Article pubs.acs.org/Langmuir
Particle Zeta Potentials Remain Finite in Saturated Salt Solutions Astha Garg, Charles A. Cartier, Kyle J. M. Bishop, and Darrell Velegol* Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States S Supporting Information *
ABSTRACT: The zeta potential of a particle characterizes its motion in an electric field and is often thought to be negligible at high ionic strength (several moles per liter) due to thinning of the electrical double layer (EDL). Here, we describe zeta potential measurements on polystyrene latex (PSL) particles at monovalent salt concentrations up to saturation (∼5 M NaCl) using electrophoresis in sinusoidal electric fields and high-speed video microscopy. Our measurements reveal that the zeta potential remains finite at even the highest concentrations. Moreover, we find that the zeta potentials of sulfated PSL particles continue to obey the classical Gouy− Chapman model up to saturation despite significant violations in the model’s underlying assumptions. By contrast, amidinefunctionalized PSL particles exhibit qualitatively different behaviors such as zero zeta potentials at high concentrations of NaCl and KCl and even charge inversion in KBr solutions. The experimental results are reproduced and explained by Monte Carlo simulations of a simple lattice model of the EDL that accounts for effects due to ion size and ion−ion correlations. At high salt conditions, the model suggests that quantitative changes in the magnitude of surface charge can result in qualitative changes in the zeta potentialmost notably, charge inversion of highly charged surfaces. These findings have important implications for electrokinetic phenomena such as diffusiophoresis within salty environments such as oceans, geological reservoirs, and living organisms.
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INTRODUCTION The zeta potential (ζ) of a colloidal particle determines the particle’s stability in solution and its motion in an electric field. Physically, this potential difference derives from the separation of charge bound to the particle’s surface (physically or chemically) from that of mobile counterions in solution. Charge separation over a finite length scale is driven by thermal fluctuations and gives rise to the so-called electric double layer (EDL). In the classical Gouy−Chapman (GC) model, this length scale is identified as the Debye screening length, κ−1 = (εkBT/2e2n0)1/2, which depends on the dielectric permittivity (ε), thermal energy (kBT), elementary charge (e), and salt concentration (n0) (here assuming a 1:1 electrolyte). Despite their widespread application and effectiveness, the GC model and its derivatives make a variety of assumptions that break down at high salt concentrations1 that approach saturation. Notably, these continuum descriptions neglect the finite size of ions and molecules,2−4 which are actually larger than the predicted screening length in concentrated electrolytes (Figure 1b,c). Furthermore, the mean-field approximations used to describe electrostatic interactions fail to account for ion−ion correlations4−6 that ultimately guide crystallization from saturated solutions. At high salt concentrations, the predicted screening length becomes smaller than the Bjerrum length, λ = e2/4πεkBT, over which ion−ion correlations are significant7 (Figure 1b). For particles in salty media, one might therefore anticipate that the structure of the EDL and the corresponding zeta potentials will differ significantly and qualitatively from expectations built largely on the study of dilute electrolytes. It is critical to identify and understand these differences in order to predict the behaviors of colloids in salty environments such as © 2016 American Chemical Society
human blood (≈150 mM), seawater (≈600 mM), wastewater (≈2 M during reverse osmosis), and geological reservoirs (≈5 M). The standard methods used to determine zeta potentials of particles at low ionic strengths are often inapplicable at high salt concentrations. Methods based on electrophoresis measure the velocity of particles in an applied electric field. Application of such fields within concentrated electrolytes results in large electric currents that polarize electrodes and rapidly heat the sample. The resulting temperature variations can induce significant convective flows and complicate the estimation of fluid properties needed to infer particle zeta potentials (e.g., dielectric constant and viscosity). Because of such challenges, few studies reliably report zeta potentials for colloidal particles at high ionic strength. Dilute latex particle dispersions have been examined by phase angle light scattering (PALS) for salt concentrations up to 3 M8,9 and by a combination of optical tweezers and high-speed microscopy up to 1 M.10 Concentrated inorganic mineral particle systems have been examined using the electroacoustic method11−14 at salt concentrations as high as 3 M. These studies found finite zeta potentials for particles in 1−3 M monovalent salt solutions as well as concentration-dependent shifts in the isoelectric point.12,14 However, we know of no reports that investigate particle zeta potentials at salt concentrations at or near saturation (5.4 M for NaCl) where finite size effects and ion−ion correlations are most pronounced. In strongly interacting charged systems (e.g., Received: July 29, 2016 Revised: October 14, 2016 Published: October 21, 2016 11837
