TheoryForces on the fluidACETThe temperature gradient induced by Joule heating in an inhomogeneous electric field generates local variations in conductivity and permittivity. The interaction between the inhomogeneity of conductivity and permittivity and electric field induces the fluid flow. For aqueous systems with high conductivity (such as seawater), a strong driving force can be generated to induce ACET vortices under a small input voltage. The volumetric ACET force can be expressed as14$$ {\mathbf{F}}_{{{\text{ET}}}} = \frac{1}{2}\frac{{\varepsilon_{{\text{m}}} \left( {\alpha – \beta } \right)}}{{1 + \left( {\omega \tau } \right)^{2} }}\left( {\nabla T \cdot {\mathbf{E}}} \right){\mathbf{E}} – \frac{1}{4}\varepsilon_{m} \alpha \left| {\mathbf{E}} \right|^{2} \nabla T, $$
(1)
where \({\mathbf{E}} = – \nabla \varphi\) is the electric field vector (effective) and \(\varphi\) is the electric potential, \(\omega = 2\pi f\) is the angular frequency of the AC electric field, \({\text{f}}\) is the frequency of the electric field, \(\tau = \varepsilon_{{\text{m}}} /\sigma_{{\text{m}}}\) is the charge relaxation time, \(\varepsilon_{{\text{m}}} = \varepsilon_{{\text{r, m}}} \varepsilon_{0}\) and \(\sigma_{{\text{m}}}\) are the permittivity and conductivity of the medium, respectively, \(\varepsilon_{{\text{r, m}}}\) is the relative permittivity of the medium, \(\varepsilon_{0} = 8.854 \times 10^{ – 12} {\text{F/m}}\) is the permittivity of vacuum, \({\text{T}}\) is temperature of the system, and \(\alpha\) and \(\beta\) are the two thermal diffusion coefficients. For aqueous solutions, the numerical values of \(\alpha\) and \(\beta\) can be approximated as \(\alpha = (\partial \varepsilon_{{\text{m}}} /\partial T)/\varepsilon_{{\text{m}}} \approx\) − 0.04 (K−1) and \(\beta = (\partial \sigma_{{\text{m}}} /\partial T)/\sigma_{{\text{m}}} \approx\) 0.02 (K−1)24.BuoyancyIn addition to ACET, Joule heating (which occurs in highly conductive aqueous solutions) can cause local density differences in the fluid and further trigger buoyancy-driven natural convection. In an ACEK system with characteristic dimensions (i.e., electrode width or spacing) greater than 1 mm, the buoyancy body force drives flow recirculation, which surpasses ACET flow and dominates the overall fluid motion25. The expression of buoyancy force per unit volume is$$ {\mathbf{F}}_{{\text{b}}} = \Delta \rho_{{\text{m}}} {\mathbf{g}} = \left( {\partial \rho_{{\text{m}}} /\partial T} \right)\Delta T{\mathbf{g}}, $$
(2)
where \({\uprho }_{{\text{m}}}\) is the density of the medium, and \({\text{g}}\) is the gravitational acceleration26.Forces on particlesDEPWhen suspended particles are subjected to a non-uniform electric field, their induced dipole moment interacts with the electric field, leading to DEP motion. The direction of the particle motion is determined by the difference of the polarizability between particle and suspension medium. If particles are more polarizable than the suspension medium, they will experience positive DEP forces (pDEP) and move towards the region of high electric field intensity. Conversely, particles which are less polarizable than the suspension medium will experience negative DEP forces (nDEP) and thus are repelled by the high electric field intensity region. In conductive aqueous suspension, many solid particles show nDEP due to their lower polarizability. The time-averaged DEP force acting on a spherical particle can be calculated by15$$ {\mathbf{F}}_{{{\text{DEP}}}} = 2\pi a^{3} \varepsilon_{{\text{m}}} {\text{Re}} \left[ {K\left( \omega \right)} \right]\nabla \left| {\mathbf{E}} \right|^{2} , $$
(3)
where a is the radius of the spherical particle, \({\text{K}}\left( {\upomega } \right)\) is the frequency-dependent Clausius–Mossotti factor27$$ K\left( \omega \right) = \frac{{\varepsilon_{{\text{p}}}^{*} – \varepsilon_{{\text{m}}}^{*} }}{{\varepsilon_{{\text{p}}}^{*} + 2\varepsilon_{{\text{m}}}^{*} }}. $$
(4)
