Quantification of Zeta-Potential and Electrokinetic Surface Charge

Article Cite This: J. Phys. Chem. C 2018, 122, 4437−4453

pubs.acs.org/JPCC

Quantification of Zeta-Potential and Electrokinetic Surface Charge Density for Colloidal Silica Nanoparticles Dependent on Type and Concentration of the Counterion: Probing the Outer Helmholtz Plane Alaa H. Jalil and Ute Pyell* Department of Chemistry, University of Marburg, Marburg, Germany S Supporting Information *

ABSTRACT: Electrokinetic data were measured for dilute aqueous dispersions of amorphous silica nanoparticles of various size via capillary electrophoresis with borate buffers containing either Li+, Na+, K+, or guanidinium as a counterion. Taking the mobility-dependent relaxation effect into account (modified analytic approximation developed by Ohshima), reliable values are obtained for the electrokinetic potential and the electrokinetic charge density dependent on the type of cation and the concentration of buffer. The reliability was confirmed by comparison of the results obtained for the nanoparticles with those values obtained for the planar-limiting case (fused-silica capillary inner wall/electrolyte interface). Regarding the inner part of the electrical double layer as a (mono)layer of unspecifically adsorbed counterions, we calculate (together with data gained by Brown et al. on the same type of nanoparticles via in situ photoelectron spectroscopy and potentiometric titration) the charge density at the outer Helmholtz plane and the fraction of charge included in the Stern layer for electrolytes containing the alkali ions Li+, Na+, or K+. This approach explains differences in the electrokinetic charge density as a result of differences in the properties of the Stern layer due to differences in the size of the hydrated cation and the hydration state of the silica surface.

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INTRODUCTION In our previous paper,1 we investigated with aqueous dispersions of amorphous silica nanoparticles of various size (Ludox TM, Ludox HS, and Ludox SM) whether electropherograms recorded from capillary electrophoresis experiments, employing a sodium borate buffer as background electrolyte, can be converted directly into exact number-based particle radius distributions. The results of this conversion procedure were compared to number-based particle radius distributions obtained from a large set of transmission electron microscopy (TEM) data. We succeeded in showing that the presented conversion method provides with high accuracy the width of the number-based size distribution, confirmed by its agreement with that gained by TEM. In addition, the presented method correctly determined the sign of the skewness of the particle size distribution. This method requires the precise determination of the zeta potential ζ from electrophoretic mobility data of nanoparticles with a high electrokinetic charge density σζ. Double layer distortion due to the relaxation effect and electrophoretic retardation must not be neglected.2,3 While this task can be fulfilled by numerical calculations,4,5 we have shown for coated gold and bare silica nanoparticles in sodium borate buffers of varied ionic strength that reliable results can also be obtained by an analytic approximation originally presented by Ohshima,6,7 modified by us with regard to the effective ionic drag coefficient.1,8 Taking bare silica nanoparticles as an © 2018 American Chemical Society

example, we are now interested in investigating whether this approach can be extended to different electrolytes with varied size (and degree of hydration) of the positively charged counterion. Variation of the buffer cation will influence the electrophoretic mobility of the charged nanoparticles via two different principles: (i) via their influence on the effective ionic drag coefficient, because the relaxation effect is dependent on the limiting equivalent conductance of the cation, and (ii) via their influence on |ζ|. Numerous investigations have shown that the type of the cation (even at fixed effective ionic drag coefficient) induces effects, which cannot be explained by classical continuum approaches that assume the ions to be point charges. These effects were first described by Hofmeister in a series of papers published during the 1880s and 1890s.9 Hofmeister studied the critical coagulation concentration (CCC) of blood plasma and hen’s egg globulin dependent on the type of salt dissolved in aqueous solution. He found that the impact of the dissolved salt on the CCC is ion-specific. With a fixed type of anion, the CCC becomes cation-specific and vice versa, enabling the cations or the anions to be arranged, respectively, in a series according to the determined CCC. Interestingly, the same order of “influence strength” was found Received: December 20, 2017 Revised: January 24, 2018 Published: January 29, 2018 4437

DOI: 10.1021/acs.jpcc.7b12525 J. Phys. Chem. C 2018, 122, 4437−4453

Article

The Journal of Physical Chemistry C

of zero charge. Exact values are obtained by repeating the titration with a solution not containing the silica sol and taking the volume difference (at identical pH). However, the quantity σ0 does not provide any information whether the cations (that are replaced during titration) are present in the (original) twophase system as ions adsorbed on the surface of the particles or as ions desorbed in solution. The fraction of counterions that are adsorbed (or fixed in any form) on the surface of the particles effects a difference in the values between that of the surface charge density σ0 and that of the effective or electrokinetic charge density σζ. The latter is defined to be the charge density that can be deduced from electrokinetic data, whereas (in the absence of overcharging25) we would expect that |σ0| ≥ |σζ|. This difference in charge densities is accompanied by a difference in the values between that of the total charge of a particle and that of the so-called kinetic charge26 or effective charge27 of a particle that refers to the charge deduced from conductance measurements or electrokinetic data. Interestingly, the effective charges deduced from conductance measurements and those deduced from electrokinetic measurements have been shown to be identical within experimental error,26 which indirectly confirms the theoretical concept of a “bound” layer. According to double (triple) layer theory,28 the total charge of a particle is compensated in part by a “bound” (inner) layer of fixed counterions, (co-ions) and solvent molecules (denoted Stern layer, SL) and in part by a diffuse (outer) layer of mobile counterions, co-ions, and solvent molecules (denoted Gouy− Chapman layer, GCL). For the static particle, ion densities in the GCL are described by the nonlinearized Poisson− Boltzmann equation (PBE). The boundary between the SL and the GCL is traditionally denoted the outer Helmholtz plane (OHP, the plane defined by the centers of unspecifically adsorbed (hydrated) counterions), while the inner Helmholtz plane (IHP) refers to the zone of specifically adsorbed (mostly dehydrated) ions and solvent molecules. In addition, the description of electrokinetic effects requires the definition of a shear plane that is assumed to be very close to the OHP.26,27 The difference between σ0 and σζ can be simply related to the phenomenon of counterion condensation following a non-ionspecific electrostatic model assuming a uniformly charged surface.29 However, this model will not explain Hofmeister effects. Simultaneously, it can be deduced from the experimentally observed ion-specificity of the electrokinetic charge density σζ that it is obviously required to employ a more sophisticated model that not only accounts additionally for ion−ion repulsions due to the volume of the ions but also for surface hydration, dispersion interactions, or the existence of discrete adsorption sites.16 As the charge density distribution in the GCL is known from solving the nonlinearized PBE, σζ can be directly calculated from electrokinetic data, if the sphere radius, the ionic strength, and the electrophoretic mobilities of the electrolyte ions are known.1,8 This can be done numerically30 or with high accuracy via analytic approximations.31,32 The comparison of σ0 with σζ then provides direct access to the sign and magnitude of the charge fixed in the SL.33 Possibly, the most simple approach to gain semiquantitative information about ζ and σζ is the determination of the CCC or the gelation kinetics at fixed salt and particle concentration. In an early study, Allen and Matijevic34 determined the CCC of Ludox HS and AM particles in alkali halide solutions dependent on the pH. While the CCC at fixed salt concentration at pH 9

by him with other colloidal materials such as isinglass and iron(III) oxide. The principles behind these phenomena are obviously of universal nature, although unravelling the true nature of these effects is still under progress. Traditionally, Hofmeister effects are related to the hydration of the ions. One quantitative approach to measure the “hydration strength” is the influence of dissolved ions on the viscosity of the solution quantified by the so-called “Jones Dole viscosity B coefficient”.10 Those ions with positive B (being classified as structure-forming “kosmotropes”) increase the viscosity, while ions with negative B (being classified as structure-breaking “chaotropes”) effect its reduction. Modern spectroscopy methods combined with computer simulations have gained further insight into the structure-modifying effects of these ions.11 Hofmeister effects are described for ion−ion and for ion−surface interactions.12 The “law of matching water affinities” (LMWA) based on hydration enthalpy differences13 provided a very successful path in the understanding of Hofmeister effects (and reversals of the Hofmeister series) with respect to ion−ion interactions (ion pair formation). However, with respect to ion−surface interactions, very obviously more approaches have to be taken into account. Following the interpretation of Salis et al.,14 the ion-specific results reported by many researchers can be understood with a model that accounts for effects of hard sphere repulsion, hydration, and dispersion forces. Due to their wide range of applications, colloidal silica nanoparticles had been the object of a large number of investigations. Mostly, potentiometric acid−base titration was performed in the presence of different salts (chlorides) at different concentrations.14−21 These titration results can be converted into surface charge densities σ0 dependent on the type and concentration of the cation in solution. We will restrict the discussion of the results to monovalent cations. These are the alkali cations plus guanidinium (Gdm+). For all investigated silica particles (including amorphous silica nanoparticles, pyrogenic silica particles, and mesoporous silica), the following sequence of surface charge densities was found for fixed salt concentration and fixed pH (pH ≥ 7): |σ0(Cs+)| > |σ0(Rb+)| > |σ0(K+)| > |σ0(Na+)| > |σ0(Li+)|. Salis et al.14 determined |σ0(Gdm+)| > |σ0(K+)|. It is also interesting to note that σ0 obtained with NaCl and with NaSCN was independent of the type of the anion.14 Generally (at constant pH), |σ0| increased with increasing salt concentration. This phenomenon can be understood as a result of charge regulation in the electrical double layer.22 Via multisite proton adsorption modeling, Hiemstra et al.23 succeeded in modeling the surface charge density (experimentally determined in aqueous NaCl solution24) dependent on the pH and on the ionic strength for Ludox particles having a mean particle diameter of 15 nm. Their model employs a so-called basic Stern model,22 which combines a constant plane parallel capacitance (its magnitude (here 2.9 F m−2) is one of the fitting parameters) with the space capacitance of a diffuse layer, both connected in series so that the charge in the diffuse layer compensates the surface charge. The silica surface was described by the dissociation reaction of the surface silanol groups, assuming a surface concentration of 8 nm−2 and a dissociation constant of pK = 7.5. Ion-specific effects might be included in this model as variation in the constant plane parallel capacitance. The surface charge density σ0 of a silica sol is calculated from the amount of substance of protons “reacted” per unit surface, e.g., when titrating a solution containing a silica sol to its point 4438