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mobility using the Smoluchowski equation. For negatively charged, sulfated polystyrene latex particles (sPSL), we find finite zeta potentials of ζ ≈ −20 mV at saturated salt conditions. Moreover, the measured zeta potentials and their dependence on salt concentration are found to be consistent within experimental uncertaintywith predictions of the GC model despite violating many of its foundational assumptions. By contrast, positively charged, amidine-functionalized PSL (aPSL) exhibit more complex behaviors including charge reversal in KBr, LiCl, and NaCl solutions. To guide the interpretation of our experimental results, we investigate a lattice model of the EDL that incorporates effects due to finite ion size and ion−ion correlations. By comparing the results of this model to those of the standard GC model, we explain how the latter can make successful predictions at even the highest salt concentrations for weak ion−ion coupling but breaks down when such coupling is strengthened. Moreover, we show that breakdown of the GC model is accompanied by charge inversion, in qualitative agreement with experimental observations for aPSL. Our results indicate that electrokinetic phenomenaparticularly diffusiophoresis17−20are present even in saturated salt solutions and may therefore contribute to the many transport processes occurring in oceans, geological reservoirs, and living organisms.
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METHODS AND MATERIALS
Measuring Zeta Potentials. We used the sinusoidal response of particles subject to a sinusoidal electric field to measure the electrophoretic mobility and thereby infer the zeta potential. In a typical experiment (see Figure 1a), a dilute suspension of polystyrene latex microparticles (diameter a ≈ 3 μm) in an aqueous electrolyte was flowed into a glass capillary. The particles were subject to an oscillating electric field of magnitude E0 and frequency f 0 = ω0/2π directed along the length of a capillary, E(t) = E0 sin(ω0t). The motion of a single particle was captured using high-speed video microscopy, and its dynamic trajectory x(t) was reconstructed using particle tracking algorithms (see below). For particles much larger than the screening length (κa ≫ 1), the electrophoretic particle velocity u is related to the applied field by the Smoluchowski equation21
Figure 1. (a) Schematic illustration of the experimental setup showing a polystyrene latex (PSL) particle in a glass capillary subject to an oscillating electric field. The electrolyte solution is characterized by its viscosity (η), conductivity (ke), permittivity (ε), and salt concentration (n0). (b) Debye length (κ−1) decreases as a function of NaCl concentration becoming smaller than the Bjerrum length (λ) at 0.3 M and the ions themselves (ca. 0.3 nm) above 2 M. (c) Illustration of the EDL at two different salt concentrations highlighting the relevant length scales such as ion size, Debye length (shown on left and right edges for low and high salt, respectively), ion−ion spacing, and surface charge spacing (red semicircles).
u = μE =
εζsm E η
(1)
where μ is the electrophoretic mobility, η is the fluid viscosity, and ζsm is the Smoluchowski zeta potential. Integrating this equation, we obtain the following expression for the anticipated particle trajectory:
multivalent electrolytes), ion−ion correlations are known to drive counterintuitive phenomena such as charge inversion (also called charge reversal), whereby an excess of counterions binds to a charged surface to reverse its polarity.15 Similar effects have been hypothesized for monovalent electrolytes near saturation. Additionally, it has been suggested that the EDL can never be thinner than the finite size of the counterions, leading to the failure of GC predictions at high ionic strength.16 Within such thin double layers, changes in the structure of interfacial water may also contribute to charge separation and the associated zeta potentialeven for uncharged surfaces.13 Investigating these and other interesting hypotheses requires experimental data on particle zeta potentials at saturated salt conditions. Here, we describe a method for measuring particle mobilities up to saturated salt conditions and present data for polystyrene latex particles in several monovalent salt solutions (KCl, KBr, and NaCl). We use electrophoresis with a sinusoidal electric field (300 Hz) to drive oscillatory particle motions, which are captured and quantified using high-speed video microscopy. An apparent zeta potential is then inferred from the measured
