AC Electrokinetic Templating of Colloidal Particle Assemblies: Effectof Electrohydrodynamic FlowsJeffery A. Wood and Aristides Docoslis*

Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada

ABSTRACT: The use of spatially nonuniform electric fields for the contact-freecolloidal particle assembly into ordered structures of various length scales is aresearch area of great interest. In the present work, numerical simulations areundertaken in order to advance our understanding of the physical mechanisms thatgovern this colloidal assembly process and their relation to the electric fieldcharacteristics and colloidal system properties. More specifically, the electric-fielddriven assembly of colloidal silica (dp = 0.32 and 2 μm) in DMSO, a near indexmatching fluid, is studied numerically over a range of voltages and concentrationby means of a continuum thermodynamic approach. The equilibrium (uf = 0)and nonequilibrium (u f ≠ 0) cases were compared to determine whether fluidmotion had an effect on the shape and size of assemblies. It was found that thenonequilibrium case was substantially different versus the equilibrium case, in bothsize and shape of the assembled structure. This dependence was related to therelative magnitudes of the electric-field driven convective motion of particles versusthe fluid velocity. Fluid velocity magnitudes on the order of mm/s were predicted for 0.32 μm particles at 1% initial solidscontent, and the induced fluid velocity was found to be larger at the same voltage/initial volume fraction as the particle sizedecreased, owing to a larger contribution from entropic forces.

■ INTRODUCTIONThe assembly of colloidal particles into ordered structuresof larger characteristic dimensions (for example, on micro orlarger scale) is an active topic of research for the creation ofnovel materials or materials with enhanced functionality, suchas photonic bandgap crystals (PBG), high sensitivity sensors,or microelectronics.1 Numerous techniques exist in order tofacilitate the creation of ordered colloidal structures, rangingfrom self-assembly, template-assisted assembly (topological/geometric assisted assembly), and external field assistedassembly (gravitational, electric, or magnetic fields).2−4 Thesetechniques vary in complexity and in scale of assembly, that is,in the final characteristic dimension of the usable device/material. Self-assembled monolayers can be used in order todirect the placement of colloidal and other micro/nanoscalematerials to specific substrate locations, such as carbon nano-tubes or gold nanoparticles.3 Template-assisted assembly can beconsidered as complementary to all the other listed techniques.Electric and magnetic fields offer a wider variety of controlforces versus relying primarily on gravity and can be integratedwith template-assisted assembly. These types of fields are typi-cally generated using planar microelectrodes made by micro-fabrication techniques (i.e., small characteristic dimensions) togenerate very high field intensities.5

Specifically, the use of dielectrophoresis (force on a particledue to induced dipole in a spatially nonuniform electric field)to construct structures of varying size, order, and geometryfrom micrometer and nanosized building blocks is of significantresearch interest. This is due to the large degree of control param-eters available for determining the final assembled structure,

such as applied voltage, frequency, choice of medium, andparticle electrical property contrast and overall electrode geom-etry.5 Docoslis and Alexandridis first demonstrated the use ofdielectrophoresis for the assembly of three-dimensional (3D)colloidal structures, using 100 μm gap quadrupolar electrodesto assemble silica and latex colloids.6 Abe et al. examined theuse of 400 μm gap hyperbolic electrodes for assembly of largepolystyrene (PS) colloids (2−10 μm) and were able to achievesingle or multiple layer structures by applying an AC and DCfield simultaneously.7 Lumsdon et al. formed reversible 2dcolloidal crystals of monodisperse polystyrene using appliedelectric fields generated by coplanar electrodes, with the assem-blies diffracting light parallel, perpendicular or both to theassembly depending on the assembly time.8,9 Reversible two-dimensional (2D) colloidal crystals fabricated by electric fieldshave also been shown by numerous authors.10−13 Similarly,binary PS colloidal crystals and aggregates have been formedusing application of DC and AC sources respectively across asimple coplanar electrode structures.14,15 Large-scale colloidalcrystals 200 μm in size have been formed without significantgrain-boundaries by successive application and relaxation of anapplied electric field in a hexapolar electrode system.16 Morerecently, three-dimensional structures of varying complexity hasbeen demonstrated using dielectrophoresis and induced-chargeelectroosmotic flows.17 For a comprehensive overview of theuse of dielectrophoresis for colloidal assembly, and for colloidal

Received: December 13, 2011Revised: February 4, 2012Published: February 10, 2012

Article

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assembly in general, the articles of Velev and Bhatt, and Velevand Gupta are recommended.18,19