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reference point, φ0 dependent on the type of counterion is directly accessible by XPS and follows (at fixed salt concentration and pH) the series |φ0(Li+)| > |φ0(Na+)| > |φ0(K+)| > |φ0(Cs+)|. This order follows what had been measured for ζ:44 |ζ(Li+)| > |ζ(Na+)| > |ζ(K+)| > |ζ(Cs+)|. However, it is opposite to the order determined for σ0 by potentiometric titration:43 |σ0(Cs+)| > |σ0(K+)| > |σ0(Na+)| > |σ0(Li+)|. It should be emphasized that this order for σ0 is in accord with the results reported in the past by many other authors.14−21 It is also important that the XPS data exclude the presence of an additional gel-type layer.44 On the basis of these data, Brown et al.43 developed a model of the electrical double (triple) layer (EDL) assuming the nonspecific electrostatic interaction of the hydrated counterions with the uniformly charged (hydrated) silica surface. According to this model, the SL consists of a monomolecular layer of water, which hydrates the deprotonated silanol groups plus a monolayer of hydrated cations. The OHP is located in a distance dSL apart from the surface, which is given by the monomolecular layer of water plus the radius of the hydrated cation. This distance dSL is identical to the thickness of the SL and increasing with the radius of the hydrated counterion from 0.4 to 0.8 nm (calculated from the experimentally determined potential drop and the measured surface charge density assuming a fixed dielectric constant within the SL; refer to subsequent section). Recently, Gmür et al.45 made additional XPS studies with microjets to determine EDL parameters for the silica/ electrolyte interface under variation of the type of the counterion and the type of the co-ion. As before, these studies were made with Ludox SM particles. Gmür et al. combined the results obtained by XPS, attenuated total reflection Fourier transform infrared (ATR-FT-IR) spectroscopy, and potentiometric titration. These three methods unequivocally confirmed that with Na+ as counterion, under the conditions of investigation, the result is independent of the type of co-ion (Cl−, Br−, I−, HCOO−, or NO3−), while with Cl− as co-ion it is strongly dependent on the type of counterion (Li+, Na+, K+, and Cs+). These data are in accord with the results reported before by other authors.14,31 They confirm experimentally that (at the silica/electrolyte interface) the co-ions are completely excluded from the SL (and depleted in the GCL28). Against this background, we will investigate with colloidal silica nanoparticles of varied size whether with tetrahydroxyborate as a common buffering (co)ion (under conditions of pH = pKA of boric acid) the method presented by us in refs 1 and 8 permits the accurate determination of ζ and σζ of the studied nanoparticles under variation of the type and concentration of the counterion. In contrast to most of the previous studies, our investigation will not only include monovalent alkali ions (Li+, Na+, and K+) but also the monovalent guanidinium (Gdm+) ion, which has a limiting equivalent conductance very similar to that of Na+, whereas its Hofmeister effects have been reported to be very different from those of Na+.14 Neutron diffraction experiments46 revealed that Gdm+ is planar, of triangular shape, and extremely weakly hydrated. As all of our experiments are based on the determination of the electrophoretic mobility of the nanoparticles in fused-silica capillaries (at constant electric field strength), we will continuously additionally measure the electroosmotic mobility for the silica/electrolyte interface; i.e., we gain simultaneously quantitative data for the spherical and for the planar limiting case (for identical electrolyte and the same type of surface

followed the series Li+ > Na+ > K+ > Cs+, it was independent of the type of halide ion (Cl−, Br−, or I−). These results were confirmed by Van der Linden et al.,35 who also determined the gelation kinetics for Ludox HS particles. For fixed salt (chloride) concentration, the gelation time followed the order Li+ > Na+ > K+ > Rb+ > Cs+. The experimentally determined order of stability is reverse to what we would have expected from σ0 data, which suggests that the order of ζ values (with respect to the type of cation present in solution) can be expected to be reverse to the order of σ0 values. This conclusion has been supported by a large set of data. Microelectrophoresis measurements36 of the electrophoretic mobility μep of colloidal silica particles (diameter about 500 nm) in alkali chloride solutions result for fixed salt concentration and pH 7.55 in the series |μep(Li+)| > |μep(Na+)| > |μep(K+)| (derived from published zeta potentials). Electroacoustic measurements were made with micrometer sized silica particles.37 The resulting zeta potentials follow the series |ζ(Li+)| > |ζ(Na+)| > |ζ(K+)| > |ζ(Cs+)| (for fixed salt concentration and fixed pH, pH > 5). AFM measurements give further evidence of the correctness of this series.38 Here charge densities (at the OHP) were obtained that follow the series |σ(Na+)| > |σ(K+)| > |σ(Cs+)|. Recently, new jet techniques allowed the area of investigation of X-ray photoelectron spectroscopy (XPS) to be extended to the aqueous−solid interface of colloidal silica.39 With a liquid microjet, the structure of colloidal silica particles (Ludox SM) was elucidated (with a sample having a high particle density). The authors concluded that the surface layer containing deprotonated silanol groups has a thickness of less than 0.2 nm. Solvation takes place only in the outermost layer of the nanoparticle, while the core retains the electronic structure of bulk SiO2. This result seems to be in clear contradiction to that of Allison,40 who modeled Ludox HS particles dispersed in aqueous solution (based on experimental data from ref 41) as particles having a solid core and a charged gel-type layer. In KCl solution (0.3 mmol L−1) and at high pH, the gel-type layer is estimated to have a thickness of about 4 nm. However, it must not be overlooked that according to the results of Allison40 the thickness of this gel-type layer decreases drastically with increasing salt concentration in the electrolyte, so that the difference in the reported values regarding the thickness of the gel-type layer might be simply due to the varied ionic strength. This conclusion is in accord with the results reported by Leroy et al.,42 who confirmed for colloidal silica immersed in an electrolyte containing NaCl at a concentration of 100 mmol L−1 that the zeta potential corresponds to the electrostatic potential at the OHP and concluded that the shear plane must be located very close to the OHP, while the assumption of a stagnant diffuse layer (which might be equalized with a gel-type layer) is not required within the theoretical modeling. XPS with microjets further allowed one to measure uniquely for colloidal silica (Ludox SM particles) immersed in an electrolyte containing an alkali chloride directly the absolute surface potential φ0 at the silica/electrolyte interface.43 With fixed salt concentration, the Si 2p binding energy (BESi 2p) is decreasing linearly with increasing hydrated ion radius, resulting in the series BESi 2p (Cs+) > BESi 2p (K+) > BESi 2p (Na+) > BESi 2p (Li+). The BE shift ΔBESi 2p directly corresponds to the change Δφ0e (with e = elementary charge). As φ0 corresponds to 0 mV at the point of zero charge, BESi 2p at the point of zero charge can be taken as the direct reference point. With this 4439


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The Journal of Physical Chemistry C material). Direct comparison of the results obtained for different geometries will shed light on the accuracy of the proposed method. As ζ can be assumed to be quasi-identical to φOHP (the electrostatic potential at the OHP) and σζ can be assumed to be very close to the sum charge density at the OHP, these data directly give access to the properties of the OHP and (indirectly in comparison with φ0 and σ0, parameters which have been gained by43 in situ photoelectron spectroscopic and potentiometric titration data) permit a characterization of the SL with regard to included charge and associated voltage drop dependent on the type and concentration of the counterion. In accordance to the procedure proposed by Brown et al.43 assuming an approximate value for the dielectric constant within the SL, we quantify the thickness of the SL assuming that no charges are within this inner layer. In addition, we will be able to determine the charge density σOHP at the OHP in the Stern layer and the fraction of charge (FOC) included in the SL in comparison to the FOC included in the GCL. It will be investigated whether this method permits the determination of quantities that might become useful in the understanding of Hofmeister effects.

Figure 1. Schematics of the underlying EDL model: Φ0 = electrostatic potential at surface, σ0 = surface charge density, ΦOHP = electrostatic potential at OHP, σOHP = charge density at OHP, OHP = outer Helmholtz plane, ζ = electrokinetic potential (=electrostatic potential at shear plane), σζ = electrokinetic charge density, SL = Stern layer, GCL = Gouy−Chapman layer, Φ∞ = electrostatic potential in the bulk, σ∞ = charge density in the bulk, xOHP = location of outer Helmholtz plane, xSP = location of shear plane, dSL = thickness of Stern layer, λD = Debye length.

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THEORY The beginning of the theory of the electrical double (triple) layer (EDL) dates back to the end of the 19th century with the first idea to model the EDL as a plane parallel capacitance.47 This model was replaced later by the concept to use the Poisson−Boltzmann equation (PBE) to describe the distributions of ion densities near a charged surface,48,49 while modern theory builds on the combination of these two concepts (Gouy−Chapman−Stern (GCS) model).50 Further refinements have been made, e.g., by taking into account specific adsorption,51 nonelectrostatic hydration repulsion,52 finite size of ions, and dependence of the dielectric constant on the electric field strength.53 Different names can be found to denote the two capacitances of the GCS model: (1) SL capacitance or inner-layer capacitance and (2) GCL or outer-layer capacitance.54 While the first capacitance is regarded to be a plane parallel capacitance (with no charges between the two oppositely charged surfaces that can be curved), the second capacitance corresponds to the diffuse layer specific capacitance (involving volume charge densities) resulting from numerically solving the PBE for a curved surface or solving the PBE analytically in the planar limiting case. Neither the charge included in each of the two capacitances nor the voltage drop over each of the two capacitances is kept constant. The “outer” capacitance can be best modeled by regarding the charged surface and the Stern layer (the charged surface and the OHP are the two planes that form the “inner” capacitance) as an entity forming the charged shear plane (refer to Figure 1). The excess charge on this shear plane (the charge of the surface that is not screened by the SL) corresponds to the effective charge or electrokinetic charge, which is equivalent to the charge (of opposite sign) included in the GCL (Figure 1). We can divide the total countercharge QT (opposite to that of the surface charge) into the excess charge QSL included in the SL and the excess charge QGCL included in the GCL:50 QT = Q SL + Q GCL

where σ0 is the surface charge density, σζ is the electrokinetic charge density, i.e., the effective charge normalized on the shear plane, and σOHP is the charge density at the OHP. Here we neglect that for a curved charged surface the area of the OHP is larger than the area of the curved charged surface, while curvature effects may not be neglected within the GCL.31 It is useful to regard the dependence of the electric field strength Ε on the distance x (Figure 1, discussion exclusively for the planar limiting case). As there are no charges between the surfaces x = 0 and xOHP, the derivative d2Φ/dx2 is zero (homogeneous electric field, constant electric field strength ΕSL). Within the GCL, the electrostatic potential Φ is described by the PBE and consequently also Ε (Ε = dΦ/dx). From these considerations follows for the electric field strength ΕSL within the SL ΕSL =

(3)

where dSL = thickness of the SL. From the Coulomb law follows for the homogeneous field within the SL σ0 ΕSL = ε0εr,SL (4) where ε0 is the electric field constant and εr,SL is the dielectric constant within the SL. Simple rearrangement gives dSL = ε0εr,SL

(Φ0 − ζ ) σ0

(5)

If we compare this equation with the specific capacitance CPPC of an ideal plate-parallel capacitor (CPPC = ε0εr1/d, where εr = dielectric constant between the plates and d = distance between the plates), we might conclude for the specific capacitance CSL of the SL σ0 CSL = Φ0 − ζ (6)

(1)

If we normalize on the area of the charged surface, we obtain charge densities σ0 = (σ0 − σζ ) + σζ = − σOHP + σζ

(Φ0 − ζ ) dSL

This derivation ignores that the charge density σOHP screens only a fraction of the (total) surface charge density σ0.50

(2) 4440


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solution, and 0.5 mL of SNP22 stock solution and filling up to 50 mL with borate buffer (of identical type of counterion as that of the background electrolyte) having a counterion concentration of 10 mmol L−1.

