Article pubs.acs.org/Langmuir
Molecular Origins of the Zeta Potential Milan Předota,*,† Michael L. Machesky,‡ and David J. Wesolowski§ †
Institute of Physics and Biophysics, Faculty of Science, University of South Bohemia, Branisovska 1760, 370 05 Ceske Budejovice, Czech Republic ‡ University of Illinois, Illinois State Water Survey, 2204 Griffith Drive, Champaign, Illinois 61820-7495, United States § Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6110, United States S Supporting Information *
ABSTRACT: The zeta potential (ZP) is an oft-reported measure of the macroscopic charge state of solid surfaces and colloidal particles in contact with solvents. However, the origin of this readily measurable parameter has remained divorced from the molecular-level processes governing the underlying electrokinetic phenomena, which limits its usefulness. Here, we connect the macroscopic measure to the microscopic realm through nonequilibrium molecular dynamics simulations of electroosmotic flow between parallel slabs of the hydroxylated (110) rutile (TiO2) surface. These simulations provided streaming mobilities, which were converted to ZP via the commonly used Helmholtz-Smoluchowski equation. A range of rutile surface charge densities (0.1 to −0.4 C/m2), corresponding to pH values between about 2.8 and 9.4, in RbCl, NaCl, and SrCl2 aqueous solutions, were modeled and compared to experimental ZPs for TiO2 particle suspensions. Simulated ZPs qualitatively agree with experiment and show that “anomalous” ZP values and inequalities between the point of zero charge derived from electrokinetic versus pH titration measurements both arise from differing co- and counterion sorption affinities. We show that at the molecular level the ZP arises from the delicate interplay of spatially varying dynamics, structure, and electrostatics in a narrow interfacial region within about 15 Å of the surface, even in dilute salt solutions. This contrasts fundamentally with continuum descriptions of such interfaces, which predict the ZP response region to be inversely related to ionic strength. In reality the properties of this interfacial region are dominated by relatively immobile and structured water. Consequently, viscosity values are substantially greater than in the bulk, and electrostatic potential profiles are oscillatory in nature.
1. INTRODUCTION
Because ZP is related to the charge state of surfaces, it is used to infer and inform many important physical and chemical properties of interfacial systems, such as aqueous particle suspensions. This includes colloidal stability and aggregation, where higher |ZP| values are generally indicative of more stable suspensions and hence facilitated transport.4−7 In fact, engineered nanoparticles are often “capped” with various charged species, thereby increasing |ZP| and making their suspensions less sensitive to solution composition and pH, thus minimizing aggregation.8−10 Zeta potentials can also provide clues about surface composition and/or adsorbate coverage. For example, the formation of natural organic matter coatings or protein coronas on nanoparticulate surfaces often result in large ZP changes relative to particles without such coatings.11−13 The surface composition and charge of nanoparticles also affect cellular inflammation and uptake, and consequently ZP is often used as a measure of toxicological potential.14−16 In one such study the ability of amine-functionalized silica nanoparticles to lyse red blood cells was significantly greater
Most solid surfaces acquire charge in aqueous systems, with that charge balanced by oppositely charged counterions in solution, forming what is typically termed the electrochemical double layer (EDL).1 EDLs profoundly affect physical and chemical behavior of such heterogeneous systems. Hence, methods capable of characterizing the charge state of solid surfaces with respect to important solution variables such as pH, ionic strength, or solute composition assume vital importance. One class of commonly used techniques involves the measurement of electrokinetic phenomena which arise from the tangential movement of an electrolyte fluid relative to the charged surface.2 For suspended particles, their movement through a static fluid under an applied electrical field is the electrokinetic property of most relevance and is termed electrophoresis. Similarly, the potential-driven flow of solution past charged surfaces is termed electroosmosis. The directly measured quantities are potential-driven velocities, which when divided by the applied field strength provide electrophoretic or electroosmotic mobilities which can then be converted to zeta or streaming potentials via various theoretical relations, most commonly the Helmholtz-Smoluchowski equation.2,3 © 2016 American Chemical Society
Received: July 6, 2016 Revised: September 15, 2016 Published: September 19, 2016 10189
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
Article
Langmuir
dynamics of a few layers of water sorbed on (110)-dominated rutile nanoparticle surfaces measured by quasi-elastic neutron scattering. Thus, rather than a discrete slipping plane, with a distance from the surface predicted to be dependent on the ionic strength of the bulk solution, we observe a gradational increase in water mobility over a region very close to the crystal surface. Furthermore, the thickness of this gradational region of interfacial fluid is rather independent of surface charge and ionic strength as also noted by Lyklema.23 Our approach for simulating electrokinetics involves applying an electric field tangentially across an immobile rutile (110) surface and monitoring the resulting electroosmotic fluid (water + ions) movement via nonequilibrium molecular dynamics (NEMD). This is the counterpart of electrophoresis and only a change of sign is necessary to calculate ZP via the Helmholtz-Smoluchowski equation2,38 from the simulated electroosmotic mobilities.
