Alternating Current Electrokinetic Properties of Gold-Coated

Article pubs.acs.org/Langmuir

Alternating Current Electrokinetic Properties of Gold-Coated Microspheres Pablo García-Sánchez,*,† Yukun Ren,‡,§,∥ Juan J. Arcenegui,† Hywel Morgan,§ and Antonio Ramos† †

Department of Electrónica y Electromagnetismo, Facultad de Física, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012, Sevilla, Spain ‡ School of Mechatronics Engineering, Harbin Institute of Technology, West Da-zhi Street 92, Harbin, Heilongjiang, P. R. China 150001 § School of Electronics and Computer Science, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom ∥ State Key Laboratory of Fluid Power Transmission and Control, Zhe Jiang University, Hang Zhou, P. R. China 310027 S Supporting Information *

ABSTRACT: We present dielectrophoresis (DEP) and electrorotation (ROT) measurements of gold-coated polystyrene microspheres as a function of frequency and for several electrolyte conductivities. Particle rotation was counterfield with a maximum rotation rate observed at a single characteristic frequency. Negative DEP was observed for frequencies lower than this characteristic frequency and positive DEP for signal frequencies higher than this. These experimental observations are in agreement with predictions for the force and torque on the induced dipole of a perfectly polarizable metal sphere. We present a theoretical model for this case, and good agreement is found for both ROT and DEP measurements if we take into account the viscous friction for a spherical particle near a wall. From the characteristic frequency for rotation, we obtain the capacitance of the electrical double layer at the electrolyte−particle interface. Remarkably, no effect of induced charge electroosmosis around the particles can be inferred from DEP measurements. the electrolyte concentration around the charged particle.7 Both the DEP and the ROT spectra are governed by the real and imaginary parts of the particle polarizability which, in turn, are related through the Kramers−Kronig relations.8 This implies that any relaxation observed in the real part of the dipole has its mirror image in the imaginary part, and therefore, both spectra are not independent. By the same token, dielectric spectroscopy measurements produce results which are, generally, consistent with electrorotation experiments since the former measurements result from a combination of the real and the imaginary parts of the induced dipole moment of the particles. Interestingly, the low-frequency response of charged dielectric spheres does not seem to satisfy this requirement and rotation in the same direction as the electric field is observed (cofield rotation),9,10 whereas rotation in the opposite direction is expected both from dielectric spectroscopy measurements and from theory (counterfield rotation). Grosse and Shilov11 explained this inconsistency by considering the electroosmotic fluid flow generated by the action of the rotating electric field on the ionic cloud around the particle. This electroosmotic flow dominates over the rotation that would be expected from the

I. INTRODUCTION Alternating current electrokinetic phenomena such as electrorotation and dielectrophoresis are well-established methods to manipulate and characterize small particles in suspension such as micro- and submicrometer particles and cells.1,2 Dielectrophoresis (DEP) refers to the motion of particles in a nonuniform electric field,3 while electrorotation (ROT) describes the asynchronous rotation of a particle in a rotating electric field.2 Also, for nonspherical particles, electroorientation, which characterizes the alignment of particle in a steady ac field, can be used.2,4,5 These phenomena arise through the action of an applied electric field on the electrical dipole that is induced on the particles, which depends on the electrical properties (conductivity and permittivity) of the particle, the surrounding liquid, and the solid−liquid interface. Both DEP and ROT spectra (i.e., behavior as a function of the frequency of the ac signal) are commonly used to characterize bioparticle properties. Different polarization mechanisms give rise to an induced dipole. For example, at frequencies of megahertz the interfacial polarization dominates, the so-called β-relaxation, and a Maxwell−Wagner mechanism describes this induced dipole.1 It is well known from dielectric spectroscopy measurements that charged particles exhibit an additional large dispersion at low frequencies, the so-called αrelaxation.6 This polarization has its origin in the polarization of © 2012 American Chemical Society

Received: June 13, 2012 Revised: August 16, 2012 Published: August 29, 2012 13861

dx.doi.org/10.1021/la302402v | Langmuir 2012, 28, 13861−13870

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Figure 1. Electric field lines around the conducting sphere. (a) For periods of the applied signal (T) much larger than τ = aε/σλD, the EDL is fully charged and electric field lines go around the particle. (b) For periods of the applied signal much smaller than τ, the charge accumulated in the EDL is negligible and field lines intersect the particle surface perpendicularly.