EXPERIMENTAL SECTION Materials. Colloidal dispersions of Ludox TM-40 (SNP22), Ludox HS-30 (SNP12), and Ludox SM-30 (SNP7) were supplied by Sigma-Aldrich (Taufkirchen, Germany). The data provided by the manufacturer report for SNP22 a particle concentration (mass fraction) of 40% (w/w), a nominal particle diameter of 22 nm, and a pH of 9.0, for SNP12 a particle concentration (mass fraction) of 30% (w/w), a nominal particle diameter of 12 nm, and a pH of 9.8, and for SNP7 a particle concentration (mass fraction) of 30% (w/w), a nominal particle diameter of 7 nm, and a pH of 10. In all cases, Na+ is the counterion of the electrostatically stabilized particles in aqueous dispersion. Two different batches of SNP (B1 = batch 1 and B2 = batch 2) were investigated. Borate buffers were prepared by dissolving either dilithium tetraborate (p.a., Merck, Darmstadt, Germany), disodiumtetraborate decahydrate (p.a., Merck, Darmstadt, Germany), or dipotassium tetraborate tetrahydrate (puriss., Acros Organics, Geel, Belgium) in Milli-Q water (18 MΩ cm, Merck MilliPore, Darmstadt, Germany). Guanidinium borate buffer was prepared by neutralizing diguanidinium carbonate with boric acid:55

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RESULTS AND DISCUSSION

Size Distributions. SNP7, SNP12, and SNP22 (B1) had been extensively characterized by transmission electron microscopy (TEM, histograms from 1533 (SNP7), 3665 (SNP12), and 3777 (SNP22) particles), dynamic light scattering (DLS), and Taylor dispersion analysis (TDA) in our previous publication.1 For the smallest nanoparticles (SNP7), there is a symmetrical size distribution, which is in accord with a Gaussian function, while, for the larger nanoparticles (SNP12), there is some asymmetry (the size distribution is left-skewed), which is considerably increased for the largest nanoparticles (SNP22). Fitting the obtained histograms to a Gram−Charlier series of type A56 and data evaluation by moment analysis (determination of the (statistical) central moments57) resulted in the following data regarding the distribution of the diameter: SNP7 (μ1 = 11.2 nm, σ = 1.9 nm, κ3 = 0.25), SNP12 (μ1 = 15.8 nm, σ = 2.4 nm, κ3 = −0.42), SNP22 (μ1 = 29.6 nm, σ = 3.3 nm, κ3 = −0.57), with μ1 = first moment, σ2 = second central moment, κ3 = skewness. The parameter κ3 for SNP7 is associated with a large standard error, so that is regarded here to be insignificant. The significance of κ3 for the other two nanoparticle populations had been confirmed. As the electrophoretic data for the two batches taken for these investigations are identical within experimental error, we assume that the nanoparticle populations of the two batches of the same type can be characterized by an identical size distribution. Electrophoretic Mobility. Electropherograms were recorded for varied ionic strength Ι (Ι = 20−120 mmol L−1) and varied type of counterion (Li+, Na+, K+, or Gdm+) at fixed pH (9.2) and fixed temperature (25 °C). In a first measurement series, we recorded electropherograms for samples containing either SNP12 or SNP22 (B1). In a second measurement series, we recorded electropherograms for samples containing either SNP7 or SNP12 or SNP22 (B2, Ι = 20 mmol L−1) or for samples containing SNP7, SNP12, and SNP22 in a mixed sample (B2, Ι = 40−120 mmol L−1). Repeatability of the measurements was confirmed by three to five repeated runs. The resulting electropherograms are shown in Figures S3 (Supporting Information). Typical electropherograms are shown in Figure 2. Drift of the baseline, seen in several electropherograms, stems from a drift in the intensity of the D2lamp employed for absorbance detection. This drift can be easily corrected and is not interfering with the developed data evaluation scheme. The small relative standard deviation for the migration times recorded with consecutive runs indicates the absence of time shifts due to adsorption of the particles onto the inner capillary wall. As already stated in our previous publication,1 adsorption of particles onto the wall is regarded to be effectively suppressed by electrostatic repulsion at the negatively charged fused-silica wall. From these electropherograms, electrophoretic mobilities μep were determined assuming a constant (homogeneous) electric field strength in the capillary given by the total length of the capillary and the applied voltage

2H 2O [C(NH 2)3 ]2 CO3 + 4H3BO3 ⎯⎯⎯⎯⎯⎯⎯→ 2[C(NH 2)3 ]+ + 2B(OH)4 − + 2H3BO3 + CO2 ↑ + H 2O

In a beaker, 4 mmol of diguanidinium carbonate (Alfa Aesar, Kandel, Germany) and 16 mmol of boric acid (Microselect, Fluka, Seelze, Germany) were dissolved in the appropriate amount of Milli-Q water. The solution was heated to a boil for 1 h to remove dissolved CO2. After cooling to room temperature in a closed vessel, the solution was filled up with Milli-Q water to 100 mL. From this stock solution (c(Gdm+) = 80 mmol L−1), buffers with a lower concentration of Gdm+ were prepared by dilution with Milli-Q water. Capillary Electrophoresis. All CE measurements were done with a Beckman (Fullerton, CA, USA) P/ACE MDQ CE system equipped with a UV-absorbance detector. Temperatures of the capillary and the sample tray were kept at 25 °C. Data were recorded with the Beckman 32 Karat software (v.5.0). Further data treatment was done with Origin 8.5 (Northampton, MA, USA). The electrokinetic potential was calculated employing a Matlab (MathWorks, Natick, MA, USA) procedure. InoLab pH 720 (WTW, Weilheim, Germany) was used for pH measurements. Fused silica capillaries (76 μm I.D., 375 μm O.D.) were obtained from Polymicro Technologies, Phoenix, AZ, USA. New capillaries were conditioned by rinsing them first with NaOH solution (0.1 mol L−1) for 60 min, water for 60 min, and electrolyte for 10 min. Between runs, the capillaries were rinsed with electrolyte for 5−10 min. The total length of the capillary was 395 mm and the length to the detector 292 mm. In all cases, the separation voltage was 7 kV. The electroosmotic mobility was determined by either using thiourea as a marker or measuring the migration time of a negative system peak. The injection parameters were 6 s at 690 Pa. The detection wavelength was set to 214 nm with B1 and to 200 nm with B2. Sample solutions were prepared from B1 or B2 by taking 1 mL of the delivered stock solution and filling up to 50 mL with borate buffer (of identical type of counterion as that of the background electrolyte) having a counterion concentration of 10 mmol L−1. From B2, also mixed samples were prepared by mixing 2 mL of SNP7 stock solution, 1 mL of SNP 12 stock 4441


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migration times were determined from the maximum of the recorded trace of the analyte zone. The calculated electrophoretic mobilities are given in Table 1. There is no systematic deviation of the results obtained with B1 from those obtained with B2. We therefore consider these two batches to be equivalent concerning size distribution, electrokinetic potential ζ, and electrokinetic charge density σζ. Comparison of the electrophoretic mobility data in Table 1 confirms that for the SNP populations investigated (Ι ≥ 40 mmol L−1) |μep| is increasing with the particle radius. As expected, |μep| is decreasing with increasing Ι (see Figure 3a). In addition, |μep| is strongly dependent on the type of counterion. Comparison of these data with those obtained for the electroosmotic mobility μeo (capillary wall and nanoparticles are of the same material: amorphous silicon dioxide) confirms that for these objects of very different size and geometry we have a similar solid phase/electrolyte interface. As with |μep| for the silica nanoparticles, the determined |μeo| is decreasing with increasing Ι and strongly dependent on the type of counterion (see Figure 3b). However, this comparison also illustrates the strong influence of the relaxation effect on |μep| of the highly charged nanoparticles. With the wide capillaries used by us, for |μeo|, we are in the validity range of the Helmholtz− Smoluchowski equation; hence, here the relaxation effect is expected to be negligible. This does not hold for |μep| of the nanoparticles; therefore, we observe |μep| ≪ |μeo| and d|μep|/dΙ ≪ d|μeo|/dΙ. Besides this, the dependence of |μeo| on the type of counterion follows the same order as that observed for |μep|. No systematic deviation of the results obtained with B1 from those obtained with B2 gives an additional indication that effects due to an inhomogeneity of the electric field strength in the capillary can be neglected, because for Ι ≥ 40 mmol L−1 results with B2 were obtained with mixed samples having a higher mass fraction of particles, while those with B1 were obtained with samples that contained only one type of SNP, resulting in a lower mass fraction of particles; hence, the results are independent of the mass fraction of particles in the sample. However, very obviously, effects due to particle−particle interaction might not be neglected. A comparison of the recorded electropherograms dependent on the size of the nanoparticles, the ionic strength, and the type of counterion (refer to Figures S1−S3, Supporting Information) shows the strong influence of unavoidable inhomogeneities of the electric field strength and/or induced aggregation processes on the recorded peak shape. In cases of insufficient colloidal stability, distorted, skewed, or even bimodal peaks were obtained. Peak distortions, which we attribute to aggregation processes, are visible with Li+ for SNP7 at Ι ≥ 80 mmol L−1 (Figure S3i) and for SNP22 at Ι = 120 mmol L−1 (Figure S1f2), with Na+ for SNP7 at Ι ≥ 80 mmol L−1 (Figure S3f), with K+ for SNP7 at Ι ≥ 40 mmol L−1 (Figure S3c), for SNP12 at Ι ≥ 60 mmol L−1 (Figure S1o1), and for SNP22 at Ι ≥ 80 mmol L−1 (Figures 2 and S1p), and with Gdm+ for SNP7 at Ι ≥ 40 mmol L−1 (Figure S3d) and for SNP12 at Ι ≥ 20 mmol L−1 (Figure S2g). It is difficult to distinguish peak distortion due to electrophoretic aggregation from peak distortion due to the formation of stabilized moving boundaries (stabilized boundaries separating zones of different electric field strength as described in isotachophoresis), which are clearly visible in the electropherograms recorded for SNP7 at Ι = 20 mmol L−1 (Figure S2a−c). Generally, peak distortions point to either pronounced electrophoretic particle aggregation58 under conditions of low ζ potential2,3 (associated with an over-

Figure 2. Cumulated superpositions of electropherograms obtained for SNP22(B1) in four to five consecutive runs with a separation electrolyte containing (a) 20 mmol L−1, (b) 40 mmol L−1, (c) 60 mmol L−1, and (d) 80 mmol L−1 K+. Experimental conditions: T = 25 °C, total length of capillary = 395 mm, capillary length to detector = 292 mm, inner diameter of fused-silica capillary = 76 μm, electrolyte 10 mmol L−1 borax in water (pH 9.2), voltage 7 kV, sample injection 690 Pa 6 s, data rate 16 Hz, moving average 50 points, absorbance detection 214 nm.

