Figure 2D(ii)) and keeps growing with a constantly increasing D T , but the total intensity does not change much.Finally, a hollow oval-like vortex-pair array with almost invariable size forms (S4).Notice that the vortex-pair array arises from the increase of particle density, indicating the importance of the particle density in the formation of vortices.We speculated that the non-covalent forces between NPs play a crucial role in the formation of vortices.Therefore, the NPs themselves are not just passive tracers.

Formation mechanism of the vortex-pair array
For theoretical studies on vortex formation, [32,33] researchers have designed a series of ratcheted microchannels, in which the presence of sharp wedges or sawtooth edges could induce inhomogeneity in the fluids, leading to a temperature gradient (electrothermal effect) or an electric field gradient (EHD effect).These ratcheted microchannels provide a simple and effective way to manipulate the motion of the suspended particles.Since the sawtooth electrode pair used here appears like the ratcheted microchannels, we speculated initially that the emergence of the vortex-pair array is an electrothermal hydrodynamic phenomenon.The electric current flows through the confined liquid around a sharp tooth and induces a strong Joule heating flow (due to a high electric field near the sharp tooth), which entrains the particles and promotes the formation of microvortices.However, this hypothesis is ruled out since the electric conductivity of the mixture of glycerol and H 2 O is very low, and the current detected in the circuit was just ∼0.1 uA.Additionally, using an infrared thermal imaging camera to monitor the temperature variation, no temperature fluctuation was observed (Figure S2).
Another explanation is electric field gradient-induced EHD flow.Previous studies showed that when a tangential electrokinetic flow passed through a wedge with a finite dielectric permittivity (compared to that of the electrolyte), a long-range vortex flow would be generated due to the wedge-induced normal leakage field. [34]Although the initial flow direction of the fluids in our system faces the sharp teeth, the microstructure is similar to the wedge, therefore, we conjecture that there is an EHD effect induced by the sharp tooth-mediated uneven charge distribution.An extremely high charge density forms at the tip of the sharp tooth (Figure 3A), but the charge density of the region a few micrometers away from the tip sharply drops to a very small value.The inhomogeneous charge distribution leads to a nonlinear electric field gradient, causes a convergent flow around the sharp tooth, and induces the formation of vortices.To verify this hypothesis, we used the suspension with extremely low AuNR concentration (0.175 pM), in which the interaction between NRs is almost negligible and a single NR can be regarded as a tracer.No vortex can be discerned with the naked eye after applying the dc signal.However, with SPT, the cumulative trajectories of single AuNRs over time depict some blurred and irregular vortice profiles (Figure 3B).Based on the sharp teeth mediated nonlinear EHD effect, we carried out a simplified COMSOL simulation with the premises of pure fluids and 2D electrode pairs.With high voltage applied, we could obtain the 3D vortex-pair array generated near the electrodes (Figure S3).Regardless of the oversimplification of the topography of the electrodes and the fluid properties, the simulation result is largely consistent with the experiments.
Besides vortex flow induced by the sharp teeth, the emergence and maturation or stabilization of the NP microvortices require a certain level of local particle density, which inevitably leads to material exchange with the surrounding environment.Close examination indicates that after applying the voltage, individual AuNRs with initial positions far away from the vortices could be drawn into the vortices due to the fluid shear-driven migration.With time increases, more and more AuNRs at a distance are captured into the vortices, while some AuNRs inside the vortices could also be ejected out, resulting in both high-particle-density regions inside the vortices and adjacent depletion zones in the surroundings.Since the influence of the interparticle interactions (e.g., van der Waals forces, hydrogen bonds, and electrostatic interaction) as well as the particle-field interactions (e.g., dielectrophoresis and shear stress), cannot be neglected, the aggregation state of the particles would change dynamically, and the local fluid field reshaped gradually.According to theoretical studies by others, [35] the increase in particle density escalates the shear-induced cross-streamline migration of particles, where a is the particle size, γ is the shear rate within the vortex, R is the radius of the vortex and Φ is the particle volume fraction.It also brings the high-concentration-enhanced field effect, [36] E = E 0 (1 where α is an unknown screening constant, which is related to the particle-field interaction and the interparticle interactions. [37]As particle density rises, the hydrodynamic shear force acting on NPs also increases.Additionally, when NPs concentrate at sharp tips on an electrode, they can enhance the local electric field by concentrating and redistributing charges in the surrounding solution.This amplifies the nonlinear electric potential difference at the sharp corners, which can overcome the stochastic force on the NPs in highviscosity solution and drive them into ordered vortex-pair arrays. To further examine the influence of the particle density, we performed a series of experiments with AuNR concentrations of 14, 7, 3.5, 1.4, 0.35, and 0.175 pM in 60% glycerol solution under an applied voltage of 3 V.At the highest concentration of 14 pM, we directly observed an ordered hollow oval-like vortex-pair array with the naked eye (Figure 4A).At 7, 3.5, and 1.4 pM, we observed the solid lung-lobe-like vortexpair array that gradually became smaller or shattered.At even lower concentrations, we discerned no vortex (Figure 4).The results indicate that the concentration of particles or the amount of their total supply is crucial to the size and morphology of the vortices formed.Therefore, the emergence of vortex pairs could be attributed to the combined effects of the sharp teeth-mediated large-scale EHD field and the high NP density-induced local reshaping effect, where the high viscosity of the solution plays a stabilization role by restriction the stochasticity of the NP motion.This mechanism lays the basis for further spatiotemporal control of the vortices by adjusting the geometric parameters of the electrodes or regulating the interparticle interactions and particle-fluid interactions.