ments provides a measurement of the double-layer capacitance of the spheres. The theoretical model predicts negative DEP at low frequencies and positive DEP at high frequencies, also in agreement with experiments, showing that the DEP and ROT spectra are then consistent. Experiments show that the limiting particle speed for negative DEP is one-half the value found for positive DEP, in accordance with dipole theory. Although this is expected for DEP motion, the fluid flow induced by the applied field can alter the measurements. Particle motion is governed by both the force on the induced dipole and the induced fluid flow around the particle, a phenomenon called dipolophoresis by Shilov and Simonova.17 They predicted no net particle motion of a perfectly polarizable sphere with a very thin electrical double layer and in a nonuniform dc electric field. In this situation, they argue that the electrical force on the particle is compensated by the fluid flow induced around it, and no net motion is expected. This theoretical prediction was also stated recently by several authors.18,19 Recent work on dipolophoresis has been performed for perfectly polarizable particles with an arbitrary electrical double-layer thickness20,21 and for polarizable dielectric objects.22,23 However, although fluid flow was observed around our gold-coated particles, we did not observe any effect of induced fluid flow on the DEP spectra. Several factors could reduce the amplitude of the electrokinetic flow without significantly affecting the induced dipole, for example, the presence of a compact (or Stern) layer,24,25 steric effects,26 or surface roughness.27

torque applied on the induced dipole, and the net rotation becomes cofield.10 Dielectrophoresis has been used by several groups as a tool for the assembly and manipulation of metallic (gold) particles in electrolytes. Examples include nanowires and rods,12 nanospheres,13 and micrometer-sized Janus particles.14 However, the commonly used models to predict and analyze the behavior of metal particles suspended in electrolytes in ac fields are incorrect because they do not take into account metal− electrolyte interfacial polarization. In other words, the particles are treated as solid homogeneous lossy dielectrics suspended in homogeneous liquids. This leads to the incorrect assumption that the DEP behavior of metals in electrolyte is always positive even at low frequencies. In the present work we investigate the ac electrokinetic behavior of conducting (metal) microspheres and develop a first-order model that predicts the behavior of particles of micrometer size with thin double layers. A Letter on the electrorotation behavior of gold-coated spheres was recently published by us.15 This paper extends that work, providing further experimental data together with an analytical model that takes into account the interfacial polarization of the particle. We use Au-coated polymer spheres as conducting particles and present experimental measurements of DEP and ROT spectra for different particle diameters and electrolyte conductivities. A theoretical model is developed, based on the assumptions that the particles are uncharged (or with very low charge) and the polarization mechanism is the charging of the electrical doublelayer capacitance at the particle−electrolyte interface. Grosse et al.16 studied the permittivity of suspensions of metal particles in electrolytes, and their experimental data were in agreement with a simple model in which the impedance of the metal− electrolyte interface is represented by a capacitance. The microparticles are heavier than water, so they rest on the bottom glass substrate. To account for this, viscous friction near a wall is considered when computing both the rotation and the translation velocities. We show that for the case of an electrical double layer induced by the applied field, no rotating electroosmotic velocity is expected in electrorotation experiments, in contrast to the case of dielectric charged spheres mentioned above. The model predicts that the conducting particles rotate counterfield with a typical frequency of rotation that scales with the inverse of particle size, in agreement with experimental results. A quantitative comparison with experi-

II. THEORY Consider a conducting sphere of radius a immersed in an aqueous electrolyte of conductivity σ.28 The sphere is assumed to be perfectly polarizable, i.e., there are no Faradaic currents (charge transfer reactions) at the particle−electrolyte interface. There is an electric field applied from infinity E which drives the ions in the electrolyte. These ions accumulate at the electrolyte−particle interface, inducing an electrical double layer (EDL) of typical thickness λD = (εkBT/∑inoi z2i e2)1/2, the so-called Debye length, where ε is the electrolyte permittivity, kB is the Boltzmann constant, T is the temperature, e is the proton charge, and noi and zi are, respectively, the bulk concentration and valence of ionic species in the electrolyte. The applied electric field is low enough so that the voltage induced in the EDL is below the threshold for Faradaic 13862


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can write the instantaneous induced particle dipole as p(t) = Re[p(ω)exp(iωt)], where p(ω) is the dipole phasor. Introducing the Clausius−Mossotti factor K(ω), the dipole phasor is written as p(ω) = 4πεE0a3K(ω), with E0 being the electric phasor. Given a nonuniform electric field with constant phase, the expression for the time-averaged force on the particle is2