μep =

L DL T L L − D T Utmig Ut0

(7)

with LD = capillary length to detector, LT = total capillary length, U = applied voltage, tmig = migration time of analyte zone, and t0 = migration time of neutral marker. In all instances, 4442


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Table 1. Electrophoretic Mobility μep Calculated from Migration Times Obtained in Consecutive Runs Compared to Electroosmotic Mobility μeo (Mean Values, Standard Deviations in Brackets, Single Values Are Listed in Tables S1a−r, for Number of Consecutive Runs Refer to Tables S1a−r, Supporting Information, B1 = Batch 1, B2 = Batch 2) cation Li+

Na+

K+

Gdm+

a

c (mmol L−1) 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80

μep (SNP7/B2) (mm2 kV−1 s−1)





μeo (B2) (mm2 kV−1 s−1)

−32.96 −30.89 −29.94 −29.04 −28.17 −27.54 −32.63 −30.53 −29.39 −28.23 −27.21 a −32.49 −29.89 −28.72 −27.25 a a a −27.47 −25.50 −23.90

−33.18 −31.93 −31.55 −30.91 −30.14 −29.55 −30.71 −31.93 −30.39 −29.74 −28.44 −27.23 −32.36 −31.15 a a a a a a a a

−33.13 −32.24 −31.72 −30.93 −30.15 −29.54 −32.79 −31.82 −31.00 −29.92 −28.92 a −32.22 −30.93 −29.78 −28.79 a a −30.35 −28.71 −26.90 −25.36

−33.51 −33.33 −34.16 −33.56 −32.64 −31.84 −34.20 −34.01 −32.53 −32.00 −30.96 a −34.80 −34.22 −33.28 −32.53 a a a a a a

−35.16 −34.88 −34.47 −33.54 −32.63 −31.81 −34.72 −34.33 −33.58 −32.36 −31.15 a −33.67 −32.19 −31.35 −30.46 a a −32.04 −30.72 −28.70 −26.88

69.05 58.73 52.21 47.66 44.36 41.53 63.98 53.67 47.58 42.78 39.03 36.36 59.87 51.86 45.35 40.34 37.41 34.77 55.78 45.95 38.94 34.46

(±0.03) (±0.03) (±0.04) (±0.03) (±0.07) (±0.06) (±0.02) (±0.03) (±0.01) (±0.01) (±0.04) (±0.02) (±0.08) (±0.11) (±n.d.)

(±0.03) (±0.18) (±0.05)

(±0.10) (±0.01) (±0.05) (±0.01) (±0.01) (±0.10) (±0.13) (±0.01) (±0.03) (±0.01) (±0.08) (±0.04) (±0.03) (±0.02)

(±0.04) (±0.02) (±0.03) (±0.02) (±0.05) (±0.08) (±0.03) (±0.02) (±0.01) (±0.02) (±0.03) (±0.01) (±0.05) (±0.08) (±0.19)

(±0.02) (±0.05) (±0.15) (±0.30)

(±0.07) (±0.03) (±0.07) (±0.03) (±0.08) (±0.04) (±0.11) (±0.06) (±0.04) (±0.05) (±0.05) (±0.13) (±0.06) (±0.19) (±0.01)

(±0.03) (±0.02) (±0.05) (±0.05) (±0.05) (±0.08) (±0.08) (±0.02) (±0.04) (±0.04) (±0.04) (±0.05) (±0.03) (±0.02) (±0.21)

(±0.02) (±0.11) (±0.19) (±n.d.)

(±0.06)b (±0.05)b (±0.04)b (±0.13)b (±0.05)b (±0.04)b (±0.04)c (±0.38)c (±0.06)c (±0.04)c (±0.05)c (±0.04)c (±0.02)d (±0.13)d (±0.08)d (±0.24)d (±0.13)d (±0.04)d (±0.02)e (±0.11)e (±0.04)e (±0.15)e

Not determined. bRefer to Table S1b. cRefer to Table S1g. dRefer to Table S1l. eRefer to Table S1q.

estimated electrophoretic mobility with regard to the electrophoretic mobility of an isolated single sphere) or an induced inhomogeneity of the electric field strength in the separation capillary (also associated with a small systematic deviation of the estimated electrophoretic mobility from the correct value). The influence of peak distortion on the measured σζ will be discussed in a subsequent section. Estimation of the Zeta Potential. Figure 3b gives experimental evidence that, for nanoparticles with |ζ| > 25 mV, the potential ζ can only be determined from μep via numerical calculations4,5 or via an analytic approximation taking the relaxation effect into account (e.g., the equation developed by Ohshima6,7). For electrolytes containing monovalent cations and anions differing largely in their electrophoretic mobility, we suggested a modified approach:1,8

f4 (κa) =

mcounter =

f3 (κa) =

1 2[1 + 2.5/{κa(1 + 2e−κa)}]3

κa(κa + 1.3e−0.18κa + 2.5) 2(κa + 1.2e−7.4κa + 4.8)3

2εrε0kTNA 3ηλ+°

(9)

where NA = Avogadro’s number. For 1 < κa < 10 and |ζ| > 100 mV, the possible error of this approximate analytic expression can become very large. At T = 25 °C, κa = 5, m+ = m− = mcounter = 0.184, and ζ = 154 mV, the difference Δμep between the result of the numerical calculation and the result of the approximate analytic expression is 35 mm2 s−1 kV−1, while with |ζ| = 103 mV the maximum error is less than 4 mm2 s−1 kV−1 and at |ζ| = 77 mV less than 1.5 mm2 s−1 kV−1 (at T = 25 °C, κa = 5, and m+ = m− = mcounter = 0.184).59 With exception of the data gained for c(Li+) = 20 mmol L−1, all results we obtained for the nanoparticles were in the range of |ζ| < 75 mV, hence in the validity range of the approximate analytic expression. The sphere radius a (=the distance between the center of the sphere and the shear plane, which is generally considered to be corresponding to the solid particle radius plus the thickness of the Stern layer) is approximated by us with the radius corresponding to the maximum of the size distribution histogram obtained from the TEM data.1 The following

(8)

where εr = relative electric permittivity, ε0 = electric permittivity of vacuum, η = viscosity, κ = Debye−Hückel screening parameter, a = sphere radius, k = Boltzmann constant, T = absolute temperature, e = elementary electric charge, and f1, f 3, and f4 are a function of κa and are given by f1 (κa) = 1 +

(8c)

The original equation presented by Ohshima in 20016,7 uses the average dimensionless ionic drag coefficient (m+ + m−)/2, which was replaced by us with the dimensionless ionic drag coefficient of the counterion mcounter.8 With negatively charged nanoparticles, the dimensionless ionic drag coefficient of the counterion mcounter is calculated from the limiting conductance of the cation λ°+

⎛ eζ ⎞2 2ε ε ζ ⎡ μep = r 0 ⎢f1 (κa) − ⎜ ⎟ f3 (κa) − mcounter ⎝ kT ⎠ 3η ⎣ ⎤ ⎛ eζ ⎞2 ⎜ ⎟ f ( κa )⎥ 4 ⎝ kT ⎠ ⎦

9κa(κa + 5.2e−3.9κa + 5.6) 8(κa + 1.55e−0.32κa + 6.02)3

(8a)

(8b) 4443


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Figure 4. Calculated μep for a negatively charged sphere with fixed ζ (ζ = −100 mV) at T = 25 °C for varied reduced sphere radius κa with electrolytes containing different counterions (see inset).

mcounter in the region that does not fall into the validity region of the Henry equation. In this region, |μep| is not proportional to |ζ|. Intersecting lines indicate the presence of a maximum regarding the function μep = f(ζ) (at fixed mcounter and κa). Figure 4 shows the lines representing μep = f(κa) (ζ = −100 V) with varied mcounter (representing electrolytes containing Li+, Na+, K+, or Cs+ as counterions). Neglecting the influence of the mobility of the counterion on the electrophoretic mobility of the nanoparticle at this high |ζ| introduces a significant error. If we take the result obtained for Na+ as counterion at κa = 4 as a reference (μep = 34.1 mm2 kV−1 s−1), the result obtained for Li+ (κa = 4) is 6.6% lower (μep = 31.9 mm2 kV−1 s−1), while the result obtained for K+ (κa = 4) is 7.1% higher (μep = 36.6 mm2 kV−1 s−1). With |ζ| = 75 mV and κa = 4, we obtain for Li+ μep = 32.5 mm2 kV−1 s−1, for Na+ μep = 33.5 mm2 kV−1 s−1, and for K+ μep = 34.5 mm2 kV−1 s−1. Here the relative difference is about 3%, having a significant influence on the determined magnitude of |ζ|. According to the limiting ion conductance of Cs+, the expected result for an electrolyte containing Cs+ as counterion will be somewhat higher than that for an electrolyte containing K+ as counterion (Figure 4). It can be further concluded from this figure that for 1 ≤ κa ≤ 100 and 75 mV ≤ |ζ| ≤ 100 mV calculation of |ζ| from electrokinetic data via the Smoluchowski equation might introduce an error of more than 50% decrease with regard to the correct value. The highest negative deviation is obtained with the counterion having the lowest limiting ion conductance, i.e., Li+. The calculated values can be compared to the experimental values obtained by capillary electrophoresis (refer to Tables S1a−r). Figure 5 shows the superposition of calculated and experimental data. All experimental data exceed the threshold value defined for the validity range of the Henry equation. Apparently (taking the influence of mcounter on μep into consideration), ζ is strongly dependent on the type of counterion. Independent of the type of counterion, the estimated values for ζ of SNP7 and SNP12 are very close, while those for SNP22 seem to be somewhat lower. More precise data than those obtainable with a graphical procedure were gained from an iterative scheme described in ref 8 that calculates ζ from the measured electrophoretic mobility (refer to Tables S1a−r) via eqs 8−10. This iterative procedure is based on a procedure in which μ is calculated with varied ζ at fixed κa and mcounter (refer to Tables S2a−c). The results are given in Table 2. Table 2 also contains those values

