kT-Scale Colloidal Interactions in High-Frequency Inhomogeneous AC

ARTICLE pubs.acs.org/Langmuir

kT-Scale Colloidal Interactions in High-Frequency Inhomogeneous AC Electric Fields. II. Concentrated Ensembles Jaime J. Juarez, Brian G. Liu, Jing-Qin Cui, and Michael A. Bevan* Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, Maryland 21286, United States ABSTRACT: We report nonintrusive optical microscopy measurements of ensembles of polystyrene colloids in inhomogeneous AC electric fields as a function of field frequency and particle size. By using an inverse Monte Carlo (MC) simulation analysis of time-averaged particle microstructures, we sensitively measure induced dipole
dipole interactions on the kT energy scale. Measurements are reported for frequencies when the particle polarizability is greater and less than the medium, as well as the crossover between these conditions when dipole
dipole interactions vanish. By using measured single dipole
field interactions and associated parameters from Part I1 as input in the inverse analysis, the dipole
dipole interactions in this work are accurately modeled with no adjustable parameters for conditions away from the crossover frequency (i.e., |fCM| > 0). As dipolar interactions vanish at the crossover, a single frequency-dependent parameter is introduced to account for microstructures that appear to result from weak AC electro-osmotic flow induced interactions. By connecting quantitative measures of equilibrium microstructures and kT-scale dipole
field and dipole
dipole interactions, our findings provide a basis to understand colloidal assembly in inhomogeneous AC electric fields.

’ INTRODUCTION The interaction of induced dipoles with each other in homogeneous AC electric fields is a well-known phenomenon leading to dipolar chain formation.2 Likewise, the interaction of induced dipoles with inhomogeneous AC electric field gradients is also a commonly known phenomenon, most commonly as dielectrophoresis, which is the transport of dipoles between spatially separated high and low field regions.3 These interactions are commonly exploited in applications such as electrorheological fluids4 and separation methods.5 More recently, electric field induced particle interactions have been used to assemble a variety of colloidal microstructures including, for example, chain6,7 and crystal8,9 configurations. The majority of studies on electric field mediated colloidal interactions have focused on interactions much stronger than the thermal energy, kT, often based on application requirements. For example, high yield stress electrorheological fluids require strong dipole
dipole attraction, and dielectrophoretic separations can be maximized through strong dipole
field interactions. On the basis of these applications, many modeling efforts have focused on potential energy calculations in the high field limit,10 rather than free energy calculations for low field mediated interactions on the order of kT. Experiments on electric field mediated colloidal assembly have also focused on suppressing Brownian motion via interactions .kT to produce static particle configurations. However, by assembling particle configurations in high electric fields, the observed microstructures generally do not correspond to equilibrated low free energy configurations (which may be important for making defect-free single crystals). A number of studies have measured or modeled various aspects of the equilibrium behavior of dipolar particles in inhomogeneous r 2011 American Chemical Society

electric fields. For example, equilibrium phase behavior and microstructures of dipolar particles in homogeneous electric fields have been predicted for induced dipoles,11 permanent dipoles,12 and induced dipoles in oscillatory shear flows.13 Equilibrium phase behavior of induced dipoles in inhomogeneous electric fields have been predicted based on an osmotic pressure balance14 and experimentally verified when particles are both more polarizable9 (i.e., fCM > 0) and less polarizable15 (i.e., fCM < 0) than the medium. However, this continuum approach does not explicitly include microstructure and is based on isotropic, symmetric distributions that allow dipole
dipole interactions to effectively be ignored. Each of these approaches has limitations for connecting electric field mediated kT-scale potentials to problems involving microscopic equilibrium colloidal assembly. To understand near-equilibrium colloidal self-assembly in highfrequency AC electric fields, it is necessary to know kT-scale dipole
field and dipole
dipole interactions and their connections to equilibrium microstructures. In Part I,1 we measured the frequency, amplitude, size, and material property dependent interactions of single induced dipoles with inhomogeneous electric fields. These measurements were inherently on the kT-scale by monitoring Brownian excursions of single particles about their mechanical equilibrium positions and obtaining potentials via a Boltzmann inversion. In concentrated systems, multiparticle packing effects complicate the process of extracting interaction potentials from microscopy measurements.16 To analyze kT-scale interactions in quasi-2D colloidal dispersions, we have previously Received: April 21, 2011 Revised: June 14, 2011 Published: June 15, 2011 9219