reactions at the metal/electrolyte interface. At scales much larger than the Debye length, the electrolyte remains electroneutral. The typical time scale for charging the metal/ electrolyte double layer is τ = aε/σλD.29,30 This is the RC time for charging the double-layer capacitance on the metal surface (for a sphere, of the order of C = 4πa2ε/λD using the Debye− Hückel approximation of diffuse layer capacitance) through the resistor of the bulk electrolyte (for a sphere, of the order of R = 1/4πσa). For periods of the applied signal much larger than τ, the electrical double layer is fully charged, the current normal to the particle surface is zero, and the electric field lines go around the particle. From the perspective of an observer, the situation is equivalent to that of an insulating sphere, see Figure 1a. For periods of the applied signal much smaller than τ, the charge accumulated in the EDL is negligible, field lines intersect the particle surface perpendicularly, and the situation is equivalent to a conducting sphere, see Figure 1b. We are interested in finding the solution of the electric potential in the bulk electrolyte in order to obtain an expression for the induced dipole. The Debye length is on the order of tens of nanometers or smaller for typical conductivities in our experiments. At length scales much larger than λD, the electrolyte is electroneutral and, therefore, the potential in the bulk electrolyte satisfies Laplace’s equation (▽2ϕ = 0). The solution can be written as ϕ = −E0r cos θ +

A cos θ r2

FDEP =

ΓROT =

i Ω − 1/2 with Ω = ωC DLa /2σ iΩ + 1

(1)

Fdrag = 6πηa(vFt + aΘt Fr )

(7)

Ft = −(8/15)ln(δ /a) + 0.9588

(8)

Fr = (2/15)ln(δ /a) + 0.2526

(9)

where δ is the distance of the particle to the wall. In our DEP experiments translation occurs without rotation (see Supporting Information). It is possible that the corrugated particle surface forces this condition, although we do not have a theoretical explanation for it. Thus, Θt = 0 and the viscous drag is

(2)

(3)

Fdrag = 6πηav( −(8/15)ln(δ /a) + 0.9588)

The term A cos θ/r in eq 1 corresponds to the potential created by a dipole of magnitude p = 4πεA, where ε is the dielectric constant of the electrolyte. We then write the dipole moment induced on the particle as i Ω − 1/2 iΩ + 1

(6)

where v is the particle translational velocity and Θt the angular velocity of the particle rotation during translation, which is along the horizontal direction, perpendicular to the direction of particle motion. Ft and Fr are coefficientes given by

2

p = 4πεA = 4πεE0a3

1 Re[p × E*0 ] = −4πεa3Im[K ]E02ez 2

where ez is a unit vector normal to the polarization plane of the electric field. When the real part of K is positive, the particle moves to regions of higher electric field intensity (positive DEP). If, on the other hand, Re[K] < 0 the particle moves to regions of lower field intensity (negative DEP). Also, if Im(K) < 0, the particle rotates in the same direction as the electric field, known as cofield rotation, and if Im(K) > 0, the rotation is in the opposite direction (counterfield rotation). In the steady state, the DEP force and ROT torque will each be balanced by the fluid viscous drag and viscous torque, respectively, on the particle. The gold-coated microparticles move near a glass substrate, and therefore, wall effects must be accounted for in the viscous drag and torque. When a sphere moves on a wall in a viscous fluid, it can translate and rotate. For creeping flow conditions, Goldman et al.31 found that the viscous drag on a spherical particle translating near a wall and immersed in a liquid of viscosity η is given by

where ω is the angular frequency of the applied electric field. Solving for A in eq 1 we obtain A = E 0a 3

(5)

where E0* is the complex conjugate of E0. For a rotating electric field of the form E(t) = E0Re[(ex − iey)exp(iωt)], the expression for the time-averaged torque on the particle is2

where A is a constant and E0 is the amplitude of the applied electric field. We use spherical coordinates, where r is the distance to the center of the sphere and θ is the angle between the position vector and the applied electric field vector. Charge conservation at the surface of the perfectly polarizable metal sphere provides the required boundary condition. The electrical current arriving at the interface charges the double layer j·n = CDL(∂(V−ϕ)/∂t). Here n is a unit vector normal to the particle surface in the outward direction, CDL is the specific capacitance of the electrical double layer, V is the electric potential in the conducting sphere, and ϕ is the electrical potential just outside the electrical double layer. Therefore, (V − ϕ) is the voltage drop across the electrical double layer. If the potential of the conducting sphere is taken as zero, V = 0, and using phasors, the boundary condition for the potential is ∂ϕ σ = iωC DLϕ at r = a ∂r

1 Re[(p·∇)E*0 ] = πεa3Re[K ]∇E02 2

(10)

Also, in creeping flow conditions, a sphere near a wall and rotating around an axis perpendicular to the plane experiences a viscous torque given by Γviscous = −8πηa3ζ(3)Θ

(4)

(11)

where Θ is the angular velocity, ζ is Riemann’s ζ function, and ζ(3) = 1.20206, see refs 32 and 33. Note that viscous drag on the particle diverges logarithmically as the particle approaches the wall (δ → 0). The viscous torque, however, remains finite, although the sphere is in perfect contact with the wall.