In terms of simulations, most of the work to date indescribing the assembly/patterning of colloids with dielec-trophoresis has focused on the force on an isolated particle todetermine trajectories, assuming no interparticle electric fieldinteractions and generally relying on the point-dipole or seriesexpansion of multipoles to account for particle perturbation ofthe local electric field.5,20−22 Some more involved numericalmethods, such as using distributed lagrange multipliers (DLM)have also been used, but also often rely on simplified electricalinteraction equations and have been limited to date to solvingthe case with a small number of particles.23 More recently,Juarez et al. showed that inverse Monte Carlo (MC) basedsimulations could be used to reconstruct colloidal densityprofiles in nonuniform electric fields, specifically 2Dassembly in finger electrodes and 3d assembly in quadrupolarelectrodes.24,25 This approach is not predictive, the experimentalradial distribution functions of an assembled colloidal structure weremeasured via confocal microscopy and used for updating aseries of MC steps to obtain a fitted frequency correctionfactor. However, this approach was able to reconstruct veryaccurately assemblies of large colloidal particles (1.5 μm for the3D case) under the action of applied nonuniform electric fields,as well as crystallinity. The question of how well this approachwill be able to capture other phenomena of interest, such asdistortions to the electric field by particles, change in thedipole-coefficient due to multiparticle effects, and so forth, isyet to be explored, although overall it does represent a veryinteresting framework. The effect of persistent fluid flows wasnot considered in this work, as the authors were interested inan equilibrium solution. For smaller particles, which will expe-rience higher relative diffusional fluxes as well as smaller electricfield induced convective motion, the contribution of fluid flowsmay not be negligible. The effect of fluid flow in nonuniformelectric field driven colloidal assembly has generally not beenparticularly discussed in literature to date and is generallyassumed to be negligible compared to field−dipole and dipole−dipole interactions on particles, or assumed dominated by anexternally driven-flow field.26 Given the difference in character-istic time-scales for diffusion and convective (electric field,gravity, etc.) driven motion, the influence of fluid flow on theresulting structure may not be negligible; that is, potentiallydifferent structures may be arrived at by allowing particles tosettle and then applying an electric field versus direct applicationof a field. In the equilibrium case, no difference would beexpected between allowing particles to settle before applying thefield versus applying the field directly, as the chemical potentialgradient of the system is zero for the equilibrium case.In this paper, the simulation of electric-field induced colloidal

pattern formation by quadrupolar electrodes was considered,specifically for the case of silica particles (dP =0.32 and 2 μm) inDMSO, a near index-matching fluid. As a theoretical basis, thecontinuum thermodynamic approach for describing the freeenergy of a colloidal suspension subjected to an electric fielddeveloped by Khusid and Acrivos was chosen.27 With thisapproach, the free energy, as well as electric field and physicaltransport properties of the suspension, are treated as functionsof volume fraction. This framework accounts for interparticleinteractions, is more numerically tractable in the sense ofsolving a single PDE for volume fraction vs multiple ODEs forindividual particles and has been successfully applied to predictthe formation of electric field driven volume fraction fronts.26,28

The primary aim of this work was to examine the influence offluid flow resulting from particle motion on electric-fieldinduced assembly of monodisperse colloids into structures ofvarious shape/size, specifically to look at the “equilibrium” orquiescent fluid flow case (uf = 0) versus the “nonequilibrium”or nonquiescent fluid flow case (u f ≠ 0). As mentioned, theeffect of any potential fluid flows in this type of system has beengenerally either neglected or not considered in previoussimulation work to date and therefore quantifying any impacton electric-field induced assemblies is quite important.Including the influence of fluid flow is particularly straightfor-ward in this framework, as the fluid is treated as an incom-pressible liquid with volume fraction dependent viscosity anddensity that experiences a volumetric force directly proportionalto the gradient of chemical potential. Simulations of theinfluence of particle size, voltage and initial volume fraction onthe predictions of the shape and size of the resulting colloidalassembly were performed with comparison between the equilib-rium and nonequilibrium case. Some validation by comparison toexperimental data in the form of optical microscopy images wasalso attempted.

■ THEORETICAL BACKGROUNDThe framework developed by Khusid and co-workers has been used bythese same authors in later work to describe electric field inducedformation of volume fraction fronts quite successfully as a predictivemodel with no fitting parameters.26,28 To describe the motion ofparticles, the chemical potential of the suspension is developed byadopting a hard-sphere approach, where entropic contributionsto chemical potential are given by classical hard sphere results andthe electrical contributions are derived from a cell model (whichcorresponds to Maxwell−Wagner type polarization).27 The attractivepart of the potential is assumed to be dominated by electric-fieldinduced interactions. In this work, a similar approach is used, withsome modification to account for additional volume fraction-dependent effects (diffusion coefficient), for describing the assemblyprocess of monodisperse silica colloids with a nonuniform electric fieldgenerated by quadrupolar microelectrodes with a gap spacing of100 μm. We also assumed that the fluid flows resulting from particlemotion are not necessarily negligible at the conditions studied for thisframework, with the idea of testing this assumption through compa-rison of the equilibrium and nonequilibrium cases. The time-averagedchemical potential (μp) and osmotic pressure (Πp) of a colloidalsuspension under the influence of an external electric field at highfrequencies are given by eqs 1 and 3, respectively28

μ = −ε ε

⟨| | ⟩⎛⎝⎜

⎞⎠⎟

k Tv

f

c cE

d

d 2ddp

B

p

0 0 s 2

(1)