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developed inverse Monte Carlo (MC) methods to obtain interaction potentials between particles,17
21 surfaces,22,23 and external fields.24 In relation to electric field mediated colloidal interactions, we previously used an inverse MC analysis to obtain amplitude and concentration dependent dipole
dipole potentials from measured microstructures at 1 MHz.24 This previous work did not explore the frequency or particle size dependence of such interactions in concentrated systems. In the present paper, we adapt and extend these nonintrusive methods to measure frequency-dependent kT-scale interactions and equilibrium microstructures of particle ensembles in inhomogeneous AC electric fields. By using the same experimental configuration as the single particle measurements in Part I,1 we are able to use the resulting single dipole
field potentials and associated parameters without adjustment in the analysis of particle ensembles. To extract frequency-dependent dipole
dipole interactions, we use an inverse MC analysis of the time-averaged structure of particle ensembles (i.e., local angular and spatial distribution functions), which has been adapted from our previous methods to now include frequency-dependent effects. Our results demonstrate quantitative connections between equilibrium microstructures and kT-scale dipole
field and dipole
dipole interactions for particle polarizabilities greater and less than the medium and the crossover between these conditions when dipolar interactions vanish. Understanding these connections provides a basis to quantitatively design, control, and optimize highfrequency AC electric field mediated colloidal assembly processes at near-equilibrium conditions.

’ THEORY Net Interaction Potential. For concentrated colloidal particles in a nonuniform electric field near a planar wall surface, the net potential energy for particle i, as a function of particle
particle separation, ri,j, orientation, θi,j, elevation above the wall, zi, and relative position within the field xi, is given by pw

pf

unet i ðrij , θij , xi , zi Þ ¼ ui ðzi Þ + ui ðxi , zi Þ +

pp ui ðrij , θij , xi , zi Þ ∑ j6¼ i

ð1Þ where particle
wall and particle
field interaction potentials are given in Part I.1 In the limit where van der Waals interactions are negligible,25 superposition gives the net particle pair interaction as the sum of an electrostatic interaction due to static surface charge and a dipole
dipole interaction due to an applied electric field as pp

upp ðrij , θij , xi , zi Þ ¼ upp e ðrij Þ + ud ðrij , θij , xi , zi Þ

ð2Þ

where εm is the medium dielectric constant, k is Boltzmann’s constant, T is absolute temperature, e is the charge of an electron, ψp is the particle electrostatic surface potential, and k
1 is the Debye length. The dipolar interaction in concentrated, quasi-two-dimensional dispersions is13 pp ud ðrij , θij , xi , zi Þ

ð5Þ where θij is the angle formed between i and j relative to the direction of the electric field, P2(cos θij) is the second Legendre polynomial, E*(x, z) = E(x, z)/E0 is the local normalized electric field (from our previous paper24), E0 = 8
0.5Vpp/dg, Vpp is the applied peak-to-peak voltage, dg is the distance between electrodes, and λ is the relative dipolar and Brownian energies, which are described in detail in Part I.1 Area fraction, ϕ, dependent modifications of the local electric field (in quasi-two-dimensional dispersions) are accounted for by fϕ, given as24

2 fϕ ¼ 1
fcm ϕð1 + IÞ

ð6Þ

which has a functional form that has been suggested for bulk systems,11,13,27 where I is a fitting constant in this and our previous measurements24 (motivated by a term consisting of an integral over the pair distribution function in the threedimensional theory).13

’ SIMULATIONS AND ANALYSIS Concentrated particle experiments were simulated using canonical (NVT) MC.28 These simulations (1) employed periodic boundary conditions, (2) used a cutoff of the long-range dipolar potential at 5a (which is less than half the simulation box size),29 (3) were initialized using experimental concentrations and configurations, and (4) included image resolution limiting effects and polydispersity for comparing results to experiments.18,19 An inverse MC algorithm was implemented to extract the frequency-dependent correction, fω, from the angular dependence of measured radial distribution functions at contact, Fm(2a, θ). In the inverse MC algorithm, the initial guess for upp0(2a, θ), was obtained using the analytical expression in eq 2 with independently measured parameters and a starting value of fω = 1 in eq 5. A revised estimate of the potential (from the initial potential or previous guess), uppi+1(2a, θ), was obtained from a MC simulation using uppi(2a, θ) to generate a simulated radial distribution functions at contact, Fs(2a, θ), which is used in an update algorithm as