x(t ) = −
εζsm E0 cos(ω0t ) ω0η
(2)
The reported zeta potentials were obtained by linear regression of particle trajectories using eq 2. Experimental Details. We collected data on three different types of surfactant-free, polystyrene latex particles (density, ρ = 1055 kg/ m3): sulfated PSL (radius, a = 2.9 μm; surface charge density, σ = −9.7 μC/cm2), carboxylated PSL (a = 3.2 μm; σ = −11.9 μC/cm2), and amidine-functionalized PSL (a = 3.3 μm; σ = 34.9 μC/cm2) henceforth denoted sPSL, cPSL, and aPSL, respectively. The particles were suspended at low volume fractions (10−5) into freshly prepared, aqueous solutions of ACS grade NaCl, KCl, KBr, LiCl, and CsCl. The particle dispersions were drawn into an RCA-1 cleaned, square glass capillary (1 × 1 × 50 mm) and positioned horizontally on a glass slide for imaging. Gold electrodes were inserted at each end, and the capillary was sealed as described previously.22 The suspension was subjected to a sinusoidal voltage with frequency f 0 = 300 Hz (Figure S1) for 0.5−2 s. The measured current i, conductivity ke, and cross-sectional area of the cell A were used to determine the applied electric field as E = i/keA (E = 300−1000 V/m). 11838
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Figure 2. Filtered position x(t) vs time for a 2.9 μm sPSL particle in (a) 10 mM NaCl for E = 1120 V/m and (b) 5.4 M NaCl for E = 733 V/m. Magnitude of the Fourier components for particle velocity ω|X(ω)| as a function of frequency for (c) 10 mM and (d) 5.4 M NaCl. The red dot highlights the peak at the applied frequency f 0 = 300 Hz. At high salt concentrations, an additional peak is seen at 600 Hz and attributed to fieldinduced thermal convection (Figure S6). Simultaneous to applying the voltage, the motion of a single particle was observed with a microscope using a 50× objective focused near the center of the capillary (scale of 0.25 μm per pixel). A high-speed camera (Phantom v310) synchronized with the voltage source recorded the particle’s motion at 6000 frames/s. Gravitational forces induced particle motions perpendicular to the applied field; however, the particles remained in focus for at least 1 seven in the densest electrolyte (4 M CsCl, ρ = 1498 kg/m3). Our experimental setup was designed to mitigate a variety of challenges that arise when measuring zeta potentials at high salts. The use of ac fields was essential to avoid the rapid polarization of the electrodes at high salt concentrations, at which the electric current density can exceed 30 mA/mm2. To prevent oscillatory electroosmotic flows in the center of the channel, the applied frequency was chosen to be much faster than the rate of momentum diffusion over the width W of the capillary; that is, ω ≫ ν/W2 ∼ 1 s−1 where ν is the kinematic viscosity.23 At the same time, the frequency was chosen to be small enough that field-induced particle displacements were clearly distinguishable from those due to Brownian motion. The cell was not thermostated; however, the temperature increase was mitigated by applying the voltage for short times to reduce heat generation and by using a narrow capillary to increase the rate of heat transfer to the surroundings. At the highest power input of 1.3 W, the measured current and hence the conductivity increased by 3.9% in 2 s, which corresponds to a temperature increase of 1.9 °C using a typical coefficient of conductivity variation.24 Further details of the experimental setup are provided in the Supporting Information. Particle Tracking and Analysis. Dynamic particle trajectories x(t) along the length of the capillary were reconstructed from the recorded images at 10 nm resolution using the Trackmate plugin in Fiji.25 Low-frequency translational motions were removed by subtracting a smoothing spline fit to the noisy trajectory (Figure S2). We then decomposed the filtered particle trajectories into Fourier series of the form, x(t) = ∑ωX(ω)eiωt, and examined the (complex) Fourier components X(ω) to confirm the validity of eq 2 (see below). The mobility was obtained by performing a least-squares fit of the filtered position data directly to eq 2 (Figure S3a). In deriving the zeta potential from the electrophoretic mobility, it is important to recognize that the viscosity and the dielectric constant of concentrated electrolytes differ significantly from that of pure water. For example,