at ZP > 30 mV, suggesting that exceeding a surface charge threshold facilitated cell membrane breakdown.17 Finally, ZP measurements are a useful complement to the sorption isotherm and pH titration approaches for characterizing surface charge and ion adsorption phenomena.18−20 Despite the utility of ZP, its true origin remains obscure and is usually equated with the potential at an ill-defined slip or shear plane at the boundary between the bulk fluid and a layer of fluid near the solid surface that either “rides along” with moving particles in the case of suspended particles or the bulk fluid “slides past” in the case of fluid flow along a fixed solid surface. In terms of classical EDL theory this boundary is thought to be near the Outer Helmholtz Plane. Although this classical notion of a slip plane is known to be “an abstraction of reality”,21 few studies have attempted to uncover the true molecular origins of the ZP. In a pioneering classical molecular dynamics study, Lyklema et al.22 probed the movement of water and ions near a solid wall under tangential shear. They determined that the so-called stagnant layer was at most 2−3 water layers thick, and that ions within this layer can move rather freely parallel to the wall but not perpendicular to it because viscosity perpendicular to the wall was 4−5 times greater than viscosity parallel to it. Lyklema later asserted that the water structuring near surfaces is responsible for stagnant layers and that to unravel the origin of ZP, fluid shear near surfaces would have to be understood at the molecular level.23 Other studies have expanded upon these findings by considering factors such as surface charge distribution,24 surface roughness,25,26 and fluid flow screening by adsorbed ions.27 English and Long28 determined the electrophoretic mobility of rutile nanoparticles for two positive surface charge values and calculated ZP using the Helmholtz-Smoluchowski equation. Huang et al.29 determined ZP from electroosmotic flows with five values of surface charge density. Additionally, several reviews summarize recent progress.30,31 However, we are aware of no molecular-level simulations of a specific mineral surface which have been directly related to experimental ZP measurements over a wide range of ionic compositions and pH values relevant to environmental and in vivo conditions. Here, we extend our previous studies of the rutile (110) surface in aqueous solutions to probe the molecular-level origins of electrokinetic phenomena. As is typical of metal oxides in contact with aqueous solutions, surface charge on rutile16 arises from the pH-dependent protonation of surface oxygens and deprotonation of chemisorbed water molecules, which is also dependent on solution ionic strength and composition, due to shielding of positive or negative net surface charge by counterions in solution. Our previous work investigated the equilibrium structures and diffusional dynamics of interfacial ions and water molecules32−35 and determined the axial profiles of their distance-dependent diffusivities near the rutile (110) surface. The results revealed that the first layer of water molecules does not exhibit translational diffusive motion on the time scale of molecular simulations, and that the diffusivity of the second water layer is very low, about 10% of the bulk value, at ambient temperature. Beyond the first two layers of water, which exhibit clear structural ordering related to the Ti and O atomic structure of the (110) surface, water molecule diffusivity continually increases up to about 15−20 Å from the surface, where bulk water values are attained. Mamontov et al.36,37 also demonstrated that the reduced water diffusional dynamics in the first few layers of pure bulk water at the rutile (110) surface are very similar to the
2. METHODS 2.1. Simulation Setup and Systems. We used the same interaction models of the surface and ions as in our previous works,32−34,39,40 i.e., SPC/E water,41 ions modeled as charged Lennard-Jones spheres and a five-layer TiO2 slab terminated by flexible bridging and terminal OH groups and a selected number of bare bridging oxygens (to generate negative surfaces) or fivefold coordinated terminal Ti atoms (to generate positive surfaces) to define the surface charge. All surface atoms except the terminal and bridging OH or O groups were kept rigid in the ab initio-obtained relaxed geometries. We used simulation cells composed of two identically charged, periodic rutile surfaces bounding the aqueous phase (planar slab geometry) of lateral dimensions 39.0 Å × 35.5 Å and width of the aqueous part (measured from the position of the last TiO layer if unrelaxed) of about 50 Å; the width was adjusted to yield ambient pressure. As found in our previous studies, at each of the two surfaces an approximately 15-Å-thick inhomogeneous interfacial region was formed, followed by a bulk region, which occupied about 20 Å in the center of the simulation cell. The width of the bulk region was sufficient to ensure that bulk water properties were established. For computational efficiency, a wider bulk region was not desired or necessary, as that would have only increased the number of atoms in the cell (already about 10 000). Moreover, all interesting phenomena occur in the interfacial inhomogeneous region. Selected simulations of wider slabs (about 80−100 Å) were carried out to confirm that indeed a wider bulk region does not modify the results. We consider that distances larger than 15 Å from the surface represent the bulk solution. Since the slab features two equally charged surfaces, the presented results are averages over both surfaces. The total number of ions, water molecules, and terminal hydroxyls is always 2048. The numbers of cations and anions for each simulation were chosen to neutralize the whole simulation cell, including surface charge, which dictates the difference between the numbers of cations and anions in each simulation. The bulk ionic strengths for our simulations range from 0.28 to 0.44 M, and are presented and discussed in detail elsewhere,40 and are tabulated in the Supporting Information (Tables SI1−SI3). Unlike in experiments where typically the interfacial volume to bulk volume is very low, the opposite is true in our simulations, where most ions interact with the surfaces and are found in the interfacial region and only a few are found in the bulk region, resulting in limited statistical accuracy of the bulk concentrations, with averages oscillating up to 0.05 M within 5 ns periods. This fact also prevented finer tuning of the simulation setup, i.e., further adjusting the number of ions to bring the resulting bulk concentrations closer to the same value for all differently charged surfaces. Note that the difference 0.05 M corresponds to a difference of less than one cation/anion pair in the bulk region of thickness 20 Å, so determining and adjusting the average bulk concentrations to decrease this variation would be possible only at the expense of even longer simulations or larger systems. 10190
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
Article
Langmuir
Figure 1. Snapshots of simulated systems with bulk RbCl, NaCl, and SrCl2 concentrations about 0.44, 0.41, and 0.28 M, respectively. The green lines at the distance about 15 Å from the TiO2 surface indicate the approximate boundary between the interfacial and bulk regions. The experimental charging curves obtained by pH titrations of rutile powder suspensions in aqueous solutions of varying ionic strength and cation type42,43 enable us to relate the surface charge densities at which our MD simulations are performed to corresponding pH values observed during macroscopic charging experiments. Establishing this correspondence to experimental data is essential because we cannot model the effect of pH on the interface directly in our computer simulations since the nondissociative SPC/E model of water is used, which does not allow any proton transfer. We studied aqueous solutions containing Rb+, Na+, or Sr2+, which represent large monovalent, small monovalent, and divalent cations, respectively. The anion was Cl− in all cases. RbCl and NaCl solutions were studied at surface charge densities from −0.2 to +0.1 C/m2, corresponding to a pH range of about 9.4 to 2.8. SrCl2 solutions were also studied at −0.4 C/m2 charge density because the strong adsorption Sr2+ results in much steeper charging curves with the result that the corresponding pH value is about 7.3. These systems mirror those of a previous study in which the effects of temperature and pH on the adsorption of Rb+, Na+, Sr2+, and Cl− were investigated.40