where the potential between colloids with thin electrostatic double layers (kR . 1) is26
 pp upp ð3Þ e ðrij Þ ¼ B exp
kðrij
2aÞ where k
1 is the Debye length, rij is center-to-center separation between particles i and j, a is the particle radius, and Bpp for a 1
1 electrolyte valence is  2   eψp kT pp 2 B ¼ 32πεm a tanh ð4Þ e 4kT

!3 2a  ¼
kTλfϕ fω P2 ðcos θij Þ E ðxi , zi Þ2 rij

pp ui + 1 ð2a, θÞ

¼

pp ui ð2a, θÞ + 0:5

Fs, i ð2a, θÞ Fm ð2a, θÞ
1

! ð7Þ

and uppi+1(2a, θ) was used to obtain a new estimate of fω,i+1 using (rearrangement of eq 5) N
1 fω, i+1 ¼ 9220

∑θ ½ui+1 ð2a, θÞ


λkTfϕ P2 ðcos θÞEðxm , hm Þ2

ð8Þ

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Figure 1. (left, a) Video microscopy images of equilibrated PS colloids within a 60 μm coplanar Au thin film electrode gap as a function of frequency, ω, and particle size, a, for a constant field amplitude of 300 mV. (right, b) Monte Carlo simulated configurations from potentials obtained via inverse analysis described in the text.

which is repeated iteratively until Fs(2a, θ) is within some tolerance, ε, of Fm(2a, θ) using " χi ¼

∑θ ε


2



Fs, i ð2a, θÞ
Fm ð2a, θÞ

2

#1=2 ð9Þ

where uniqueness of the converged solution cannot be guaranteed in the presence of experimental noise and uncertainty (e.g., polydispersity, image limitations, etc.); however, in practice, rapid convergence to physically realistic solutions in all cases did not suggest the need for more elaborate inverse algorithms that could be implemented to improve robustness.

’ RESULTS AND DISCUSSION Frequency and Particle Size Dependent Microstructures. With a clear understanding of the frequency and amplitude dependent interactions of single induced dipoles with inhomogeneous electric fields in Part I,1 we now explore interactions at relatively high particle concentrations. The measurement geometry is identical except the electrode gap spacing has been doubled from ∼30 μm to ∼60 μm. This reduces the field amplitude by a factor of 2 everywhere but allows room for the addition of more particles. The most important change associated with the introduction of finite particle concentrations is that induced dipoles interact with each other while still interacting with the inhomogeneous electric field. As a result, these measurements provide additional information on how particle concentration and configuration influence the frequency dependence of both dipole
dipole and dipole
field interactions. As in the single particle measurements, the raw data in our measurements and simulations is quasi-2D PS particle coordinates, which is depicted for representative experiments in Figure 1. Figure 1a shows a matrix of static images extracted from optical microscopy videos of dynamic equilibrium particle configurations within the inhomogeneous AC electric fields. Figure 1b shows the MC simulated renderings that correspond to the experiments in Figure 1a as obtained through an inverse analysis. The matrix in Figure 1a serves as a sort of visual state diagram for the frequency and size-dependent quasi-2D ordering of PS particles in inhomogeneous AC electric fields. The representative results reported in Figure 1 are shown for experimental conditions including the same three PS particle

