Transition from Dilute to Concentrated Electrokinetic Behavior in the

Jun 22, 2012 - Dielectric spectroscopy of colloidal dispersions: Comparisons between experiment and theory. Langmuir. Rosen, Saville. 1991 7 (1), pp 3...
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Transition from Dilute to Concentrated Electrokinetic Behavior in the Dielectric Spectra of a Colloidal Suspension Peter J. Beltramo and Eric M. Furst* Department of Chemical and Biomolecular Engineering and Center for Molecular and Engineering Thermodynamics, University of Delaware, 150 Academy St., Newark, Delaware 19716, United States S Supporting Information *

ABSTRACT: Dielectric spectroscopy is used to measure the complex permittivity of 200 and 100 nm diameter polystyrene latex suspended in potassium chloride (KCl) solutions over the frequency range 104−107 Hz as a function of particle volume fraction (ϕ) and ionic strength. Dilute suspension dielectric spectra are in excellent agreement with electrokinetic theory. A volume fraction dependence of the dielectric increment is observed for low electrolyte concentrations (0.01, 0.05, and 0.1 mM) above ϕ ≈ 0.02. This deviation from the dilute theory occurs at a critical frequency ω* that is a function of volume fraction, particle size, and ionic strength. The dielectric increment of suspensions at the highest salt concentration (1 mM) shows no volume fraction dependence up to ϕ = 0.09. Values of ω* are collapsed onto a master curve that accounts for the length and time scales of ion migration between neighboring particles. The measured conductivity increment is independent of volume fraction and agrees with theory after accounting for added counterions and nonspecific adsorption.

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heuristic can only be applied at a fixed frequency; a key aim of this work is to establish the frequency dependent limits of electrokinetically dilute behavior. Many electrokinetic models of colloidal suspensions are valid in the infinite dilution limit.21−26 Additional assumptions of these models include considering the spherical particle as homogeneous in dielectric constant and surface charge, ions as point charges, and applying uniform, macroscopic solvent properties such as dielectric constant, viscosity, and density. As the applied field perturbs the static, equilibrium ionic double layer around a charged particle, the standard model solves for the conductivity and dielectric increments (among other quantities) from the far-field decay of the electrostatic potential and ionic concentrations as a function of frequency. This treatment is appropriate when particle−particle correlations are absent, satisfying the infinite dilution assumption. When the particle concentration increases beyond the dilute limit, interparticle interactions alter the dielectric spectra and agreement between the dilute theory and experiment becomes poor.27−29 As a result, there are few systematic dielectric spectroscopy studies of concentrated suspensions.30,31 Recent work has expanded the range of electrokinetic theory to nondilute volume fractions using cell models. Cell models account for finite volume fraction by modifying the far-field boundary conditions of a single particle by incorporating global constraints on the volume averaged electric field, ion transport,

INTRODUCTION The electric field responsiveness of colloidal dispersions has become increasingly useful in the development of materials with novel rheological,1 mechanical,2 and optical properties3 that can be exploited in the design of coatings,4 biosensors,5 or electronic inks.6 These technologies rely on the applied field directing the self-assembly of colloidal particles into new nanoand microstructures.7 Despite significant empirical progress in electric field directed self-assembly,8,9 there remains a need to understand the underlying electrokinetic processes that drive assembly.10,11 Specifically, optimal self-assembly conditions are expected to depend on a combination of properties, including electric field frequency and magnitude, solution pH and ionic strength, particle size and shape, and dielectric properties, giving rise to a potentially enormous parameter space. This motivates a more fundamental understanding of the electrokinetic properties of concentrated and ordered suspensions under ac fields in order to rationally design field directed selfassembly. Dielectric spectroscopy has been primarily used to study dilute colloidal suspension electrokinetics,12−16 although the electrokinetics of concentrated colloidal suspensions are of increased practical and technological importance.17,18 The transition between “dilute” and “concentrated” is not precisely defined and may occur at different volume fractions depending on the characteristics of the suspension. In general, a suspension is considered dilute when the electrophoretic mobility is independent of volume fraction19 or until an intrinsic property, such as the conductivity or dielectric increment, becomes nonlinear in volume fraction.20 The first © 2012 American Chemical Society

Received: May 8, 2012 Revised: June 20, 2012 Published: June 22, 2012 10703

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Figure 1. Schematic of the polarization frequency regimes. (d) In the absence of electric field, a uniform equilibrium double layer forms around a negatively charged particle. (c) At high frequencies, ions do not have time for transport and the only contribution to the polarizability is from the dielectric contrast mismatch between the particle and solvent (oriented antiparallel to the applied field when εp < ε∞). (b) Double layer polarization (oriented parallel with the applied field for highly charged particles) develops as the frequency approaches the time scale for ion transport across the double layer, causing the overall polarization to be parallel with the electric field. (a) Tangential ionic fluxes increase the magnitude of the parallel polarizability as ion transport occurs over the particle length scale. Only counterions shown for clarity.