∫= + −f c

cc

Z cc

clne

( ) 1d

c0 0 (2)

Π = +ε

ε −ε

⟨| | ⟩⎡⎣⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥

k Tv

cZ cc

E2

ddp

B

p

0s

s 2

(3)

where kBT is thermal energy, c is particle volume fraction, Z is thesuspension compressibility factor, εs is the real part of suspensionpermittivity and (dεs/dc) is the derivative of the real part of sus-pension permittivity with respect to particle volume fraction. f 0represents the entropic contributions to the free energy of the systemand is determined via an equation of state. The imaginary componentof complex permittivity is negligibly small compared with the real partat the frequencies of interest in this work (MHz range), and this highfrequency will minimize the influence of conductivity on the polariz-ability of the particle. The suspension compressibility factor can becalculated using the Carnahan−Starling equation, with the divergingvolume fraction chosen as that of a random close-packed suspension,

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eq 4, and the suspension permittivity from the Maxwell−Wagnerexpression, eq 5:28

=

+ + −−

< ≤

−< <

⎧

⎨⎪⎪

⎩⎪⎪

Z

c c cc

c

cc

1

(1 )0 0.5

1.850.64

0.5 0.64

2 3

3

(4)

ε = ε + β− β

β =ε − ε

ε + ε

⎡⎣⎢

⎤⎦⎥

cc

1 21 2s m

p m

p m (5)

where in eq 5, εi is the permittivity of the suspension, medium, orparticle (subscript s, m, or p respectively), and β is the real part of theClausius−Mossotti factor. More complicated models accounting forvolume fraction-dependent effects on polarization could be used, suchas the model proposed by Sihvola and Kong, but Maxwell−Wagnertype polarization was assumed for simplicity, as the interest in thiswork was in testing the influence of fluid flows resulting from thechemical potential gradients arising in the system.29

Along the coexistence curve for the single phase region of thesuspension versus a two-phase region induced by the electric fieldeffects, the value of chemical potential and osmotic pressure of eachphase are equal. The volume fraction of particles in each phase isdenoted by c1 and c2, where 1 refers to the low solids volume fractionphase and 2 to the high solids volume fraction phase respectively, asshown in eq 6:27

μ = μ Π = Πc c c c( ) ( ) and ( ) ( )p 1 p 2 p 1 p 2 (6)

For electric-field induced aggregation the value of the spinodalcomposition, or metastable limit of the system, can also be useful forillustrating pattern formation. The spinodal composition is found fromsolutions to eq 7:

+ −ε ε

⟨| | ⟩ =⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟⎛⎝⎜⎜

⎞⎠⎟⎟Z c

Zc

cv

k T cE

dd 2

d

d0

p

B

02

s2

2

(7)

The solution of eqs 6 and 7 allows for the phase diagram (coexistenceand spinodal curves) of volume fraction versus applied field strengthto be determined for a given particle−medium combination and thecritical point common to both indicates the minimum value of electricfield strength required to drive particle aggregation.The electric field profile of the suspension can be determined using

Gauss’ law with the permittivity described by eq 5:

∇· ε ∇φ = = −∇φc E[ ( ) ] 0 ands (8)

The particle volume fraction profile evolves according to eq 9

∂∂

+ ∇· + = = −π η

−∇μct

cu j jc c

r c[ ] 0 and

(1 )6 ( )

[ ]f p p

2

p sp

(9)

where uf is the fluid velocity and jp represents particle flux.

Equation 9 can be rewritten in terms of the electrical and entropic(diffusional) contributions, specifically as

∂∂

+ ∇· − ∇ + + =ct

D c u u c[ ( ) ] 0f elec (10)

where the diffusivity, D, and electric-field induced velocity, u elec, aregiven by eqs 11 and 12

=−

π ηD

k T cr c

cZc

(1 )6 ( )

d( )d

B2

p s (11)

=−

π ηε

∇ε

⟨| | ⟩⎡⎣⎢⎛⎝⎜

⎞⎠⎟

⎤⎦⎥u

c v

r c cE

(1 )

6 ( ) 2ddelec

2p

p s

0 s 2

(12)

The factor (1 − c)2 is referred to as the hindrance function, andcombined with the particle volume fraction dependent viscosity istaken to account for all hydrodynamic interactions of particles in thesuspension. The suspension viscosity is treated using the Leighton-Acrivos equation:28

η = η − −c c c( ) (1 / ) cs m max

2.5 max (13)

Finally, a momentum balance on the fluid yields

ρ∂ ∂

+ ·∇ = −∇ + ∇· ∇ + ∇ − ∇μ

∇· =

⎛⎝⎜

⎞⎠⎟

⎡⎣ ⎤⎦ut

u u p u u c

u

and

0

Ts

ff f f f p

f (14)

The Navier−Stokes form of the momentum balance was choseninitially, as although the expected Reynolds number is quite low,inertial effects were not necessarily negligible due to the large gradientsthat can potentially arise in this system. Subsequent simulationsshowed in fact that the inertial term was negligible and Stokes’ flowcan be assumed. Solution of eqs 8, 9, and 14 allows for the time-evolution profile of particle volume fraction within the system to bedetermined. Examination of the force term in eq 14 shows that in thelimit of dilute suspensions (c→ 0) the fluid velocity becomes zero andthe steady-state solution will be equal to the equilibrium solution.

