# Analytical Guidelines for Designing Curvature-Induced Dielectrophoretic Particle Manipulation Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**E**is the externally applied electric field, f is the frequency of the applied voltage and ${f}_{CM}$ is the Clausius-Mossotti factor of the particle-fluid system. Based on the method of inducing electrical energy gradients, i.e., $\mathbf{\nabla}\left(E\xb7E\right)$, dielectrophoretic concentration process can be broadly classified as electrode-based (i.e., eDEP, where the gradients are generated by a set of patterned [22,23,24,25,26,27,28] or virtual electrodes [29]), insulator-based (i.e., iDEP, where the gradients are generated by non-uniform cross sections within the microfluidic circuit [30,31,32,33,34,35]), or curvature-induced (i.e., C-iDEP, where the curvature of the microfluidic channel produces unequal electric field intensities across a channel cross section [36]). Simply making the channel curved is sufficient to generate DEP, and hence, the cross section of the microfluidic channel need not be reduced, thereby rendering C-iDEP systems much less susceptible to localised Joule heating effects that are more prevalent in their insulator-based counterparts [37,38,39,40]. Similarly, the typical fluid-reservoir-based electrode insertion outside the flow channel not only renders the fabrication simple, but also keeps these systems relatively safe from electrolysis compared to the metallic microelectrode-based eDEP. Additionally, the curvature facilitates the use of a longer channel length within a given area compared to a straight channel, and this greatly increases the area utilisation of lab-on-a-chip systems implementing C-iDEP [41].

## 2. Theory and Analysis

#### 2.1. Dielectrophoretic Particle Dynamics in a Circular Arc Microchannel

_{c}. The arc subtends an angle $\beta $ at the centre of the curvature of the channel. Assuming (a) a thin electric double layer limit, (b) negligible displacement currents and (c) uniform liquid properties, the applied DC voltage drop generates an electric field ${E}_{\mathrm{DC}}$ inside the microchannel, which is governed by the Laplace equation. In cylindrical coordinates, this can be expressed as

**n**normal to the wall (i.e., the radial direction in this case). Neglecting the surface conduction effects, the force ${\mathbf{F}}_{\mathrm{w}}$ acting on a particle sufficiently smaller than the channel curvature can be estimated up to a first order approximation as [50]

^{*}and a dimensionless angular co-ordinate θ

^{*}as ${r}^{*}=\left(r-{R}_{i}\right)/W$ and ${\theta}^{*}=\theta /\beta $ where $W={R}_{o}-{R}_{i}$ is the channel width. In addition, the mean radius of curvature ${R}_{C}$ of the channel is introduced by using the relation ${R}_{C}={R}_{i}+\left(W/2\right)$. With these substitutions, the non-dimensional form of Equation (15) can be written as