the viscosity of saturated NaCl is 70% higher than that of water while the relative permittivity (εr = ε/ε0 where ε0 is the permittivity of free space) is 50% lower. Failure to account for these changes leads to errors in the predicted zeta potential of more than 300%. We used literature data for the concentration-dependent viscosity and dielectric constant for monovalent electrolytes summarized in the Supporting Information (Figure S3b,c) to best approximate the Smoluchoswski zeta potential from the measured mobilities (Figure S3a). For each condition (i.e., particle surface chemistry, salt, and concentration), the reported zeta potentials were obtained by averaging at least six measurements taken on different particles. The 90% confidence intervals reported in Figure 3 represent the larger of two errors: that obtained from linear regression on each particle and that derived from the distribution of zeta potentials measured for various particles. Variations in the zeta potential from particle to particle were typically smaller than the 90% confidence interval for each particle, indicating good repeatability. Method Validation. To confirm the validity of eq 2, we examined the Fourier components X(ω) to assess their dependence on the applied frequency and field strength. Equation 2 implies that X(ω) should be finite and real only at the driving frequency, corresponding to a pure cosine component at ω0. At low salt concentrations (10 mM NaCl), the trajectories appear roughly sinusoidal, and the dominant Fourier component was indeed that of the driving frequency f 0 = 300 Hz (Figure 2a,c). By contrast, at high salt concentrations (5 M NaCl), the sinusoidal motion of the particle is hardly visible above the noise; however, the Fourier component at the driving frequency can still be clearly resolved (Figure 2b,d). We further confirmed (i) that the phase of X(ω0) was near 0 or ±π (Figure S1), (ii) that the particle velocity ω0X(ω0) was linearly proportional to the field strength, and (iii) that the inferred zeta potential was independent of the driving frequency (Figure S3). Interestingly, an additional component at 2 times the driving frequency 2f 0 was observed at high salt concentrations. The magnitude of this component increased linearly with the electric power supplied, indicating a second-order dependence on the applied field (Figure S6a,b). As detailed in the Supporting Information (Section 1.2, Figure S2), we attribute these observations to thermally induced convective flows, which depend on the rate of Joule heating within the capillary. Importantly, these effects are second-order in the field and do not 11839
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Figure 4. (a) Schematic illustration of the lattice model of a charged surface in a symmetric electrolyte. (b) Electric potential within the simulation cell averaged over the 1 and 2 directions. The solid markers represent the average of 12 independent Monte Carlo simulations; the solid curves are predictions of the GC model; the x markers denote the zeta potentials. The simulation parameters are charge density, σ = 0.0625e/S2 and Bjerrum length, λ = 2S within a cell of dimensions L1 = L 2 = 8S and L3 = 32S ; the salt concentrations are nc = 0.005S −3 (low salt) and nc = 0.32S −3 (high salt). (c) Zeta potential vs concentration for different three different λ corresponding to conditions similar to experiments on sPSL. The markers are results of the MC simulations; solid curves are the GC predictions of eq 3. Simulation parameters are the same as in (b) unless stated otherwise.
Figure 3. (a) Zeta potentials of 2.9 μm sPSL particles vs concentration in five different monovalent salts. (b) Zeta potentials of 3.3 μm aPSL particles vs concentration in four different monovalent salts. Only data for high salt concentrations ≥100 mM are shown; see Supporting Information for a larger salt concentration range (Figure S7). The error bars represent 90% confidence intervals. The green curves show the prediction of the GC model assuming different surface charge densities. The solid curves are fits obtained for data in NaCl. contribute to the first-order, electrophoretic motions of interest here (nonlinear electrophoretic effects are third-order in the field as required by symmetry26). Finally, we performed an independent validation of our measurement technique by comparing its predictions to those of a commercial PALS-based instrument (Malvern Zetasizer Nano ZS90) at lower salt concentrations. For 2.9 μm sPSL particles in 1 mM NaCl, our setup gave a zeta potential of −110 ± 2.4 mV compared with −107 ± 4.8 mV from the commercial instrument. Monte Carlo Simulations of the EDL. We performed Monte Carlo simulations of the EDL at high salt concentrations using the lattice restricted primitive model, which accounts for effects due to ion size and ion−ion correlations (Figure 4a). In the model, ions are represented by point charges (±e) positioned within a uniform dielectric medium onto discrete lattice sites separated by a distance, S . Physically, the lattice spacing mimics effects due to ion size, and the parameter S can be interpreted as the diameter of a solvated ion (typically, 0.2−0.5 nm). Initially, M surface charges, M counterions, and N ion pairs are distributed onto a rectangular simulation cell of dimensions L1 × L2 × L3. The surface charges are distributed at random within the x3 = 0 plane (Figure 4a) to create an average