■ MATERIALS AND METHODSSimulation Details. Numerical solution of the volume fraction,

velocity, and electric field profiles was achieved through solving thetime-dependent PDEs described previously using FEM techniques inComsol Multiphysics (Burlington, MA). A 2D representation of theelectrode geometry was drawn, with the system solved using a one-eighth symmetry in terms of the mirror about the 45° line extendingfrom the origin, as is illustrated in Figure 1. Quadrilateral elementswere used for improving resolution of the nonlinearities present inthe system of PDEs, which arise due to volume fraction-dependenceof physical properties of the suspension. as well as to improvetime-stepping stability.30 The electric field solution was obtainedthrough solving for potential, with the potential treated using quadratic

Figure 1. Left: Top-down system view divided into eighths. Right: Expanded one-eighth view.

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Lagrange elements. For the fluid velocity, the Navier−Stokesequations were solved using quadratic Lagrange elements for thevelocity components and linear for pressure, while the volume fractionutilized linear Lagrange elements to handle the sharp gradients/discontinuities in volume fraction and for better stability in terms oftime-stepping. Numerical stabilization of the particle volume fractionprofile was achieved through the use of artificial diffusion, in the formof an O(h2) isotropic stabilization. Artificial diffusion was requiredfor solution of this convection-dominated PDE, particularly in the caseof larger particle sizes/higher applied voltages, to avoid spuriousoscillations and nonphysical results.31 Mesh-independence tests wereperformed to ensure decoupling of the solution with mesh quality andadditionally mass conservation in the suspension versus time was alsoconsidered and found to be negligible (less than 1e-10% relativedeviation from initial to final integration time for conditions studied).Experimental Details. Aqueous suspensions of silica particles

(dP = 0.32, 1, and 2 μm) were purchased from Bangs Laboratories(Fishers, IN). These suspensions were directly diluted to desiredvolume fractions using DMSO (Sigma Aldrich, Canada) and containedsmall amounts of water (<2 vol % maximum). DMSO is a near-indexmatching solvent for silica, which will allow for minimizing the van derWaals interactions, and also will result in negligible acid−base (polar)interactions between particles as determined via calculations usingXDLVO theory.32,33 DMSO also is a spreading liquid, meaning it is avery good candidate as a suspending fluid for silica particles due to thethin liquid, flat liquid films which will spread on the electrode surfacewhich will minimize settling effects and any surface tension drivenflows. Use of DMSO, a high boiling point solvent, allowed for a verythin, nonevaporating film of liquid to be placed on a chip (500 μmthick silicon with a 0.5 μm layer of SiO2 separating the substrate fromthe microelectrodes). The suspension is not perfectly index matched inorder to allow for sufficient contrast for observation by opticalmicroscopy. As DMSO is a spreading liquid on silica surfaces, a 1/8″(3.175 mm) diameter circular well was constructed using a double-sided press to seal adhesive with parafilm on top. This circular edgeacted to constrain any droplet on the chip. Then 0.5 μL aliquots ofsuspension were pipetted for each experimental run. Nonuniform ACelectric fields were generated by using gold microelectrodes (200 nmthick) fabricated via photolithography on a SiO2 surface (0.5 μm thick)deposited on top of a silicon wafer as previously mentioned. Thefrequency of all experiments was chosen as 1 MHz, to eliminate theconductivity effects on particle polarization. The tip-to-tip distancebetween opposite electrodes ( c) was 100 μm. Power to the micro-electrodes was supplied by a signal generator (BK Precision 4040A).Microelectrodes were connected to the source in an alternating fashion(180° phase difference between adjacent electrodes). The value ofthe applied voltage (V, peak-to-peak) and applied frequency ( f) weremonitored by an oscilloscope (Tektronix 1002B). A top-down opticalmicroscopy image of the electrodes is provided in Figure 2. Opticalmicroscopy was performed using an Olympus BHM microscope, witha digital camera for image capture, to observe the suspension behaviorupon application of an electric field.

■ RESULTS AND DISCUSSION

Simulation Conditions. To determine the effectiveness ofusing the Khusid−Acrivos framework for predicting shapeand size of electric field induced structures, a number of simula-tion conditions were chosen and compared with experimentalcounterparts. The behavior of silica particles of 0.32 and 2 μmdiameter at an initial (uniform) suspension of 0.1 and 1%particles by volume, respectively, were simulated at variousvoltages, ranging from 5 to 20 V (peak to peak) for 0.32 μmand 0.5 to 5 V for 2 μm, by solving for volume fraction, velocity,and electric field profiles simultaneously versus time for 2 h(0.32 μm) and 10 min (2 μm). 2 and 0.32 μm were chosen asparticle sizes in order to examine the influences of entropic andelectrical contributions on the final assembly profile (volumefraction of particles), as well as on the dielectrophoretic and fluid

velocity profiles in the system. Higher voltages for 2 μm simu-lations were not considered due to experimental results showingthat multilayer structures clearly arose well before 20 V. Based on

Figure 2. Top-down view of 100 μm gap spacing hyperbolic micro-electrode chip [scale bar is 100 μm].