Here \(\varepsilon_{{\text{p}}}^{*}\) and \(\varepsilon_{{\text{m}}}^{*}\) are the complex permittivity of the particle and medium, respectively, \(\varepsilon^{*} = \varepsilon – j\sigma /\omega\) with the permittivity \(\varepsilon = \varepsilon_{{\text{r}}} \varepsilon_{0}\), the relative permittivity \(\varepsilon_{{\text{r}}}\) and the electrical conductivity \(\sigma\).DragWhen relative motion exists between fluid and solid, the fluid exerts a drag force on the surface of the solid. For spherical particles, the drag force can be expressed as follows28:$$ {\mathbf{F}}_{{{\text{Drag}}}} = \frac{1}{2}\pi a^{2} \rho_{{\text{m}}} C_{{\text{D}}} \left( {{\mathbf{u}} – {\mathbf{v}}} \right)^{2} , $$
(5)
where CD is the particle Reynolds number-dependent drag coefficient, u is the fluid velocity, and v is the particle velocity.GravityIn a gravitational field, a solid is subject to the buoyant forces exerted by surrounding fluid. For spherical particles, this buoyant force is$$ {\mathbf{F}}_{{\text{g}}} = \frac{4}{3}\pi a^{3} \left( {\rho_{{\text{p}}} – \rho_{{\text{m}}} } \right){\mathbf{g}}, $$
(6)
where \(\rho_{{\text{p}}}\) is the density of the particle.Design principlesThe design principle of the proposed underwater surface antifouling system is illustrated in Fig. 1. Note that this proposed technology is intended to be used for the practical cleaning of underwater surfaces. In this work, we only explore the feasibility and dependencies in a lab-based setup as a proof-of-concept study. Specifically, an array of asymmetric interdigitated electrode is designed. Transparent ITO is chosen as electrodes on the surface of underwater devices to generate DEP force and fluid flows. Many particles have lower polarizability compared to seawater and therefore subject to nDEP, such particles will be moved away from the surface. Meanwhile, the Joule heating induced flows, i.e., ACET and buoyancy, will generate asymmetric vortices in the vicinity of the electrode surfaces and unidirectional fluid flow away from electrodes. As a result, through the synergistic interaction between the asymmetrical vortex flow and the nDEP effect, pollutant particles deposited on the surface of optic devices are lifted upwards and drifted out of the system by the unidirectional fluid flow. To better illustrate the ability of the surface antifouling strategy, we evaluate the antifouling performance of the proposed system by a combined effect of the net fluid flow velocity on surfaces and the nDEP velocity of pollutant particles.Figure 1(a) Sketch of the proposed underwater surface antifouling system with (b) computational domain and boundary conditions for the numerical simulations.Numerical simulationGoverning equationsThe finite element software package COMSOL Multiphysics 5.5 was utilized to conduct numerical simulations and assess the efficacy of an ACEK-based surface antifouling system (see Fig. 1). To simplify the system, we reduce variables by letting the distance between electrodes (d1) equal to the size of small electrode (w1), and the distance between electrode pairs (d2) equal to the size of large electrode (w2). The aspect ratio r is introduced to characterize the degree of asymmetry of the electrode array, given as:$$ r = w_{2} /w_{1} . $$
(7)
The simulation framework consists of a two-dimensional model that incorporates electrostatics, heat transfer, and laminar flow. Additionally, the particle tracking module is used to investigate the efficiency of the system on particle trajectories. In a system with strong temperature variations, the flow dynamics of the vortex generated by the electrothermal force deviates significantly from those predicted by conventional models using a small temperature approximation. In light of this, Loire et al.29 modified the Poisson equation to solve the electric field and arrived at a convection–diffusion style equation:$$ \nabla^{2} \varphi = \gamma \cdot \nabla \varphi , $$
(8)
where the factor \(\gamma = – \beta \nabla T\) if \(\omega \tau \ll 1\) and \(\gamma = – \alpha \nabla T\) if \(\omega \tau \gg 1\).The energy balance equation solves the Joule heating effect under AC electric field by coupling the electric field and the thermal field30:$$ \nabla \cdot \left( {k_{{\text{m}}} \left( T \right)\nabla T} \right) + \frac{{\sigma_{{\text{m}}} \left( T \right)}}{2}\left| {\mathbf{E}} \right|^{2} = 0, $$