and pressure gradients.32,33 Carrique et al.32 analyze the double layer relaxation frequency and low frequency permittivity as a function of volume fraction in place of the entire frequencydependent dielectric spectra. Their numerical solutions are in qualitative agreement with experimental data but overestimate the relaxation frequency and underestimate the low frequency permittivity. Bradshaw-Hajek et al.33 present a solution with improved quantitative agreement with experiments using 500 nm polystyrene spheres up to the volume fraction ϕ = 0.50 over a relatively small range of high frequencies (1−15 MHz). In this work, we report an experimental study of colloidal suspension dielectric spectra, spanning the dilute to concentrated regimes by varying the ionic strength and volume fraction of the suspensions. Over this regime, the suspension structure also changes, from disordered dispersions to iridescent, ordered crystals. We begin by first presenting a brief outline of the standard electrokinetic model of dilute suspensions before discussing our dielectric spectroscopy experimental methods. In the results and discussion, we demonstrate that the transition from dilute to concentrated dielectric spectra manifests as a decrease in the dielectric increment at low frequencies. The frequency dependence of the transition collapses onto a single master curve using an effective volume fraction and scaled frequency based on the particle and double layer length scale. The results give important insight into how neighboring particles produce additional sources and sinks of ionic fluxes to alter the permittivity increment.

much faster than the particle movement. This results in several characteristic frequency regimes over which the dielectric spectra varies depending on the dominant process of ion transport (Figure 1). The characteristic frequency for ion transport across the length of the double layer and the particle radius, a, are given by

ωκ = Dκ 2 2π and

ωa D = 2 2π a

THEORETICAL BACKGROUND Here we review the electrokinetic properties and polarization of a single charged spherical particle suspended in an infinite electrolyte and subjected to a uniform ac electric field. In the absence of the applied field, an electric double layer develops around the particle to screen the surface charge. The characteristic dimension of the double layer, given by the Debye length kBTε∞ε0 /(2Ie 2)

(3)

respectively, where D = kBT/(λNa) is the mean ion diffusion coefficient (D = 1.99 × 10−9 m2/s for KCl) based on the limiting ion mobility, λ. At frequencies ω > ωκ, ions do not have sufficient time to respond to the applied field. This is shown in Figure 1c, where the double layer retains its equilibrium structure and polarization of particles is due to the dielectric contrast with the solvent, as is the case for a polystyrene sphere (εp = 2.6) suspended in water (ε∞ = 78.54).34 The electric double layer polarizes as the frequency decreases to ωκ. At this frequency, counterions (co-ions) follow the direction of the field lines, accumulating (depleting) at one end of the particle and depleting (accumulating) at the other, resulting in an additional ionic polarization. Although Figure 1 only shows one-half cycle of an ac field, the polarization has components both in and out of phase with the oscillation as the dipole inverts by ion diffusion and electromigration. The magnitude of ionic polarization increases as the frequency decreases in this regime to produce an overall polarization in phase with the applied field. A concentration gradient of counterions between the poles of the particle is now present, and an additional flux of ions around the particle over this gradient develops as the frequency decreases to ωp. This is supplemented by a gradient of neutral electrolyte that also develops further away from the particle, causing an additional relaxation in the dielectric spectra known as concentration polarization.18 Quantitatively, perturbations to the double layer surrounding individual particles are apparent in the complex polarizability of the suspension, which has components in and out of phase with the applied field. From the complex polarizability, P, of the particles, the dielectric and conductivity increments of the suspension are

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κ −1 =

(2)

(1)

/2∑Ni=1(zi2n∞ i ),

1

depends on the ionic strength, I = of the electrolyte. Here, kB is the Boltzmann constant, T is the absolute temperature, ε∞ is the electrolyte dielectric constant, ε0 is the permittivity of free space, e is the fundamental charge, zi is the ion valence, and n∞ i is the concentration of each ionic species. The oscillating ac field will perturb this equilibrium double layer because ion transport due to electromigration is 10704

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ε′s (ω)/ε∞ − 1 = 3[R(P) − ω̂ −1I(P)] ϕ

(4)

σs(ω)/σ∞ − 1 = 3[R(P) + ω̂ I(P)] ϕ

(5)

did not alter the spectra; the applied voltage is below the threshold voltage required to form particle chains and crystal microstructures in the suspension.8 The measured complex impedance Z* is corrected using open/ short circuit compensation to remove test fixture residuals in parallel and in series with the sample impedance. The complex admittance Y* = 1/Z* is related to the frequency dependent conductivity σ(ω) and apparent dielectric constant ε′app(ω) by

and Δσ(ω) =

where ω̂ = ωε∞ε0/σ∞ is the scaled frequency, σ∞ is the conductivity of the suspending fluid (electrolyte), ε′s(ω) and σs(ω) are the dielectric constant and conductivity of the suspension, respectively, and ϕ is the particle volume fraction. We expect that as the particle concentration increases, the dielectric spectra will change depending on how the ionic transport is altered by the presence of neighboring particles.33