The systems simulated are summarized in the Supporting Information (Tables SI1−SI3). We first investigated the surface charge dependence of systems where the total amount of ions was adjusted to keep concentrations in the bulk region approximately constant for a given cation and all surface charge conditions. Representative snapshots are shown in Figure 1 (and Figure SI1 for Sr2+, σ = −0.4 C/m2). To study the concentration dependence of the ZPs, additional simulations were carried out. We also conducted simulations which approximated zero bulk concentrations by constructing systems containing only counterions to compensate surface charge and no co-ions or a very small number of cations and anions in the case of neutral surfaces. These simulations are labeled in graphs as 0 M, since the resulting bulk concentration was either zero for the entire simulation length (strong adsorption of counterions), or below 0.1 M if some counterions entered the bulk region at any time during the simulation. Representative snapshots are shown in Figure SI2 (and SI1 for Sr2+, σ = −0.4 C/m2). 2.2. Nonequilibrium Molecular Dynamics. We model electroosmotic flow using NEMD simulations, which closely mimic the electroosmotic experimental setup. An external electric field parallel to the surface was applied, which acted on water, ions, and flexibly 10191
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
Article
Langmuir bonded surface atoms. All the reported and discussed results are from simulations with field Ex = 2.1 × 108 V/m. We also ran several simulations at a three times stronger field (Ex = 6.4 × 108 V/m), which resulted in similar and statistically more accurate mobilities. However, the density profile of ions from some simulations with this stronger field showed an increased population of binding sites further from the surface (bidentate) and decreased population of sites closer to the surface (tetradentate) in comparison with density profiles for the 2.1 × 108 V/m field and no electric field (equilibrium simulations). This effect is easy to understand as the very strong field causes ions to jump from site to site over the potential and height barriers, increasing their average height with respect to the situation where they adopt the equilibrium distribution. Therefore, the results from simulations with the field 6.4 × 108 V/m were not used in our analysis. We also ran a few simulations with a lower 6.4 × 107 V/m field but in this case the fluctuations in the distance-dependent streaming velocities were large. The electric field Ex = 2.1 × 108 V/m proved to be the best choice in terms of providing a good signal-tonoise ratio from the limited number of atoms and preserving the equilibrium distributions of ions. The average streaming velocities on the order of 10 m/s are significantly smaller than the magnitude of the average thermal velocity of water at 25 °C, which is 371 m/s, indicating that the field, though many orders of magnitude larger than in real experiments, can still be treated as a perturbation of the zerofield behavior with the streaming velocities proportional to the applied field, i.e., in the upper limit of the linear response regime. This is in agreement with similar conclusions from other simulation studies; e.g., Huang et al.29 observed linear response for fields Ex = 0.5 × 108 V/m to 4 × 108 V/m, while Hartkamp et al.44 used a field of 2 × 108 V/m. Botan et al.45 used a field of 1 × 108 V/m, and also observed that bulk viscosity was affected when Ex was increased from 0.5 × 1010 V/m to 1.5 × 1010 V/m. To thermostat the system at the desired temperature (T = 25 °C) and to avoid problems with nonzero streaming velocity in the direction of the field, we used the same thermostat as in our studies of viscosity34,35 applying Poiseuille flow, i.e., thermostatting of only the y component of the translational velocity perpendicular to both the field and the surface (the z component might be thermostated as well) and no thermostatting of the rotational motion. The length of production runs was at least 10 ns for Rb+, while simulations with Na+ and Sr2+ were extended to at least 20 ns due to strong interactions of these cations with the surface and their reduced mobility.40 An alternate method of simulating ZP values utilizing equilibrium molecular dynamic simulations (EMD) to determine interfacial charge densities, combined with a single NEMD simulation to determine the viscosity profile of water at a neutral surface is presented in the last section of the Supporting Information.
ZP only the electroosmotic mobility of water far from the surface is needed. The electroosmotic mobilities of water, summarized in Figure 2 for bulk ionic concentrations of 0.28−0.44 M,
Figure 2. Electroosmotic mobilities of water in 0.44 M RbCl (top), 0.41 M NaCl (middle), and 0.28 M SrCl2 (bottom) solutions from NEMD simulations (points). The horizontal dashed lines indicate the averaged bulk mobilities at the various surface charge conditions simulated.
strongly depend on the surface charge. The electroosmotic mobilities for positively charged surfaces are negative (water moves with the anions) for all cations studied, while for negatively charged surfaces the mobilities are either positive (water moves with Rb+ cations) or negative (water moves with Cl− anions in simulations with Na+ or Sr2+ cations). Note also that the distance over which both ions and water transition from essentially zero mobility to bulk-like mobilities (∼12−15 Å) is nearly independent of surface charge density. This distance coincides with the distance where the electrostatic potential ϕ(z) approaches the bulk value, in accord with the eq (1.13) of ref 3. (rewritten in our notation)
3. RESULTS AND DISCUSSION 3.1. Distance-Dependent Mobilities of Water. The electroosmotic mobility of water, which is our main interest, is averaged in 0.125-Å-wide bins parallel to the surface directly as μH2O(z) = ⟨vHx 2O(z)⟩/Ex from the streaming velocity of water in each 1 fs MD integration step. We report the mobilities as the ratio of streaming velocity and external field including the sign, yielding negative values for anions. This notation is particularly convenient for mobilities of layers with mixed charges as well as for water, which is charge-neutral. The mobilities of ions are discussed in detail in Supporting Information. From the streaming velocities of water molecules and ions in each layer one can calculate the streaming velocities of the center of mass of a layer, which can be termed the streaming velocity of the fluid in that layer. As discussed in Supporting Information, the differences between the streaming velocities of water molecules and the whole fluid layer containing those molecules away from the interface are negligible. Therefore, we discuss in detail only the electroosmotic mobility of water, since for the prediction of
v x (z ) = −
ε0εr Ex [ϕ(z) − ζ ] η
(1)
where v (z) is the electroosmotic streaming velocity of the fluid at distance z from the surface, ε0 and εr are the permittivity of a vacuum and relative permittivity, η denotes dynamic viscosity, ϕ(z) electrostatic potential at distance z, and ζ is the ZP at a distance where vx(z) = 0 The limitations of this equation are further discussed in the Supporting Information and Conclusions. x
10192
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
Article
Langmuir
Figure 3. ZPs (ζ) from NEMD simulations. pH dependence for higher and lowest bulk concentrations of ions (left) and concentration dependence (right). Reference experimental data for rutile in NaCl are given as black symbols (left). The lines connect the data points from NEMD simulations. The experimental pHiep for the reference rutile in NaCl data is 6.2,51 and the experimental pHznpc for our rutile powder is 5.4.52
electrophoretic mobilities to ZPs. The resulting values are plotted in Figure 3 vs pH for low and high ionic strength (left column), and vs ionic concentration (right column). Also plotted are experimental electrophoretic data51 for rutile powder in 0.02 and 0.2 M NaCl solutions at 25 °C (left column, middle panel). Although these data do not exactly match our simulated systems, they are useful for comparative purposes. The ZP for RbCl changes from positive values at low pH to negative values at high pH. The simulated pHiep increases with ionic strength (from about 6 to 6.5) but is nonetheless somewhat higher than the experimental pHznpc of 5.4.52 However, given the discrepancy between the mobilities of the ions vs experimental mobilities mentioned in the Supporting Information, which affects the accuracy of the simulated electroosmotic mobilities, this difference is acceptable. Moreover, published rutile pHznpc and pHiep values vary between studies, with one literature compilation reporting average pHznpc and pHiep values of 5.4, with a standard deviation of 0.8 pH units.53 Negative electroosmotic mobilities and positive ZPs throughout the explored surface charge density (and related pH) range are observed in 0.41 M NaCl solution, conflicting with our reference experimental data for NaCl at 0.02 and 0.2 M. The results for 0.28 M SrCl2 solution are qualitatively similar to those of NaCl at 0.4 M, with the ZP staying positive throughout the pH range and even becoming more positive as negative surface charge increases.