(9)
where \(k_{{\text{m}}}\) is the thermal conductivity of the medium related to temperature.The solution of the flow field is based on the continuity equation and the Navier–Stokes equation31:$$ \rho_{{\text{m}}} \left( T \right){\mathbf{u}} \cdot \nabla {\mathbf{u}} = – \nabla P + \nabla \cdot \left( {\eta \left( T \right)\nabla {\mathbf{u}}} \right) + {\mathbf{F}}_{{{\text{ET}}}} + {\mathbf{F}}_{{\text{b}}} , $$
(10)
$$ \nabla \cdot \left( {\rho_{{\text{m}}} \left( T \right){\mathbf{u}}} \right) = 0, $$
(11)
where \(\eta\) is the dynamic viscosity of the medium, which is related to the fluid temperature; and P is the pressure of the flow field.Trajectories of particles affected by drag, gravity, and DEP forces are solved by Newton’s second law:$$ \frac{{{\text{d}}\left( {m_{{\text{p}}} v} \right)}}{{{\text{d}}t}} = {\mathbf{F}}_{{{\text{Drag}}}} + {\mathbf{F}}_{{\text{g}}} + {\mathbf{F}}_{{{\text{DEP}}}} , $$
(12)
where mp is the particle mass, and t is the time.Computational domain and boundary conditionsIn this study, the calculated system can be simplified to a two-dimensional geometry since the electrodes are long compared to their height and width (Fig. 1). The computational domain is composed of three groups of asymmetric plate electrodes embedded on a 1.1 mm thick glass substrate, with periodic boundary conditions applied to both the left and right boundaries. For the electric field, the device surface (excluding the electrode) and the top fluid boundary are electrically insulating with \({\text{n}} \cdot {\text{J}}\) = 0, where \({\text{n}}\) is the unit normal vector and \({\text{J}} = \sigma_{{\text{m}}} {\mathbf{E}}\) is the current density. For the flow field, the fluid velocity at the top boundary is assumed to be stationary with a velocity of u = 0, while the fluid velocity on both surface of electrodes and glass substrate are given as slip boundary with \(- {\text{n}} \cdot {\text{u}}\) = 0 due to the smoothness of material surfaces. Additionally, since the device is in an infinitely large natural fluid environment (in comparison to the size of the device itself), the upper boundary of the fluid and the lower boundary of the device are considered to be at room temperature (T = 293.15 K).Model and mesh independence studyThe model used in this study was validated by comparing numerical simulations of the field distributions with literature values by Williams32. As presented in Fig. S1, the simulated electric field and flow patterns are in good agreement with those in the literature, indicating good accuracy of our model. In addition, the fitted relationship between velocity and applied voltage (see Fig. S2) indicates that the flow rate is proportional to the power of 5.28 of the voltage, which conforms to the enhanced model proposed by Loire et al.29.A mesh-independence study was conducted for the model system (see Fig. S3) to obtain reliable simulation results independent of mesh size. The maximum relative deviations of the flow field and electric field between simulation results using different mesh numbers of 426,938 and 329,865 are 1.22% and 1.03%, respectively. Further increasing the mesh number gives a negligible effect on the accuracy of the calculations while significantly increasing computational costs. Therefore, a total mesh number in the computational domain was selected as 329,865 in this study.Experimental methodElectrodes fabricationThe substrate material consists of 1.1 mm thick glass slides (25 × 75 × 1.1 mm3) which are coated with a 650 nm thick ITO layer (BIOTAIN CRYSTAL CO., LIMITED, CHN), as shown in Fig. S4. The surface resistivity is 2 ± 0.5 Ohm/sq. The surface electrodes were created by selective laser ablation. A 100 W Nd:YAG laser (FA Clean-Lasersysteme GmbH) with wavelength 1064 nm was used to scan the samples at a focal length of 160 mm in a 2D scanning system. The power of the laser was set to 5 W, resulting in a circular ITO ablation of 