Microparticle transport along a planar electrode array using moving dielectrophoresis

Abstract

We present a device that can achieve controlled transport of colloidal microparticles using an array of micro-electrodes. By exciting the micro-electrodes in regular sequence with an AC voltage, a time-varying moving dielectrophoretic force-field is created. This force propels colloidal microparticles along the electrode array. Using this method, we demonstrate bidirectional transport of polystyrene micro-spheres. Electromagnetic simulation of the device is performed, and the dielectrophoretic force profile around the electrode array is mapped. We develop a Brownian dynamics model of the trajectory of a particle under the influence of the time-varying force-field. Numerical and experimental results showing controlled particle transport are presented. The numerical model is found to be in good agreement with experimental data. The developed numerical framework can be useful in designing and modeling lab-on-a-chip devices that employ external non-contact forces for micro-/nanoparticle manipulation.

I. INTRODUCTION

Trapping and manipulation of micron and submicron sized particles/objects using dielectrophoretic forces are of great interest in many scientific fields including biotechnology,^1,2 physics,³ and medicine.⁴ As lab-on-a-chip (LOC) devices and micro-total-analysis-systems ( $\mu$TASs) become more elaborate, the need for complex controlled particle manipulation platforms has risen. Therefore, design and analysis of such platforms are of significant interest.

Dielectrophoresis (DEP) is the motion of a polarizable particle under the influence of a non-uniform electric field.^5–8 The interaction between polarized charges on a particle surface and an external AC electric field creates dielectrophoretic force which drives this motion. The non-uniform field is usually created by applying AC voltage of radio/microwave frequencies across metal micro-electrodes. The electric field gradient produces DEP forces that can be used for trapping and manipulating microparticles/samples. Similar effects can be achieved using patterned insulators⁹ or illuminated photoconductors¹⁰ instead of metal electrodes. As DEP force arises from non-uniform field distribution across the particle surface, the structures (e.g., electrodes) generating the field distribution should have dimensions of the same order as the particle/sample. The relevant dimensions are usually in the range of micrometers to tens of micrometers. These microstructures can be integrated with microfluidic systems using standard photo-lithography and micro/nano-fabrication techniques. Thus, DEP techniques are often used in many LOC and $\mu$TAS devices.

DEP trapping can be accomplished using a single pair of electrodes. However, more complex and dynamic manipulation requires multi-electrode systems. One such system is traveling wave dielectrophoresis (TwDEP).^11–13 TwDEP methods apply AC voltages of different phases simultaneously to a set of electrode arrays to create a time-varying force profile. As a result, a continuous translatory motion of microparticles across the electrode array can be achieved. An alternative approach to TwDEP is a moving DEP system,^14–17 where the electrodes are excited one at a time in a specific sequence. The excitation sequence can be controlled by a computer or a microcontroller. The switching order can easily be reprogrammed, making it possible to create arbitrary motion sequences. Also, unlike TwDEP, the complexities associated with maintaining phase-offset synchronization between multiple electrodes are not present in moving DEP systems. Despite these advantages, experimental and/or theoretical works related to moving DEP methods are limited in the literature. A few notable papers have reported using moving DEP techniques for applications such as cell motion,¹⁶ cell fractionation, and cell transportation.¹⁴ The scheme can potentially be used for other applications as well.

Since moving DEP schemes are usually employed for particle transport, theoretical and experimental work on the trajectory of a particle in a moving force-field can be of significant interest. While several impressive works have been published on the trajectory of colloidal microparticles, they usually focus on DEP trapping setups in a microfluidic environment.^18–21 The DEP force profile is usually static for such cases. Particle motion under time-varying DEP forces, as is the case for moving DEP setups, has not been thoroughly investigated. Theoretical/simulation work related to moving DEP is usually limited to modeling the force-field^15,22 only. In this paper, we present a study on the trajectory of a microparticle in a moving DEP system. For the study, we design and fabricate a five electrode moving DEP device. We show experimental results of microparticle transport using the fabricated device. Electromagnetic simulation of the structure is performed, and the DEP force profile is mapped. We develop a Brownian motion model with time-varying external force term to simulate the microparticle motion. We implement and solve the model numerically using a custom computer code that we wrote in Python programming language. Using this code, a numerical study of the Brownian dynamics is presented. The numerical model is found to be in good agreement with the experiment. This suggests that the model can be used to accurately simulate particle behavior in a dynamic force-field. We have made the developed numerical tool available on a GitHub repository²³ for others to use. Such a tool can be very useful for the design process of DEP particle manipulating systems. Since Python is open source and freely available, we hope that the tool can be used by a wide range of researchers. To the best of the authors’ knowledge, similar works have not been reported in the literature.