sizes investigated in part I1 (i.e., 2a = 3, 4, 5 μm) and frequencies of ω/2π = 100, 350, 600 kHz. In all cases, the applied voltage was fixed at 300 mV, which corresponds to nondimensional field strengths on the order of λ = 1 (∼0.1
3 for all conditions investigated) or interactions on the order of kT. Interactions comparable to kT allow particles to sample different configurations in dynamic equilibrium. This allows us to use inverse MC simulation methods and also directly connects these interactions and microstructures to self-assembly processes. Numerous frequencies were investigated in the range ω/2π = 100
1000 kHz where these PS particle sizes can be expected to show a cross over behavior based on the single particle results in Part I.1 The average area fraction in each case is determined by the number of particles within the electrode gap divided by the electrode gap area. These values remain essentially constant as frequency is changed with some minor variations due to several particles entering or leaving the gap at the electrode edges. For all cases where fCM < 0, the number of particles trapped at the electrode center remains unchanged, whereas for fCM > 0, some particles initially levitated over the gold film electrode can be pulled into the gap at the edge to slightly increase the number of particles. In the analysis of all cases, the exact number of particles is monitored directly and used in MC simulations. The average area fractions for the 3, 4, and 5 μm data are ϕ = 0.39, 0.32, and 0.36. To motivate a quantitative analysis of the results in Figure 1a, we briefly discuss some effects that cannot be easily anticipated from the single particle results in Part I.1 As expected, particles form condensed structures in the middle of the electrode gap for fCM < 0 (e.g., 600 kHz) and near the electrode edges for fCM > 0 (e.g., 100 kHz). However, understanding the assembly of nonclose-packed microstructures for fCM < 0 and fCM > 0 requires knowing the relative balance of dipole
dipole and dipole
field interactions that produce a competition between particle chaining and compression. In addition, an unexpected result is that isotropic fluid configurations are not observed in the vicinity of fCM = 0 in Figure 1a. This can be seen in the 350 kHz results close to the single particle cross over frequencies of 365
385 kHz. Instead, residual chaining in the gap center is observed even though both dipole
dipole and dipole
field interactions should have vanished. For particles experiencing only electrostatic repulsion at ϕ ≈ 0.35, Brownian motion should drive random dispersion of particles into isotropic fluid configurations. 9221

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Figure 2. (left, a) Equilibrated angular distributions from 0° to 90° averaged over four quadrants including symmetry considerations based on field orientation for experiments (points) and simulations (lines) of 3 μm (top), 4 μm (middle), and 5 μm (bottom) PS for fCM < 0. (right, b) Density profiles across the electrode gap averaged over the same configurations as in Figure 2a. In all plots, colors indicate ω/2π = 1 MHz (black), 600 kHz (red), 500 kHz (green), 450 kHz (yellow), and 400 kHz (blue).

Frequency-Dependent Induced Dipolar Interactions. To understand frequency and particle size dependent dipolar interactions and microstructures in Figure 1, we employ an inverse MC simulation analysis.24 Because of many particle packing effects, the results in Figure 1 cannot be interpreted using a simple Boltzmann inversion, which was appropriate for the single particle results in Part I.1 In brief, the inverse MC analysis matches measured and simulated microstructures by adjusting interaction potentials. We previously developed this method to study field amplitude dependent measurements at a fixed frequency.24 We now extend this approach in the present study to frequency and particle size dependent dipole
dipole interactions. With this overview of the approach, we describe the measured distribution functions to be used in the analysis. Figures 2 and 3 report frequency-dependent distribution functions that characterize the average angular distribution of particles within the nearest-neighbor coordination shell, F(θ)2a, and the spatial distribution of particles across the electrode gap, F(x). Figure 2 shows these results for the three PS particle sizes when the particles are confined to the electrode gap center for fCM < 0 at frequencies of ω/2π = 400, 450, 500, 600, and 1000 kHz. Figure 3 shows the same type of results when particles are confined near the electrode edges for fCM > 0 at frequencies of ω/2π = 100, 150, 250, 300, and 350 kHz. In each case, the points correspond to distribution functions constructed from particle centers located from image analysis. Distributions are averaged over all particles, spatial positions within the electrode gap, and observation times of ∼20 min (temporal averaging is the same as in single particle measurements; i.e., 30 000 images acquired at 28/s). The lines correspond to MC simulation results that are