Figure 3. Comparison of (a) electrophoretic mobility determined for SNP12/B2 and (b) electroosmotic mobility (closed symbols) together with absolute electrophoretic mobility determined for SNP12/B2 (open symbols) dependent on concentration and type of counterion. For experimental conditions, refer to Figures 2 and S1.

parameters were taken: a(SNP7) = 5.75 nm, a(SNP12) = 8.25 nm, a(SNP22) = 15.25 nm. The Debye−Hückel parameter κ is given by κ=

e 2NA ∑ zi 2ci εrε0kT

(10)

where zi = charge number (valence) of the ith component and ci = molar concentration of the ith component. The dimensionless ionic drag coefficients are dependent on the type of the counterion (and the temperature). We obtained the following values for T = 25 °C: m(Li+) = 0.333; m(Na+) = 0.257; m(K+) = 0.175; m(Cs+) = 0.161; and m(Gdm+) = 0.250. These values correspond to the following limiting equivalent conductancies (T = 25 °C): λo(Li+) = 38.69 S cm2 mol−1; λo(Na+) = 50.11 S cm2 mol−1; λo(K+) = 73.50 S cm2 mol−1; λo(Cs+) = 79.91 S cm2 mol−1; λo(Gdm+) = 51.45 S cm2 mol−1.60,61 With the analytic approximation given in eqs 8−10, we calculated μep dependent on mcounter, κa, and ζ (refer to Figures 4, 5, and S4). For all counterions, there is a distinct minimum of the function |μep| = f(κa) (at fixed |ζ|) above a threshold value that marks the validity range of the Henry equation. The influence of the varied dimensionless ionic drag coefficient is reflected by a decrease in |μep| (at fixed |ζ|) with increased 4444


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calculated for ζ at the inner capillary wall/electrolyte interface from the experimentally measured electroosmotic mobilities μeo (refer to Tables S1a−r) via the Helmholtz−Smoluchowski equation: μeo = −

ε0εrζ η

(11)

We regard this comparison as a possibility to confirm the validity of the employed approximate analytic approach, because nanoparticles and capillaries are both of amorphous silicon dioxide, so that we would expect similar values for ζ independent of size and geometry (if curvature effects may be neglected). Figure 6 compares ζ calculated from the electroosmotic mobility to ζ calculated from the electrophoretic mobility determined for SNP12/B2 dependent on concentration and type of counterion. There is good agreement of ζ calculated for the fused-silica wall and the silica nanoparticles. The difference in |ζ| obtained for the nanoparticles and the capillary wall particularly at lower ionic strength (refer to Figure 6 and Table 2) can be expected from a difference in the diffuse layer specific capacitance, which is curvature-dependent. Wang and Pilon53 reported for a surface potential of 10 mV, an ionic strength of 10 mmol L−1, T = 25 °C, and εr = 78.5 that the electrode curvature of ultramicroelectrodes has a measurable effect on the predicted diffuse layer specific capacitance for sphere radii ≤40 nm. Another possible reason for the observed difference in ζ observed for the spherical and for the planar limiting case might be the neglect of effects induced by the displacement of ions within the Stern layer. This type of effects has been reported by Zukoski and Saville62,63 and by Mangelsdorf and White64 to influence to a measurable extent the electrokinetic properties of colloidal particles (here, reduction of the electrophoretic mobility). However, charge transport in the Stern layer will not influence the electrophoretic mobility with κa ≫ 1.62,63 Consequently, comparing of results observed for the spherical and for the planar limiting case directly gives experimental access to the maximum error introduced by neglecting effects induced by the displacement of ions within the Stern layer. Hence, the comparison of the results depicted in Figure 6 permits the conclusion to be drawn that under the conditions of our measurements the calculation of ζ with a procedure that neglects the displacement of ions within the SL might be justified and is not introducing intolerable error (in contrast to neglecting effects induced by the displacement of ions within the GCL). The excellent agreement of data obtained for the two batches (Table 2) reflects the good reproducibility of the method. However, the limited accuracy of the method does not allow a dependence of ζ to be observed on the particle radius, which might be predicted from the curvature dependence of the diffuse layer specific capacitance.53 The largest variation of data (at fixed ionic strength) is observed for c(cation) = 20 mmol L−1. As can be seen in Figure 5, at this low ionic strength, the estimation of ζ is associated with the highest imprecision due to the nonlinearity of the function μep = f(ζ) under conditions of a strong impact of the relaxation effect. For Li+ as counterion, we even observe a distinct maximum of the function μep = f(ζ) that precludes a reliable estimation of ζ for c(Li+) ≤ 30 mmol L−1. Electrokinetic Charge Density. For a curved surface, a given ζ potential, and a given ionic strength, the numerical solution of the PBE30 gives access to the electrokinetic charge

Figure 5. Calculated electrophoretic mobility μep of a negatively charged sphere with electrolytes containing (a) Li+, (b) Na+, (c) K+, or (d) Gdm+ at T = 25 °C for varied reduced sphere radius κa and varied ζ together with superimposed experimental data for (■) SNP7(B2), (●) SNP12(B2), and (◆) SNP22(B2) (error bars representing the standard deviation for five consecutive measurements; for experimental parameters, refer to Tables 1 and S1a−p and Figures 2 and S1, * = single value). 4445


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Table 2. Electrokinetic Potential ζ Calculated for the Nanoparticles Investigated with the Iterative Procedure Illustrated in Table S2 Compared to ζ of the Capillary Wall/Electrolyte Interface Calculated from the Data Listed in Table 1 via the Smoluchowski Equation cation Li

c (mmol L−1)

SNP7/B2 ζ (mV)

SNP12/B1 ζ (mV)

SNP12/B2 ζ (mV)

SNP22/B1 ζ (mV)

SNP22/B2 ζ (mV)

capillary ζ (mV)

20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80

(−78) −66.6 −61.0 −56.6 −52.9 −50.4 −72.3 −63.0 −57.7 −53.2 −49.7 a −69.1 −59.4 −54.7 −49.9 a a a −53.8 −47.6 −43.0

(−83) −66.4 −61.4 −57.2 −53.7 −51.2 −69.7 −59.9 −56.5 −53.1 −49.0 −45.6 −66.9 −59.5 a a a a a a a a

(−82) −67.7 −62.0 −57.3 −53.8 −51.2 −71.8 −63.4 −58.1 −53.6 −50.1 a −66.5 −58.9 −53.8 −50.1 a a −62.2 −54.2 −47.9 −43.5

(−67) −59.7 −58.7 −55.4 −52.2 −49.8 −66.9 −60.1 −54.0 −51.4 −48.5 a −66.2 −59.2 −54.7 −51.8 a a a a a a

(−76) −64.2 −59.4 −55.3 −52.2 −49.8 −68.8 −60.9 −56.3 −52.2 −48.8 a −62.8 −54.7 −50.9 −48.0 a a −60.0 −52.3 −46.3 −41.9

−88.7 −75.4 −67.1 −61.2 −57.0 −53.3 −82.2 −68.9 −61.1 −54.9 −50.1 −46.7 −76.9 −66.6 −58.2 −51.8 −48.0 −44.7 −71.6 −59.0 −50.0 −44.3

+

Na+

K+

Gdm+

a

Not determined.

⎡ ⎛ ⎛ eζ ⎞⎢ 2εrε0κkT 1 2 ⎜ ⎟ ⎢1 + σζ = sinh⎜ ⎝ 2kT ⎠⎢ κa ⎜⎜ cosh2 eζ e ⎝ 4kT ⎣

( )

⎞ ⎟ ⎟⎟ ⎠

1/2 ⎛ eζ ⎤ ⎞⎤ ⎡ 8 ln⎣cosh 4kT ⎦ ⎟⎥ ⎜ 1 + ⎟⎥ ⎜ (κa)2 ⎜ sinh2 eζ ⎟⎥ 2kT ⎠⎦ ⎝

( ) ( )

The results of both equations include significant errors for κa < 0.5. As in all of our cases κa ≫ 0.5 (refer to Figure 5), we expect with the two equations (12 and 13) correct values for σζ with a maximum relative error much smaller than 5%. In the planar limiting case, for a given ζ potential and a given ionic strength, the PBE can be solved analytically (without linearization), which gives direct access to the electrokinetic charge density σζ (Grahame equation)3,51,65

Figure 6. Comparison of |ζ| calculated from the electroosmotic mobility (closed symbols) to |ζ| calculated from the electrophoretic mobility determined for SNP12/B2 (open symbols) dependent on concentration and type of counterion. For experimental conditions, refer to Figures 2 and S1.