The paper is organized as follows: Sec. II discusses the structure of the device. The electric field distribution and the DEP force profile around the electrode array are presented in Sec. III. Section IV discusses the mathematical framework for the Brownian dynamics model. Experimental and numerical results of the particle trajectory are presented in Sec. V. Finally, concluding remarks are made in Sec. VI.

II. DEVICE DESIGN AND GEOMETRY

The device geometry considered for this study along with the coordinate system used through out the paper are shown in Fig. 1. The structure consists of five planar gold micro-electrodes on a silicon substrate. The micro-electrodes are electrically shorted to contact pads via larger bus lines. External circuitry is connected to the contact pads to provide electrical excitation. The standard metal lift-off process is used to fabricate the micro-electrodes. A SU-8 spacer layer is fabricated to form a rectangular chamber around the electrodes. During experiment, a dilute colloidal solution of spherical polystyrene microparticles in de-ionized water is pipetted onto the SU-8 well. A standard 0.17 mm thick glass cover-slip is placed on top to seal the chamber. The cover-slip squeezes the liquid to the same thickness as the SU-8 chamber and makes the top surface flat. The reduced thickness and flat interface makes it easier to image the device. The whole device is placed on a upright optical microscope setup to image and record the experiment through the cover-slip. Sealing the chamber with the cover-slip also slows down the evaporation of the liquid.

FIG. 1.

Geometry of the moving DEP device with the coordinate system used through out the paper. (a) Top view ( $z=0$ plane) and (b) side view ( $x=0$ plane) of the device. The figure is not drawn to scale.

The characteristic geometric features of the device are the parallel micro-electrodes. The electrode geometry is defined by the electrode width, $w_e$, electrode height, $h_e$, and the separation between adjacent electrodes, $d_e$. The electrode width should be comparable to the microparticle diameter. This ensures that the resulting field distribution has a spatial gradient across the particle surface. Thus, a net DEP force would be exerted on the particle. We focus on microparticles with a radius of $r_o=10\mu \mathrm{m}$ in this paper. The electrode dimensions are selected accordingly. The numerical value of the relevant parameters are listed in Table I. The relatively small value of $h_e$ implies that the electrode geometry is mostly planar. The desired function of the device is to manipulate microparticles along the plane of the electrodes (i.e., the $xy$ plane).

TABLE I.

Geometrical parameters.

Parameter	Value (μm)
Electrode width, we	15
Electrode height, he	0.2
Electrode spacing, de	50
Spacer height, hs	80
Particle radius, ro	10

III. ELECTROMAGNETIC ANALYSIS

During device operation, an AC voltage is applied to one of the micro-electrodes (referred to as the active electrode), while the others are electrically grounded. This creates an electric field gradient in the direction perpendicular to the orientation of the electrodes. As the electrodes run parallel along the $x$ direction, the field gradient is expected to be mostly along the $y$ direction. The generated dielectrophoretic gradient force depends on the relevant geometrical and material properties as well as the electromagnetic field profile. Thus, it is necessary to simulate the electromagnetic field distribution around the electrode array.

The field distribution depends on the geometry of the electrodes, the amplitude and frequency of the applied voltage, and the material properties of the particle and the suspension medium. We consider polystyrene particles suspended in water for this study. The electrical excitation and material parameters are listed in Table II. It should be noted that the material properties of polystyrene micro-beads are altered when they are suspended in water due to surface charge accumulation. This is taken into account within the electrical conductivity values.²⁴ The frequency of 20 kHz is selected to ensure that the particle experience positive DEP force. This frequency falls within the low-frequency regime where the conductivity effect dominates over the dielectric polarization. Thus, having a medium with a lower conductivity results in a positive Clausius–Mossotti factor and positive DEP force on the particle. Additional details are included in the supplementary material.