#### 2.2. A Simplified Exact Solution

#### 2.3. The Full Solution

#### 2.4. Data Analysis

#### 2.5. Numerical Model

## 3. Results and Discussions

#### 3.1. Regime 1: $\delta \le 5\%$

#### 3.2. Regime 2: $5<\delta \le 12\%$

#### 3.3. Regime 3: $\delta >12\%$

#### 3.4. Utility for Multiple Arcs

## 4. Conclusions and Future Work

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Psaltis, D.; Quake, S.R.; Yang, C. Developing optofluidic technology through the fusion of microfluidics and optics. Nature
**2006**, 442, 381–386. [Google Scholar] [CrossRef] - Yager, P.; Edwards, T.; Fu, E.; Helton, K.; Nelson, K.; Tam, M.R.; Weigl, B.H. Microfluidic diagnostic technologies for global public health. Nature
**2006**, 442, 412–418. [Google Scholar] [CrossRef] [PubMed] - Pratt, E.D.; Huang, C.; Hawkins, B.G.; Gleghorn, J.P.; Kirby, B.J. Rare cell capture in microfluidic devices. Chem. Eng. Sci.
**2011**, 66, 1508–1522. [Google Scholar] [CrossRef] [Green Version] - Karimi, A.; Yazai, S.; Ardekani, A.M. Review of cell and particle trapping in microfluidic systems. Biomicrofluidics
**2013**, 7, 021501. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kale, A.; Patel, S.; Xuan, X. Three-dimensional reservoir-based dielectrophoresis (rDEP) for enhanced particle enrichment. Micromachines
**2018**, 9, 123. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gossett, D.R.; Weaver, W.M.; Mach, A.J.; Hur, S.C.; Kwong Tse, H.T.; Lee, W.; Amini, H.; DiCarlo, D. Label-free cell separation and sorting in microfluidic systems. Anal. Bioanal. Chem.
**2010**, 397, 3249–3267. [Google Scholar] [CrossRef] [Green Version] - Li, N.; Kale, A.; Stevenson, A.C. Axial acoustic field barrier for fluidic particle manipulation. Appl. Phys. Lett.
**2019**, 114, 013702. [Google Scholar] [CrossRef] - Laurell, T.; Petersson, F.; Nilsson, A. Chip integrated strategies for acoustic separation and manipulation of cells and particles. Chem. Soc. Rev.
**2007**, 36, 492–506. [Google Scholar] [CrossRef] - Pamme, N. Magnetism and microfluidics. Lab Chip
**2006**, 6, 24–38. [Google Scholar] [CrossRef] - Gijs, M.A.; Lacharme, F.; Lehmann, U. Microfluidic applications of magnetic particles for biological analysis and catalysis. Chem. Rev.
**2010**, 110, 1518–1563. [Google Scholar] [CrossRef] - Liang, W.; Wang, S.; Dong, Z.; Lee, G.-B.; Li, W.J. Optical spectrum and electric field waveform dependent optically-induced dielectrophoretic (ODEP) micro-manipulation. Micromachines
**2012**, 3, 492–508. [Google Scholar] [CrossRef] [Green Version] - Wang, M.M.; Tu, E.; Raymond, D.E.; Yang, J.M.; Zhang, H.; Hagen, N.; Dees, B.; Mercer, E.M.; Forster, A.H.; Kariv, I.; et al. Microfluidic sorting of mammalian cells by optical force switching. Nat. Biotechnol.
**2005**, 23, 83–87. [Google Scholar] [CrossRef] - Kale, A.; Song, L.; Lu, X.; Yu, L.; Hu, G.; Xuan, X. Electrothermal enrichment of submicron particles in an insulator-based dielectrophoretic microdevice. Electrophoresis
**2018**, 39, 887–896. [Google Scholar] [CrossRef] [Green Version] - Liu, C.; Hu, G. High-throughput particle manipulation based on hydrodynamic effects in microchannels. Micromachines
**2017**, 8, 73. [Google Scholar] [CrossRef] [Green Version] - Kale, A.; Lu, X.; Patel, S.; Xuan, X. Continuous flow dielectrophoretic trapping and patterning of colloidal particles in a ratchet microchannel. J. Micromech. Microeng.
**2014**, 24, 075007. [Google Scholar] [CrossRef] - Fernandez, R.E.; Rohani, A.; Farmehini, V.; Swami, N.S. Review: Microbial analysis in dielectrophoretic microfluidic systems. Anal. Chim. Acta
**2017**, 966, 11–33. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bazant, M.Z.; Squires, T.M. Induced-charge electrokinetic phenomena. Curr. Opin. Colloid Interface Sci.
**2010**, 15, 203–213. [Google Scholar] - Kale, A. Joule Heating Effects in Insulator-Based Dielectrophoresis Microdevices. Ph.D. Thesis, Clemson University, Clemson, SC, USA, 7 August 2015. [Google Scholar]
- Lu, C.; Verbridge, S.S. Microfluidic Methods for Molecular Biology, 1st ed.; Springer: Manhattan, NY, USA, 2016. [Google Scholar]