Figure 3. Visualization planes for simulation [scale bar is 100 μm].

Figure 4. Electric field strength at Vapplied = 20 V peak-to-peak at t = 0[scale bar is 100 μm].

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order of magnitude estimates for transit and aggregation timeof particles under the influence of electric fields and at thefield strengths chosen 2 h for 0.32 μm and 10 min for 2 μmrepresents a more than sufficient time for the system to be at∼steady-state.27 This was confirmed by observing that the finalshape of the simulated volume fraction profile ceased to changewell before the final simulation time was reached, as wellas through optical microscopy observations of experimentalequivalents where possible. Both the quiescent (equilibrium)and nonquiescent (nonequilibrium) cases were solved at eachvoltage, initial volume fraction and particle size in order to

determine what impact the fluid flows resulting from electricaland diffusional driven particle fluxes have on the steady-stateparticle volume fraction (c) profile.Visualization of the resulting volume fraction profiles was

focused on a 100 μm × 100 μm box at the center betweenelectrodes, shown in Figure 3 as region A, while fluid anddielectrophoretic velocity profiles are visualized on a one-quartercutout of the entire plane in Figure 3 bounded by the dotted-line.A representative electric field profile at t = 0, that is, uniformvolume fraction, with 20 V applied peak to peak is shown inFigure 4. At uniform volume fraction, the suspension permittivity

Figure 5. Particle volume fractions with and without fluid flow for dP = 0.32 μm in DMSO, c0 =0.1%: (a) 5 V equilibrium case, (b) 5 V with fluidflow, (c) 10 V equilibrium case, (d) 10 V with fluid flow, (e) 15 V equilibrium case, (f) 15 V with fluid flow, (g) 20 V equilibrium case, and (h) 20 Vwith fluid flow [scale bar is 50 μm].

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is constant throughout the system and eq 8 becomes the linearLaplace equation. The electric field maximum occurs at thesmallest gap spacing between adjacent electrodes, which is ap-proximately at the location labeled B in Figure 3. The maxi-mum electric field intensity in the system is approximatelyconstant versus time at the volume fractions studied, as particlesare pushed away from region B meaning the suspension per-mittivity is essentially constant in the gap between electrodes.For 0.32 μm particles, the simulated volume fraction profilesafter 7200s are shown in Figure 5 (c0 = 0.1%) and Figure 6(c0 = 1%) for both the equilibrium and nonequilibrium casesat 5, 10, 15, and 20 V. As can be seen from comparing the

equilibrium to nonequilibrium cases, at both initial volumefractions there is an impact on the final volume fraction profileobtained. For c0 = 0.1%, as voltage increases the shape of theevolved structures for both equilibrium and nonequilibriumbecomes similar, with the nonequilibrium case assembliesbeing slightly larger (for example, at the 20 V case, ∼ 24 versus∼20 μm). Additionally, the equilibrium case simulationsachieve a more “diamond” shape at earlier voltages, whilethe nonequilibrium case remains more rounded until 15 V,Figure 5h). The maximum volume fraction in the system is alsohigher in the equilibrium case versus the nonequilibrium untilthe system reaches 20 V. For the c0 = 1% case, there is a more

Figure 6. Particle volume fractions with and without fluid flow of dP = 0.32 μm silica in DMSO, c0 =1%: (a) 5 V equilibrium case, (b) 5 V with fluidflow, (c) 10 V equilibrium case, (d) 10 V with fluid flow, (e) 15 V equilibrium case, (f) 15 V with fluid flow, (g) 20 V equilibrium case, and (h) 20 Vwith fluid flow [scale bar is 50 μm].

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dramatic impact on the assembly profile, as can be seen fromcomparing equilibrium to nonequilibrium cases in Figure 6.Comparing the equilibrium and nonequilibrium case at c0 = 1%,the former yields a much larger predicted assembly which tendsfrom circular shape at lower voltages to a more roundeddiamond at higher voltages, while the latter is far smaller in sizeand maximum volume fraction as well as progressing frommore inwardly rounded diamond/hyperbola-like shapes to adiamond as voltage is increased. At the higher initial volumefraction in the system, there is somewhat counterintuitively asmaller steady-state assembly and lower maximum volumefraction versus the c0 = 0.1% case when the influence of fluidflow is considered. However, examination of the electrical andfluid velocities for 0.32 μm particles at the given volumefractions and voltages helps to explain this effect.Figure 7 depicts the dielectrophoretic and fluid velocity