TABLE II.

Electrical excitation and material parameters.

Parameter	Value
Applied AC voltage, Va	1 V
Frequency of the AC voltage, f	20 kHz
Permittivity of the particle, εp	2.55
Permittivity of the water, εm	78.5
Conductivity of the particle, σp	1 × 10−2 S/m
Conductivity of the water, σm	1 × 10−3 S/m
Dynamic viscosity of water, η	8.9 × 10−4 Pa s

With the relevant electrical, material, and geometrical parameters defined, the field distribution around the electrodes is calculated using Comsol Multiphysics®  which is a commercially available finite element solver. Additional details are included in the supplementary material. For the simulation, the electrode at $y=0$ is assumed to be the active electrode (i.e., connected to the AC voltage), while the other electrodes are connected to the electrical ground. The electric field intensity distribution is shown in Fig. 2. The distribution is normalized with respect to a reference value of $|{\mathbf{E}}_0|=0.02\mathrm{V}/\mu \mathrm{m}$. The intensity is focused along the $y$ direction in the region between the active electrode and adjacent grounded electrodes. The maximum intensity is located at the two sides of the active electrode. The field non-uniformity in the $x$ direction (parallel to the electrodes) is negligible.

FIG. 2.

Electric field intensity distribution near the micro-electrodes when the center electrode (at $y=0$) is active. At (a) the $z=h_e$ plane and (b) the $x=0$ plane. Here, $|{\mathbf{E}}_0|=0.02\mathrm{V}/\mu \mathrm{m}$ and $f=20\mathrm{kHz}$. The dashed lines represent the outlines of the electrodes.

In a moving DEP system, the location of the active electrode changes with time. The field distribution shown in Fig. 2 assumes that the center electrode (at $y=0$) is active. For an arbitrary active electrode position, the field profile would be a spatially shifted version of the same distribution. This is due to the symmetry of the electrode geometry. Thus, it is not necessary to simulate the field profile for each active electrode position separately. A simple coordinate transformation applied to the $y=0$ active electrode data can give field distribution for the cases when a different electrode is active.

The DEP force exerted on a microparticle near the micro-electrode array can be calculated from the field distribution using the Maxwell stress tensor method.^25,26 The equations for the stress tensor and the corresponding force are

$$\overleftrightarrow{\mathbf{T}}={ϵ}_m(\mathbf{E}\mathbf{E}-\frac{1}{2}|\mathbf{E}{|}^2\overleftrightarrow{\mathbf{I}})+{\mu }_m(\mathbf{H}\mathbf{H}-\frac{1}{2}|\mathbf{H}{|}^2\overleftrightarrow{\mathbf{I}}),$$ (1)

$$\mathbf{F}={\int }_S⟨\overleftrightarrow{\mathbf{T}}{⟩}_t\cdot \hat{\mathbf{n}}\mathrm{d}S.$$ (2)

Here, $\overleftrightarrow{\mathbf{T}}$ is the Maxwell stress tensor, $\mathbf{E}$ is the electric field, $\mathbf{H}$ is the magnetic field, ${\mu }_m$ is the permeability of the surrounding medium (water), $\overleftrightarrow{\mathbf{I}}$ is the identity tensor, $\mathbf{F}$ is the net electromagnetic force (i.e., DEP force) acting on the particle, $S$ is the outer surface of the particle, and $\hat{\mathbf{n}}$ is the surface normal to $S$. $⟨\cdot {⟩}_t$ represents the time-averaged value. At low-frequency range ( $<100\mathrm{MHz}$) and a system with no magnetic materials, the magnetic field in Eq. (1) can be ignored.²⁵ For time harmonic fields, the time-averaged stress tensor can be expressed as²⁵