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Y *(ω) = G[σ(ω) + iωε0ε′app (ω)]

(6)

where ω = 2πf is the angular frequency and G is the cell constant measured for each spacer using the electrolyte solution. We verified that G is within ±3% of the expected value based on the spacer geometry. The electrode polarization contribution to the apparent dielectric constant is corrected following the method outlined by Hollingsworth and Saville.36 While there are other methods to correct for electrode polarization,37,38 this method was chosen because it is based on a theoretical model that has been successfully applied to suspensions containing 1−1 electrolytes similar to those studied in this work.15 The low-frequency dielectric constant is fit to

EXPERIMENTAL SECTION

Materials. Monodisperse polystyrene latex microspheres (Invitrogen, Eugene, OR) with nominal diameter 2a = 200 nm (batch #1312,1, lot #638478) and 2a = 100 nm (batch #923,3, lot #512131) are used in this work. The hydrodynamic diameter from dynamic light scattering measurements is 197 ± 8 and 117 ± 4 nm, respectively, in agreement with the diameter determined by transmission electron microscopy (210 ± 7 and 110 ± 5 nm). The particles are stabilized by sulfate groups that give the particle a titratable surface charge of −1.3 and −1.0 μC/cm2, respectively, according to the manufacturer. We refer the reader to the Supporting Information for electrophoretic mobility measurements as a supplementary measure of the particle surface charge density. To obtain reliable electrokinetic data, the suspensions are prepared using an established cleaning protocol consisting of at least nine centrifugation−decantation−resuspension steps.9,15,35 As-received particles are diluted from a volume fraction of ϕ = 0.08 to 0.01 in ultrapure water (Millipore Direct-Q, Billerica, MA, resistivity ρ ≥ 18.2 MΩ·cm) and washed four times using repeated centrifugation (7200g for 90 min for 200 nm and 15000g for 120 min for 100 nm) and redispersion to remove potential contaminants. The particles are then washed a minimum of five times in the potassium chloride solution of interest (Alfa Aesar Puratronic, Ward Hill, MA; 99.997% pure, metals basis) to ensure accuracy in the solvent electrolyte concentration. The total washing procedure is carried out over ∼3 days, during which the background electrolyte reproducibly equilibrates with atmospheric carbon dioxide, resulting in an additional 3.2 μM of carbonate ions in the suspension. An additional centrifugation step concentrates the particles to the highest tested volume fraction, from which additional samples are prepared by dilutions. The suspensions are sonicated for 30 min and immediately loaded into the dielectric spectrometer. Methods. Dielectric Spectroscopy. A dielectric spectrometer based on the design by Hollingsworth and Saville36 was fabricated and used in this work. Here we present a brief description of the apparatus and experimental techniques. For a detailed description of the cell construction, calibration, and validation procedures the reader is referred to ref 36. The spectrometer consists of an impedance analyzer (Agilent 4294A-1D5, Santa Clara, CA; 40 Hz ≤ ω/2π ≤ 110 MHz) connected to a custom fabricated parallel plate dielectric cell via four-terminal pair BNC connectors. The dielectric cell is made of two stainless steel electrodes clamped around an acrylic spacer to create the sample chamber. Five acrylic spacers of widths 2h = 1.40, 2.92, 3.18, 4.24, and 6.08 mm and varying sample areas exposed to the electrodes are designed to minimize fringing effects. Multiple spacers are used to compensate for electrode polarization by extrapolating multiple measurements to infinite electrode separation.15,36 Silicone oil circulates (ThermoFisher Scientific SC150-A10, Waltham, MA) through vertical bores in the electrodes to maintain a constant temperature of 25 ± 0.1 °C for all experiments. Logarithmic frequency sweeps are made from 110 MHz to 40 Hz at a nominal oscillator voltage level of 0.5 V and the maximum bandwidth to optimize the signal-to-noise ratio. Changing the voltage

ε′app (ω) ε∞

=β

1 + β(ω /κ 2D)2 1 + β 2(ω /κ 2D)2

(7)

using D, the average ion diffusion coefficient, as an adjustable parameter. Here β = κh is a geometric parameter. For wide electrode spacing and low frequencies, eq 7 reduces to

ε′app (ω) ε′(ω)