30 µm at the laser focus.To obtain interdigitated electrodes on the ITO glass slides, a CAD model was created and transferred to the laser. The electrodes were rounded in a semicircle at their ends to avoid electrical voltage peaks. The ITO ablation finally results in the interdigital spaces between the electrodes. With a laser focus distance along the scan direction and between adjacent laser scan lines of 2 µm, sharp-edged electrodes are obtained in the Keyence VHX-600 light microscope shown in Fig. S4. In this way, areas of 1 × 1 cm2 (Fig. S4) on the glass slides were generated with electrodes and contact areas being cut out with the laser. The resistance between the electrodes at the contact points was approx. 1.5 ± 0.5 kΩ. The smaller electrode distance w1 was chosen as 40 µm and w2 = 200 µm achieve sufficient electrical resistances (> 100 Ω) between adjacent electrodes. The resulting electrode array is shown in Fig. S4.Materials and methodsFluoresbrite® yellow green polystyrene (PS) particles (Polysciences, Germany) with diameter of 0.75 µm (1.08 × 1011 particles/mL, excitation max. = 441 nm, emission max. = 486 nm) were used in experiments. Sodium chloride (NaCl) (Merck Germany) was mixed with water at a certain mass concentration for synthesizing salty water with a defined conductivity, which is measured and determined using a portable conductivity meter (WTW Cond 3110) at room temperature. The concentration of particles in NaCl suspension was 8.64 \(\times\) 109 particles/L. 0.005 vol% of Tween 20 (Merck Germany) was added to reduce the interactions between particles and homogenize the suspension. The identical suspension was used in all experiments in this work.A high precision linear stage (LIMES150, OWIS) allows precisely moving in the y direction to defined height by the installed motor and controlled by software OWISoft v2.90 (OWIS, Germany). It was used to hold, control, and adjust the height of electrokinetic cell (Fig. S4), which was fixed on the bottom of a sample container (85 \(\times\) 130 \(\times\) 20 mm3) (3 in Fig. 2a). Suspension was added into the sample container with its height maintained at 10 mm from the bottom of sample container. The sample container was mounted on two translation stages (THORLABS, DTS50/M) for adjusting positions in x and z directions (Fig. 2b). A fluorescence microscope (ZEISS Scope A1) installed with camera (FLIR Grasshopper 3, GS3-U3-51S5C-C), objective lens (EC Epiplan 10×/0.25 M27) and fluoresce filter set (GFP) was used to observe, acquire and save the videos of particles’ motion trajectories to a computer. Function generator (RIGOL DG4062) is applied to provide electrokinetic cell electrical signal with maximal peak-to-peak voltage of 20 V and frequency up to 40 MHz. The signal output from the function generator is monitored using an oscilloscope (RIGOL DS2072A).Figure 2Schematic diagram (a) and picture (b) of experimental setup, with linear stage (1), fluorescence microscope (2), electrokinetic cell (3), computer (4), oscilloscope (5), and function generator (6).After filling particle suspension, the height of zero (y = 0) is first found and fixed by focusing the objective of fluorescence microscope on the surface of ITO electrodes. The zero-height position of the setup is defined for the linear stage in its controlling software OWISoft v2.90. By adjusting the relative height of the linear stage through OWISoft and the motor, we can record particle trajectories at different heights from the ITO array. The function generator is switched on to generate electric field at a defined frequency, 1 kHz in this work. The oscilloscope measures the output signal and then record the effective voltage (Ueff) and frequency. The video clips of particles motion are recorded and saved using software FlyCapture 2. The velocity of particles is measured and evaluated using software Tracker, a free video modelling tool built on the opensource physics (OSP) Java framework (https://physlets.org/tracker/).
A universal AC electrokinetics-based strategy toward surface antifouling of underwater optics