σς =

density σζ, which is defined as the effective electric charge Qeff normalized on the area of the shear plane, which might be approximated to be 4πa2.4 For a 1:1 electrolyte, σζ can also be calculated with good accuracy employing an approximate empirical formula suggested by Loeb et al.:30,31 σζ =

Q eff 4πa

2

=

(13)

⎛ eζ ⎞ 2εrε0κkT ⎟ = sinh⎜ ⎝ 2kT ⎠ e

⎛ eζ ⎞ ⎟ 8nεrε0kT sinh⎜ ⎝ 2kT ⎠ (14)

where n = number density of buffer ions. Equation 14 is the special case for a 1:1 electrolyte. The resulting values for σζ (nanoparticles and planar limiting case) are given in Table 3. Results obtained via eq 12 do not deviate from those obtained via eq 13 (results not shown). The comparison of the data (Table 3) clearly shows that σζ (nanoparticles and planar limiting case) is dependent on the type of the counterion. These data are a direct confirmation of observed Hofmeister effects. We can predict that at fixed ionic strength the colloidal stability of negatively charged colloidal nanoparticles will decrease in the order Li+ > Na+ > K+ > Gdm+. This prediction was confirmed by us indirectly by the

⎛ eζ ⎞⎤ ⎛ eζ ⎞ εrε0kT ⎡ 4 ⎟ + ⎟ tanh⎜ κ ⎢2 sinh⎜ ⎝ ⎠ ⎝ 4kT ⎠⎥⎦ ⎣ e 2kT κa (12)

For κa > 0.5, the maximum deviation from the result of the numerical treatment is only 5% independent of ζ. For a 1:1 electrolyte, Ohshima et al.32 derived a more accurate analytic approximate expression: 4446


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Table 3. Electrokinetic Charge Density σζ Obtained via eq 13 for the Nanoparticles Investigated Compared to That of the Capillary Inner Wall (Grahame Equation) cation Li+

Na+

K+

Gdm+

c (mmol L−1)

SNP7/B2 σζ (C m−2)





meanb σζ (C m−2)

capillary σζ (C m−2)

rel. dev.c (%)

20 40 60 80 100 120 meand SDd RSDe (%) 20 40 60 80 100 120 meand SDd RSDe (%) 20 40 60 80 100 120 meand SDd RSDe (%) 20 40 60 80 meand SDd RSDe (%)

(−0.0443) −0.0468 −0.0494 −0.0506 −0.0512 −0.0521 a a a −0.0395 −0.0434 −0.0459 −0.0467 −0.0473 a a a a −0.0371 −0.0401 −0.0427 −0.0432 a a a a a a −0.0353 −0.0359 −0.0361 a a a

(−0.0455) −0.0444 −0.0477 −0.0494 −0.0504 −0.0515 −0.0487 0.0028 5.6917 −0.0352 −0.0385 −0.0427 −0.0448 −0.0448 −0.0446 −0.0431 0.0027 6.2876 −0.0333 −0.0382 a a a a −0.0382 a a a a a a a a a

(−0.0446) −0.0457 −0.0484 −0.0495 −0.0504 −0.0515 −0.0491 0.0022 4.5152 −0.0368 −0.0417 −0.0443 −0.0453 −0.0460 a −0.0443 0.0019 4.2502 −0.0329 −0.0377 −0.0400 −0.0416 a a −0.0398 0.0020 4.9299 −0.0301 −0.0338 −0.0346 −0.0350 −0.0345 0.0006 1.7728

(−0.0311) −0.0363 −0.0428 −0.0454 −0.0467 −0.0480 −0.0438 0.0046 10.5676 −0.0309 −0.0366 −0.0383 −0.0412 −0.0425 a −0.0397 0.0027 6.7752 −0.0354 −0.0304 −0.0358 −0.0390 a a −0.0351 0.0043 12.3954 a a a a a a a

(−0.0371) −0.0401 −0.0436 −0.0453 −0.0467 −0.0479 −0.0447 0.0030 6.7980 −0.0322 −0.0373 −0.0405 −0.0419 −0.0429 a −0.0407 0.0024 6.0040 −0.0283 −0.0322 −0.0354 −0.0377 a a −0.0351 0.0028 7.8696 −0.0265 −0.0304 −0.0315 −0.0320 −0.0313 0.0008 2.6151

(−0.0396) −0.0416 −0.0456 −0.0474 −0.0486 −0.0497 −0.0466 0.0032 6.7804 −0.0338 −0.0385 −0.0415 −0.0433 −0.0441 −0.0446 −0.0424 0.0025 5.8139 −0.0325 −0.0346 −0.0371 −0.0394 a a −0.0370 0.0024 6.4907 −0.0283 −0.0321 −0.0331 −0.0335 −0.0329 0.0007 2.1735

−0.0451 −0.0482 −0.0490 −0.0495 −0.0500 −0.0501 −0.0494 0.0008 1.5862 −0.0393 −0.0418 −0.0428 −0.0426 −0.0421 −0.0423 −0.0423 0.0004 0.9363 −0.0352 −0.0396 −0.0400 −0.0394 −0.0399 −0.0399 −0.0398 0.0003 0.6313 −0.0314 −0.0333 −0.0326 −0.0322 −0.0327 0.0006 1.7027

(−12.3) −13.6 −6.9 −4.2 −2.9 −0.7

−14.1 −7.8 −3.2 1.6 4.6 5.4

−7.7 −12.6 −7.3 0.1 a a

−9.9 −3.6 1.4 4.0

Not determined. bArithmetic mean of values determined for SNP12 and SNP22. crel. dev. = relative deviation in percent = {[σζ(capillary) − σζ(mean)]/σζ(capillary)}(−100). dWithout value for c = 20 mmol L−1. eRSD = relative standard deviation. a

reached in our investigations, λD = Debye length) the effect of particle size on surface charge density can be neglected, regardless of the pH and the ionic strength. Hence, comparison of σζ obtained for colloidal nanoparticles with σζ obtained for the inner capillary wall (planar limiting case) should provide an unbiased confirmation of the correctness of the applied approach (with regard to the calculation of ζ and σζ), provided that the measured values for μep and μeo do not show significant deviations from the true values. However, this prerequisite is not given in all cases due to particle aggregation during the electrophoretic run resulting in distorted peaks (as discussed in a previous section). We expect that the data obtained for μeo are very accurate, while those data obtained from distorted electrophoretic zones are systematically higher than the true values. Due to the decreased colloidal stability with increased ionic strength, we expect a more pronounced (upward) shift due to aggregation effects under conditions of higher ionic strength. It is also of interest to study the dependence of σζ on the ionic strength. As we expect the data obtained from the measurement of μeo (σζ of the inner capillary wall) to be

recorded traces. If we compare for fixed electrolyte concentration the recorded peaks dependent on the type of counterion, we find distortion increasing in the order Li+ < Na+ < K+ < Gdm+. In accord with the manufacturer’s information on colloidal stability of different populations, peak distortion is increasing in the order SNP22 < SNP12 < SNP7 (at fixed type of counterion and ionic strength). These considerations allow us to predict the order of the critical coagulation concentration (CCC) for a given nanoparticle population to be Li+ > Na+ > K+ > Gdm+. This order corresponds to the Hofmeister series9 (or lyotropic sequence33) and exactly matches those data determined experimentally for silica nanoparticles by other workers.34,35 Our data confirm that the observed differences in CCC can be ascribed to differences in ζ induced by differences in σζ. In accordance with the results of Barisik et al.,66 who studied for fixed ionic strength and pH the decrease in the surface charge density with an increase in particle size, we would expect σζ to be invariant with the particle diameter (and identical with that of the planar limiting case). Barisik et al. reported for silica nanoparticles that in the range of λD/dp < 0.2 (which was 4447


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results obtained for the interface fused-silica capillary/buffer is varied with Ι, reaching positive values with higher Ι in the case of Na+, K+, and Gdm+. This observation can be attributed to the influence of aggregation that is more pronounced with higher Ι and lower σζ. This result confirms that a correct measurement of μep by CE requires the elimination of effects due to aggregation during the electrophoretic run. In addition, comparison of the results obtained for different batches gives an indication of the precision of the method. Variation of results obtained for different nanoparticle populations reflects differences in the colloidal stability but also inaccuracies due to the imprecision in the determination of ζ. Calculation of the Thickness of the Stern Layer. In a very recent paper, Brown et al.43 reported the possibility to determine the surface potential Φ0 of silica nanoparticles (Ludox SM) by measuring the Si 2p binding energy via XPS with microjets. They suggested modeling the specific capacitance of the SL with that of an ideal parallel-plate capacitance (with no charges between the plates):

accurate and unbiased, we will start our discussion with these data. Here we confirm the results of our previous publication,1 in which we concluded for Na+ as counterion that σζ is (within the selected parameter range, Ι = 40−120 mmol L−1) invariant with ionic strength, temperature, and particle diameter. If we restrict our considerations to this range, we find for the inner capillary surface with all counterions no measurable dependence of σζ on the ionic strength (the relative standard deviation of σζ is 0.7−2%; refer to Table 3). However, it is very obvious that the result for Ι = 20 mmol L−1 is significantly lower than that for the mean calculated within the range Ι = 40−120 mmol L−1. Very obviously, for a larger parameter range, σζ is increasing with Ι, which directly reflects the increase in the (total) surface charge density σ0 with increasing Ι (at fixed pH), which is a result of charge regulation in the EDL.22−24 The degree of dissociation of acidic groups attached to a surface is a function of the pH and the ionic strength. In our previous publication,1 we concluded that the comparison of the calculated σζ values shows that there is no significant difference between the results obtained for the nanoparticles investigated and the results obtained for the interface fused-silica capillary/buffer, although there is a significant difference for ζ at lower ionic strength (refer to Table 2). We regarded this result as an indirect confirmation of the validity of the modified analytic approximation presented by us in ref 8. We have now extended our considerations to a set of counterions differing in their limiting equivalent conductance. Also, in this case, we find only a small relative deviation between the results obtained for the nanoparticles investigated and the results obtained for the interface fusedsilica capillary/buffer (Figure 7 and Table 3). This result

Φ0 − ζ =

dSL σ0 εr,SLε0

(15)

This equation is equivalent to that derived for the planar limiting case (refer to eq 5). While the surface potential Φ0 of silica nanoparticles can be determined from spectroscopic data,43,67 σ0 has to be determined via potentiometric titration. In addition, the parameter ζ has to be determined from the results of electrokinetic measurements (and also σζ), which requires for nanoparticles to take the relaxation effect into consideration (see Figure 4) and to measure the electrophoretic mobility precisely in a medium of exactly known composition. It must be emphasized that the parameter εr,SL is the crucial parameter of the whole concept. Sverjensky54 derived for silica from correlation studies with data taken from the experimental results of several authors an approximate value of 43 for εr,SL. He applied a model that assumes the division of dSL into a distance parameter characteristic for each solid and a distance parameter corresponding to the size of the hydrated metal ion on the surface. Neither temperature nor ionic strength were taken into account. Principally, we would expect εr,SL < εr,bulk as the dielectric constant of matter formed by permanent dipoles is dependent on the electric field strength.68−71 In contrast to dSL, the parameter CSL can be obtained from quantities that can be determined with high accuracy from experimental data without the uncertainty about the value of εr,SL (refer to eqs 1−6). Hence, this parameter might be regarded to be more accurate and reliable than the parameter dSL, which is in our calculations inversely proportional to CSL, because εr,SL is assumed to be constant independent of the local electric field strength. Those data obtained for SNP7 (Ludox SM, T = 25 °C, pH 9−10, Ι = 50 or 100 mmol L−1) with different counterions are given in Table 4 (this work and data taken from ref 43 or ref 67). It is important to see that the order of σ0 is reversed (at fixed Ι and fixed pH) with regard to the order of σζ due to the reversed order of CSL (in accord with ref 45). Hence the quantity that is governing Hofmeister effects is primarily the specific capacitance of the Stern layer, as already outlined by Gmür et al.45 In accordance with the studies performed by Brown et al.43 (assuming εr,SL = 43), the thickness of the SL is directly correlated to the hydrated cation radius (at fixed ionic