profiles for 0.32 μm silica for both c0 = 0.1% and 1% at 5 V in aone-quarter cutout of the visualization plane, Figure 3. Even atlower voltages, the fluid velocity is substantially larger versusthe dielectrophoretic velocity at both initial volume fractionsand this trend continues throughout all voltages studied. Thefluid velocity is several orders of magnitude larger than thedielectrophoretic velocity. As well, from examining both figures,a recirculation zone can be seen in the bottom-left corner of thefigures, meaning near the center of the system significant fluidrolls are predicted. These fluid rolls are responsible for the shiftin shape and size of the structures between the equilibrium andnonequilibrium cases, with this effect more pronounced athigher initial solids volume fraction as the fluid velocityincreases while the dielectrophoretic velocity is slightlydecreased in fact. Increasing the initial solids volume fractionalso leaves the dielectrophoretic and fluid velocity spatialprofiles nearly unchanged, meaning the maxima/minima occurat nearly the same locations. For c0 = 1% and 5 V, the fluid

velocity maximum is approximately 2 orders of magnitudegreater than the dielectrophoretic maximum, whereas it iscloser to 1 order of magnitude at c0 = 0.1%, and a similar resultis obtained at higher voltages. This dominance of fluid flow andthe resulting change in the assembly profile illustrates theimportance of accounting for fluid flow effects for smallerparticles, for which electric-field driven convection effects onparticles are weaker. For 0.32 μm particles at the voltagesstudied, the maximum fluid velocity in all simulation cases wasfound to be on the order of mm/s (20 V, c0 = 1%), but even thesmallest value was on the order of ∼10 μ m/s, which issignificantly higher than ∼0.1 μm/s to 1 μm/s range for thedielectrophoretic velocity.The 2 μm silica case is shown in Figure 8 (c0 = 0.1%) and

Figure 9 (c0 = 1%) after an assembly time of 600 s, for boththe equilibrium and nonequilibrium cases at 0.5, 2.5, and 5 V.For the 0.1% c0 case, simulations show that fluid flow has aninfluence on both shape and size of assembled volume fractionprofiles and that as for 0.32 μm particles, this effect is morepronounced at larger initial volume fractions. Unlike 0.32 μmparticles, the nonequilibrium case for 2 μm particles also hashigher volume fractions within the assembly as well as a largerassembled size vs the equilibrium case and the increase in size isalso more drastic (∼ 18 μm versus ∼10 μm diameter at 5 V).Increasing solids volume fraction for 2 μm particles also in-creases the size of the assembly and maximum volume fractionfor both the equilibrium and nonequilibrium cases, which isunlike the results for 0.32 μm particles where increasing solidsvolume fraction led to a decrease in both size and maximumvolume fraction. These results are attributed again to the inter-play between fluid and dielectrophoretic convection, and to thesmaller influence of diffusion for 2 μm vs 0.32 μm particles.Figure 10 shows the dielectrophoretic and fluid velocity profilesof 2 μm silica for 5 V. At 0.5 V, the dielectrophoretic velocity of

Figure 7. Dielectrophoretic and fluid velocities (m/s) for 0.32 μm silica in DMSO at 5 V: (a) DEP velocity, c0 =0.1%, (b) DEP velocity, c0 =1%, (c)fluid velocity, c0 =1%, and (d) fluid velocity, c0 =1% [scale bar is 50 μm].

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the c0 = 0.1% case is larger than that of the fluid velocity, andthis holds true as the voltage is increased up to 5 V. For the1% c0 case, the fluid velocity is larger at any voltage vs thedielectrophoretic velocity but this difference is closer to a factorof 4×, which is much less than the contrast that was observedfor 0.32 μm particles (closer to 100×). This reduction is due tothe fluid velocity at the same voltage/initial volume fractionbeing lower for 2 μm particles versus 0.32 μm, as illustrated bycomparing the 5 V cases for both particles sizes Figure 7d withFigure 10d. For 2 μm particles at 5 V, the dielectrophoreticvelocity is larger versus fluid velocity at 0.1% while the fluidvelocity becomes larger at 1% initial solids loading. 0.32 μmparticle simulations are attributed to have a larger fluid velocityat similar conditions versus 2 μm due to increased entropiccontributions to the chemical potential (increases as particlesize decreases) dominating over decreased electrical contribu-tions. The fluid velocity for 2 μm particles was on the order of0.01 to 100 μm/s and the dielectrophoretic velocity ∼0.1 to∼10 μm/s, although over a smaller voltage range versus thestudied 0.32 μm conditions.The maximum values of electrically driven (DEP) and

fluid velocity magnitudes versus voltage at 0.1 and 1% byvolume for both 0.32 and 2 μm particles are shown in Figure 11