Nontrivial behavior is observed at the neutral surface. Even though the charges of cations and anions in neutral systems are equal, the electroosmotic mobility is nonzero. In other words, we observe that for a neutral surface corresponding to pH of zero net proton charge, pH = pHznpc, the mobility and subsequently ZP is nonzero, i.e., pHznpc ≠ pHiep, where pHiep denotes the isoelectric point pH, which might also be termed the point of zero electroosmosis. While this inequality is not predicted by theories treating fluids as a continuum,46 it is easily explainable using molecular arguments, as will be discussed below, and as has been observed in other simulation studies. Nonzero electroosmotic flow was observed in simulations of NaI solutions in contact with uncharged hydrophobic nanochannels,29,47 while NaCl solutions resulted in negligible mobilities. Nonzero electroosmotic flows of NaCl aqueous solutions were observed in uncharged hydrophobic nanochannels made of LJ particles,48 and in KCl aqueous solutions between uncharged quartz slabs.49 Experimentally, nonzero ZP has been measured for neutral alumina surfaces.50 3.2. Dependence of Zeta Potential Values on pH. From the average bulk mobilities of water in a region further than 15 Å from the surface, we calculated the ZP (ζ) using the ημ Helmholtz-Smoluchowski equation, ζ = − ε ε , where η = 8.9 0 r
× 10−4 Pa.s and εr = 79 are the experimental bulk water values of dynamic viscosity and relative permittivity,36 since these bulk values are also commonly used to convert experimental
10193
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
Article
Langmuir
Figure 4. Experimental ZPs (ζ) of anatase vs pH for different NaCl concentrations (data from ref 55) (left); experimental ZPs for anatase vs pH for different CaCl2 concentrations (data from ref 56) (right).
results from two such studies for the anatase form of TiO2 are presented in Figure 4 and show trends similar to our high concentration simulations, as follows: With increasing concentration of NaCl, ZP values decrease in magnitude. At high enough NaCl concentrations ZPs remain positive throughout the pH range. This trend is apparent for both the simulated (Figure 3, middle graphs) and experimental (Figure 4) ZP data, although the NaCl concentration above which ZPs remain positive is lower for the simulations. For divalent Ca2+, which is expected to behave similarly to Sr2+ studied computationally by us, the experimental ZPs remain positive throughout the pH range for all concentrations between 0.01 and 1 M CaCl2. This is very similar to the trend observed in our simulations of SrCl2 solutions where positive ZPs prevail above about 0.04 M SrCl2 over the entire pH range studied (Figure 3, lower right plot). 3.5. Molecular Origins of the Zeta Potential. Having obtained detailed information on structure and dynamics of water and ions as a function of distance from a prototypical oxide surface, the (110) surface of rutile, we can discuss the molecular-level origins of electrokinetic phenomena and the observed dependence of ZP on pH, as well as on the electrolyte concentration and ionic composition. The total external electric force acting on any water molecule in our simulations is zero because the imposed external field is homogeneous and water molecules are uncharged. Likewise, there is no external driving force for electroosmotic motion of a charge-neutral bulk electrolyte. Consequently, the electroosmotic mobility of water originates exclusively from water− ion interactions and viscous drag among adjacent layers of fluid at a charge-imbalanced interface. The surface does not move the fluid, but plays a fundamental role in creating a double-layer environment which is sensitive to the external electric field through a very complex and delicate interplay of driving forces (dependent on charges of the inhomogeneous layers) and retarding forces (viscous and surface interactions). The origin of the electroosmotic mobility of the whole fluid (including the bulk region far from the surface) is the inhomogeneous interfacial region, where streaming mobilities build. The ion-specific density profiles of cations and anions result in charge-imbalanced interfacial layers. The total electric force acting on a charged layer is nonzero and generates a nonzero transfer of momentum. If the system was not viscous, that would lead to an unhindered acceleration of such a layer; however, the viscous forces, connecting all layers mutually and eventually with the immobile surface, ensure that the velocity of
The simulations approaching zero concentration of ions result in ZPs which change sign with surface charge and agree reasonably well with experimental data for 0.02 and 0.2 M NaCl. For Sr2+ the ZP approaches zero for very negative surfaces. A similar trend is also observed for Rb+ and Na+ at low concentrations with ZP being less negative for the −0.2 C/m2 surfaces (highest pH) compared to data for the less negatively charged surfaces. 3.3. Concentration Dependence of Zeta Potential Values. The concentration dependence of ZP for all surface charge densities (right column of Figure 3) shows that with increasing bulk concentration of ions all the curves approach each other and tend toward positive ZPs of about 15−25 mV, with the exception of the RbCl curve for −0.2 C/m2, which remains close to zero and negative. Solutions with only Na+ and Sr2+ cations at negative surfaces generate negative ZPs, but at concentrations above about 0.1 M for NaCl and 0.04 M for SrCl2 the curves switch to positive ZPs. At uncharged surfaces, ZPs approach zero values in the limit of zero concentration of ions, as indicated by asterisks in Figure 3. 3.4. Comparison of Simulated and Experimental Zeta Potentials. Considering the approximations inherent in the NEMD simulations, as well uncertainties intrinsic to the experimental data, the agreement of simulated ZPs with experimental data is rather good. Most notably, many of the general trends observed in experimental data are captured by the simulations, as follows: The ZP values are of the correct magnitude, i.e., tens of mV. Considering that the profiles of interfacial electrostatic potential (presented in the Supporting Information) reveal damped oscillations of the order of volts that decay with distance,32 the present NEMD approach to determine ZPs proved to be successful. The NEMD results at low concentrations follow the experimental data obtained in 0.02 and 0.2 M NaCl, i.e., transition from positive ZPs at low pH to negative values at high pH. Moreover, the pHiep from the low concentration simulations (pHiep ≈ 6.2) is in reasonable agreement (within 1 pH unit) with the pHznpc value for our rutile (5.4), and within one standard deviation of reported literature values.53 At negative surfaces (higher pH) for higher concentrations of NaCl and SrCl2, ZPs increase with pH and are positive throughout the pH range. These results agree qualitatively with experimental results obtained using electroacoustics,2,54 which permits ZP measurements in concentrated solutions. The 10194