surface charge density of σ = eM/2L1L2 (the factor of 2 corrects for the fact that the surface is bounded by the electrolyte on two sides and not just one, as in experiment). Similarly, the 2N + M ions are distributed at random onto bulk lattice sites (x3 ≠ 0) to give a nominal salt concentration of nc = N /L1L 2(L3 − S) within the cell. The positions of the ions are equilibrated using the Metropolis Monte Carlo algorithm;27 those of the surface charges are fixed throughout the simulation. During each Monte Carlo move, two sites are selected at random and their contents swapped. If the move lowers the electrostatic energy of the system, it is accepted unconditionally. If it raises the energy of the system, it is accepted with probability p(Accept) = exp(−ΔU/kT), where kT is the thermal energy and ΔU is the energy increase accompanying the change in ion configuration. For lower salt concentrations, several such lattice swaps are conducted during each Monte Carlo move to achieve an acceptance frequency of around 50%. During each simulation, the system is equilibrated for 5 × 105 attempted moves; the resulting distribution is then sampled over 11840
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Langmuir Table 1. Zeta Potentials of SPSL Particles Averaged over 1−5.5 M for Each Salta salt ζavg (mV) a
KCl −18.8 ± 7.0
KBr −29.5 ± 7.4
CsCl −28.9 ± 7.6
NaCl −30.4 ± 8.0
LiCl −40.1 ± 7.4
The uncertainties are calculated as root mean square of the 90% confidence intervals for the measurement for each salt concentration, for each salt.
the course of an additional 5 × 105 moves. Each condition is simulated 12 times using different realizations of the surface charge distribution; the final results are obtained by averaging over these realizations. The electrostatic energy U is computed using the Ewald method,27 which is described in detail in the Supporting Information. The behavior of this model is fully specified by three dimensionless parameters: (i) the salt concentration within the periodic cell nc (scaled by S −3), (ii) the surface charge density σ (scaled by e/S2 ), and (iii) the Bjerrum length, λ = e2/4πεkBT (scaled by S ), which characterizes the strength of ionic interactions. At room temperature (25 °C), λ ranges from 0.71 nm in pure water to ∼1.3 nm in 5 M NaCl, which is 1−4 times the characteristic ion diameter (S = 0.4 nm).
By contrast to sPSL, zeta potentials of positively charged, aPSL particles exhibit qualitatively different behaviors at high salt concentrations (Figure 3b). For each of the salts investigated (KCl, KBr, NaCl, and LiCl), the zeta potentials decrease to zero at high concentrations despite following closely the predictions of the classic electrokinetic model at low salt concentrations.30 Additionally, for KBr, LiCl, and NaCl, the zeta potentials change sign to become negative at concentrations above ∼1 M. Comparing the different salts, charge inversion is significantly more pronounced for KBr than KCl, suggesting that Br− counterions have higher affinity for the aPSL surface that Cl−. This trend is consistent with the Hofmeister series, which orders ions based on their ability to precipitate proteins32 and correlates with the extent of ion hydration.33 Interestingly, the co-ions also appear to influence the degree of charge inversion, with the magnitude of the effect increasing as Li+ > Na+ > K+. These observations indicate that the detailed structure of the EDL at high salt depends on the chemical identity of all ions present in the electrolyte. By contrast, the zeta potentials at low salt depend only on the ionic strength (independent of chemical identity) and perhaps the identity of some potential determining ions21 (e.g., hydroxide). Our reported zeta potentials are based on the Smoluchowski equation; however, we also considered the more rigorous treatment of Stout and Khair2 that accounts for ion steric effects3 and ion−ion correlations.34 Using their analysis (see Supporting Information for details), we find that the Stout and Khair zeta potential (ζSK) is consistently larger than the Smoluchowski zeta potential (ζsm). For example, in 4 M NaCl, ζSK can be up to 2 times larger than ζsm depending on the choice of the correlation length. Importantly, we are operating in a regime where the Stout and Khair model predicts a significant mobility correction but no mobility reversal as observed in experiment. We chose to present our results in terms of the Smoluchowski zeta potential rather than ζSK or the raw mobilities for several reasons. First, the Smoluchowski equation normalizes the mobilities by a concentration-dependent factor ε/η that provides numbers of comparable magnitude for a broad range of salt concentrations. Second, ζsm is considerable simpler than the Stout and Khair zeta potential and involves no unknown quantities such as the correlation length. Finally, the surface charge density implied by the zeta potential (eq 3) can be compared to that obtained by titration to provide additional information on the arrangement of ions in the EDL. Those readers who prefer to consider the measured mobilities directly can do so (Figure S3a). To better interpret our experimental findings, we performed Monte Carlo simulations of the EDL at high salt concentrations using the lattice restricted primitive model (see Methods section and Figure 4a). Despite its simplicity, this model offers insights as to how the continuum GC model breaks down (or not) at high salt concentrations due to finite ion size and ion− ion correlations. We first consider the case of small surface charge densities (σ < 4εkBT /e S ) characteristic of sPSL particles (Figure 4b,c). At low salt concentrations, when the Debye screening length is greater than the ionic radius (κ −1 > S /2 ), the electric potentials predicted by the lattice