$$⟨\overleftrightarrow{\mathbf{T}}{⟩}_t=\frac{1}{4}\mathrm{\Re }({ϵ}_m)(\stackrel{\sim }{\mathbf{E}}\stackrel{\sim }{\mathbf{E}}\ast +\stackrel{\sim }{\mathbf{E}}\ast \stackrel{\sim }{\mathbf{E}}-|\stackrel{\sim }{\mathbf{E}}{|}^2\overleftrightarrow{\mathbf{I}}),$$ (3)

where $\stackrel{\sim }{\mathbf{E}}$ is the phasor representation of the electric field,²⁷ $\stackrel{\sim }{\mathbf{E}}\ast$ is the complex conjugate of $\stackrel{\sim }{\mathbf{E}}$, and $\mathrm{\Re }(\cdot )$ is the operator that gives the real part of its argument.

Equation (2) gives the force value for a given position of the particle. Thus, the force is a function of particle coordinates. Note that the coordinates $(x_o,y_o,z_o)$ are used to express particle position dependent quantities (e.g., force profile), whereas $(x,y,z)$ are used for general space coordinates (e.g., for expressing geometry and field distribution). To map out the spatial profile of the force, the position of the particle is varied and the force is calculated for each position using Eqs. (1) and (2). It should be noted that the fields $\mathbf{E}$ and $\mathbf{H}$ depend on the position of the particle. Thus, it is necessary to recalculate the field distribution for every particle position during the mapping process of the force-field.

Since the geometry of the parallel micro-electrodes is uniform along the $x$ direction (within the area of interest), there is no significant field gradient along that direction. Thus, no DEP force is expected to be along the $x$ direction. Therefore, we focus on the $y$ and $z$ components of the force, $F_y$ and $F_z$, respectively. Taking the electrode at $y=0$ is active, the calculated profiles of $F_y$ and $F_z$ are shown in Fig. 3. To generate Fig. 3(a), the particle position was swept along the straight line $x_o=0$, $z_o=r_o+400\mathrm{nm}=10.4\mu \mathrm{m}$ which is parallel to the $y$ axis. Similarly, for Fig. 3(b), the particle position was swept along the $z$ axis. Figure 3 shows that the equilibrium trapping position of the particle is near the center of the active electrode (at $y_o=0$ in this case). Particle located to either side of this position is pulled toward the center, indicated by the sign of $F_y$. The peak magnitude of the force is above 15 pN, which is sufficient for the manipulation of microparticles. Figure 3(b) shows that $F_z$ is always negative, suggesting the particle is pulled toward the surface where the micro-electrodes are located. This is consistent with the nature of positive DEP. Due to this pulling force, the particle motion is expected to occur very near the substrate plane. For this reason, the $F_y$ component was calculate at a plane very close to the substrate.

FIG. 3.

The dielectrophoretic force profile. (a) The $y$ component of the force, $F_y$, along the line $x_o=0$, $z_o=10.4\mu \mathrm{m}$ and (b) the $z$ component of the force, $F_z$, along the line $x_o=0$, $y_o=0$.

IV. PARTICLE DYNAMICS

Once the force profile is calculated, the motion of a microparticle can be modeled using Brownian dynamics. The Langevin equation is a well-known approach for modeling such phenomenon. The Langevin equation for a colloidal particle in a low Reynolds number environment is given by ^28–30

$$\dot{\mathbf{r}}(t)=\frac{\overleftrightarrow{\mathbf{D}}(\mathbf{r})}{k_{B}T}{\mathbf{F}}_{\mathrm{ext}}(\mathbf{r},t)+\sqrt{2}{\overleftrightarrow{\mathbf{D}}}_{\frac{1}{2}}(\mathbf{r})\mathbf{W}(t).$$ (4)