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
Article
Langmuir each layer reaches the observed steady-state values. By “layer” we mean here any volume of fluid parallel to the surface and not necessarily related to structural layers, e.g., as small as our bin width (0.125 Å) used for analysis of distance-dependent variables. This phenomenon occurs not only at charged surfaces, but also at neutral surfaces, because differential sorption and/or approach of cations and anions, even to uncharged surfaces, can generate charge-imbalanced layers parallel to the surface. Therefore, even at the point of zero net proton charge, pHznpc, a nonzero electrokinetic response is generated, due to the unequal distribution of cations and anions in the interfacial region, i.e., pHznpc ≠ pHiep. However, if cations and anions are sorbed equally at charge-neutral surfaces, then pHznpc = pHiep, which agrees with classical notions of these measures.57,58 The total external force acting on any fluid layer in the bulk is zero and generates zero net momentum change, since bulk layers are charge-neutral. The charge-balanced bulk fluid therefore has no external reason to move faster or slower than the neighboring layers and its speed is thus dictated exclusively by the viscous forces and velocities of interfacial neighboring layers. In particular, if the interface generates a nonzero streaming velocity of a distant layer adjacent to chargeneutral (bulk) layers, all the bulk layers will quickly (within picoseconds for our systems) start to move at the same velocity. Regardless, the ZP does not arise from the existence of a slipping plane between a “static” fluid layer at the interface and a mobile bulk solution further from the surface, but rather precisely from the electrokinetically driven motion of ions within the whole inhomogeneous interfacial region due to cation−anion separation into charged-imbalanced layers within 15−20 Å of the surface. In other words, the bulk fluid is actually “dragged along” by the electroosmotic motion of the chargeimbalanced electrolyte region very near the surface, which exhibits a gradationally increasing velocity away from the charged surface. 3.6. Simulated Zeta Potential Values. The simulation results in the limit of zero bulk concentration, with only counterions in the interfacial region and no co-ions, are easy to interpret. Next to the negative surface (high pH), there are layers of excess cations. Even though many of them are strongly adsorbed and have a limited capacity to move and drag water with them, at least some of them move in the direction of field and generate a positive streaming velocity of near-surface layers which corresponds to a negative ZP. The absent anions cannot modify this result. Conversely, a positive surface (low pH) and the presence of only negatively charged counterions generates a negative streaming mobility and positive ZP. Simulated ZPs in higher concentration RbCl solutions are explained similarly. The only difference from the zero bulk concentration situation is that at a negative surface there is a surplus of cations and the minority anions reduce the positive streaming mobility induced by cations. However, the majority cations dominate the minority anions and dictate positive mobilities. The opposite occurs at positive surfaces. Since Rb+ does not specifically adsorb on the positively charged surface, the negative mobility behavior is dictated by Cl−, as it is also for NaCl and SrCl2 solutions. The behavior of NaCl and SrCl2 solutions at negative surfaces resulting in a positive ZP is an interesting and counterintuitive result. Both Na+ and Sr2+ cations adsorb strongly on the rutile (110) surface, as we have discussed previously.40 In fact, the strong adsorption of Na+ and Sr2+
overcompensates the negative surface charge, which leads to positively charged fluid layers out to about the outer extent of inner-sphere adsorption for Na+ (4.1 Å), and to about the end of the first peak of outer-sphere adsorption for Sr2+ (6.1 Å), (see Figure SI6). Although these fluid layers carry positive charge, their mobility and contribution to positive streaming velocities is limited by strong adsorption. On the other hand, all layers in the region between the overcompensating layers and the bulk carry net negative charge, compensating the excess adsorption of cations. These mobile layers negatively contribute to the streaming velocity of the interface more than the positive overcompensating layer, resulting in a net negative streaming velocity and positive ZP. In other words, if surface charge is overcompensated, as is often observed for strongly adsorbed ions (e.g., Na+ and Sr2+ in our systems), it is the more mobile ions with charge of the same sign as that of the surface charge (co-ions), which are attracted to the oppositely charged overcompensating layers closer to the surface, that are responsible for the “anomalous” ZP being of opposite sign to the surface. This phenomenon does not occur for RbCl at negative surfaces or in RbCl, NaCl, or SrCl2 solutions at positive surfaces, because for these cases the surface charge is not overcompensated. Thus, in our systems, it is specific adsorption rather than ion correlation effects which are responsible for the ZP trends we observe.21,59 Charge inversion and flow reversal in NaCl solutions in contact with negatively charged silicon surfaces has been observed computationally more than a decade ago60 and later for CaCl261 and SrCl2, NaCl, KCl, and CsCl44 solutions interacting with negative silica surfaces. Charge inversion has been also reported experimentally for fused silica and divalent ions at high concentrations.62