The net force acting on the induced dipole, F = (p·▽)E, gives rise to motion of the particle (dielectrophoresis). In the case of rotating electric fields, it also causes a torque on the induced dipole, Γ = p × E, which is responsible for particle rotation. We 13863


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III. EXPERIMENTAL DETAILS

The DEP velocity and ROT angular velocity can be written as vDEP

a 2ε Re[K ]∇E02 = 6η((8/15)ln(a/ δ) + 0.9588)

ΘROT = −

εE02 Im[K ]ez 2ηζ(3)

A. Particle Preparation. A reducing solution was prepared by adding 12-hydroxymethyl phosphonium chloride with 1 mL of 1 M KOH solution to 45 mL of deionized water. A gold plating solution of 25 mM gold(III) chloride hydrate (Sigma, 27988-77-8) in 2 mL of deionized water was prepared separately. Rapid injection of freshly prepared gold plating solution into the reducing solution with vigorous stirring led to a dark brown but translucent colloidal gold suspension. Gold-coated polystyrene particles were prepared using a seeding method, similar to that in ref 34. Twenty microliters of a 2% volume fraction (w/v) suspension of polystyrene particles (Polysciences, Inc.) was incubated with 1 mL of colloidal gold suspension (15 nm) overnight. Gold nanoparticles adsorbed to the plain polystyrene particle surfaces by van der Waals forces, Figure 3. After two washes in

(12)

(13)

For the present case of a conducting sphere, the real and imaginary parts of the Clausius−Mossotti factor are obtained from eq 3 Re[K ] =

Ω2 − 1/2 3Ω/2 ; Im[K ] = 2 2 Ω +1 Ω +1

(14)

Figure 2 shows a plot of the real and imaginary parts of the Clausius−Mossotti factor versus nondimensional frequency.

Figure 3. Gold coating process: plain polystyrene microspheres are immersed in a suspension of colloidal gold. Gold nanoparticles are deposited onto the microspheres surface, forming a conducting shell around it with a thickness of, approximately, 50 nm. deionized water, 20 μL of these gold seeded particles was mixed with 600 μL of gold plating solution, which consisted of 34 mM gold sodium thiosulfate (Alfa Aesar, 039741) and 57 mM ascorbic acid (Sigma, A5960-25G) in deionized water. A 300 μL aliquot of 1 M NaOH was added to the solution, which initiated reduction of gold from solution to metal, with the adsorbed colloidal gold acting as nucleation centers. After 5−10 min, the gold coating was complete and particles were washed twice in deionized water and resuspended in water. Figure 4 shows SEM images of gold-coated particles, indicating Figure 2. Real and imaginary parts of the Clausius−Mossotti factor versus nondimensional frequency for a conducting sphere. Theory predicts counterfield rotation, negative DEP at low frequencies, and positive DEP at high frequencies.

The model predictions are as follows: (a) the rotation in ROT experiments is counterfield with a single peak at a characteristic angular frequency ωc = 2σ/CDLa; (b) there is positive DEP at high frequencies and negative DEP at low frequencies, with the relaxation frequency equal to ωc; (c) the torque ΓROT and force FDEP are both proportional to the square of the voltage applied to electrodes. Together with the force and torque on induced dipoles, we should take into account the fact that induced charge electrosmotic flow can occur at the particle surface and affect particle motion. As mentioned in the Introduction, Shilov and Simonova17 predicted no net motion of polarizable metal particles in a dc nonuniform field, because in the thin-doublelayer approximation the induced electrosmotic flows would balance out the dielectrophoretic motion. However, electrorotation is not affected by induced charge electroosmosis. The Appendix I provides an illustration of this, together with a solution for the induced flow in the Debye−Hü c kel approximation. Note that the present case of an uncharged conducting sphere is different from that mentioned in the Introduction for a dielectric charged sphere. In the latter case, Grosse and Shilov11 predicted that electrosmotic flow would be important for particle rotation.

Figure 4. SEM micrograph of the 10 μm diameter gold-coated polystyrene microspheres. (Inset) Micrography of a 45 μm diameter sphere. that almost complete coverage occurs, with a thickness of approximately 50 nm. The final particles appeared dark in color, and as shown in the SEM image, the surfaces were highly corrugated and each particle therefore had a large effective surface area. B. Experimental Setup. Gold-coated microspheres were immersed in an aqueous solution of KCl with different conductivites. A small amount of liquid containing a few microspheres was placed in the central region of a coplanar quadrupolar electrode array, see Figure 5. Electrodes are made of platinum on glass and have a hyperbolic 13864


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Figure 5. (a) Scheme of the experimental setup. Electrodes are energized with a four-phase function generator. Videos of the particles motion are captured and analyzed with a PC. (b) Five hundred micrometer gap electrode array. Figure 7. ROT speed versus amplitude squared of the applied signal for a gold-coated microsphere (45 μm diameter) at a single frequency of 1 kHz. Microspheres rotation is counterfield. Angular velocity increases with the square of the voltage for all electrolyte conductivities, in agreement with theory.