Spatiotemporal regulation of the vortex-pair array
With an initial understanding of the mechanism, we investigated the factors that could influence the formation of the vortex-pair array, including the voltage amplitude, the viscosity, the surface properties of particles, and the electrode morphology.
Firstly, we carried out a series of experiments under different voltages ranging from 0 to 12 V and different viscosities including 0% (0.8 cP), 40% (3.4 cP), 60% (10 cP), 80% (48 cP), 90% (147 cP), 100% (612 cP) volume fractions of glycerol.As expected, the voltage amplitude is a decisive inducing factor for the emergence of vortices, and the viscosity is an important stabilizing factor.Taking the 7 pM AuNRs dispersed in 90% glycerol as an example, it exhibits four dynamic states as the applied voltage increases, including gas-like motion, single vortex-shaped motion, vortex-pairarray-shaped motion, and chaotic motion (Figure 5).When the voltage is lower than the critical value (U dc < 1.9 V), the AuNRs present gas-like stochastic motion (Figure 5A(i) and Movie S3).The ensemble average D T is ∼0.03 μm 2 /s, which is the smallest one of the four states (Figure 5B), and the ensemble average dy is almost 0 (Figure 5C).Once the voltage reaches the critical value (1.9 V ≤ U dc < 2.3 V), the AuNRs undergo directed motion and assemble into a longcylindric vortex (Figure 5A(ii) and Movie S4), and the |dy| of ∼1.0 μm are the largest one among the four states.Further increasing the voltage (2.3 V < U dc ≤ 7 V), the AuNRs continuously enrich around the sharp teeth of electrodes until the particle density exceeds a certain threshold, and then an ordered vortex-pair array gradually emerges (Figure 5A(iii) and Movie S5).In this state, the most characteristic kinetic parameter is dy of ∼0, because the diffusional directions of each vortex pair are almost mirror symmetrical.As the voltage continues to increase (U dc > 7 V).The diffusional states of the AuNRs become extremely unstable, showing two possible behaviors.One is the chaotic state (Figure 5A(iv)), in which the D T of ∼23.8 μm 2 /s is the largest one among the four states and ∼800 times that of without applying the voltage.The violent collisions among the vortices lead to large-scale material redistribution.The morphology, spatial position, and diffusional direction of the vortices are continuously changing (Movie S6).The other state is water electrolysis at the time of applying the high voltage, which results in the generation and rupture of bubbles as well as the perturbation to the suspension and promotes AuNRs to collide and aggregate (Movie S7).
Besides the voltage, the viscosity also regulates some dynamic features of the NP vortices, such as their response sensitivity, speed, and size.Specifically, under the same voltage, the smaller the viscosity, the more sensitive the AuNRs respond to the electric stimulation, with higher vortex rotation speed and larger vortex size (Figure S4).The phase diagram summarizing both the voltage and viscosity effect is shown in Figure 5E.We note that, however, as the proportion of water increases, the interaction between NPs and the formation of vortices tends to be more uncontrollable.With the viscosity ranging from 60% glycerol (10 cP) to 100% glycerol (612 cP), the reproducibility of the four dynamic states is pretty well.When the viscosity decreases to the range of 0% glycerol (0.80 cP) to 60% glycerol (10 cP), although the four states can still be carefully reproduced, we occasionally obtain some weird patterns, such as a uniaxial-vortex array, linear vortex outside the insulating space, and irregular vortex-pair array (data not shown).Therefore, a relatively high viscosity is required to achieve the controlled and stable formation of vortices.
Since the surface modification of particles not only determines the local interparticle interaction, but also affects the particle-field interaction, we also studied the effect of surface modifications on the formation of vortices, including positively charged cetyltrimethylammonium bromide (CTAB) coating, Pt nanodots decoration, and bovine serum albumin (BSA) protein adsorption.Transmission electron microscopy images indicate that these surface-modified AuNRs were monodispersed (Figure S5).Under the typical condition in 60% glycerol solution with a voltage of 3 V, we observed the emergence of vortex-pair arrays in all the cases, but their fine structures are slightly different (Figure 6), indicating that the surface modification undertakes a certain regulating role, and more diverse vortex-pair array may be obtained by modifying the particles with other ligands.
To obtain more diverse vortice arrays (Figure 7), we designed more electrodes (Figure S6), including the interdigital sawtooth electrode-pair array, the spiral interdigital sawtooth electrode-pair array, and the parallel plate electrode pair.In the interdigital electrodes, the ground electrodes and positive electrodes are arranged alternately.Compared with the ITO sawtooth electrode pair, the angle and period of the sharp tooth were changed from 130 • to 90 • , from ∼75 to 200 μm, respectively, and the electrode material was changed from ITO to gold.We still obtained the ordered vortex-pair arrays in which each vortex pair could be mapped onto each sharp tooth of the simple or spiral interdigital sawtooth electrode-pair array, including the vortex-pair rectangular arrays (Figure 7J and Movie S8) and the spiral-type vortex-pair array (Figure 7K and Movie S9).However, in the parallel plate electrode without any sharp-tooth substructure, we can just see vague outlines of vortices (Movie S10), rather than ordered vortex-pair arrays.All these results further verified the coupled field-driven mechanism and imply that more patterns of vortices may be generated by fabricating electrodes with more complex sharp-teeth substructures.