4. CONCLUSIONS We have shown that NEMD simulations of electroosmotic flow between parallel slabs of hydroxylated rutile (110) surfaces at various surface charge and pH states, as well as aqueous electrolyte type and concentration, predict ZP values which mirror those observed experimentally. The molecular-level origin of the electroosmotic flow, including situations when the zeta potential and surface charge are of opposite sign has been revealed in detail. The response arises gradationally over the narrow interfacial region which is anchored by the surface at one end, where ion and water mobilities are greatly reduced, and by the bulk solution at the other end, where ion and water mobilities attain bulk values. This interfacial region, which also corresponds to the region of strongly ordered solution structure differing from that of the bulk fluid, extends only about 15−20 Å from the surface, and this distance is nearly independent of surface charge or ion concentrations in the bulk solution. The change in experimentally observed ZP with solution concentration is shown to arise solely from competing effects of cation and anion motions in the charge-imbalanced layers immediately adjacent to the surface. Motion of the bulk solution region is solely driven by the motion induced in the very near-surface fluid layers, through viscous drag. The inequality between the pH of zero net surface charge and the isoelectric pH (zero electrokinetic effect) is shown to be a consequence of the differing distributions and mobilities of cations versus anions in the interfacial region. The RbCl, NaCl, and SrCl2 solutions studied yielded distinctly different ZPs with respect to concentration, surface charge, and pH, which nonetheless mimic trends observed experimentally. Simulated ZPs changed from positive to 10195
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
Langmuir negative values with increasing negative surface charge and pH at all RbCl and low NaCl and SrCl2 concentrations. However, they remained positive at higher NaCl and SrCl2 concentrations because adsorbed Na+ and Sr2+ overcompensate the negative charge of the rutile surface and the Cl− anions govern the ZP response, which consequently remains positive at these conditions. The anomalous ZP values (opposite in sign to the surface charge) often observed experimentally for solutions containing small or highly charged ions at high concentration are thus shown to be a result of overcompensation of surface charge by the strongly sorbing counterions which are relatively immobile compared to the more weakly sorbing co-ions whose motion then determines the ZP. The notion of the ZP as an electrostatic potential at an illdefined slip plane near a surface is qualitatively correct but pinpointing its location is not possible, which prevents matching the ZP with a particular point on the electrostatic potential profile. Moreover, continuum-based expressions which relate electroosmotic mobilities and ZP, such as the Helmholtz-Smoluchowski equation, assume constant solvent dielectric constant and viscosity, although these properties vary significantly at the interface. Interfacial viscosities actually increase rapidly from bulk values within about 4 Å of the surface34 (Figure SI7), while interfacial permittivities exhibit more complex behavior.63 Consequently, the relation between the electroosmotic mobility of bulk fluid layers and the ZP is more complex than is captured by continuum descriptions of the underlying interfacial phenomena. Still, the ZP is a very useful concept for expressing electrokinetic effects directly in terms of electroosmotic or electrophoretic mobilities because it captures all the interfacial properties behind these phenomena in integrated fashion. Therefore, deciphering the molecular origins of the ZP will ultimately make it possible to quantitatively interpret these oft-reported measures in a molecularly comprehensive manner including extracting local diffusivities, solvent rearrangements, and ion sorption processes, thereby providing far more information than can be extracted from continuum models. Such information is critically important for developing conceptual and predictive approaches for phenomena such as capacitive energy storage and desalination using nanoporous electrode materials, in vivo biological processes, colloidal transport through porous media, and mineral dissolution/precipitation rates and mechanisms.
■
■
ACKNOWLEDGMENTS
■
REFERENCES
M.P. was supported by the Czech Science Foundation Project 13-08651S. Access to the CERIT-SC computing and storage facilities provided under the programme Center CERIT Scientific Cloud, part of the Operational Program Research and Development for Innovations, reg. no. CZ. 1.05/3.2.00/ 08.0144, is greatly appreciated. M.L.M. and D.J.W. were supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division.
(1) Lyklema, J. In Fundamentals of Interface and Colloid Science; Academic Press, 1995; Vol. 2, pp 1−232. (2) Delgado, A. V.; Gonzalez-Caballero, E.; Hunter, R. J.; Koopal, L. K.; Lyklema, J. Measurement and interpretation of electrokinetic phenomena - (IUPAC technical report). Pure Appl. Chem. 2005, 77, 1753−1805. (3) Qian, S.; Ai, Y. Electrokinetic Particle Transport in Micro-/ Nanofluidics: Direct Numerical Simulation Analysis; Taylor & Francis, 2012. (4) Hunter, R. J. In Zeta Potential in Colloid Science; Academic Press, 1981; pp 219−257. (5) Williams, O. A.; Hees, J.; Dieker, C.; Jäger, W.; Kirste, L.; Nebel, C. E. Size-Dependent Reactivity of Diamond Nanoparticles. ACS Nano 2010, 4, 4824−4830. (6) Keller, A. A.; Wang, H.; Zhou, D.; Lenihan, H. S.; Cherr, G.; Cardinale, B. J.; Miller, R.; Ji, Z. Stability and Aggregation of Metal Oxide Nanoparticles in Natural Aqueous Matrices. Environ. Sci. Technol. 2010, 44, 1962−1967. (7) Chowdhury, I.; Mansukhani, N. D.; Guiney, L. M.; Hersam, M. C.; Bouchard, D. Aggregation and Stability of Reduced Graphene Oxide: Complex Roles of Divalent Cations, pH, and Natural Organic Matter. Environ. Sci. Technol. 2015, 49, 10886−10893. (8) Song, W.-J.; Du, J.-Z.; Sun, T.-M.; Zhang, P.