≈1+

κ 3D2 ω2h

(8)

where ε′(ω) is the electrode polarization corrected dielectric constant (equal to ε∞ for electrolyte solutions). We confirm that the apparent dielectric constant varies linearly with κ3D2/h for both the electrolyte and suspensions and find ε′(ω) for each frequency by extrapolating to infinite separation (h → ∞). The linearity in the extrapolation procedure for the electrolyte, dilute suspensions, and crystal suspensions is verified in the Supporting Information. Finally, the suspension spectra are converted to dielectric and conductivity increments via eqs 4 and 5, respectively, for comparison with electrokinetic theory. Standard Electrokinetic Theory. We use the standard electrokinetic model (SEKM) to calculate the dielectric and conductivity increments for interpreting our experimental results.26,39,40 The numerical solution scheme developed by Hill et al.,26 MPEK, provides accurate calculations over a wide frequency range and is capable of incorporating charged39 or uncharged40 polymer layers on the particle surface. Using MPEK, the SEKM has been applied successfully in recent dielectric spectroscopy studies.15,41 The particle and electrolyte properties are determined from the experimental conditions, and the surface charge density is used as an adjustable parameter to fit the suspension spectra. A carbonate ion concentration of 3.2 μM (HCO3− and H3O+) is added to the background electrolyte concentration in all calculations to account for dissolved carbon dioxide. Previous studies note that the presence of dissolved carbon dioxide alters the electrohydrodynamic forces on a colloidal particle near an electrode in ac electric fields.42 Our sample preparation protocol ensures a constant, equilibrium concentration of dissolved carbon dioxide, and we are able to explicitly account for it in the SEKM calculations. Additional details regarding parameters used in the model calculations and calculations incorporating a thin layer of uncharged/charged polymer on the particle surface are given for comparison in the Supporting Information.

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RESULTS Suspension Phase Diagram. We first identify the fluid− crystal transition in suspensions of 100 and 200 nm spheres as a function of salt concentration and particle volume fraction

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Dielectric Spectroscopy. Dilute Suspensions as a Function of Salt Concentration. We first present dilute dielectric spectroscopy results, focusing on the 200 nm particle suspensions at varying electrolyte concentrations before analyzing the concentration dependence of suspension dielectric spectra. These results illustrate the “electrokinetically ideal” nature of the suspensions in the dilute limit through their quantitative agreement with the standard electrokinetic model (SEKM) calculations using a constant particle surface charge density. The dielectric and conductivity increments are shown for ϕ = 0.01 for 0.01 mM and 0.1 mM KCl and ϕ = 0.018 for 1 mM KCl (Figure 3). Excellent quantitative agreement with the dielectric increment using a surface charge density q = −1.6 μC/cm2 is found in all cases, confirming that in the low volume fraction limit these suspensions behave electrokinetically dilute. The dielectric increment spectra highlights the changing magnitude of the dispersion relaxations as the salt concentration increases. The characteristic frequency for ion transport across the double layer is Dκ = 2.9 × 105, 2.2 × 106, and 2.2 × 107 Hz for κ−1 = 84, 30, and 9.6 nm, respectively, while the characteristic frequency for ion transport across the particle radius is Da−2 = 2.0 × 105 Hz. As the frequency decreases, the dielectric increment switches from negative to positive as the double layer polarization magnitude exceeds the polarization charge due to the dielectric mismatch. The specific frequency where the overall polarization changes sign and where the various dispersions occur is a function of electrolyte concentration, as expected. Above 200 kHz (Da−2), as the electrostatic screening length increases the double layer polarization increases, causing the dielectric increment to be greater for weaker ionic strength suspensions. Below 200 kHz, the dielectric increment increases with increasing ionic strength due to the higher concentration of ions in both the double layer and bulk suspension that leads to an increase in the concentration polarization magnitude. The conductivity increment exhibits quantitative agreement when an offset is included with SEKM calculations. The increment decreases as the electrolyte concentration increases since it is scaled by σ∞. The conductivity increment calculated by the SEKM in Figure 3 does not include the contribution of added counterions and nonspecific adsorption, which may contribute to the offset between the experimental data and the calculations. Saville45 originally considered these frequencyindependent contributions to the conductivity by noting that the dissociated groups on the particle surface produce changes in the counterion concentration in both the double layer and

before comparing their dielectric spectra. Particle sizes and electrolyte concentrations were chosen such that κa ∼ O(1), where the repulsion between charged spheres leads to a disorder−crystal transition at relatively low volume fractions compared to the hard-sphere values.43,44 Figure 2 shows that

Figure 2. Phase diagram showing the turbid (open symbols) and iridescent (filled symbols) 100 nm (□) and 200 nm (△) suspensions studied in this work. Dashed and dotted line is the fluid−crystal boundary (ϕeff = 0.545, |ζ| = 100 mV) predicted using a screened Coulomb potential for 100 and 200 nm diameter spheres, respectively.