shape. The distance between opposite electrodes (electrode gap) was either 200 or 500 μm. The chip was connected to a 4-phase function generator (Thurlby Thandar Instruments), and the phase shift between adjacent electrodes was set at either 90° (for ROT) or 180° (for DEP). Figure 6 shows the signals applied to each electrode as well as the domains used for computing the electric field (see next section). For DEP experiments and for the 10 and 25 μm particles, the electrode gap was 200 μm, energized with a 5 Vpp signal amplitude; for the 45 μm particles the gap was 500 μm, energized with an amplitude of 12.5 Vpp. For the ROT experiments, all particles were measured using the same electrodes with a gap of 500 μm and a voltage amplitude of 4 Vpp. Images of particle movement were recorded on video and analyzed with custom software in Labview. ROT and DEP spectra of the same particle were recorded together. Data shown in the Experimental Results are the average spectrum for at least 5 different particles.

conducting sphere. Also, the velocity increases linearly with the square of the applied voltage, as predicted by eq 13. Figure 8 shows ROT and DEP spectra of microspheres of three different sizes and, for each particle size, for three different conductivities. Bulk fluid flow due to induced charge ac electroosmosis can occur for ac signals with similar amplitude and frequency as used in these experiments,35 which could influence the DEP velocity measurements. However, induced charge ac electrosmosis is driven on top of the electrodes, and there is negligible fluid flow in the central part of the electrode array, where particle motion is recorded. Small fluorescent tracer particles were used to observe fluid flow during experiments. Fluid flow was observed near and on the electrodes, at low frequencies, which is characteristic of induced charge ac electroosmosis. Virtually no flow was observed at the center of the electrode array (which is in fact a stagnation point because of the symmetry of the electrode array). Additionally, the substrate on which the particles rest is probably charged. Classical electroosmotic flow could occur, and this might influence the motion of the particle. However, this flow (if any) would oscillate at the frequency of the applied

IV. EXPERIMENTAL RESULTS Figure 7 shows the rotation speed of a gold-coated microsphere (45 μm diameter) versus the square of the applied signal for different electrolyte conductivities. Particles were placed at the center of the quadrupolar electrode array, and the frequency was kept constant at 1 kHz for all conductivities. Note that these measurements, which are made at a single frequency, do not give the complete picture of the conductivity dependence because the frequency of the maximum electrorotation speed increases with electrolyte conductivity as shown in Figure 8. Particle rotation was counterfield, as expected for the case of a

Figure 6. Electrode arrays and signal applied to each electrode: (a) electrorotation experiments and (b) dielectrophoresis experiments. 3D domains shown in this figure are used in the next section for computing the electric field. 13865


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Figure 8. Electrorotation velocity (left) and dielectrophoresis velocity (right) versus frequency for gold-coated microspheres of three different diameters. (Insets) Particle diameter, amplitude signal, and microelectrode gap in each case.

solution for the potential did not change, so h = 1000 μm was chosen in all cases. In order to check the accuracy of the simulation for a large surface area electrode array, we also increased the area of the base of the 3D domain and the potential solution did not significantly change. At the center of the quadrupolar electrode array and for a ROT signal of 4 Vpp applied to electrodes with a gap of 500 μm we obtained E0 = 3.9 kV/m, giving a theoretical value for the peak electrorotation speed of ΘROT(max) = 3.3 rad/s. The maximum angular speeds found from experiments are 20% lower than the theoretical value. Part of this deviation is due to the influence of the glass substrate on the electrorotational torque. As shown in Appendix II, the peak electrorotational torque for a sphere resting on a wall is 6.5% lower than for a sphere in the liquid bulk. Measurements of the DEP velocity show negative DEP at low frequencies, positive DEP at high frequencies with a

field with an amplitude on the order of l = vslip/ω = εEζ/ωη. Using typical values for the permittivity and viscosity of water, together with ω = 1000 rad/s, ζ = 25 mV and E = 4000 V/m, gives a fluid displacement amplitude of only 0.07 μm. Figure 8 shows that the applied signal frequency for maximum electrorotation speed increases with liquid conductivity and decreases with particle size, in agreement with theory. The theoretical value of the peak electrorotation velocity occurs for Ω = 1 (Im[K] = 0.75) and does not depend on either the particle size or the electrolyte conductivity, ΘROT(max) = 3εE20/ 8ζ(3)η. Experimental measurements do not depend on particle size, but for the electrolyte with highest conductivity (15.9 mS/ m) the data show that the rotation speed is clearly smaller. The electric field and gradient was calculated using COMSOL (commercial finite element solver) to determine the 3D electric field in the domains of Figure 6. The domain height h was systematically varied, and it was found that for h > 2d the 13866