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RESULTS AND DISCUSSION Our experimental zeta potential measurements are summarized in Figure 3 for negatively charged, sPSL particles (Figure 3a) and for positively charged, aPSL particles (Figure 3b) dispersed in a series of monolavent electrolytes. The measured zeta potentials of sPSL particles are negative at all salt concentrations and remain finite up to saturated NaCl concentration of 5.4 M (Figure 3a). Apart from fluctuations due to experimental uncertainty, the zeta potentials are roughly constant for each salt beyond 1 M (Table 1), ranging from ca. −20 mV for KCl to −40 mV for LiCl and comparable to the thermal potential of kBT/e = 25 mV. The maximum salt concentration of 5.4 M represents the solubility of NaCl but is considerably lower than that of LiCl (≈13 M). We attempted measurements in saturated LiCl; however, the signal was overwhelmed by noise due to the ca. 13-fold decrease in the ratio ε/η and thereby the particle mobility. A tabulated summary of all of our zeta potential measurements can be found in the Supporting Information. For comparison, Figure 3a also shows the zeta potentials predicted by the standard GC model, which relates the zeta potential to the surface charge density σ as ⎛ eζ ⎞ σ 2kT = sinh⎜ ⎟≈ζ εκ e ⎝ 2kBT ⎠
(3)
where the approximate equality is appropriate for small zeta potentials (ζ < kBT/e).21 The charge density is estimated to be σ = −9.7 μC/cm2 based on independent conductometric titrations;28 however, this method is known to overestimate σ as it neglects ion conduction in the Stern layer.29,30 Therefore, we derived an additional estimate of σ = −5.1 ± 0.16 μC/cm2 for NaCl by fitting the GC model to the experimental data using nonlinear least-squares regression (see Supporting Information Section 2.5 for details). Only zeta potentials obtained at salt concentrations larger than 10 mM were used in the fitting process, as the assumption of constant surface charge is known to fail at lower concentrations for “hairy” particles31 such as PSL. Even at 5 M salt concentrations, at which the predicted Debye length κ−1 = 0.12 nm is smaller than the ions themselves (e.g., 0.37 nm for Na), the continuum GC model provides a reasonable description of the experimental data. GC model fits for aPSL, sPSL, and cPSL particles over a larger range of salt concentrations can be found in the Supporting Information (Figure S7). 11841
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Langmuir model agree well with those of the continuum GC model (Figure 4b, low salt). At high salt, however, the lattice model presents a qualitatively different picture of the EDL, wherein an excess of counterions accumulates at the surface resulting in damped oscillations in the potential (Figure 4b, high salt). Nevertheless, the zeta potentials predicted by the two models are remarkably similar. Here, the zeta potential is defined somewhat arbitrarilyas the electric potential at x3 = 0.5S , the plane separating the surface charge from the first layer of counterions. Figure 4c shows these zeta potentials as a function of salt concentration for three different values of the Bjerrum length λ spanning the range found in experiment. Notably, there is strong agreement between the predictions of the MC simulations and the continuum GC modeleven at concentrations above the critical value where κ −1 = S /2 (black x’s). Intuitively, one might guess that the zeta potential could never fall below ζ = σ S /2ε, which corresponds to all counterions located in the single layer closest to the surface. In fact, the overadsorption of counterions in this layer leads to an even smaller zeta potentialone more closely in line with predictions of the GC model, ζ = κσ/ε. These simulation results provide one possible explanation for the surprising success of the GC model in describing the experimental findings for sPSL particles. For highly charged surfaces (σ > 4εkT /e S ) such as that of aPSL particles, the zeta potentials predicted by the lattice model at high salt concentrations are significantly greater than the thermal potential (Figure 5a). Moreover, the electric potential within the plane of the first counterion layer shows significant heterogeneity, with peaks and valleys separated by many times the thermal potential (Figure 5b). Consequently, it is reasonable to expect that the ions in this layer will not move freely in the direction parallel to