Here, $\mathbf{r}=(x_o,y_o,z_o)$ is the position of the center of the particle, ${\mathbf{F}}_{\mathrm{ext}}$ is the external trapping/manipulation force acting on the particle, $k_B$ is the Boltzmann constant, $T$ is the temperature, $\overleftrightarrow{\mathbf{D}}$ is the diffusion tensor, and $\mathbf{W}(t)$ is a vector white noise term. The tensor ${\overleftrightarrow{\mathbf{D}}}_{\frac{1}{2}}$ is defined as the element-wise square root of $\overleftrightarrow{\mathbf{D}}$. Each Cartesian component of $\mathbf{W}(t)$ is a random process with zero mean and unit variance. The term represents the random thermal vibration experienced by colloidal particles. This Brownian vibration is usually prominent for sub-micrometer particles. However, it is possible to use the same formulation for micro-sized particles as well. In such cases, the deterministic force, ${\mathbf{F}}_{\mathrm{ext}}$, is usually larger than the white noise term. Therefore, the random thermal vibrations are less noticeable. It should be stated that Eq. (4) does not take into account inertial effects and particle sedimentation motion due to gravity. For the current analysis, these effects can be neglected. We include a brief discussion on these components in the supplementary material. We also outline how to incorporate some of these effects into the model if needed.

To simulate the particle trajectory, each term in Eq. (4) must be defined and related to the physics of the micro-electrode device under consideration. We start with the hydrodynamic interaction between the microparticle and the liquid layer near the solid bottom surface of the device ( $z=0$ plane). This interaction can be incorporated in the model through the diffusion tensor. The diffusion tensor, $\overleftrightarrow{\mathbf{D}}(\mathbf{r})$, is a diagonal tensor with $D_{11}(\mathbf{r})=D_{22}(\mathbf{r})=D_{\parallel }(\mathbf{r})$ and $D_{33}(\mathbf{r})=D_{\mathrm{\perp }}(\mathbf{r})$. Here, the $\parallel$ and $\mathrm{\perp }$ subscripts indicate in plane ( $xy$ plane) and perpendicular to the plane directions, respectively. These components are defined as^31,32

$$D_{\parallel }(\mathbf{r})=\frac{k_{B}T}{6\pi \eta r_o}(1-\frac{9r_o}{16z_o}+\frac{r_o^3}{8z_o^3}-\frac{45r_o^4}{256z_o^4}-\frac{r_o^5}{16z_o^5}),$$ (5)

$$D_{\mathrm{\perp }}(\mathbf{r})=\frac{k_{B}T}{6\pi \eta r_o}\left(\frac{6z_o^2+2r_o{z}_o}{6z_o^2+9r_o{z}_o+2r_o^2}\right).$$ (6)

Here, $\eta =8.9\times {10}^{-4}\mathrm{Pa}\mathrm{s}$ is the dynamic viscosity of the medium (water), $r_o=10\mu \mathrm{m}$ is the radius of the nanoparticle, and $z_o$ is the $z$-coordinate of the center of the particle. As the separation from the $z=0$ surface increases, the hydrodynamic interaction becomes less prominent. So, for large $z_o$, $D_{\parallel }(\mathbf{r})$ and $D_{\mathrm{\perp }}(\mathbf{r})$ converge to the free space diffusion coefficient of $k_{B}T/6\pi \eta r_o$.

The external force term, ${\mathbf{F}}_{\mathrm{ext}}(\mathbf{r},t)$, in Eq. (4) drives the motion of the particle. It consists of the DEP force and any other external force components. The term can be modeled as

$${\mathbf{F}}_{\mathrm{ext}}(\mathbf{r},t)=\mathbf{F}(\mathbf{r},t)+{\mathbf{F}}_{\mathrm{irr}}(\mathbf{r},t),$$ (7)

where $\mathbf{F}(\mathbf{r},t)$ is the DEP force calculated from Eqs. (1) and (2) and ${\mathbf{F}}_{\mathrm{irr}}(\mathbf{r},t)$ is an irregular force term. As geometrical features fabricated using photo-lithography have some imperfections such as ragged sidewalls and surface roughness, they can create some irregular force components. The term ${\mathbf{F}}_{\mathrm{irr}}(\mathbf{r},t)$ accounts for such spurious forces arising from geometrical imperfections and other factors. It is modeled using a zero-mean Gaussian random process with peak amplitude of one-tenth of the peak DEP force. The amplitude is empirically selected based on experimental observations. It should be noted that this term is not related to the Brownian motion term, $\sqrt{2}{\overleftrightarrow{\mathbf{D}}}_{\frac{1}{2}}(\mathbf{r})\mathbf{W}(t)$, which arises from interactions between the fluid molecules and the colloidal particle. Other external force terms (e.g., gravitational force) can be incorporated within Eq. (7) as stated in the supplementary material.