averaged at the same spatial resolution as experiments and also include particle size polydispersity.18,19 The measured results for fCM < 0 in Figure 2a show that 3 μm PS particles sample all angles in F(θ)2a in a relatively uniform manner. This is consistent with the dipole
dipole interaction being comparable to kT so that isotropic fluid configurations are observed rather than dipolar chains. The results in Figure 2b show that 3 μm PS particles also sample wider distributions within the gap center in the F(x) data. This observation is consistent with the dipole
field interactions being comparable to kT so that particles are only weakly confined within the electrode gap. As the particle sized is increased in the 4 and 5 μm PS cases, the results in Figure 2 show stronger confinement at the electrode gap center in the F(x) data and more chaining based on an enhancement of F(θ)2a near 0° (i.e., dipole chaining and alignment with the field perpendicular to the electrode gap). The results are consistent with the single particle dipole
field results in Part I1 including the a2 size dependence of the nondimensional field strength, λ. In addition, the results in Figure 2 show the role of dipole
dipole interactions in controlling microstructures in concentrated quasi-2D dispersions. The measured results for fCM > 0 in Figure 3a and b shows that the 3 μm PS particles remain as an inhomogeneous fluid near the electrode edges for all frequencies including those well below the crossover frequency for single 3 μm PS particles. This indicates that dipole
dipole interactions are not strong enough compared to kT to form chains, and dipole
field interactions are not strong enough to compress particles into crystalline configurations. In contrast, peaks near 30° in the F(θ)2a data for the 4 and 5 μm PS particles show that these larger particles are compressed into 9222

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Figure 3. (left, a) Equilibrated angular distributions from 0° to 90° averaged over four quadrants including symmetry considerations based on field orientation for experiments (points) and simulations (lines) of 3 μm (top), 4 μm (middle), and 5 μm (bottom) PS for fCM > 0. (right, b) Density profiles across the electrode gap averaged over the same configurations as in Figure 2a. In all plots, colors indicate ω/2π = 350 kHz (black), 300 kHz (red), 250 kHz (green), 150 kHz (yellow), and 100 kHz (blue).

crystalline configurations near the electrode edges. Peaks corresponding to hexagonal packing appear at 150 kHz for the 4 μm particles and 250 kHz for the 5 μm particles, indicating that a stronger compression is obtained for the larger particle size. The findings for fCM > 0 all qualitatively make sense with knowledge of the behavior of single dipole
field interactions in terms of whether the particle ensemble is observed in the electrode gap center or at the electrode edge. However, understanding the formation of the observed fluid, chain, and crystal microstructures requires understanding how dipole
dipole interactions that drive chaining compete with dipole
field interactions that compress particles into crystals and thermal energy that drives configurations toward random dispersions. To quantify the frequency dependence of the dipole
dipole interactions and its contribution to the observed dynamic equilibrium microstructures in Figures 1
3, we now use the inverse MC algorithm we previously developed for amplitude and concentration dependent interactions at fixed frequencies.24 Because the dipole
field interactions were already measured for single particles in the absence of dipole
dipole interactions in Part I,1 the inverse analysis of many particle measurements is well posed by using the single particle interaction as a nonadjustable input. Because the dipole
field and dipole
dipole potentials contain many of the same parameters (listed in the tables of Part I1), we also fix these parameters in the dipole
dipole potentials. The dipole
dipole potentials in eq 5 include a concentration-dependent factor13 that we model based on our previous amplitude and concentration dependent measurements (with I =
0.18 in eq 6).24 Finally, the electrostatic interactions share all of the parameters in the dipole
field potentials measured

in Part I,1 so none of these parameters are adjustable. Ultimately, this leaves no adjustable parameters, and as we discuss in the following, we introduce a single new adjustable parameter to produce agreement between our measurements and simulations. To match measured and simulated distribution functions in Figures 2 and 3, a scalar correction factor, fω, was introduced in the dipole
dipole potential in eq 5. This correction factor was added to account for apparent chaining observed in the vicinity of the crossover frequency that is suggestive of a residual dipolar attraction. The single scalar factor, fω, in the dipole
dipole potential in eq 5 when fit at each frequency produces good agreement between the measured and simulated distribution functions in Figures 2 and 3. Matching the angular distribution functions, F(θ)2a, in Figures 2a and 3a shows that the local microstructure and dipole
dipole potentials are well captured. Matching the density profiles across the electrode gap, F(x), in Figures 2b and 3b shows the distribution within the electrodes, and the dipole
field potentials are well captured. In all cases, excellent agreement was observed between the measured and simulated images/renderings in Figure 1 and distribution functions in Figures 2 and 3. The values of the correction factors for all cases are reported in Figure 4. Values of fω are unity for frequencies away from the crossover regime, showing that the dipolar interaction is well described by eq 5 using the parameters obtained from the dipole
field interaction in the single particle measurements. This is consistent with our previous amplitude-dependent measurements at 1 MHz.24 However, the apparent chaining of particles through the crossover (from both high- and low-frequency sides) causes this quantity to become much greater than unity in the vicinity of 9223