- Pohl, H.A. Dielectrophoresis: The Behavior of Neutral Matter in Nonuniform Electric Fields (Cambridge Monographs on Physics); Cambridge University Press: Cambridge, UK, 1978. [Google Scholar]
- Pohl, H.A. The motion and precipitation of suspensoids in divergent electric fields. J. Appl. Phys.
**1951**, 22, 869–871. [Google Scholar] [CrossRef] - Park, B.Y.; Madou, M.J. 3-D electrode designs for flow-through dielectrophoretic systems. Electrophoresis
**2005**, 26, 3745–3757. [Google Scholar] [CrossRef] - Park, S.; Beskok, A. Alternating current electrokinetic motion of colloidal particles on interdigitated microelectrodes. Anal. Chem.
**2008**, 80, 2832–2841. [Google Scholar] [CrossRef] - Demierre, N.; Braschler, T.; Linderholm, P.; Seger, U.; van Lintel, H.; Renaud, P. Characterization and optimization of liquid electrodes for lateral dielectrophoresis. Lab Chip
**2007**, 7, 355–365. [Google Scholar] [CrossRef] [PubMed] - Natu, R.; Martinez-Duarte, R. Numerical model of streaming DEP for stem cell sorting. Micromachines
**2016**, 7, 217. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Morgan, H.; Hughes, M.P.; Green, N.G. Separation of submicron bioparticles by dielectrophoresis. Biophys. J.
**1999**, 77, 516–525. [Google Scholar] [CrossRef] [Green Version] - Tang, S.-Y.; Zhang, W.; Soffe, R.; Nahavandi, S.; Shukla, R.; Khoshmanesh, K. High resolution scanning electron microscopy of cells using dielectrophoresis. PLoS ONE
**2014**, 9, e104109. [Google Scholar] [CrossRef] - Tang, S.-Y.; Zhang, W.; Baratchi, S.; Nasabi, M.; Kalantar-zadeh, K.; Khoshmanesh, K. Modifying dielectrophoretic response of nonviable yeas.t cells by ionic surfactant treatment. Anal. Chem.
**2013**, 85, 6364–6371. [Google Scholar] [CrossRef] [PubMed] - Smith, A.J.; O’Rorke, R.D.; Kale, A.; Rimsa, R.; Tomlinson, M.J.; Kirkham, J.; Davies, A.G.; Walti, C.; Wood, C.D. Rapid cell separation with minimal manipulation for autologous cell therapies. Sci. Rep.
**2017**, 7, 41872. [Google Scholar] [CrossRef] [Green Version] - Hawkins, B.G.; Smith, A.E.; Syed, Y.A.; Kirby, B.J. Continuous-flow particle separation by 3D insulative dielectrophoresis using coherently shaped, DC-biased, AC electric fields. Anal. Chem.
**2007**, 79, 7291–7300. [Google Scholar] [CrossRef] - Romero-Creel, M.F.; Goodrich, E.; Polniak, D.V.; Lapizco-Encinas, B.H. Assessment of sub-micron particles by exploiting charge differences with dielectrophoresis. Micromachines
**2017**, 8, 239. [Google Scholar] [CrossRef] [Green Version] - Chen, K.P.; Pacheco, J.R.; Hayes, M.A.; Staton, S.J.R. Insulator-based dielectrophoretic separation of small particles in a sawtooth channel. Electrophoresis
**2009**, 30, 1441–1448. [Google Scholar] [CrossRef] [PubMed] - Sanghavi, B.J.; Varhue, W.; Rohani, A.; Liao, K.T.; Bazydlo, L.A.L.; Chou, C.-F.; Swami, N.S. Ultrafast immunoassays by coupling dielectrophoretic biomarker enrichment in nanoslit channel with electrochemical detection on graphene. Lab Chip
**2015**, 15, 4563–4570. [Google Scholar] [CrossRef] [PubMed] - Cao, Z.; Zhu, Y.; Liu, Y.; Dong, S.; Chen, X.; Bai, F.; Song, S.; Fu, J. Dielectrophoresis-based protein enrichment for a highly sensitive immunoassay using Ag/SiO2 nanorod arrays. Small
**2018**, 14, 1703265. [Google Scholar] [CrossRef] [PubMed] - Zellner, P.; Shake, T.; Hosseini, Y.; Nakidde, D.; Riquelme, L.V.; Sahari, A.; Pruden, A.; Behkam, B.; Agah, M. 3D insulator-based dielectrophoresis using DC-biased, AC electric fields for selective bacterial trapping. Electrophoresis
**2015**, 36, 277–283. [Google Scholar] [CrossRef] [PubMed] - Zhu, J.; Xuan, X. Particle electrophoresis and dielectrophoresis in curved microchannels. J. Colloid Int. Sci.
**2009**, 340, 285–290. [Google Scholar] [CrossRef] [PubMed] - Xuan, X. Joule heating in electrokinetic flow. Electrophoresis
**2008**, 29, 33–43. [Google Scholar] [CrossRef] [PubMed] - Hawkins, B.J.; Kirby, B.J. Electrothermal flow effects in insulating (electrodeless) dielectrophoresis systems. Electrophoresis
**2010**, 31, 3622–3633. [Google Scholar] [CrossRef] - Kale, A.; Patel, S.; Qian, S.; Hu, G.; Xuan, X. Joule heating effects on reservoir-based dielectrophoresis. Electrophoresis
**2014**, 36, 721–727. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Prabhakaran, R.A.; Zhou, Y.; Patel, S.; Kale, A.; Song, Y.; Hu, G.; Xuan, X. Joule heating effects on electroosmotic entry flow. Electrophoresis
**2017**, 38, 572–579. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Xuan, X. Curvature-Induced Dielectrophoresis (C-iDEP) for Microfluidic Particle and Cell Manipulations. In International Conference on Nanochannels, Microchannels, and Minichannels; Paper No: ICNMM2012-73042; American Society of Mechanical Engineers: New York, NY, USA, 2012; pp. 681–685. [Google Scholar]
- Church, C.; Zhu, J.; Xuan, X. Negative dielectrophoresis-based particle separation by size in a serpentine microchannel. Electrophoresis
**2011**, 32, 527–531. [Google Scholar] [CrossRef] [PubMed] - Zhu, J.; Xuan, X. Curvature-induced dielectrophoresis for continuous separation of particles by charge in spiral microchannels. Biomicrofluidics
**2011**, 5, 024111. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dubose, J.; Zhu, J.; Patel, S.; Lu, X.; Tupper, N.; Stonaker, J.M.; Xuan, X. Electrokinetic particle separation in a single-spiral microchannel. J. Micromech. Microeng.
**2014**, 24, 115018. [Google Scholar] [CrossRef] - Zhu, J.; Canter, R.C.; Keten, G.; Vedantam, P.; Tzeng, T.R.J.; Xuan, X. Continuous-flow particle and cell separations in a serpentine microchannel via curvature-induced dielectrophoresis. Microfluid. Nanofluid.
**2011**, 11, 743–752. [Google Scholar] [CrossRef] - Zhu, J.; Tzeng, T.R.J.; Xuan, X. Continuous dielectrophoretic separation of particles in a spiral microchannel. Electrophoresis
**2010**, 31, 1382–1388. [Google Scholar] [CrossRef] - Li, M.; Li, S.; Li, W.; Wen, W.; Alici, G. Continuous manipulation and separation of particles using combined obstacle- and curvature-induced direct current dielectrophoresis. Electrophoresis
**2013**, 34, 952–960. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cummings, E.B.; Griffiths, S.K.; Nilson, R.H.; Paul, P.H. Conditions for Similitude between the Fluid Velocity and Electric Field in Electroosmotic Flow. Anal. Chem.
**2000**, 72, 2526–2532. [Google Scholar] [CrossRef] [Green Version] - Kirby, B.J.; Hasselbrink, E.F. Zeta potential of microfluidic substrates: 2. Data for polymers. Electrophoresis
**2004**, 25, 203–213. [Google Scholar] [CrossRef] [PubMed] - Yariv, E. “Force-free” electrophoresis? Phys. Fluids
**2006**, 18, 031702. [Google Scholar] [CrossRef] - Martinez-Duarte, R. Carbon-Electrode Dielectrophoresis for Bioparticle Manipulation. ECS Trans.
**2014**, 61, 11. [Google Scholar] [CrossRef] - Islam, M.; Natu, R.; Larraga-Martinez, M.F.; Marntinez-Duarte, R. Enrichment of diluted cell populations from large sample volumes using 3D carbon-electrode dielectrophoresis. Biomicrofluidics
**2016**, 10, 033107. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Schematic of a circular arc microchannel explaining the dimensions and the r-θ-z coordinate system, and framework used for a theoretical analysis of the underlying physics of curvature-induced dielectrophoresis (C-iDEP). The 2D projection of the r-θ section plane shows the components of the particle velocity, ${\mathbf{u}}_{\mathrm{P}}$, inside the microchannel, consisting of the stream-wise electrokinetic (${\mathbf{u}}_{\mathrm{EK}}$) component and the cross-stream dielectrophoretic deflection (${\mathbf{u}}_{\mathrm{Def}}$) component. The electric field contour (the darker colour the larger magnitude) and lines (equivalent to the fluid streamlines [48]) are superimposed to explain the orientation of the velocity components relative to the electric field. A situation of negative DEP is shown, where the particle will move from the inner channel wall to the outer wall. For a positive DEP, the resultant velocity and the deflection component will reverse their directions. Note that the underlying physics of C-iDEP are two dimensional because of the application of the voltage over the entire r-z plane.