(0.32 μm) and Figure 12, respectively. Both fluid velocity andDEP (electrically driven) velocity follow near quadratic depen-dence on voltage with respect to maximum intensity, with themagnitude of DEP velocity slightly affected by volume fraction(more so for the 2 μm case versus the 0.32 μm case) and thechanges in fluid velocity magnitude nearly linearly proportionalto changes in initial volume fraction. This quadratic depend-ence on voltage for both fluid and DEP velocities is expectedbased on the nature of the chemical potential and matches thedilute-case/point-dipole result for the dielectrophoretic force,which depends on the gradient of the electric field intensitysquared. Unlike electrothermal flows induced by Joule heating,where the voltage dependence is near-quartic, in this case anypermittivity gradients in the suspension are caused by volumefraction gradients and there is no direct dependence on appliedvoltage.22 The expression for calculating the force on the fluid isexact and given the laminar nature of the flow profile is verylikely to be physically accurate. The slight decrease in dielectro-phoretic maximum velocity with increasing initial volumefraction is also expected, as can be seen by examining eq 15 forthe dielectrophoretic force on a single particle, correcting forparticle volume fraction, derived previously using the chemicalpotential approach taken in this work.26 This equation reduces

Figure 8. Particle volume fractions with and without fluid flow of dP = 2 μm silica in DMSO, c0 = 0.1%: (a) 0.5 V equilibrium case, (b) 0.5 V withfluid flow, (c) 2.5 V equilibrium case, (d) 2.5 V with fluid flow, (e) 5 V equilibrium case, and (f) 5 V with fluid flow [scale bar is 50 μm].

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to the classic case of the dielectrophoretic force in the limit ofzero volume fraction but for nonzero volume fraction and anegative value of β (negative dielectrophoresis), increasingvolume fraction will lead to a decrease in the dielectrophoreticforce a particle experiences. Viscosity effects from increasingmaximum volume fraction and assembly size (2 μm case) willalso cause a decrease in the dielectrophoretic velocity, andadditionally perturbations of the electric field near theminimum gap between electrodes/electrode edges (locationof maximum DEP force) could also contribute to this changealthough this would only be an issue for the larger assemblysizes. The change in fluid velocity with initial volume fraction(order of magnitude increase in initial volume fraction led toorder of magnitude increase in fluid velocity) is also expected tobe nearly linearly dependent, as the force on the fluid isproportional to particle volume fraction. These trends are alsopresent for 2 μm particles, although fewer voltages were studiedfor that case. The volume fraction proportionality of maximumfluid velocity is still linear, although with a higher constant ofproportionality. This can be attributed to the large size of theassembled structure at c0 = 1% and high volume fraction withinthe assembly significantly affecting the viscosity of the

suspension over a large area, illustrated quite clearly in the5 V case in Figure 10d.

=ε ε

∇⟨| | ⟩ =π ε ε β

− β∇⟨| | ⟩

⎛⎝⎜

⎞⎠⎟F

v

cE

r

cE

2dd

2

(1 )DEP

0 p s 2 p3

0 m2

2

(15)

For 0.32 μm particles at any initial volume fraction inDMSO, no clear structure could be discerned using opticalmicroscopy during assembly, although the larger accumula-tion of materials near the center of the microelectrodes aftermedium evaporation indicated that in fact there was somedegree of assembly. This is attributed to the small size ofparticles, low refractive index contrast, and low overall volumefraction of particles in the region of interest. Additionally, thenonequilibrium simulation results indicate that even for highervoltages the overall assemblies formed at both low and highinitial solids loading compared with the much larger structuresthat would be expected under equilibrium conditions. The2 μm particle case in DMSO was observable, however, andonset of electric field induced aggregation was observed whenswitching between 0.25 and 0.5 V, which is consistent with the

Figure 9. Particle volume fractions with and without fluid flow of dP = 2 μm silica in DMSO, c0 = 1%: (a) 0.5 V equilibrium case, (b) 0.5 V with fluidflow, (c) 2.5 V equilibrium case, (d) 2.5 V with fluid flow, (e) 5 V equilibrium case, and (f) 5 V with fluid flow [scale bar is 50 μm].

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predicted critical electric field strength required for aggregationas calculated via the Khusid/Acrivos framework. Theexperimental results for a 0.5 μL droplet with c0 = 0.1% inDMSO are shown in Figure 13, for 0.5, 2.5, 5, and 20 V. As canbe seen from this figure, the experimentally obtained assembliesare all larger than the simulated results for c0 = 0.1%, (Figure 8)and are in fact closer size-wise to the results obtained fromassuming an initial volume fraction of 1% (Figure 9), althoughthis still underpredicts the size of the assembled structure.Furthermore, looking at the 5 V case for 2 μm particles (Figure 13c),it can be observed that there are in fact multiple layers ofparticles stacked in the assembly. Even at lower voltages, the

only confirmed monolayer-sized assembly happened in the 0.5V case (Figure 13a); beyond this voltage, the assembly wouldalways be at least a few layers thick. The overall structure tran-sitioned from a more square shape (2.5 V) to more roundedshapes (5−10 V) and finally to a diamond (20 V). These resultsare a clear indication that sedimentation is a relevant force inthe system, as well as that the z-component of the electric fieldforce also plays a role in shaping the assembly (gz, Ez). Both ofthese effects are neglected when dealing with a 2D assemblyprocess assumed to take place on the microelectrode surface. Inspite of the very low suspending liquid volume and spreadingnature of DMSO, this system cannot be accurately represented

Figure 10. Dielectrophoretic and fluid velocities (m/s) for 2 μm silica in DMSO at 5 V: (a) DEP velocity, c0 = 0.1%, (b) DEP velocity, c0 = 1%, (c)fluid velocity, c0 = 1%, (d) fluid velocity, c0 = 1% [scale bar is 50 μm].