Dynamic evolution of vortex pairs
In many cases with high AuNR concentration using the ITO sawtooth electrodes, we obtained an ordered hollow oval-like vortex-pair array whose spatial position of each vortex did not match the position of each sharp tooth on the electrodes.We conjecture that there is some specific vortex growth mechanism akin to Ostwald ripening.By monitoring the vortex transformation process in real-time, we observed that at high particle density, some vortices could trap adjacent particles and grow continuously, until becoming large enough to collide with the neighboring vortex pairs.As a result, a specific growth mode of "collision-swallow" could occur occasionally, in which the "strong" vortex pair swallows the adjacent "weak" pair to grow larger.In the example shown in Figure S7 and Movie S11, two vortex pairs are adjacent to each other initially.The "strong" vortex pair (yellow dashed box at t = 0 s) quickly merged with the left vortex in the "weak" vortex pair (blue dashed box) in about 4 s initially and grew slightly larger in size.Subsequently, it continued to squeeze the remaining right vortex in the "weak" pair, and "swallowed and digested" it gradually.A material transport channel (green dashed box) was temporally formed with an abnormally high concentration of AuNRs, and the "weak" vortex, after being engulfed, gradually deformed, shrunk, and completely disappeared.Finally, the "strong" vortex pair returned to their normal shape with a considerably larger size.Since the emergence of the vortices is the result of EHD-mediated dynamic assembly and redistribution of NPs, similar vortex evolution, and morphological change would continue until a metastable state is reached between the shear force-induced NP transport and electric field gradient-induced NP diffusion.
This "collision-swallow" growth mode or inverse cascades of NP vortices are a higher level of living NP assembly.It can be not only regulated by the particle density which dominates the initial size of each vortex pair but also adjusted by the period of the sharp tooth which controls the initial distance between vortex pairs.Such property could be exploited to obtain more complicated NP assembly patterns.As a preliminary experiment, we produced a pair of sawtooth ITO electrodes with a higher tooth density, whose period is ∼19 μm (Figure S8).Using this electrode pair, we observed a distinct dynamic vortex emergence path (Figure S9).At the initial stage of applying the voltage, the particles enrich and assemble into a uniaxial vortex array arranged at almost equal intervals (Figure 7E and Movie S12).Over time, a col-lapse point appears near the middle of the array (Figure 7F and Movie S13).The collapse point divides the single vortex array into two shorter vortex arrays.Subsequently, these new arrays deformed like springs, squeezing toward the center of the array from both sides.Finally, a pair of vortices with a large gap formed.In another example, we constructed a three-electrode pair system with high sharp-teeth density (Figure S8B).Interestingly, a fountain-like multi-vortex complex emerges (Figure 7G and Movie S14), and we observed the hierarchical structures at different focal planes by Z-axis optical sectioning (Movie S15).Moreover, after the voltage was applied, if we changed the initial two-electrode system to the three-electrode-pair system, the initial cylindrical vortex would gradually compress, forming a dumbbell-like vortex, and it would further deform to form a half fountain-like vortex (Figure 7H,I and Movie S16).All results indicate that the complexity of the vortex structure increases as the complexity of the electrode increases.Therefore, if we introduce more complex sharp-teeth enrichment effects or replace the three-electrode-pair system with a more complex microelectrode array, it is possible to obtain various complicated emergence paths, growth modes, and interesting structures of the vortices.The high controllability of the electric-driven NRs lays the basis for producing rich vortice families and provides a huge imagination space for the design of NP functional materials.