the observed transition from a disordered, turbid suspension to an iridescent, ordered suspension agrees with the fluid−crystal boundary predicted for both 100 and 200 nm particle suspensions using a screened Coulomb potential with an appropriate zeta potential for the corresponding surface charge densities. For 200 nm particles at 0.01 mM KCl (κ−1 = 83.7 nm, including dissolved CO2, κa = 1.19), Bragg-reflecting crystallites form when the volume fraction of particles is ϕ ≥ 0.08. Increasing the salt concentration to 0.1 mM (κ−1 = 29.9 nm, κa = 3.34) causes the crystal transition to occur at higher volume fractions, between 0.10 ≤ ϕ ≤ 0.17, because the higher salt concentration screens the particle charge. Crystallization is not observed at the highest ionic strength, 1 mM KCl (κ−1 = 9.60 nm, κa = 10.4), in the range of the suspension concentrations studied. For 100 nm particles suspended in 0.05 mM KCl (κ−1 = 41.7 nm, κa = 1.20) crystallization occurs at ϕ = 0.04. Before analyzing the volume fraction dependence of the concentrated suspensions, the dielectric spectra of the most dilute particle concentration will be presented next to identify the surface charge density using the SEKM .

Figure 3. Experimental measurements and SEKM calculations of the (a) dielectric and (b) conductivity increments for 200 nm diameter polystyrene spheres suspended in 0.01 (□), 0.1 (○), and 1 mM KCl (△). All SEKM calculations incorporate 3.2 μM of dissolved CO2 and use a particle surface charge density q = −1.6 μC/cm2.40 10706

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Figure 4. (a) Permittivity and (b) conductivity increments for 100 nm diameter polystyrene particles suspended in 0.05 mM KCl (κa = 1.20). Symbols are experimental results at increasing volume fractions, and the lines represent SEKM solutions for several surface charge densities.

Figure 5. Dielectric and conductivity increments for 200 nm diameter polystyrene particles suspended in 0.01 mM KCl (κa = 1.19; a, b), 0.1 mM KCl (κa = 3.34; c, d), and 1 mM KCl (κa = 10.4; e, f). Symbols are experimental results at varying volume fractions, and lines are SEKM calculations at varying surface charge densities.

reveal the upper concentration limit for the electrokinetically dilute regime and identify the critical length scale for ion transport to explain the observed concentration dependence. First, we will present the results for 100 nm diameter particles before returning to results for 200 nm diameter suspensions. The dielectric spectra of 100 nm particle suspended in 0.05 mM KCl, which exhibit iridescence when ϕ > 0.04, are shown for 0.01 ≤ ϕ ≤ 0.125 (Figure 4). Also shown are SEKM calculations for three surface charge densities: q = −1.0 (the

bulk solution. A more detailed analysis of the conductivity spectra in terms of this offset will follow in the Discussion section. Concentrated Suspensions. Standard electrokinetic model calculations are strictly valid in the dilute (noninteracting) limit, therefore increasing the particle concentration illustrates how interparticle interactions alter the dielectric properties. By measuring a wide range of volume fractions as a function of particle size and background electrolyte concentration, we 10707

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titratable surface charge density), −1.3, and −1.6 μC/cm2. Calculations using the titratable surface charge density underpredict the observed dielectric increment over the entire frequency range, while calculations with −1.6 μC/cm 2 overpredict the increment at high frequencies. At q = −1.3 μC/cm2, we find the best quantitative agreement between the SEKM and the dielectric increment and qualitative agreement in the conductivity increment. Between the frequencies 10 and 20 kHz, the measured dielectric increment is less than the SEKM calculation in all suspensions. As the frequency increases, the experimental spectra eventually agree with the calculated spectra at a critical frequency, ω*/2π. We define the critical frequency as the point where Δεexp < ΔεSEKM by more than 5% and demonstrate several points in Figure 4a. This critical frequency increases as the particle concentration increases. The dielectric and conductivity increments for 200 nm particle suspensions measured by dielectric spectroscopy along with SEKM calculations are given in Figure 5 for three ionic strengths. From the range of surface charge densities considered, q = −1.6 μC/cm2 yields the best agreement. For all ionic strengths, the suspension with the most dilute volume fraction quantitatively agrees with theory for the dielectric increment and qualitatively agrees with the conductivity increment. Similar to the previous case for 100 nm particles, systematic deviations from theory occur for the dielectric increment when ϕ ≥ 0.02 at the lowest salt concentration, 0.01 mM KCl. The dielectric increment agrees with theory above a critical frequency that increases as the volume fraction of particles increases. As the ionic strength is increased to 0.1 mM KCl, the dielectric spectra again exhibit systematic differences with theory. The disagreement occurs over a broader frequency range as the particle volume fraction increases. Although only the most concentrated suspension is iridescent, Δεexp < ΔεSEKM in the 10 kHz frequency regime for ϕ ≤ 0.10. Remarkably, there is no dependence of the permittivity increment on volume fraction for the 1 mM KCl dielectric increment over an identical frequency range. Even when ϕ = 0.087, the spectra at this salt concentration agree with dilute theory. Clearly, the transition from “dilute” to “concentrated” in these suspensions is as dependent on the double layer thickness as it is on the volume fraction of particles. For all salt concentrations, the measured conductivity increment is greater than that calculated by the SEKM by a constant value across all frequencies. There is no identifiable trend with volume fraction in this offset; however, the difference with the measured values is greatest for the 0.01 mM KCl suspensions. This is expected because at low conductivities the increment calculated by eq 5 is more sensitive to small changes in the background electrolyte concentration. Finally, the data show that the dielectric increment is most sensitive to surface charge density at high ionic strengths, in contrast to spectra measured at lower ionic strengths. On the basis of the dielectric spectroscopy results, the 100 and 200 nm particles surface charge density is slightly higher than the titratable surface charge reported by the manufacturer. This discrepancy is expected based on previous literature11,15,16 and may be attributed to the sample preparation protocol and differences between the surface and electrokinetic charge of a colloidal particle.46