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crossover frequency around the frequency of peak velocity in the ROT spectrum, as expected from theory. An important feature of the DEP measurements is that the magnitude of the velocity at low frequencies is approximately one-half of the magnitude at high frequencies. Some deviations appear depending on particle size and electrolyte conductivity, but this points to the fact that particle movement is mainly driven by DEP. Induced-charge electroosmosis, if any, only plays a minor secondary role. We also computed the electric field for DEP experiments. In this electrode configuration the magnitude of ∇E20 in the central region increases linearly with the distance to the center of the array. The DEP velocity was measured as determined from the average velocity between two fixed positions. The average magnitude of ∇E20 was determined within that boundary. For example, 10 μm particles were tracked between positions 10 and 30 μm radially away from the array center, and for this region ∇E20 = 3.17 × 1012 V2/m3. From eq 12 the theoretical value of the maximum positive DEP velocity (Re[K] = 1) can be estimated if the distance of the particle to the wall, δ, is known. Following ref 36 we take the particle surface roughness in our experiments (∼15 nm) as δ, and the maximum theoretical DEP velocity for 10 μm particles is 2.98 μm/s. Microspheres with diameter 25 μm were tracked between positions 25 and 50 μm; the theoretical maximum DEP velocity is 34.0 μm/s. Microspheres with 45 μm diameter were tracked between 25 and 80 μm with a maximum DEP expected velocity of 19.8 μm/s. For all particle sizes, the theoretically predicted maximum velocity is around the maximum value observed in experiments, which confirms that the estimate of δ as the particle roughness is a good approximation However, a more reproducible DEP measurement would be desirable to draw firmer conclusions. Note that for the highest value of conductivity, the DEP velocity is reduced. One explanation for this is that the interaction between the electrical double layer of the particle and the wall can play an important role in the value of δ. In particular, the repulsion of electrical double layers is smaller for increasing conductivity, and this could lead to a reduction of δ and an increase in the viscous friction. For this to be true the particles cannot be completely uncharged. This effect is beyond the scope of the present work and not included in the theoretical model. ROT and DEP spectra were fitted to the imaginary and real part, respectively, of a single Debye relaxation K0 +

(ωτ )ΔK ΔK ΔK = K0 + −i 2 1 + iωτ 1 + (ωτ ) 1 + (ωτ )2

Table 1. Frequencies for Peak ROT Velocity According to the Fit of Data to a Debye Relaxation 10 μm 25 μm 45 μm

0.7 mS/m

3.6 mS/m

15.9 mS/m

0.37 kHz 0.19 kHz 0.19 kHz

1.33 kHz 0.37 kHz 0.26 kHz

5.6 kHz 1.4 kHz 1.05 kHz

Figure 9. Frequency for maximum rotation speed ( fc) versus the reciprocal of the particle radius for three different conductivities. Continuous lines show the result of the fit.

πaCDL), and the fitting parameters are used to obtain values for CDL, shown in Table 2. Table 2. Values of the Double-Layer Capacitances Obtained from Figure 9 and the Debye−Hückel Model conductivity CDL (fitting) CDL (Debye−Hückel)

0.7 mS/m

3.6 mS/m

2

2

0.179 F/m 0.0159 F/m2

0.160 F/m 0.0361 F/m2

15.9 mS/m 0.164 F/m2 0.0760 F/m2

These values for the double-layer capacitance are greater than those obtained from the Debye−Hückel theory CDL = ε/λD. This is not surprising since it is well known that surface roughness and/or the existence of intrinsic surface charge increases the double-layer capacitance (the differential capacitance is minimum at the point of zero charge).

V. CONCLUSIONS Experiments on dielectrophoresis and electrorotation of goldcoated microspheres are in agreement with the theoretical expectation for conducting spheres in an ac field; the direction of rotation is counterfield: positive DEP is found at high frequencies and negative DEP at low frequencies. The ratio of positive DEP to negative DEP velocity limits is 2, and there is no evidence of induced-charge electroosmosis around the spheres. As recently suggested in ref 27, surface roughness can severely reduce induced-charge electroosmotic flows. A quantitative comparison shows that the peak velocities for ROT experiments are close to the theoretical prediction when viscous effects near a wall are considered. DEP measurements are also in agreement with theoretical values if we consider that the particles slide over the glass surface at a distance on the order of the particle roughness. The capacitance of the electrical double layer was derived from the frequency of peak ROT velocity, and they were greater than the values given by Debye−Hückel theory. Surface roughness and/or intrinsic surface charge could explain this point. We also note that a