the surface upon application of an external field. These ions may instead form a tightly bound Stern layer that does not contribute to electrophoretic motion. To approximate this scenario, we introduce a second potential denoted ζ2 and defined as the electric potential at x3 = 1.5S , the plane separating the first and second ion layers. In qualitative agreement with experiments on aPSL particles, ζ2 is comparable to the thermal potential and changes sign upon increasing the salt concentration (Figure 5c). This effect is more pronounced for larger Bjerrum lengths, which corresponds to stronger electrostatic interactions and stronger ion−ion correlations. These results suggest that electrostatic effects alone neglecting contributions due to solvation and dispersion interactionscan provide a suitable mechanism for the inversion of highly charged surfaces at high salt concentrations. At the same time, this simple model cannot explain differences among specific salts (except indirectly via the characteristic ion size, S and the permittivity, ε). As noted above, the specific salt effects observed in experiment are largely consistent with expectations based on the Hofmeister series. Poorly hydrated counterions are expected to interact more strongly with hydrophobic surfaces such as PSL,15,32 thereby reducing the magnitude of the zeta potential. Consequently, the zeta potentials of sPSL particles increase in magnitude as LiCl > NaCl > KCl at high salt concentrationsin order of increasing ion hydration. Notably, the results for CsCl deviate from this trend. A similar argument can be made to explain the behavior of aPSL particles at high salt. The adsorption of an excess of counterions (Cl− or Br−) results in a negatively charged surface to which poorly hydrated
Figure 5. (a) Electric potential near a highly charged surface averaged over the 1 and 2 directions. The solid markers represent the average of 12 independent Monte Carlo simulations; the solid curves are predictions of the GC model; the x markers denote the zeta potential. The simulation parameters are σ = 0.0188e/S2 , λ = 4S , and nc = 0.32S −3 within a cell of dimensions L1 = L 2 = 8S and L3 = 32S . (b) Electric potential in the plane x3 = S for a particular realization of the surface charge distribution; other parameters correspond to those in (a). (c) Effective zeta potentials ζ2 vs concentration for three different λ corresponding to conditions similar to experiments on aPSL. The markers are results of the MC simulations; solid curves are the GC predictions of eq 3. Simulation parameters are the same as in (a) unless stated otherwise.
co-ions adsorb more strongly (K+ > Na+ > Li+). Furthermore, we note that charge inversion of positively charged, hydrophobic particles has been observed previously at lower salt concentrations for polyanions35 and for monovalent anions such as ClO4− and SCN−.15 The salt concentrations required to induce charge reversal increases as SCN− 9,15 (200 mM), ClO4− 15 (400 mM), and Br− (1 M, this study)again consistent with the Hofmeister series. Given the extreme thinness of the double layer at high salt concentrations, it is remarkable that continuum approximations to the dielectric permittivity and the liquid viscosity are so effective. In both the GC model and the lattice model, it is assumed that the dielectric constant in the EDL is the same as that in the bulk, despite inevitable changes in water structure near the solid surface. Nevertheless, we note that the characteristic electric field within the double layer is too small to induce significant alignment of water molecules therein.36 In relating the electrophoretic slip velocity and the zeta potential, it is assumed that the continuum equations of hydrodynamics are applicable with a constant viscosity equal to that of the bulk liquid. This assumption is supported by previous experiments that reveal that the viscosity of a fluid approaches its bulk value over molecular dimensions.37 The continuum hydrodynamic treatment also assumes the validity of the no slip boundary 11842
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at salt concentrations ∼1 M. In this case, the MC simulations show that the potential in the first layer of counterions is large and heterogeneous, suggesting that the layer is likely immobile. By shifting of the plane of shear to enclose the first layer of counterions, the model reproduces the inversion of the zeta potential at high salt concentrations. Our experiments indicate that these zeta potentials, while too low to stabilize dispersions (∼kBT/e), are large enough to cause electrokinetic transport even in high ionic strength systems, notably geological reservoirs, seawater, and blood. In these systems, spontaneous electric fields may result from salt concentration gradients, which cause diffusiophoresis at charged surfaces.17,44 Thus, diffusiophoresis might yet be a useful tool to affect transport in otherwise unreachable places, as long as a concentration gradient exists or can be created.