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Figure 4. Data points for adjustable parameter, fω, in eq 5 as a function of scaled frequency for 3 μm (black), 4 μm (red), and 5 μm (green) PS. The dashed line is eq 10 plotted using a value of c = 20 and ωm/2π = 10 kHz, and the solid line is eq 10 with the same parameters smoothed using a Gaussian kernel with a bandwidth of 100 kHz.

the crossover. This occurs because fCM and the dipole
dipole and dipole
field interactions should vanish at the crossover, so the value of fω increases to capture the apparent residual dipolar chaining and apparent attraction. Physical Origin of the Single Adjustable Parameter. Introduction of a scalar factor in the dipolar interaction (eq 5) appears sufficient to capture the microstructure in the vicinity of the crossover frequency in each case, but the underlying physical mechanism is not obvious. The primary issue is that, as fCM vanishes, there should be no induced dipole, and hence no interactions of the particles with the electric field (i.e., dipole
field) or electric field mediated particle
particle (i.e., dipole
dipole) interactions. In addition, within the resolution of the single particle measurements in Part I,1 there was no indication of an induced dipole
field interaction in the vicinity of the crossover frequency. As a result, we seek alternative explanations for the origin of the microstructure in the vicinity of the crossover. Both the single particle measurements in Part I1 and the ensemble measurements in this paper are well described for nearly all conditions by simple models of induced dipoles on charged stabilized colloids. However, as these simple models predict the induced dipoles to vanish, more subtle contributions might become important. For example, the assumptions underlying the derivation of the potential in eq 5 are not all satisfied for our measurement configuration. In particular, the potential in eq 5 is generally derived for point dipoles in bulk systems rather than (1) induced dipoles formed by polarization of overlapping electrostatic double layers between adjacent particles and the underlying substrate, (2) in concentrated quasi-2D dispersions, (3) in an inhomogeneous, high-frequency AC electric field. This complex configuration is certainly capable of producing a range of electrokinetic phenomena that could become more significant as dipolar interactions vanish. We explore the possibility of AC electro-osmotic (ACEO) flows as a source of the residual microstructure and apparent dipolar attraction in the vicinity of the crossover frequency in Figures 1
3. Our motivation for exploring ACEO is based on

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numerous literature studies that have observed formation of complex configurations of charged colloids in stagnation regions between complex 3D recirculating interfacial and confined flows between electrodes.30
33 Of available electrokinetic phenomena, we note that electrophoretic transport is unimportant at such high frequencies.34 Previous measurements by us using the same electrode geometry showed ACEO flows to have a maximum effect on entrained 0.5 μm Au colloids for field frequencies around ωm/2π = 10 kHz.35 Although this is well below the 0.1
1 MHz range of our present measurements, ACEO transport effects could again contribute at the crossover where dipolar interactions vanish. ACEO flows are complex and produce nontrivial particle microstructures that in many cases do not appear to correspond to an apparent dipolar pair attraction. However, for recirculating flows between parallel thin film electrodes, the flow near the surface is normal to the electrodes and decreases as the gap center is approached. We speculate that such a flow could be manifested as an apparent “flow-induced” dipolar attraction and also localize particles near the center at a steady-state stagnation point. In other words, ACEO could push particles together in the center of the electrode gap and cause them to align with stream lines normal to the electrode edges. These types of structures have been observed before in similar electrode geometries at low frequencies and higher fields at conditions where ACEO usually dominates.33 On the basis of this assumption, we attempt to capture relative frequency dependent contributions from flow field and electric field induced dipolar interactions with a simple scaling analysis as   ωm 2 ð10Þ fω  1 + c ωfcm which is based on (1) the frequency dependence of dipolar interactions in eq 5 as fCM2 that we measured in Part I,1 (2) the frequency dependence of ACEO as (ωm/ω)2,34 far from the maximum frequency, ωm, and (3) the relative nondimensional magnitude of ACEO forces at their maximum (we introduce k as an adjustable parameter rather than predicting it based on more fundamental models34). The dashed line in Figure 4 is eq 10 plotted using a value of c = 20, and the solid line is eq 10 smoothed using a Gaussian kernel with a bandwidth of 100 kHz. Smoothing broadens the peak width beyond what is predicted based solely on the frequency dependence of fCM and could be attributed to smearing effects such as polydispersity, charge heterogeneity, and averaging interactions at different locations within the inhomogeneous electric and flow fields (where particles experience different induced interactions at different locations). There is a correlation between the smoothed version of eq 10 with the 4 and 5 μm data and to a lesser extent with the 3 μm data. The dipolar energies/forces are the weakest in the 3 μm case, so an ACEO flow effects could have a greater influence on these particles. The 3 μm particles are also the lightest and experience the greatest Brownian excursions away from the microscope slide, which could also allow them to experience greater ACEO flows. On the basis of both the simplicity of the scaling and the complexity of ACEO flows, the correspondence between the data and eq 5 is quite good. Proving that the apparent dipolar interaction at the crossover is uniquely due to an ACEO flow effect is not trivial and is beyond the scope of this study. However, the scaling analysis in Figure 4 suggests it is a 9224