**Figure 2.**% Deviation of the dimensionless parameter $\left[\frac{\left|{V}_{App}{}^{*}\right|\left|Real\left({f}_{CM}\right)\right|}{{R}_{C}{}^{*}}\right]$ generated from the curve fitting equations for pDEP (Equation (27)) and nDEP (Equation (28)) from its value predicted by the exact solution (Equation (21)) as a function of the dimensionless particle diameter ${d}^{*}$. Note that the curve fitting equations deviate increasingly from the exact solution for both pDEP and nDEP with increasing particle diameters, indicating the increasing influence of the wall repulsion forces on the C-iDEP particle dynamics with particle blockage.

**Figure 3.**Numerically predicted path-lines for two test particles of a dimensionless diameter ${d}^{*}=0.005$ under the action of (

**a**) positive DEP ($Real\left({f}_{CM}\right)=0.5$) and (

**b**) negative DEP ($Real\left({f}_{CM}\right)=-0.5$) inside a circular arc microchannel having a curvature ratio ${R}_{C}{}^{*}$ of 5. The dimensionless applied voltage $\left|{V}_{App}{}^{*}\right|$ for both the DEP directions is 400,000, as calculated from the exact solution. The ideal particle is represented by the red colour and its motion is governed by the exact solution. The realistic particle is represented by the blue colour and its motion is governed by the full solution of particle motion inclusive of wall repulsion effects. This particle size demonstrates a regime 1 behaviour, where the design parameters for the C-iDEP microchannel can be determined reliably by the exact solution. The inset images show the DEP velocity vectors superimposed over the electric field norms, confirming an agreement of the simulated particle dynamics with their theoretical predictions.