-Z.; Wang, J. Gold Nanoparticles Capped with Polyethyleneimine for Enhanced siRNA Delivery. Small 2010, 6, 239−246. (9) Mudunkotuwa, I. A.; Grassian, V. H. Citric Acid Adsorption on TiO2 Nanoparticles in Aqueous Suspensions at Acidic and Circumneutral pH: Surface Coverage, Surface Speciation, and Its Impact on Nanoparticle−Nanoparticle Interactions. J. Am. Chem. Soc. 2010, 132, 14986−14994. (10) Niu, Z.; Li, Y. Removal and Utilization of Capping Agents in Nanocatalysis. Chem. Mater. 2014, 26, 72−83. (11) Hunter, K. A.; Liss, P. S. The surface charge of suspended particles in estuarine and coastal waters. Nature 1979, 282, 823−825. (12) Casals, E.; Pfaller, T.; Duschl, A.; Oostingh, G. J.; Puntes, V. Time Evolution of the Nanoparticle Protein Corona. ACS Nano 2010, 4, 3623−3632. (13) Townson, J. L.; Lin, Y.-S.; Agola, J. O.; Carnes, E. C.; Leong, H. S.; Lewis, J. D.; Haynes, C. L.; Brinker, C. J. Re-examining the Size/ Charge Paradigm: Differing in Vivo Characteristics of Size- and Charge-Matched Mesoporous Silica Nanoparticles. J. Am. Chem. Soc. 2013, 135, 16030−16033. (14) Berg, J. M.; Romoser, A.; Banerjee, N.; Zebda, R.; Sayes, C. M. The relationship between pH and zeta potential of ∼ 30 nm metal oxide nanoparticle suspensions relevant to in vitro toxicological evaluations. Nanotoxicology 2009, 3, 276−283. (15) Jiang, J.; Oberdörster, G.; Biswas, P. Characterization of size, surface charge, and agglomeration state of nanoparticle dispersions for toxicological studies. J. Nanopart. Res. 2009, 11, 77−89. (16) El Badawy, A. M.; Silva, R. G.; Morris, B.; Scheckel, K. G.; Suidan, M. T.; Tolaymat, T. M. Surface Charge-Dependent Toxicity of Silver Nanoparticles. Environ. Sci. Technol. 2011, 45, 283−287. (17) Yu, T.; Malugin, A.; Ghandehari, H. Impact of Silica Nanoparticle Design on Cellular Toxicity and Hemolytic Activity. ACS Nano 2011, 5, 5717−5728.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b02493. Link to movies of studied systems, additional snapshots of the simulated systems, ionic mobilities, discussion of mobility of fluid vs water, profiles of electrostatic potentials, and approximate predictions of ZP from theory combined with equilibrium molecular dynamics (PDF)
■
Article
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest. 10196
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
Article
Langmuir (18) Fedkin, M. V.; Zhou, X. Y.; Kubicki, J. D.; Bandura, A. V.; Lvov, S. N.; Machesky, M. L.; Wesolowski, D. J. High Temperature Microelectrophoresis Studies of the Rutile/Aqueous Solution Interface. Langmuir 2003, 19, 3797−3804. (19) Ridley, M. K.; Hackley, V. A.; Machesky, M. L. Characterization and Surface-Reactivity of Nanocrystalline Anatase in Aqueous Solutions. Langmuir 2006, 22, 10972−10982. (20) Machesky, M. L.; Wesolowski, D. J.; Palmer, D. A.; Ridley, M. K.; Benezeth, P.; Lvov, S. N.; Fedkin, M. V. In Surface Complexation Modelling; Johannes, L., Ed.; Elsevier, 2006; pp 324−358. (21) Lyklema, J. Surface charges and electrokinetic charges: Distinctions and juxtapositionings. Colloids Surf., A 2011, 376, 2−8. (22) Lyklema, J.; Rovillard, S.; De Coninck, J. Electrokinetics. The properties of the stagnant layer unraveled. Langmuir 1998, 14, 5659− 5663. (23) Lyklema, J. Molecular interpretation of electrokinetic potentials. Curr. Opin. Colloid Interface Sci. 2010, 15, 125−130. (24) Freund, J. B. Electro-osmosis in a nanometer-scale channel studied by atomistic simulation. J. Chem. Phys. 2002, 116, 2194−2200. (25) Sendner, C.; Horinek, D.; Bocquet, L.; Netz, R. R. Interfacial Water at Hydrophobic and Hydrophilic Surfaces: Slip, Viscosity, and Diffusion. Langmuir 2009, 25, 10768−10781. (26) Zhang, C.; Lu, P.; Chen, Y. Molecular dynamics simulation of electroosmotic flow in rough nanochannels. Int. Commun. Heat Mass Transfer 2014, 59, 101−105. (27) Wu, P.; Qiao, R. Physical origins of apparently enhanced viscosity of interfacial fluids in electrokinetic transport. Phys. Fluids 2011, 23, 072005−072009. (28) English, N. J.; Long, W. F. Estimation of zeta potentials of titania nanoparticles by molecular simulation. Phys. A (Amsterdam, Neth.) 2009, 388, 4091−4096. (29) Huang, D. M.; Cottin-Bizonne, C.; Ybert, C.; Bocquet, L. IonSpecific Anomalous Electrokinetic Effects in Hydrophobic Nanochannels. Phys. Rev. Lett. 2007, 98, 177801. (30) Pagonabarraga, I.; Rotenberg, B.; Frenkel, D. Recent advances in the modelling and simulation of electrokinetic effects: bridging the gap between atomistic and macroscopic descriptions. Phys. Chem. Chem. Phys. 2010, 12, 9566−9580. (31) Rotenberg, B.; Pagonabarraga, I. Electrokinetics: insights from simulation on the microscopic scale. Mol. Phys. 2013, 111, 827−842. (32) Předota, M.; Bandura, A. V.; Cummings, P. T.; Kubicki, J. D.; Wesolowski, D. J.; Chialvo, A. A.; Machesky, M. L. Electric Double Layer at the Rutile (110) Surface. 1. Structure of Surfaces and Interfacial Water from Molecular Dynamics by Use of ab Initio Potentials. J. Phys. Chem. B 2004, 108, 12049−12060. (33) Předota, M.; Zhang, Z.; Fenter, P.; Wesolowski, D. J.; Cummings, P. T. Electric Double Layer at the Rutile (110) Surface. 2. Adsorption of Ions from Molecular Dynamics and X-ray Experiments. J. Phys. Chem. B 2004, 108, 12061−12072. (34) Předota, M.; Cummings, P. T.; Wesolowski, D. J. Electric Double Layer at the Rutile (110) Surface. 3. Inhomogeneous Viscosity and Diffusivity Measurement by Computer Simulations. J. Phys. Chem. C 2007, 111, 3071−3079. (35) Parez, S.; Predota, M. Determination of the distance-dependent viscosity of mixtures in parallel slabs using non-equilibrium molecular dynamics. Phys. Chem. Chem. Phys. 2012, 14, 3640−3650. (36) Mamontov, E.; Vlcek, L.; Wesolowski, D. J.; Cummings, P. T.; Wang, W.; Anovitz, L. M.; Rosenqvist, J.; Brown, C. M.; Garcia Sakai, V. Dynamics and Structure of Hydration Water on Rutile and Cassiterite Nanopowders Studied by Quasielastic Neutron Scattering and Molecular Dynamics Simulations. J. Phys. Chem. C 2007, 111, 4328−4341. (37) Mamontov, E.; Wesolowski, D. J.; Vlcek, L.; Cummings, P. T.; Rosenqvist, J.; Wang, W.; Cole, D. R. Dynamics of Hydration Water on Rutile Studied by Backscattering Neutron Spectroscopy and Molecular Dynamics Simulation. J. Phys. Chem. C 2008, 112, 12334− 12341. (38) Hunter, R. J. Foundations of Colloid Science; Oxford University Press, 2001.