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DISCUSSION

Two interesting observations are immediately apparent from the complete data set presented above: (1) nonlinear particle concentration dependence is only observed in the dielectric increment and is dependent on the field frequency, ionic strength and particle size and (2) the measured conductivity increment is consistently greater than that calculated from the SEKM by a linear offset. In this section, we analyze the concentration dependence of the dielectric increment using the critical frequency. This allows us to identify a scaling that collapses the critical frequency data onto a single curve. The results are rationalized by considering the interparticle ionic fluxes that cause the decrease in dielectric increment and are used to classify arbitrary suspensions as electrokinetically dilute or concentrated. We end by explaining the offset in conductivity increment in the framework of added counterions and nonspecific adsorption. Permittivity Increment. The measured permittivity increment deviates from the standard theory when the particle size and electrolyte concentration change. In general, the onset of the difference occurs at higher frequencies as the particle concentration increases. Under these circumstances, the permittivity increment is lower, implying that additional particles affect the degree of charge separation in the suspension. Here, we show that this behavior can be desrcibed in terms of a single characteristic length scale for ion transport between particles. We plot the critical frequency, ω*/2π, at which the experimental spectra diverges from electrokinetic theory as a function of volume fraction in Figure 6a. In all cases, the critical frequency increases as a function of volume fraction. 100 nm

Figure 6. (a) Critical frequency below which Δεexp < ΔεSEKM as a function of particle volume fraction. (b) Scaled critical frequency plotted versus the effective suspension volume fraction collapses the data onto a single curve, accounting for particle size (2a = 100 and 200 nm) and double layer thickness (κ = 83.7, 41.7, and 29.9 nm). 10708

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diameter particles exhibit the highest critical frequencies, while the data for 200 nm diameter particles show that ω*/2π decreases with increasing ionic strength. Different particle diameters with similar values of κa also result in different values of ω*/2π. Together, this suggests using a characteristic length scale, the effective distance between particles a + κ−1, to calculate an effective volume fraction ⎛ a + κ −1 ⎞ 3 ϕeff = ϕ⎜ ⎟ ⎝ a ⎠

(9)

which determines the transition from dilute to nondilute electrokinetic behavior. Scaling the critical frequency by ωa+κ−1 = D/(a + κ−1)2, which accounts for ion transport over this length scale, we find a master scaling which collapses the critical frequency dependence of the dielectric increment onto a single master curve, as shown in Figure 6b. The scaling captures the behavior of all three suspensions onto a single line that can be used to identify at what particle concentrations, ionic strengths, and frequencies the dielectric spectra will agree with dilute theory. At scaled frequencies ω/ωa+κ−1 > ω*/ωa+κ−1, the permittivity scales linearly with volume fraction and the dielectric increment agrees with SEKM calculations. Below ω/ωa+κ−1 < ω*/ωa+κ−1, however, the increment is nonlinear with respect to volume fraction and the measured dielectric increment is less than that calculated from standard electrokinetic theory. This result is remarkableit captures the behavior of suspensions with two particle diameters (100 and 200 nm), three double layer thicknesses (83.7, 41.7, and 29.9 nm), and two κa values (1.2 and 3.3) over a wide range of volume fractions (0.01 ≤ ϕ ≤ 0.18) and three orders of magnitude in frequency (10 kHz ≤ ω*/2π ≤ 3 MHz). We fit the volume fraction dependence of the electrokinetically dilute/concentrated threshold to the exponential function ω* = y0 + Ae(R 0ϕeff ) ωa + κ −1