CONCLUSION
In summary, using dark-field imaging and SPT analysis, we found that seemingly simple AuNRs could organize themselves into fascinating dynamic assemblies when they were energized out of equilibrium by a dc electric field.By monitoring the entire formation processes of the microvortices array in situ in real-time, the combination of the sharpteeth-mediated large-scale EHD effect and the rectification effect of high-concentration NPs were discovered to be the major driving force.The dynamic vortices of the dc electricdriven AuNRs system are highly controllable (Figure 7).The fine structure, size, rotating speed, response sensitivity, and growth mode can be regulated by adjusting the voltage amplitude, viscosity, particle density, surface properties of particles, and electrode morphology.The spatial enrichment effects and more diverse emergence path of the vortices can also be introduced by increasing the density of the sharp teeth of the electrodes.These highly controllable electric-driven NP assembly systems may help us to understand the selforganization behavior of various synthetic and living active matters.Moreover, since colloidal particles have been shown to be building blocks, or mesoscopic counterparts of atoms or ions, of many materials with fascinating physical and chemical properties, our electric-driven NP assembly strategy could provide the experimental and theoretical basis for the design of next-generation functional materials.

Chemicals
Glycerol was purchased from Aladdin and BSA was purchased from Solarbio.PEG-modified AuNRs, CTABmodified AuNRs, and AuNR@PtNDs were purchased from Nanoseedz.BSA-coated AuNRs were obtained by adding BSA (1.67 mg/mL, 60 μL) into the mixture of 20 μL PEG-modified AuNRs and 120 μL glycerol.Ultrapure water was obtained from a three-stage Millipore Milli-Q Plus 185 purification system.

Fabrication of ITO electrodes
We designed the electrode layout and had the electrodes fabricated elsewhere (Suzhou Crystal Silicon Electronic & Technology Co., Ltd.).The company specializes in using laser direct-writing technology to batch fabricate transparent ITO electrodes.Particularly, the sawtooth ITO electrode with an insulating region of ∼300 μm was manufactured by using a high-energy laser to divide a whole piece of ITO conductive glass into two independent conductive areas, in which the ITO conductive film of 185 nm ± 10 nm between two areas was peeled off completely.The resistance in the conductive area is ∼30-∼60 Ω, whereas the resistance between the two conductive areas exceeds the maximum range of 1 GΩ of the digital multimeter (DMM4040 6.5; Tektronix).The angle of the sharp teeth is ∼130 • , and the distance between the sharp teeth is ∼75 μm.The three-electrode-pair systems with insulating regions of 50 μm were fabricated by using the laser to peel off three ITO conductive films and get six independent conductive areas.The resistance values between any two conductive areas are >1 GΩ, indicating the ITO region between the electrodes is completely stripped.As shown in Figure S8, the angle of the sharp teeth is ∼100 • , and the distance between the sharp teeth is ∼19 μm.

Fabrication of gold electrodes
To obtain more diverse vortice structures, we designed several lithographic masks (Figure S6).