Now that all the terms of Eq. (4) are defined, the trajectory of a microparticle in the moving DEP force-field can be calculated. The final step is to convert Eq. (4), which is a stochastic differential equation, into a discrete form suitable for numerical solution. The discrete time finite difference form of the equation is²⁹

$${\mathbf{r}}_{i+1}={\mathbf{r}}_i+\frac{\overleftrightarrow{\mathbf{D}}({\mathbf{r}}_i)}{k_{B}T}{\mathbf{F}}_{\mathrm{ext}}({\mathbf{r}}_i,t_i)+\sqrt{2\mathrm{\Delta }t}{\overleftrightarrow{\mathbf{D}}}_{\frac{1}{2}}({\mathbf{r}}_i){\mathbf{W}}_i.$$ (8)

Here, $i$ represents the time index and $\mathrm{\Delta }t$ is the discrete time step. From a given arbitrary initial position ( ${\mathbf{r}}_0$), Eq. (8) can be used to simulate the time evolution (i.e., the trajectory) of a microparticle position. The Euler–Maruyama method³³ is used to numerically solve the equation. We implement the numerical solver using our custom developed Python code that takes into account all the relevant physics. The simulated trajectory along with experimental data are discussed in Sec. V. We have made the developed numerical analysis code available on a GitHub public repository²³ for others to use.

V. RESULTS

The particle transport mechanics of the micro-electrode array device is experimentally tested. The fabricated device is connected to a programmable micro-controller switching circuit. The circuit directed the AC voltage from a signal generator to one of the micro-electrodes (i.e., the active electrode). The circuit activates the electrodes in the following sequence: top to bottom and then bottom to top (top and bottom refers to directions along the $y$ axis). Each electrode is kept active for $t_s=1.8\mathrm{s}$ before switching. This switching sequence creates a moving DEP force in the up–down–up direction. Thus, a bidirectional controlled manipulation of microparticles can be achieved. The experiment is observed in real-time using a sCMOS camera connected to a upright microscope.

A few frames from the experimental video demonstrating microparticle transport are shown in Fig. 4. The complete video file is included as multimedia file 1. The video is recorded using a sCMOS camera (Thorlabs CS2100M-USB) connected to a standard upright microscope with Nikon CFI Plan Fluor 10X objective lens (NA = 0.3). The figure shows two microparticles being transported from the top of micro-electrode to the bottom micro-electrode. The video file shows multiple round-trip transport along the electrode array. A round trip is defined when a microparticle traverses the entire length of the electrode array and then returns to its original position. It can be observed that the microparticles follow the electrode excitation sequence. Thus, successful controlled bidirectional particle transport using moving DEP force is achieved. It should be noted that the experimental video (Multimedia View) shows a third stationary particle in addition to the two moving particles. That stationary particle was stuck to the electrode and, thus, unable to move. The initial position of that particle was very close to the electrode. Thus, when the voltage was turned ON at the start of the experiment, the temporary large force created by electrical transients could have lead to the sticking of that particular particle to the electrode.

FIG. 4.

Experimental demonstration of particle transport along the electrode array. A few sample frames of the experimental video at different time instances are shown here. The blue boxes represent tracking boxes uses by the particle tracking algorithm. Multimedia views: https://doi.org/10.1063/5.0049126.1 Download video file ^{(486.4KB, mp4)} DOI: 10.1063/5.0049126.1 ; https://doi.org/10.1063/5.0049126.2Download video file ^{(215.3KB, mp4)} DOI: 10.1063/5.0049126.2

To analyze the particle trajectory, a particle tracking algorithm is used to extract features from the experimental video. A Python implementation of the IDL tracking algorithm developed by Crocker and co-workers^34,35 is used for this purpose. The extracted position data are shown in Fig. 5. The experimental data show that the $y$ position of the particle follows the excitation sequence of the micro-electrodes. Figure 5(b) shows two complete round-trips (top–bottom–top) along the $y$ direction. The $x$ position data are expected to be relatively unchanged as there is no significant DEP force in that direction. However, some slight right to left motion along the $x$ direction is observed in the experiment. This particle drifting could be due to some initial velocity of the particles prior to the switching sequence, random flow of the suspension fluid, or the presence of some other spurious external force. Since there is no $x$ component of DEP force, any initial particle drift may persist unaffected during the experiment. In future work, different electrode geometries may be investigated that suppresses particle drifting.