Figure 7. When ω > ω*, the suspension is electrokinetically dilute because interparticle ion fluxes do not have time to develop, despite equilibrium double layer overlap in some cases. When ω < ω*, neighboring particles become sources and sinks for additional ionic fluxes antiparallel to the electric field, decreasing the dielectric increment.

in Figure 7 we only show the nearest-neighbor particles parallel to the applied field, since, based on the electric field lines, particle orientations parallel to the field will be the primary source of these ionic fluxes and not the perpendicular components of particle orientation. This is not meant to imply the particles are chaining. In a suspension there will be a distribution of particles in all radial directions, yet the ionic fluxes parallel to the field will drive the interparticle interactions. Interestingly, the volume fraction dependence reported here shows no correlation with the formation of iridescent crystallites observed in Figure 2. Concentration dependence of the dielectric increment is still observed at volume fractions below the fluid−crystal transition. The critical frequency then increases exponentially with volume fraction over the range studied. There is no discernible change in the spectra as the fluid/crystal transition is traversed, indicating that the double layer perturbation and interactions with the surrounding particles is the primary determinant of the suspension electrokinetic properties and confining the particles to a crystal lattice has little or no effect. We suspect that the distance to the nearest-neighboring particle dictates the concentration dependence of the dielectric increment, as our results suggest the length scale a + κ−1 establishes when additional ion flux alters the spectra. The average distance to the nearest adjacent particle does not change significantly as particles are confined to a crystal, explaining the lack of a sharp change in the electrokinetic properties with suspension morphology. Additionally, since the ions in suspension migrate due to the electric field much faster than the particles even in dilute conditions, hindered particle motion in the crystal state should also not effect the electrokinetic response. Conductivity Increment. To a first approximation, the contribution of added counterions and nonspecific adsorption to the conductivity of the suspension is frequency independent and scales with particle volume fraction40,45

(10)

and find y0 = −1.230, A = 0.986, and R0 = 3.328 are the best-fit parameters with a goodness of fit of 0.991. This function is shown by the black dashed line in Figure 6b. The red dashed line is the fit performed with the constraint y0 = −A, since at infinite dilution one should expect the SEKM to be valid at all frequencies. The latter expression is best fit with parameters A = 0.875, R0 = 3.428 and a goodness of fit of 0.991 and requires ω* → 0 at infinite dilution (ϕ → 0). These fits are purely empirical in nature and are given to emphasize the remarkable collapse of the data. The above results demonstrate that the transition from electrokinetically dilute to nonlinear (nondilute) behavior occurs when ions have sufficient time to migrate between neighboring particles in a suspension. As illustrated in Figure 7, the additional sources and sinks of ions from these neighbors reduce the overall charge separation around particles that would be expected by the standard electrokinetic theory, leading to a lower permittivity increment (although the permittivity in general still increases with particle volume fraction). It is noteworthy that at sufficiently high frequencies (ω > ω*) we find that experiment and theory agree regardless of volume fraction. Even in situations where significant double layer overlap occurs between particles there must be sufficient time for ions to be transported for the permittivity to become nonlinear in ϕ. Thus, the transition does not occur at a single volume fraction, double layer thickness or time scale. Note that 10709

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Figure 8. Modified conductivity increment calculated by subtracting a frequency independent Δσ′ (calculated at 100 kHz) for 200 nm particles suspended in (a) 0.01 mM, (b) 0.1 mM, (c) 1 mM KCl, and (d) 100 nm particles suspended in 0.05 mM KCl.

σs(ω) = 1 + ϕ[Δσ(ω) + Δσ ′] σ∞

(decreasing double layer thickness) (Figure 9). These results are in agreement with theories outlined previously.44,45 In

(11)

where Δσ′ is the scaled conductivity increment due to added counterions and nonspecific adsorption. The polystyrene particles studied here are stabilized by surface sulfate groups that have a pKa = 2; therefore, we expect complete dissociation at the particle surface. This contributes “added counterions” to the suspension. “Nonspecific adsorption” refers to the attraction (repulsion) of counterions toward (co-ions away) from the particle surface.44 Depending on the electrolyte concentration, this contribution can in theory increase or decrease the magnitude of the conductivity increment without changing the frequency-dependent “dipole” term.15 We can therefore account for the O(ϕ) modification to the bulk conductivity by subtracting Δσ′ from the experimental frequency-dependent conductivity increment, Δσ(ω) Δσcorr(ω) = Δσ(ω) − Δσ ′

(12)

Figure 9. Δσ′ values used to generate offset corrected plots of the conductivity increment (Figure 8) are independent of volume fraction for a given ionic strength. Representative error bars and the average value are also shown.