**Figure 4.**Numerically predicted path-lines for the ideal and realistic test particles under the action of (

**a**) positive DEP ($Real\left({f}_{CM}\right)=0.5$) and (

**b**) negative DEP ($Real\left({f}_{CM}\right)=-0.5$) inside a circular arc microchannel having a curvature ratio ${R}_{C}{}^{*}$ of 5. The dimensionless particle diameter ${d}^{*}$ is chosen as 0.08 for positive DEP and its motion is driven by a dimensionless voltage of $\left|{V}_{App}{}^{*}\right|=1404.85$. Similarly, the dimensionless particle diameter ${d}^{*}$ is chosen as 0.05 for negtive DEP and its motion is driven by a dimensionless voltage of $\left|{V}_{App}{}^{*}\right|=3569.5$. These particle sizes clearly demonstrate a regime 2 behaviour, where the design parameters for the C-iDEP microchannel can be determined reasonably by the exact solution but more reliably by the curve fitting equations.

**Figure 5.**Numerically predicted path-lines for the ideal and realistic test particles under the action of (

**a**) positive DEP ($Real\left({f}_{CM}\right)=0.5$) and (

**b**) negative DEP ($Real\left({f}_{CM}\right)=-0.5$) inside a circular arc microchannel having a curvature ratio ${R}_{C}{}^{*}$ of 5. The dimensionless particle diameter ${d}^{*}$ is chosen as 0.12 for positive DEP and its motion is driven by a dimensionless voltage of $\left|{V}_{App}{}^{*}\right|=694.44$ predicted from the exact solution. Similarly, the dimensionless particle diameter ${d}^{*}$ is chosen as 0.08 for negative DEP and its motion is driven by a dimensionless voltage of $\left|{V}_{App}{}^{*}\right|=1562.5$ as predicted by the exact solution. These particle sizes demonstrate a regime 3 behaviour, where the realistic particle motion deviate significantly from the ideal behaviour. This is evident from the fact that the realistic particles in both cases of DEP are focused substantially before they reach the microchannel outlet. The location of full focusing along the arc length is highlighted by the dotted lines, where a kink in the particle path-line is visible, and the particle is seen traveling parallel to the channel wall beyond that point. This regime is characterised by more reduced voltage requirements to focus the particles fully due to the assistance of wall repulsion forces. The inset images represent a 1-D radial profile of the net radial particle velocity component, along with the identification of the equilibrium radial co-ordinates.

**Figure 6.**Numerically predicted behaviour of the negative DEP particle dynamics ($Real\left({f}_{CM}\right)=-0.5$) inside a two-turn microchannel having a curvature ratio of ${R}_{C}{}^{*}$ of 5. (

**a**) Numerically predicted dimensionless electric potential plot for a two-turn channel with opposing turns. (

**b**) Path-lines of particles of diameter ${d}^{*}=0.02$ undergoing a DEP motion for a two-turn channel with opposing turns, under an applied voltage of $\left|{V}_{App}{}^{*}\right|=\mathrm{25,000}$, with the inset showing the final position of the particle. (

**c**) Path-lines of particles of diameter ${d}^{*}=0.02$ undergoing a DEP motion for a two-turn channel with unidirectional turns with other conditions identical to the opposing turn geometry.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kale, A.; Malekanfard, A.; Xuan, X.
Analytical Guidelines for Designing Curvature-Induced Dielectrophoretic Particle Manipulation Systems. *Micromachines* **2020**, *11*, 707.
https://doi.org/10.3390/mi11070707

**AMA Style**

Kale A, Malekanfard A, Xuan X.
Analytical Guidelines for Designing Curvature-Induced Dielectrophoretic Particle Manipulation Systems. *Micromachines*. 2020; 11(7):707.
https://doi.org/10.3390/mi11070707

**Chicago/Turabian Style**

Kale, Akshay, Amirreza Malekanfard, and Xiangchun Xuan.
2020. "Analytical Guidelines for Designing Curvature-Induced Dielectrophoretic Particle Manipulation Systems" *Micromachines* 11, no. 7: 707.
https://doi.org/10.3390/mi11070707