Electric control and darkfield imaging
Darkfield imaging was performed on a Nikon 80i (Japan) upright microscope.A 100 W halogen tungsten lamp, an oil immersion darkfield condenser (NA 1.20 ∼ 1.43), a 4× objective (NA 0.2) or a 10× objective (NA 0.45), and a color camera (DP74; Olympus) were equipped to capture the darkfield image sequence with a sampling rate of ∼50 Hz.After connecting electric wires to the ITO or gold electrodes with conductive tapes, the electrode plate is placed on the sample stage of the microscope.The AuNR dispersion was deposited on the insulating region and covered with a clean cover glass (5 × 5 mm).The dc electric signal was output from the CH1 channel of the Keysight waveform generator (33500B; Agilent) or the analog synchronous output card (PCI8250, Beijing Altaïr Technology Development Co., Ltd.).

Kinetic parameters calculation
The translational diffusion coefficient D T , exponential factor α, and the normalized net displacement along the y-axis d N y i of each NR can be extracted from the trajectories fragment R with N spots, which are obtained by using the TrackMate in ImageJ plugins to detect and link the 2D coordinates, r i = (x i , y i ), of the AuNRs within an image sequence (the time interval, τ): Here, we employ an anomalous diffusion model with a power-law dependence, as depicted in Equation (6).Unlike typical diffusion processes explained by Einstein and Smoluchowski, where the mean square displacement shows a linear relationship with time, anomalous diffusion follows a powerlaw behavior.In this context, D T represents the generalized diffusion coefficient.The exponent α is commonly employed to distinguish between normal diffusion (Brownian, α = 1) and anomalous diffusion (α ≠ 1).Subdiffusion (0 < α < 1) and superdiffusion (α > 1) are regimes that characterize different diffusion behaviors, which include external fields in a straightforward manner. [38]n our specific study, which pertains to particles in a convection flow, the diffusion characteristics of AuNRs are affected by intricate interactions with the fluid, boundaries, and other factors.The use of an anomalous diffusion coefficient is instrumental in capturing the non-standard diffusion behavior of nanoparticles within the convection flow.This information is crucial for understanding the nature of particle movement, which cannot be fully described by velocity.
Furthermore, we introduce the concept of d N y i as shown in Equation ( 4).This parameter quantifies how far an object has moved vertically concerning a reference point or its initial position.The resulting value falls within the range of −1 to 1, where a positive value indicates movement in the positive y-direction (upward), a negative value indicates movement in the negative y-direction (downward), and a value of 0 signifies no net movement along the y-axis.The normalized net displacement serves as an additional measure to understand the direction of particle movement and complements the information provided by the anomalous diffusion coefficient.
By combining the normalized net displacement along the y-axis with the anomalous diffusion coefficient, we facilitate a more comprehensive analysis of particle motion, especially in our electric-driven nanoparticle assembly system.In this system, both non-standard diffusion behavior and directional preferences of particles are of significant interest.This integrated approach allows us to gain deeper insights into the underlying processes governing particle dynamics in our unique experimental setup.

COMSOL simulation
Shown in Figure S3A is the 3D computational geometry of the sawtooth electrode pair along with the boundary conditions used in COMSOL, in which the full length and width of the geometry are L = 2 mm and W = 0.9 mm, and the fluidic height and base height are H 1 = 0.8 mm and H 2 = 0.2 mm.
The electric field distribution in the system is bounded by the following equation.
∇ ⋅ (E) = 0 ( 7 )   where ∇ is the gradient along the coordinate axis, σ is the electric conductivity of the fluid, and E is the electric field inside the fluid when the electric potential applied is Φ.
The steady-state flow field is governed by the Navier-Stokes equation along with the continuity equation.
where ρ is the mass density, u is the velocity, p is the pressure, η is the dynamic viscosity of the fluid, and F E is the additional body force induced by the electric field on the fluid.
The general form of F E is given as [33] F where ε is fluid permittivity.Based on the above equations, we could also obtain the vortices streamlines by COMSOL simulation as shown in Figure S3B,C.

A C K N O W L E D G M E N T S
Financial support was provided by the National Natural Science Foundation of China (21425519, 21621003, 91853105 and 22127807).

C O N F L I C T O F I N T E R E S T S TAT E M E N T
The authors declare no conflict of interest.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.