where Δσcorr(ω) is the corrected conductivity increment. The value for the offset is calculated at 100 kHz for all spectra, and the results are given in Figure 8. The corrected data are in quantitative agreement with electrokinetic theory over the entire frequency range, except some discrepancies in the high-frequency shoulder of the conductivity increment. Above 1 MHz, the corrected conductivity increment decreases with increasing volume fraction for the 100 nm suspensions; however, there is no systematic dependence on concentration for the 200 nm suspension data. Additionally, the offset verifies the surface charge densities of −1.6 and −1.3 μC/cm2 for the 200 and 100 nm diameter particles, respectively. There are two important observations to be made from the experimental values for Δσ′: (1) the offset is independent of the volume fraction of particles in suspension, and (2) the offset decreases with increasing suspension ionic strength

Table 1, we compare the experimental offset to the values calculated from standard electrokinetic theory. The MPEK Table 1. Experimental and SEKM Values for the Contribution to the Conductivity Increment Due to Added Counterions and Nonspecific Adsorption

10710

particle diam (nm)

KCl concn (mM)

200 100 200 200

0.01 0.05 0.10 1.00

Δσ′

Δσ′SEKM

± ± ± ±

27.6 13.9 2.49 0.60

80.0 38.0 11.4 0.52

48.4 8.7 2.9 0.27

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transition for charged spheres, this morphology change was not directly correlated with the dielectric response. The distortion of the ion distribution around the particles due to ionic fluxes from neighboring particles causes the polarization and dielectric increment to decrease at low frequencies, regardless of suspension microstructure. Such insight may be exploited in future dielectric spectroscopy and self-assembly studies.

software package used in this work only uses a single counterion species in the added counterion calculation. Since there are both hydronium and potassium counterions in suspension after the thorough washing procedure, we average the effect of each based on their relative concentrations in suspension to estimate Δσ′SEKM. This is necessary because the increased diffusivity of the hydronium ion increases Δσ′, and only taking the potassium ion into account gives negative values for the offset. The result of the calculation is in good qualitative agreement with the experimental results. Clearly, the particles contribute an O(ϕ) increase to the conductivity that is overshadowed as the background electrolyte concentration increases. There are several scenarios to account for the lack of quantitative agreement in the conductivity offset. The presence of a thin “hairy” layer of uncharged polymer or polyelectrolyte layer on the particle surface changes the offset slightly but also changes the frequency dependence of the dielectric and conductivity increments. The experimental results presented here are in good agreement with calculations based on a bare particle. However, calculations were also performed with a thin layer of charged or uncharged polymer. These are described in the Supporting Information. Second, the SEKM assumes impermeable particles; however, particle porosity would cause excess surface area and additional ionizable sites that may account for the additional counterion contribution and differences between the electrokinetic and titratable surface charge density. Recent holographic characterization of polystyrene spheres synthesized via emulsion polymerization identified particle porosities of ∼6%.47 The results for the offset in conductivity increment reveal additional avenues for further research. The effect of added counterions and nonspecific adsorption on the conductivity increment is clearly frequency independent, as predicted by Saville;45 however, the subtleties of calculating the theoretical offset based on the ionic species in suspension must be examined before quantitative agreement between experiment and theory is possible. Experiments using suspensions of particles with a well-defined (from alternative measurements) polymer layer may further elucidate this effect and validate the SEKM for those scenarios.39,40 Alternatively, suspending these particles in HCl should give a suspension with only one counterion species to afford a more rigorous comparison between experiment and theory. Several postpolymerization treatments can be made to evaluate the effect of particle porosity on the dielectric spectra, such as heat treating the particles48,49 to smooth the particle surface and collapse any small “hairy” layer of polymer chains that may be present. These are all open areas of possible future research.

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ASSOCIATED CONTENT

S Supporting Information *

Extrapolation to infinite separation method details for salt solutions, dilute suspensions and concentrated suspensions; standard electrokinetic model calculation details and additional calculations including a thin layer of uncharged/charged polymer; electrophoretic mobility measurements; critical frequency scaling analysis. 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.

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ACKNOWLEDGMENTS We thank J. McMullan, M. Mittal, and J. Swan for helpful discussions, R. Hill for supplying his MPEK code and useful discussions regarding its use, and A. Hollingsworth for fruitful discussions and assistance in designing the dielectric cell. Funding from the U.S. Department of Energy (Basic Energy Sciences Grant DE-FG02-09ER46626) is gratefully acknowledged.

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REFERENCES

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CONCLUSIONS Dielectric spectroscopy has been successfully used to analyze and interpret the electrokinetic properties of concentrated colloidal suspensions. Using dilute suspensions, the particle surface charge density was identified and the dielectric response agreed with standard electrokinetic theory. By increasing the particle concentration in relatively thick double-layer suspensions, systematic deviations from theory were found. The particle plus double layer length scale is used to collapse the data onto a master curve, which can now be used to predict at what frequencies and volume fractions a given suspension will behave electrokinetically dilute or concentrated. Although the suspensions studied here transversed the order−disorder 10711

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