(15)

where τ is the relaxation time of the polarization mechanism and K0 and ΔK are constants. We first fit the ROT data to obtain τ. In order to fit the DEP data, we constrained τ to be the value obtained from ROT and adjust the values of ΔK and K0. Fit lines are also plotted in Figure 8. Data of ROT experiments for 10 μm particles are the same as in ref 15. We refitted these data and obtained very similar results for τ. The frequency of maximum rotation speed is fc = 1/(2πτ), shown in Table 1. Note that this frequency increases with liquid conductivity and decreases with particle diameter, as expected from theory. To obtain a value for the capacitance of the double layer, Figure 9 shows a plot of the frequency of maximum rotation speed ( fc) versus the reciprocal of the particle radius for the three conductivities. Data are fitted to a linear function ( fc = σ/ 13867


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complete theoretical model should include the particle fixed surface charge. In summary, we have described a double-layer charging mechanism that is responsible for the ac electrokinetic behavior of perfectly polarizable particles with thin double layer. For micrometer-sized particles, basic theory shows reasonable agreement with experimental results. The characteristic frequency of charging is given by the reciprocal of the RC time and therefore depends on electrolyte conductivity. This contrasts with the mechanism of concentration polarization which is governed by the diffusion time of ions around the particle and is therefore independent of electrolyte conductivity. The behavior of metal particles suspended in an electrolyte subjected to ac fields is governed by the charging of the double layer and cannot be explained using the Maxwell− Wagner interfacial polarization model as is common in the literature.

■

u +i

εE02a ⎡ −9/4 Re⎢ sin θ cos θ u θ ⎣ 1 + Ω2 2η

⎤ εE02a 9/4 9/4 θ = − sin θ cos θ u θ sin u ⎥ φ ⎦ 2η 1 + Ω2 1 + Ω2 (20)

which has zero circulation around the sphere, and so, the average velocity around the particle is zero. In spherical coordinates, a Stokes streamfunction ψ can be defined by vr =

1 ∂ψ 1 ∂ψ ; vθ = − r sin θ ∂r r sin θ ∂θ 2

and ψ satisfies

(21)

38

2 ⎡ ∂2 sin θ ∂ ⎜⎛ 1 ∂ ⎟⎞⎤ ⎢ 2 + 2 ⎥ψ=0 ⎣ ∂r r ∂θ ⎝ sin θ ∂θ ⎠⎦

(22)

From the angular dependence of velocity, we look for solutions of the form Ψ = f(r)sin2 θ cos θ, which by substitution in the equation above leads to

APPENDIX I: ELECTROOSMOTIC VELOCITY AROUND THE PARTICLE FOR AN APPLIED ROTATING ELECTRIC FIELD

⎛ Aa 4 ⎞ ψ = ⎜ 2 + Ba 2⎟sin 2 θ cos θ ⎝ r ⎠

In this appendix we use the Helmholtz−Smoluchowski equation and the Debye−Hückel approximation to compute the induced-charge electroosmotic flow around a spherical conducting particle for an applied rotating electric field. The applied rotating field is given by E(t) = E0Re[(ex − iey)exp(iωt)]. The electric potential is written as the sum of the potential of the applied field and a dipole field, as in eq 1. From the results, section II, the electric potential phasor is given by ⎛ i Ω − 1/2 ⎞ x − iy ⎟ ϕ = −E0(x − iy) + E0a3⎜ ⎝ iΩ + 1 ⎠ r 3

=

(23)

where we used regular solutions at infinity. From the boundary conditions at r = a we obtain (vr = 0;vθ = veo) ⎛ a4 ⎞ εE 2 a 9 ψ = A⎜ 2 − a 2⎟sin 2 θ cos θ ; A = − 0 16η 1 + Ω2 ⎝r ⎠

(24)

Figure 10 shows the streamlines around the particle. The fluid velocity has axial symmetry, without rotation, and therefore,

(16)

where Ω = ωCDLa/2σ ⎛ a3 i Ω − 1/2 ⎞ ϕ = E0r sin θ exp( −iφ)⎜ −1 + 3 ⎟ ⎝ r iΩ + 1 ⎠

(17)

which satisfies boundary condition 2. The electrosmotic velocity on top of a metallic surface can be computed from the Helmholtz−Smoluchowski velocity.37 The time average of this velocity is given by u

=

ε( ϕ − V ) Et η

=

1 ⎡ εϕ *⎤ Re⎢ Et ⎥ 2 ⎣ η ⎦

(18)

Figure 10. Streamlines corresponding to the induced charge electroosmosis around a conducting sphere generated by a rotating electric field. The resulting flow is axisymmetric.

where V, the electric potential within the sphere, which is taken to be zero. Et is the component of the electric field tangential to the particle surface, the asterisk (*) indicates the complex conjugate, and η is the liquid viscosity. The tangential components of the electric field at r = a computed from the solution of the potential are 3/2 E0 cos θ exp( −iφ); 1 + iΩ 3/2 Eφ = −i E0 exp( −iφ) 1 + iΩ

ROT is not affected by induced-charge electroosmotic flow around the particle. Streamlines for the ICEO are modified near the wall. However, flow symmetry should not be broken when the particle is near a plane, and therefore, no rotating flow is to be expected. In other words, the ICEO will not affect the electrorotation measurements. However, particle-wall hydrodynamics might influence the interaction between double layers and therefore the equilibrium height of the particles (δ).