Figure 7. Comparison of |σζ| calculated from the electroosmotic mobility (closed symbols) to |σζ| calculated from the electrophoretic mobility determined for SNP12/B2 (open symbols) dependent on concentration and type of counterion. For experimental conditions, refer to Figures 2 and S1.

corroborates our prediction that, in the case of a buffered solution with a weak electrolyte co-ion and a strong electrolyte counterion, the effective ionic drag coefficient (refer to eqs 8 and 9) should be approximated with the ionic drag coefficient of the counterion.8 However, for the nanoparticles, we find a significant increase in σζ with Ι over the complete parameter range that is not observed in the planar limiting case (refer to Figure 7 and Table 3). For all counterions, the mean relative deviation between the results obtained for the nanoparticles investigated and the 4448


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Table 4. Stokes Radius rS of Electrolyte Cation, Hydrated Cation Radius rhyd, Calculated Thickness of the Stern Layer dSL, Surface Potential Φ0, Electrokinetic Potential ζ, Surface Charge Density σ0, Number Surface Charge Density n0, Electrokinetic Charge Density σζ, Charge Density at the OHP σOHP, Number Charge Density at the OHP nOHP, Fraction of Charge Included in the SL (FOC(SL)), and Specific Capacitance of the Stern Layer CSL for SNP7 (Ludox SM) in an Electrolyte of Ι = 50 mmol L−1 or 100 mmol L−1 and pH 9−10 at T = 25 °C rS

rhyd

dSL

dSL

Φ0

ζ

σ0

n0

σζ

σOHP

nOHP

cation

(nm)

(nm)

(nm)

(nm)

(mV)

(mV)

(C m−2)

(nm−2)

(C m−2)

(C m−2)

(nm−2)

FOC(SL)

(F m−2)

50 mmol L−1 Li+ 50 mmol L−1 Na+ 50 mmol L−1 K+ 50 mmol L−1 Gdm+ 100 mmol L−1 Na+

0.238 0.184 0.125 0.179 0.184

0.382a 0.358a 0.331a (0.18)h 0.358a

0.80b 0.74b 0.60b h 0.62b

0.78c 0.71c 0.57c h 0.60c

−415d −385d −325d h −355d

−63.8e −60.4e −57.1e −50.7e −49.7e

−0.172f −0.173f −0.179f h −0.194

−1.07 −1.08 −1.12 h −1.21

−0.048g −0.045g −0.041g −0.036g −0.047g

0.124 0.128 0.138 h 0.147

0.77 0.80 0.86 h 0.92

72% 74% 77% h 76%

0.49 0.53 0.67 h 0.64

CSL

a

Data from ref 72. bData from refs 43 and 67. cThis work, calculated with eq 5. dData from refs 43 and 67, chloride, pH 10, calculated from Si 2p binding energy. eThis work, interpolated data for c = 50 mmol L−1, SNP7, pH 9.2, refer to Table 2. fData from refs 43 and 67, potentiometric titration, chloride, pH 10. gThis work, interpolated data for c = 50 mmol L−1, SNP7, pH 9.2, refer to Table 3. hNo data available (approximation rhyd = rS for Gdm+).

us to calculate dSL in this case. The drastic reduction of σζ with Gdm+ containing electrolytes (at fixed Ι, see Table 3) compared to that obtained with alkali ion containing electrolytes (although λo(K+) > λo(Gdm+) > λo(Li+) and consequently rS(K+) < rS(Gdm+) < rS(Li+), rS = Stokes radius, see Table 4) points to an adsorption of Gdm+ at the silica/electrolyte interface that cannot be described exclusively by nonspecific electrostatic interactions. Possibly, with Gdm+ containing electrolytes, water is largely expelled from the silica surface and the SL, which would be in accordance with the observation that Gdm+ is extremely weakly hydrated.46 Specific adsorption of unhydrated Gdm+ on an unhydrated silica surface (whereas the adsorption sites are the negatively charged deprotonated surface silanol groups, corresponding to the formation of localized ion pairs) would result in a small dSL and consequently a high specific capacitance of the Stern layer. If the “degree of hydration” of the surface (and/or of the adsorbed ions) is responsible for ion-specific effects with regard to the adsorption of cations on an oxidic surface (quantified as the thickness of the hydration layer on the silica surface and/or thickness of the hydration shell of the adsorbed ions), then Hofmeister effects for the adsorption of ions on a surface might be of the same nature as those postulated for ion−ion interactions (ion pair formation). This idea was already developed by Salis et al.14 on the basis of the Collins concept of matching water affinities.12,13 In addition, the important role of surface hydration in the understanding of Hofmeister effects with regard to the adsorption of cations on silica surfaces was experimentally investigated by Morag et al.73 via AFM measurements of the force acting between silica surfaces immersed in electrolytes of varied composition. They demonstrated that the reversal of the Hofmeister series observed for aqueous electrolytes of different pH can be explained by a change in the hydration state of the silica surface that is strongly dependent on the pH. Determination of the Parameter FOC(SL). Estimated charge densities (σ0 and σζ) at different distances from the solid/liquid interface permit the calculation of the fraction of charge (with respect to the total charge within the EDL, which is identical to σ0 multiplied with the surface of the particle) included in the volume enclosed by the two surfaces

strength). From the dimensions of the SL, Brown et al.43 concluded that the alkali ions investigated are adsorbed at the silica/electrolyte interface (that is covered with a monomolecular layer of water independent of the type of cation adsorbed) via nonspecific electrostatic interactions (the hydrated surface is assumed to be uniformly charged). Variation in the specific capacitance of the Stern layer is explained exclusively by variation in the size of the adsorbed hydrated cation. However, the comparison of the data in Table 4 also shows that dSL for Na+ is significantly reduced with increased ionic strength. Consequently, the interpretation of a monomolecular layer of water independent of the type of cation adsorbed (and possibly independent of the ionic strength) has to be modified. It must also be stated that the data taken by Brown et al.43 to quantify the hydrated radius of the cations deviate significantly from those published in more recent literature.72 The values resulting from eq 5 for the thickness dSL of the Stern layer are slightly lower than those obtained by Brown et al.,43,67 because of the higher value for ζ obtained in our studies. However, simultaneously these data also point to the binding of water within the SL outside of a monomolecular layer of water on the silica surface. If we compare the calculated value for dSL with the hydrated radius rhyd of the cation, we find that the difference dSL − rhyd (for Ι = 50 mmol L−1) is highest for Li+ (dSL − rhyd = 0.40 nm), lower for Na+ (dSL − rhyd = 0.36 nm), and lowest for K+ (dSL − rhyd = 0.24 nm). This order directly follows that deduced for the water-binding ability of the cation via viscosity measurements (resulting in the classification of structure-forming “kosmotropes” and structure-breaking “chaotropes”).10,12 Possibly, the type of cation governs the hydration state of the surface, which might also explain the observations made for Gdm+ (see below). Indirectly, Brown et al.67 have supported this theory, as they find a dependence of dSL on the ionic strength for silica nanoparticles immersed in an electrolyte containing NaCl at varied concentration (pH 10). According to these authors, the dependence of dSL on the concentration of NaCl can be understood as a compression of the SL when increasing Ι. Hence, these investigations support our result of a variable distance dSL − rhyd that cannot be rationalized by a monomolecular hydration layer of fixed thickness on the silica surface. As no data are available for the surface potential Φ0 with regard to Gdm+ containing electrolytes, it was not possible for

FOC(SL) = 4449

(

σ0 − σς 1 + σ0

dSL 2 a

)

(16) DOI: 10.1021/acs.jpcc.7b12525 J. Phys. Chem. C 2018, 122, 4437−4453

Article


that the net orientation and thickness of a compact layer of water near the charged surface are different for Na+ vs Cs+, while the dimensions of the diffuse layer (including oriented water) are independent of the type of cation, which indirectly confirms the applicability of the classical continuum theory (regarding ions as point charges) for the understanding and description of the GCL. Influence of Ionic Strength on EDL Structure. In accordance with the results of Brown et al.,67 we determined for all cations investigated (pH 9.2) σζ to be quasi-invariant with Ι within the range Ι = 40−120 mmol L−1 (refer to Table 3) and confirmed previous results obtained with Na+.1 In addition, there is a clear decrease in σζ, when Ι is decreased from 40 to 20 mmol L−1 (refer to Table 3, also reported by ref 67). This result means (within the classical theory of the GCL) that the observed decrease in ζ with increased Ι (at quasi-invariant σζ for the range Ι = 40−120 mmol L−1) is due to an increase in the diffuse layer specific capacitance resulting from a decrease in 1/κ (double layer compression), which directly follows from the exact solution of the PBE.31 Simultaneously, an increase in σ0 with increased Ι was reported by many workers14,15,18,19,24 and follows from charge regulation in the EDL.22,23 According to Brown et al.,67 this increase in σ0 with increased Ι (at reduced potential drop across the SL (=Φ0 − ζ)) is due to a compression of the Stern layer resulting in an increase in CSL quantified as a decrease in dSL (obviously accompanied by a change in the hydration state). We obtain for c(Na+) 100 mmol L−1 a length difference dSL − rhyd of 0.24 nm (Table 4), which would account for a fully hydrated silica surface and a purely nonspecific ion−solid interaction maintained at higher ionic strength (expected thickness of a monomolecular layer of water on a silica surface = 0.2−0.3 nm43,78). This result is in full accord with observations made by Lovering et al.76 for the structure of the Stern layer dependent on the type of adsorbed cation. They reported that the alkali ions Na+ and Li+ are unable to displace the hydration water on the silica surface even at very high ionic strength. Obviously, increasing Ι will increase σ0 due to an increase in the degree of dissociation.22−24 This increase in σ0 is accompanied by a reduced voltage drop across the SL (=Φ0 − ζ) due to a compression of the SL.67 Simultaneously, we observe only a very moderate change in σζ when increasing Ι from 20 to 120 mmol L−1, while ζ is considerably reduced. The observed decrease in ζ at quasi-invariant σζ (refer to Tables 2 and 3 and Figures 4 and 7) is exclusively due to a compression of the GCL. Simultaneously, increasing Ι (and σ0) is followed by an increase in σOHP and FOC(SL) (Table 4) that largely compensates the increase in σ0. This last observation fully explains the moderate impact of Ι on σζ within the parameter range investigated (Table 3), being consistent with the classical model of charge condensation.29,79,80

with FOC(SL) = fraction of charge contained in the Stern layer. For thin volumes, curvature effects can be neglected: FOC(SL) =