(39) Předota, M.; Vlcek, L. Comment on Parts 1 and 2 of the Series ″Electric Double Layer at the Rutile (110) Surface″. J. Phys. Chem. B 2007, 111, 1245−1247. (40) Předota, M.; Machesky, M. L.; Wesolowski, D. J.; Cummings, P. T. Electric Double Layer at the Rutile (110) Surface. 4. Effect of Temperature and pH on the Adsorption and Dynamics of Ions. J. Phys. Chem. C 2013, 117, 22852−22866. (41) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. The missing term in effective pair potentials. J. Phys. Chem. 1987, 91, 6269−6271. (42) Zhang, Z.; Fenter, P.; Cheng, L.; Sturchio, N. C.; Bedzyk, M. J.; Předota, M.; Bandura, A.; Kubicki, O. J. D.; Lvov, S. N.; Cummings, P. T.; Chialvo, A. A.; Ridley, M. K.; Benezeth, P.; Anovitz, L.; Palmer, D. A.; Machesky, M. L.; Wesolowski, D. J. Ion Adsorption at the RutileWater Interface: Linking Molecular and Macroscopic Properties. Langmuir 2004, 20, 4954−4969. (43) Machesky, M. L.; Předota, M.; Ridley, M. K.; Wesolowski, D. J. Constrained Surface Complexation Modeling: Rutile in RbCl, NaCl, and NaCF3SO3 Media to 250 °C. J. Phys. Chem. C 2015, 119, 15204− 15215. (44) Hartkamp, R.; Siboulet, B.; Dufreche, J.-F.; Coasne, B. Ionspecific adsorption and electroosmosis in charged amorphous porous silica. Phys. Chem. Chem. Phys. 2015, 17, 24683−24695. (45) Boţan, A.; Marry, V.; Rotenberg, B.; Turq, P.; Noetinger, B. How Electrostatics Influences Hydrodynamic Boundary Conditions: Poiseuille and Electro-osmostic Flows in Clay Nanopores. J. Phys. Chem. C 2013, 117, 978−985. (46) O’Brien, R. W.; Beattie, J. K.; Djerdjev, A. M. The electrophoretic mobility of an uncharged particle. J. Colloid Interface Sci. 2014, 420, 70−73. (47) Huang, D. M.; Cottin-Bizonne, C.; Ybert, C.; Bocquet, L. Aqueous Electrolytes near Hydrophobic Surfaces: Dynamic Effects of Ion Specificity and Hydrodynamic Slip. Langmuir 2008, 24, 1442− 1450. (48) Kim, D.; Darve, E. High-ionic-strength electroosmotic flows in uncharged hydrophobic nanochannels. J. Colloid Interface Sci. 2009, 330, 194−200. (49) Joseph, S.; Aluru, N. R. Hierarchical Multiscale Simulation of Electrokinetic Transport in Silica Nanochannels at the Point of Zero Charge. Langmuir 2006, 22, 9041−9051. (50) Dukhin, A.; Dukhin, S.; Goetz, P. Electrokinetics at High Ionic Strength and Hypothesis of the Double Layer with Zero Surface Charge. Langmuir 2005, 21, 9990−9997. (51) Yotsumoto, H.; Yoon, R.-H. Application of Extended DLVO Theory: I. Stability of Rutile Suspensions. J. Colloid Interface Sci. 1993, 157, 426−433. (52) Machesky, M. L.; Predota, M.; Wesolowski, D. J.; Vlcek, L.; Cummings, P. T.; Rosenqvist, J.; Ridley, M. K.; Kubicki, J. D.; Bandura, A. V.; Kumar, N.; Sofo, J. O. Surface Protonation at the Rutile (110) Interface: Explicit Incorporation of Solvation Structure within the Refined MUSIC Model Framework. Langmuir 2008, 24, 12331−12339. (53) Kosmulski, M. The significance of the difference in the point of zero charge between rutile and anatase. Adv. Colloid Interface Sci. 2002, 99, 255−264. (54) Dukhin, A. S.; Ohshima, H.; Shilov, V. N.; Goetz, P. J. Electroacoustics for Concentrated Dispersions. Langmuir 1999, 15, 3445−3451. (55) Gustafsson, J.; Mikkola, P.; Jokinen, M.; Rosenholm, J. B. The influence of pH and NaCl on the zeta potential and rheology of anatase dispersions. Colloids Surf., A 2000, 175, 349−359. (56) Kosmulski, M.; Prochniak, P.; Rosenholm, J. B. Electroacoustic study of titania at high concentrations of 1−2, 2−1 and 2−2 electrolytes. Colloids Surf., A 2009, 345, 106−111. (57) Stumm, W. Chemistry of the Solid-Water Interface: Processes at the Mineral-Water and Particle-Water Interface in Natural Systems; WileyInterscience: New York, 1992. (58) Sposito, G. On Points of Zero Charge. Environ. Sci. Technol. 1998, 32, 2815−2819. 10197
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
Article
Langmuir (59) Jiménez, M. L.; Delgado, Á . V.; Lyklema, J. Hydrolysis versus Ion Correlation Models in Electrokinetic Charge Inversion: Establishing Application Ranges. Langmuir 2012, 28, 6786−6793. (60) Qiao, R.; Aluru, N. R. Charge Inversion and Flow Reversal in a Nanochannel Electro-osmotic Flow. Phys. Rev. Lett. 2004, 92, 198301. (61) Lorenz, C. D.; Crozier, P. S.; Anderson, J. A.; Travesset, A. Molecular Dynamics of Ionic Transport and Electrokinetic Effects in Realistic Silica Channels. J. Phys. Chem. C 2008, 112, 10222−10232. (62) van der Heyden, F. H. J.; Stein, D.; Besteman, K.; Lemay, S. G.; Dekker, C. Charge Inversion at High Ionic Strength Studied by Streaming Currents. Phys. Rev. Lett. 2006, 96, 224502. (63) Parez, S.; Předota, M.; Machesky, M. Dielectric Properties of Water at Rutile and Graphite Surfaces: Effect of Molecular Structure. J. Phys. Chem. C 2014, 118, 4818−4834.
10198
DOI: 10.1021/acs.langmuir.6b02493 Langmuir 2016, 32, 10189−10198