condition at the particle surface despite some evidence to the contrary.38 One fortunate consequence of the thin double layer is that theories developed for flat platessuch as the present lattice modelprovide an excellent approximation for the relatively large spherical particles studied here. Thus, the Smoluchowski equation should provide results very close to that of the model proposed by O’Brien and White that includes double-layer polarization.39 Our observation of negative zeta potentials for positive aPSL particles at high salt can be explained by a combination of two effects: (1) the overadsorption of counterions at the charged surface and (2) the displacement of the shear plane to enclose some of these counterions. Based on the MC simulations, the first effect is attributed to ion−ion correlations, which are increasingly significant at high salt concentrations. We attribute the second effect to the high surface charge of aPSL particles, which reduces the mobility of the most closely bound counterions. However, differences in the behavior of sPSL and aPSL particles may also result from differences in their “hairiness”31 due to dangling polyelectrolytes at the particle surface. For hairy particles, the shear plane encloses counterion charge even at lower salt concentrations, resulting in a discrepancy between surface charge estimates from titration and from zeta potential measurements.29,40−42 In our experiments, these discrepancies are more significant for aPSL particles than for sPSL particles. Consequently, the overadsorption of counterions “behind” the shear plane at high salt concentrations could result in zeta potentials of opposite sign for aPSL (more “hairy”) but not for sPSL (less “hairy”). The Ohshima model43 describes the electrokinetics of soft (i.e., hairy) particles; however, there are two key difficulties in applying this model to explain our experiments. First, we do not have a way to independently measure the frictional coefficient γ of the polyelectrolyte layer, which is necessary for a quantitative comparison. Second, the model relies on a mean-field treatment of electrostatics that does not allow for charge reversal as observed in our experiments for aPSL particles.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b02824. Supplementary methods and figures (PDF) Tabulated summary of all zeta potential measurements (XLS)
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AUTHOR INFORMATION
Corresponding Author
*E-mail
[email protected]. Present Address
K.J.M.B.: Department of Chemical Engineering, Columbia University, New York, NY 10027. Author Contributions
A.G. and D.V. came up with the question and a way to do the measurement. A.G. and C.A.C. put together the experimental setup. A.G. did the measurement and analysis. K.B. designed and carried out the simulations. The manuscript was written through contributions of all authors.
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SUMMARY AND CONCLUSIONS We have measured finite electrophoretic mobilities of negatively charged, sulfated PSL and positively charged, amdine-functionalized PSL particles up to saturation in NaCl, KCl, and KBr. Our measurements were made by applying sinusoidal electric fields in a disposable cell using high-speed microscopy to capture the particle motion. As a simple way for normalization of the mobilities, we have presented our results in terms of Smoluchowski zeta potentials. Equipped with zeta potential measurements, we attempted to test the validity of the classical mean field continuum GC model at high salt concentration, approaching saturation. At the outset we hypothesized that the continuum picture might break down as the electrostatic screening length, κ−1 becomes smaller than the hydrated ion size and the Bjerrum length, λ. However, experiments indicated that the classical pictureno change in sign of zeta potentialsis still qualitatively valid for sPSL particles. MC simulations reproduced these experimental trends for surfaces with relatively low surface charge density. Interestingly, overadsorption of counterions within the EDL allows the effective screening length to be smaller than the ionic size in agreement with the continuum model. Experiments also indicate that the more strongly charged aPSL particles undergo charge inversion in KBr, LiCl, and NaCl
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was funded in part by Penn State Materials Research Science and Engineering Centers under National Science Foundation Grant DMR-1420620. D.V. acknowledges support from NSF CBET 1603716. C.A.C. acknowledges support from the National Science Foundation under Award CBET-1351704. K.J.M.B. acknowledges support from the Center for Bioinspired Energy Science, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award DE-SC0000989.
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ABBREVIATIONS sPSL, aPSL, and cPSL, sulfated, amidine-functionalized, and carboxylated polystyrene latex, respectively; EDL, electrical double layer; GC model, Gouy−Chapman model; PALS, phase angle light scattering.
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