F I G U R E 1
Illustration of experimental devices and characterization methods.(A) Scheme of the electric control and darkfield imaging setup combination.The partially enlarged view and its insert are the indium tin oxide (ITO) electrode pair consisting of a pair of sharp sawteeth with angle θ, period d, and an insulating area with width w. (B) The pipeline of particle collective analysis (PECAN)-based single particle tracking (SPT) analysis.Scale bar, 100 μm.

F I G U R E 2
Entire formation processes of the hollow oval-like vortex-pair array.(A) The spatiotemporal cumulative movement trajectories and (B) the time-dependent variation of average d N y (the normalized net displacement along the y-axis) of all gold nanorods (AuNRs).The gray region highlights the relaxation region with a relaxation time of ∼13 s of the systems after applying the direct current (dc) signal.(C) Time-dependent variation of the scattering intensity of individual particles, either monomers or aggregates, in the black-box marked region in (A).Also shown is the average D T of all the individual particles in (A) as a function of time.Four states were identified.(D) The heat map of d N y in states S2 and S3.Scale bar, 100 μm.

F I G U R E 3
Verification of the microvortex pair formation mechanism.(A) Schematic diagram of the nonlinear electrohydrodynamic (EHD) entrainment of nanoparticles (NPs) around a single sharp tooth.(B) Some blurred vortice profiles can be extracted by single particle tracking (SPT) in 0.175 pM polyethylene glycol-coated gold nanorods (PEG-AuNRs).

F I G U R E 4
The influence of the particle density on the emergence and morphology of the vortex-pair array.(A) The hollow oval-like vortex-pair array emerging in 14 pM polyethylene glycol-coated gold nanorods (PEG-AuNRs).(B) The solid lung-lobe-like vortex-pair array occurring in 7 pM PEG-AuNRs.After long-term observation, the solid cores will turn into hollow cores as shown in the inset.(C,D) Incomplete hollow oval-like vortex-pair array and smallsized lung-lobe-like vortex-pair array emerging in 3.5 pM PEG-AuNRs.(E,F) The backbone of the vortex-pair array and the lung-lobe-like vortex-pair array with a smaller size emerging in 1.4 pM PEG-AuNRs.(G) Incomplete backbone of the vortex-pair array emerging in 0.35 pM PEG-AuNRs.(H) No characteristic structure in the vortex-pair array can be observed in 0.175 pM PEG-AuNRs.F I G U R E 5 Quantitative characterization of the four states of gold nanorod (AuNR) dynamic assembly.(A) The heat map of the ensemble average of D T of all AuNRs as a function of applied voltage.i, 0 V; ii, 2 V; iii, 3 V; iv, 12 V.The AuNRs dispersed in 90% glycerol.Scale bar, 100 μm.The corresponding time-dependent variation of (B) D T and (C) dy, respectively.(D) The scatter plot of dy-D T .The scatter points spontaneously cluster into four regions.(E) The phase diagram of voltage-viscosity (as volume fractions of glycerol), also exhibits four blocks.

F I G U R E 7
Different vortice patterns were obtained under various experimental conditions.(A) Hollow oval-like vortex-pair array, (B) solid lung-lobelike vortex-pair array.(C) doughnut-like vortex-pair array, (D) cylindrical uniaxial vortex, (E) uniaxial vortex array, (F) compressing-spring-like vortex, (G) fountain-like multi-vortex complex, (H) dumbbell-like vortex, (I) half fountain-like vortex, (J) vortex-pair rectangular arrays, and (K) spiral-type vortex-pair array obtained by regulating the voltage, the viscosity, the particle density, the surface modification of gold nanorods (AuNRs) and the electrode morphology.Scale bar, 200 μm.
Using ultraviolet lithography (URE-2000/35; Institute of Optics and Electronics, Chinese Academy of Sciences) and metal evaporation coating machine (ZHD-300; Technol Science), 36 nm gold film with 5 nm chromium as an adhesion layer was deposited on SiO 2 glass.Subsequently, the photoresist, gold, and chromium in the red area of the mask were stripped off with acetone to obtain three kinds of gold electrodes, including an interdigital sawtooth electrode-pair array, a spiral interdigital sawtooth electrode-pair array, and a parallel plate electrode.The parallel plate electrode with the insulating region of 200 μm has no sharp tooth.The distance between each pair of electrodes in the interdigital sawtooth electrode-pair array is also 200 μm, the angle of the sharp teeth is 90 • , and the distance between the sharp teeth is 200 μm.The parameters used in the interdigital spiral sawtooth electrode-pair array are the same.