Figure 11. DEP and fluid velocities (m/s) versus voltage for 0.32 μm silica in DMSO.

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by a 2D approximation. However, the relevance of inducedfluid flows on the assembly process that were determined by

these simulations is clear even from a 2D approximation. Theeffect of gravity would be the introduction of an additionalconvective force, which could potentially mitigate some of theimpact on assembly shape that diffusion/fluid flow has leadingto a result closer to that found for larger particles where theimpact is more on the assembly size vs the shape.Experimental confirmation of the fluid velocities and

presence of rolls in a refractive-index matching medium arechallenging as visualizing individual particles is also difficult,however, these types of rolls are consistent with previousexperimental data collected by Docoslis and Alexandridis.34

Upon activation of an applied electric field, fast swirlingmotions of particles can be observed for 1.5 μm silica in water.Videos of these experiments are available via the authors' Website.34 The need for accounting for flow effects and dealing withfully 3D systems is clearly illustrated for larger assemblies,namely, those where the final structure is of larger than theelectrode gap spacing. These types of structures perturb theelectric field significantly at all applied voltages, comparing withthe case of smaller assemblies formed from very dilute initialparticle concentrations. A 2D approximation is only able toprovide qualitative information on the overall shape of assem-blies, but is illustrative as to the relative influence of entropicand electrical and the induced fluid flows that result from theseforces.

■ CONCLUSIONSThe nonuniform electric field (dielectrophoretic) drivenassembly of colloidal particles using a quadrupolar micro-electrode geometry was explored by simulations based on athermodynamic framework. In this framework, forces on theparticles and fluid are calculated in terms of a chemicalpotential gradient, where electrical and entropic terms areassumed to be the two primary contributions. The stationaryfluid or equilibrium case was compared with the nonstationaryor nonequilibrium case and it was found that the nonequilib-rium solution was substantively different in both shape and size.Analysis of the dielectrophoretic and fluid velocity profiles indi-cated that at all but the very lowest of initial volume fractionsand voltages the fluid velocity dominates. The velocity profile

Figure 12. DEP and fluid velocities (m/s) versus voltage for 2 μm silica in DMSO.

Figure 13. Optical microscopy images of electric field induced assemblyprofiles for 2 μm silica in DMSO, c0 = 0.1%: (a) 0.5 V, (b) 2.5 V, (c) 5 V,(d) 10 V, and (e) 20 V [scale bar: (a) 100 μm, (b−e) 50 μm].

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of the fluid contained rolls/vortexes near the center of thespace between microelectrodes which played an important rolein shaping the assembly. Interestingly, these flows were gen-erated purely based on the electrical and entropic contributionsof particles in an applied electric field. The fluid flow behaviorwithin the system can be taken as analogous to electrothermalflows, without the need for Ohmic heating to generate tem-perature gradients and thereby permittivity and conductivitygradients.The important conclusion from this work is that the

influence of fluid flow is non-negligible over a wide range ofconditions, affecting primarily the size of the assemblieswhen electrically driven (convective) forces dominated (largerparticles/voltages) and both the shape and size of assemblieswhen entropic (diffusive) forces dominated (smaller particles/voltages). These fluid flows have been observed by previousexperiments, and given the nature of microfluidic flows (verylow Reynolds number), the order of magnitude of the predic-tion should be very accurate. This has important implicationsfor understanding the dynamics of the assembly process, aswell as for predicting the steady-state colloid profile for usingdielectrophoresis to shape colloidal structures. The frameworkpresented in this paper shows great promise for being able toenhance understanding of the interplay between entropic, elec-trical and fluids forces during nonuniform electric field drivenassembly. Preliminary 3D simulations, performed at uniformparticle concentration/volume fraction, show that there is asignificant electric-field induced fluid velocity component in thez-direction for both particle sizes studied even for smaller liquiddroplets. Future work is planned in regards to expanding thesimulation framework into three dimensions, which will greatlyincrease the numerical complexity of the problem and maynecessitate the use of more sophisticated numerical techniques.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: aris.docoslis@chee.queensu.ca.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe authors would like to acknowledge the contributions ofMr. Ian Swyer, Department of Chemical Engineering, Queen’sUniversity and the Queen’s Microfabrication Lab (QFAB) forfabrication of the electrodes used in this work. Additionalfunding for this research was provided by NSERC, CanadaFoundation for Innovation (CFI) and Queen’s University.

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