σ0 − σς σ0

(17)

Results for SNP7 in an electrolyte of Ι = 50 mmol L−1 and pH 9−10 at T = 25 °C (eq 17) are given in Table 4. It can be easily rationalized that the observed series in σζ (which might be regarded to be the quantity that is directly responsible for the observed Hofmeister effects) is the result of an increased FOC(SL) in the order Li+ < Na+ < K+. This order follows that of σ0 and reflects the increasing specific capacitance of the Stern layer. The advantage of the quantity FOC(SL) (with respect to C SL ) is its accessibility from data easily gained via potentiometric titration and electrokinetic measurements avoiding the problem of measuring the surface potential Φ0. The calculated FOC(SL) is dependent on the type of cation and increases with increasing ionic strength. Calculation of the Charge Density at the OHP. The charge density σOHP due to the arrangement of counterions at the outer Helmholtz plane at the boundary of the Stern layer (Figure 1) is different in sign to σ0 and σζ. It is accessible from the difference in σ0 and σζ under the assumption that xOHP is very close to xSP (refer to eq 2 and Figure 1): σOHP = − (σ0 − σς)

(18)

Due to the increasing FOC(SL), σOHP is increased (at fixed Ι) in the order Li+ < Na+ < K+ (refer to Table 4). We can also convert σ0 and σOHP into number charge densities (n0 and nOHP) by normalizing σ0 and σOHP on the elementary charge (refer to Table 4). This conversion confirms that the values for n0 are considerably lower than the number density of OH groups on the silica surface (4.6 ± 0.5 nm−2, obtained with 100 silica gels differing in their specific surface area, data evaluation via linear regression74). Taking a value of 0.9 for the maximum degree of coverage obtainable with circles on a plane results (for the given hydrated ion radii, regarding the hydrated ions as spheres densely packed onto an ideal plane, neglecting disorder and curvature effects) in a maximum number charge density nmax of 2.0 nm−2 for Li+, of 2.2 nm−2 for Na+, and of 2.6 nm−2 for K+. When comparing these values with nOHP, we find in all instances nOHP < nmax. Our results suggest that the adsorbed cation modifies the hydration state of the silica surface which results in a modification of CSL (and the FOC(SL)). This conclusion is in accord with observations made in spectroscopic and molecular modeling studies. It is known from time-resolved IR spectroscopy and recent computer simulations that hydration effects are important beyond the first hydration shell.11 From their experimental results on the ultrafast vibrational relaxation of water at silica/aqueous electrolyte interfaces (NaCl solutions with varied concentration), Eftekhari-Bafrooei and Borguet75 concluded that for c(NaCl) > 10 mmol L−1 there is an accumulation of cations near the silica surface with one or more layers of water between the silica and the cations. Lovering et al.76 observed that the structure of water in the SL depends on the type of cation. The cations Na+ and Li+ are reported to be unable to displace the hydration water on the silica surface under the conditions of their measurement (c = 12 mol L−1). Dewan et al.77 performed molecular dynamics simulations on aqueous alkali chloride solutions near a charged surface (modeling a silica/aqueous electrolyte interface). They found

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CONCLUSIONS The good agreement of results obtained for spherical and for planar geometry confirms for silica nanoparticles immersed in aqueous borate buffers varying in the type and concentration of the strong-electrolyte cation that the modified analytic approximation8 introduced by Ohshima6 adequately describes the mobility-dependent relaxation effect, which must not be neglected under the conditions of our measurements. We obtain reliable values for ζ and σζ by modeling the electrophoretic mobility as a function of the sphere radius 4450


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(4) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G Calculation of the Electrophoretic Mobility of a Spherical Colloid Particle. J. Colloid Interface Sci. 1966, 22, 78−99. (5) O’Brien, R. W.; White, L. R. Electrophoretic Mobility of a Spherical Colloidal Particle. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607−1626. (6) Ohshima, H. Approximate Analytic Expression for the Electrophoretic Mobility of a Spherical Colloidal Particle. J. Colloid Interface Sci. 2001, 239, 587−590. (7) Kimura, K.; Takashima, S.; Ohshima, H. Molecular Approach to the Surface Potential Estimate of Thiolate-Modified Gold Nanoparticles. J. Phys. Chem. B 2002, 106, 7260−7266. (8) Pyell, U.; Jalil, A. H.; Pfeiffer, C.; Pelaz, B.; Parak, W. J. Characterization of Gold Nanoparticles with Different Hydrophilic Coatings via Capillary Electrophoresis and Taylor Dispersion Analysis. Part I: Determination of the Zeta Potential Employing a Modified Analytic Approximation. J. Colloid Interface Sci. 2015, 450, 288−300. (9) Kunz, W.; Henle, J.; Ninham, B. W. ’Zur Lehre von der Wirkung der Salze’ (About the Science of the Effect of Salts): Franz Hofmeister’s Historical Papers. Curr. Opin. Colloid Interface Sci. 2004, 9, 19−37. (10) Jenkins, H. D. B.; Marcus, Y. Viscosity-B-Coefficients of Ions in Solution. Chem. Rev. 1995, 95, 2695−2724. (11) Paschek, D.; Ludwig, R. Specific Effects on Water Structure and Dynamics beyond the First Hydration Shell. Angew. Chem., Int. Ed. 2011, 50, 352−353. (12) Salis, A.; Ninham, B. W. Models and Mechanisms of Hofmeister effects in Electrolyte Solutions, and Colloid and Protein Systems Revisited. Chem. Soc. Rev. 2014, 43, 7358−7377. (13) Collins, K. D. Charge Density-Dependent Strength of Hydration and Biological Structure. Biophys. J. 1997, 72, 65−76. (14) Salis, A.; Parsons, D. F.; Borstrom, M.; Medda, L.; Barse, B.; Ninham, B. W.; Monduzzi, M. Ion Specific Surface Charge Density of SBA-15 Mesoporous Silica. Langmuir 2010, 26, 2484−2490. (15) Tadros, Th. F.; Lyklema, J. Adsorption of Potential-determining Ions at the Silica- aqueous Electrolyte Interface and the Role of Some Cations. J. Electroanal. Chem. Interfacial Electrochem. 1968, 17, 267− 275. (16) Abendroth, R. P. Behavior of a Pyrogenic Silica in Simple Electrolytes. J. Colloid Interface Sci. 1970, 34, 591−596. (17) Allen, L. H.; Matijevic, E. Stability of Colloidal Silica. II. Ion Exchange. J. Colloid Interface Sci. 1970, 33, 420−429. (18) Milonjic, S. K. Determination of Surface Ionization and Complexation Constants of Colloidal Silica/Electrolyte Interface. Colloids Surf. 1987, 23, 301−312. (19) Sonnefeld, J.; Gobel, A.; Vogelsberger, W. Surface Charge Density of Spherical Silica Particles in Aqueous Alkali Chloride Solutions. Part 1. Experimental Results. Colloid Polym. Sci. 1995, 273, 926−931. (20) Sonnefeld, J.; Lobbus, M.; Vogelsberger, W. Determination of Electric Double Layer Parameters for Spherical Silica Particles under Application of the Triple Layer Model Using Surface Charge Density Data and Results of Electrokinetic Sonic Amplitude Measurements. Colloids Surf., A 2001, 195, 215−225. (21) Dove, P. M.; Craven, C. M. Surface Charge Density on Silica in Alkali and Alkaline Earth Chloride Electrolyte Solutions. Geochim. Cosmochim. Acta 2005, 69, 4963−4970. (22) Trefalt, G.; Behrens, S. H.; Borkovec, M. Charge Regulation in the Electrical Double Layer: Ion Adsorption and Surface Interactions. Langmuir 2016, 32, 380−400. (23) Hiemstra, T.; de Wit, J. C. M.; van Riemsdijk, W. H. Multisite Proton Adsorption Modeling at the Solid/Solution Interface of (Hydr)oxides: A New Approach. J. Colloid Interface Sci. 1989, 133, 105−117. (24) Bolt, G. H. Determination of the Charge Density of Silica Sols. J. Phys. Chem. 1957, 61, 1166−1169. (25) Lyklema, J. Quest for Ion-ion Correlations in Electric Double Layers and Overcharging Phenomena. Adv. Colloid Interface Sci. 2009, 147−148, 205−213.

(approximated with the maximum of the solid particle radius distribution), the screening parameter κ, the electrokinetic potential ζ, and the dimensionless ionic drag coefficient mcounter (calculated from the limiting ionic conductance of the counterion), neglecting additional effects due to the displacement of ions within the Stern layer. With these data, Hofmeister effects can be rationalized to be due to differences in the electrokinetic charge density σζ induced by ion-specific effects. The model of the EDL depicted in Figure 1 permits (if Φ0, σ0, ζ, σζ, and εr,SL are known or can be approximated) the determination of the charge density at the OHP, the thickness dSL of the Stern layer, and the specific capacitance CSL of the Stern layer from electrokinetic, spectroscopic, and potentiometric titration data. This simple model explains (in accord with the results of Brown et al.43,67) differences in σζ as a result of differences in CSL (additionally quantified as differences in the FOC(SL)). With our data (for Li+, Na+, and K+), these differences cannot be fully explained by differences in the size of the hydrated cation. Obviously, there is an influence of the type of the adsorbed cation on the hydration state of the silica surface that has a large impact on observed Hofmeister effects and that might also explain the large decrease in σζ we have observed for Gdm+ vs K+ or Na+.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b12525. Superimposed electropherograms recorded for different nanoparticle populations under variation of type of counterion and ionic strength, μep calculated for a negatively charged sphere with electrolytes containing Li+, Na+, K+, or Gdm+, tables of measured μeo, μep, and calculated ζ potentials, and illustration of the iterative scheme employed for the determination of ζ (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Address: Department of Chemistry, University of Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany. E-mail: pyellu@staff.uni-marburg.de. Phone: +49 6421 2822192. ORCID

Ute Pyell: 0000-0002-5461-0707 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the Deutsche Forschungsgemeinschaft (DFG PY 6/11-1) is gratefully acknowledged. A.H.J. thanks the Iraqi Ministry of Higher Education and Scientific Research (MoHESR) for providing him with a Ph.D. scholarship (via University of Mosul, Iraq).

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REFERENCES

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Quantification of Zeta-Potential and Electrokinetic Surface Charge

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