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Langmuir reasonable possibility that ACEO flows could influence the microstructure as dipolar interactions vanish at the crossover. Finally, it should be noted that the single particle measurements in Part I would not be obviously influenced by these ACEO flows, since the particles already reside at the electrode center as the crossover is approached (where the stagnation point occurs), and two particles are not present in these measurements to observe apparent dipole
dipole interactions.

’ CONCLUSIONS Our results include nonintrusive optical microscopy measurements and a simulation based analysis that provide sensitive measures of kT-scale electric field induced dipole
dipole interactions as a function of frequency and particle size. By using the measured induced dipole
field interactions from Part I as input in the analysis, the measured dipole
dipole interactions in this work are modeled with no adjustable parameters in the limits of |fCM| > 0 and a single adjustable parameter as fCM vanishes at the crossover frequency. The adjustable parameter is introduced to account for microstructures at the crossover frequency that are not expected to occur as the result of electric field induced interactions. We speculate that these microstructures occur as a result of ACEO flow induced interactions in the absence of induced dipolar interactions. By connecting quantitative measures of equilibrium microstructures to kT-scale dipole
field and dipole
dipole potentials, our findings provide a basis to design, predict, and control colloidal self- and directed-assembly processes in inhomogeneous AC electric fields. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We acknowledge financial support provided by the National Science Foundation through a Cyber Enabled Discovery and Innovation grant (CMMI-0835549) and an unsolicited grant (CBET-0932973), and the Air Force Office of Scientific Research (FA9550-08-1-0329). ’ REFERENCES (1) Juarez, J. J.; Cui, J.; Liu, B.; Bevan, M. A. kT-Scale Colloidal Interactions in High Frequency Inhomogeneous ac Electric Fields. I. Single Particles. Langmuir 2011, in press; doi: 10.1021/la201478v. (2) Jones, T. B. Electromechanics of Particles; Cambridge University Press: Cambridge, 1995; p 265. (3) Pohl, H. A. Dielectrophoresis: the behavior of neutral matter in nonuniform electric fields; Cambridge University Press: Cambridge, 1978; p xii, 579. (4) Gast, A. P.; Zukoski, C. F. Electrorheological fluids as colloidal suspensions. Adv. Colloid Interface Sci. 1989, 30, 153–202. (5) Pethig, R. Review Article---Dielectrophoresis: Status of the theory, technology, and applications. Biomicrofluidics 2010, 4 (2), 022811. (6) Fraden, S.; Hurd, A. J.; Meyer, R. B. Electric-field-induced association of colloidal particles. Phys. Rev. Lett. 1989, 63 (21), 2373. (7) Hermanson, K.; Lumsdon, S.; Williams, J.; Kaler, E.; Velev, O. Dielectrophoretic Assembly of Electrically Functional Microwires from Nanoparticle Suspensions. Science 2001, 294, 1082–1086. (8) Yethiraj, A.; Blaaderen, A. v. A colloidal model system with an interaction tunable from hard sphere to soft and dipolar. Nature 2003, 421, 513–517.

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