FIG. 5.

Experimental and simulation data of the particle trajectory. (a) The $x$ coordinate of the particle position and (b) the $y$ coordinate of the particle position as a function of time. Multimedia files 3 and 4 show the animation of the simulated trajectory. Multimedia views: https://doi.org/10.1063/5.0049126.3Download video file ^{(376.5KB, mp4)} DOI: 10.1063/5.0049126.3; https://doi.org/10.1063/5.0049126.4 Download video file ^{(379.7KB, mp4)} DOI: 10.1063/5.0049126.4

It should be stated that although we present data from a single experimental run, we repeated the experiment multiple times and obtained positive results. A second experimental video showing successful particle transport is included as multimedia file 2. Roughly four out of five experimental attempts resulted in positive results. We also tested the device with smaller polystyrene particles (radius = 3  $\mu$m). This, however, did not work well. Although the smaller particles moved, controlled linear motion was not achieved. The spacing between the electrodes were too large compared to the diameter to maintain control over the particle during transport. We posit that scaling the overall geometry to match the electrode gap with the particle diameter would allow successful transport of smaller particles.

In addition to the experimental results, numerical simulation of the particle trajectory was also performed with our developed Python code using the approach discussed in Sec. IV. The numerical results are shown on top of the experimental data in Fig. 5. Animations of the simulated trajectory are shown in multimedia files 3 and 4. The initial particle position for simulation is selected to match the experimental initial condition. To account for the drift in the $x$ direction, a drift velocity term is incorporated in the Langevin equation. Figure 5(a) shows the simulated particle $x$ trajectory for two cases. One case considers no net drift velocity. The other case uses a net drift velocity of $-2\mu$m/s in the $x$ direction. This value is taken empirically. The simulated $x$ coordinate of the particle trajectory matches closely to the experimental data for the drift adjusted case. Figure 5(b) shows that the simulation is in good agreement with experimental data for the $y$ coordinate of the trajectory as well. The $y$ trajectory shows some stepped motion behavior. This can be observed in multimedia files 1–4. This is related to the electrode switching interval. The switching interval ( $t_s=1.8\mathrm{s}$) is slightly larger than the time it requires for the microparticle to traverse the gap between two adjacent micro-electrodes ( $d_s=50\mu \mathrm{m}$). As a result, the particle reaches a stationary position on an active electrode before it is accelerated by the next switched electrode. This results in the slight stepped nature of the trajectory. The phenomenon is captured accurately in the numerical and experimental data.

VI. CONCLUSIONS

A moving DEP device employing an array of planar micro-electrodes was presented. Numerical simulation of the electromagnetic field distribution and the dielectrophoretic force profile were calculated. The planar force was found to be mostly in the direction perpendicular to the electrode orientation. Using the force profile, the trajectory of a microparticle in the moving DEP force-field was simulated. To perform the simulation, we developed a Python code that solves the particle trajectory taking into account the relevant physics (i.e., Brownian dynamics, hydrodynamic interactions, etc.). Experimental data showing transport of microparticles were presented. The numerical trajectory model was found to be in good agreement with experimental results. In addition to moving DEP force, the developed numerical model can be used to simulate colloidal particle trajectory for any stationary or time-varying external force-field. The Brownian dynamics model is also valid for sub-micrometer sized particles. The developed Python code is available publicly on GitHub. The code can be of interest for designing/simulating lab-on-a-chip devices that incorporate any type of colloidal particle/sample trapping/manipulation.

SUPPLEMENTARY MATERIAL

See the supplementary material for experimental video and animation of the numerical simulation of particle transport. Additional details regarding the Clausius–Mossotti factor, the effect of different force components, and the field equations are also included.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.