Eθ =

(19)

and the electroosmotic velocity is given by 13868


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APPENDIX II: INFLUENCE OF THE GLASS WALL ON THE ELECTROROTATIONAL TORQUE In section II we computed the induced dipole on a conducting spherical particle suspended in the liquid bulk. However, the gold-coated spheres used in experiments rest on a glass substrate. In this appendix we show that when the wall is considered, the maximum theoretical electrorotational torque on the particle decreases by 6.5%. This explains in part the 20% discrepancy between ROT maximum velocities and the theoretical expectations. Note that the increase in viscous torque due to the wall effect is a more important effect (around 17%). The theoretical dielectrophoretic force is also affected by the presence of the wall by a factor of the same order. However, as mentioned in the main body of the manuscript, the viscous drag on a particle moving near a wall diverges logarithmically with the reciprocal of the distance to the wall, and it turns out that the wall effect on the dielectrophoretic force is a minor correction. In general, the electrical torque on a body is found from the Maxwell stress tensor (Te = εEE − (1/2)εE2I) as an integration over the body surface S, Γ = ∫ S(r × Te)·ndS, where r is the vector position of a point on that surface and n is a unit vector normal to the surface. We compute the torque for a spherical particle resting on a wall with respect to the particle center and for an applied rotating electric field in the horizontal plane. In this case, the electric potential far from the particle can be written using cylindrical coordinates as ϕ(ρ → ∞) = Re[−E0ρ exp(i(ωt − φ))], where E0 is the amplitude of the applied rotating field. The electric potential in the liquid satisfies Laplace equation (▽2ϕ = 0) and in the present case can be written in cylindrical cordinates as ϕ(ρ,z,φ) = Re[Φ(ρ,z)exp (i(ωt − φ))], where we factored out the dependence with φ. The phasor Φ satisfies the following equation ∇2 Φ(ρ , z) − Φ(ρ , z)/ρ2 = 0

Normal derivative equal to zero is imposed in all other boundaries. Figure 12 shows the nondimensional electrorotational torque versus nondimensional frequency for two cases: (a) when the

Figure 12. Nondimensional electrical torque versus nondimensional frequency for two cases: (a) a spherical particle on a wall (as in Figure 11) and (b) a spherical particle in the liquid bulk. Maximum torque on the particle is decreased by 6.5% due to the presence of the wall, and frequency is shifted to a lower value.

particle rests on a wall and (b) for a particle in the liquid bulk, which coincides with eq 6. The presence of the wall has an influence both in the peak value of the torque and on the frequency of the peak. The peak electrorotational torque decreased by 6.5%, as mentioned above, while the frequency for peak torque decreased by a 0.84 factor.

■

(25)

We make use of the axial symmetry in order to find Φ(ρ,z) in the 2D domain shown in Figure 11. We scale distances with the particle radius, and E0a is the scale for voltages. The electric potential and electrorotational torque were computed using COMSOL. The electrical torque is scaled by εa3E20. The boundary conditions for Φ are also shown in the figure. On the particle surface, eq 2 transforms to n·▽Φ = 2iΩΦ. Far from the particle, the potential equals the applied potential, Φ = −ρ.

ASSOCIATED CONTENT

S Supporting Information *

Video of a gold-coated particle undergoing positive dielectrophoresis shows no rotation of the particle during translation. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS P.G.S., J.J.A., and A.R. acknowledge financial support from Regional Government Junta de Andaluciá and the Spanish Government Ministry MEC under contracts P09-FQM-4584 and FIS2011-25161, respectively. Y.K.R. acknowledges the National Natural Science Foundation of China (Grant No. 51075087) and the State Key Lab of Fluid Power Transmission and Control, Zhe Jiang University (Grant No. GZKF-201107).

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REFERENCES

(1) Morgan, H.; Green, N. G. AC Electrokinetics: colloids and nanoparticles; Research Studies Press Ltd.: Baldock, Hertfordshire, England, 2003. (2) Jones, T. B. Electromechanics of Particles; Cambridge University Press: New York, 1995.

Figure 11. Problem domain for computing the electrical torque exerted on a sphere resting on a plane. We take the domain dimensions much larger than the sphere radius L ≫ a (not to scale in figure). 13869


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