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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 Hussein Jalil, and Ute Pyell J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b12525 • Publication Date (Web): 29 Jan 2018 Downloaded from http://pubs.acs.org on February 2, 2018

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The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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The Journal of Physical Chemistry

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, Ute Pyell* University of Marburg, Department of Chemistry, Marburg, Germany *Corresponding author: Ute Pyell, University of Marburg, Department of Chemistry, Hans-Meerwein-Strasse, D-35032 Marburg, Germany E-mail: [email protected], Telephone: +49 6421 2822192

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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 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 surface 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 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.



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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; (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 fixed type of anion the CCC becomes cation-specific and vice versa, enabling to arrange the cations or the anions, respectively, in a series according to the determined CCC. Interestingly, the same order of “influence strength” was found 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


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Hofmeister effects are described for ion-ion and for ion-surface interactions. 12 The “law of matching water affinities” (LMWA) based on hydration enthalpy differences

13

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 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 modelling, Hiemstra et al.

23

succeeded in modelling the surface charge density (experimentally determined in aqueous



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NaCl solution 24) 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 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) two phase 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 overcharging

25)

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 charge 26

or effective charge 27 of a particle that refers to the charge deduced from conductance


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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, (coions) and solvent molecules (denoted Stern layer, SL) and in part by a diffuse (outer) layer of mobile counterions, coions 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 outer Helmholtz plane (OHP, the plane defined by the centres 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-ion-specific 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



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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. be done numerically

30

1,8

or with high accuracy via analytic approximations.

This can 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 semi-quantitative 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 Matijevic

34

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 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 measurements

36

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+)


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> 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 to extend the area of investigation of X-ray photoelectron spectroscopy (XPS) 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 modelled Ludox HS particles dispersed in aqueous solution (based on

experimental data from 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 Allison

40

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 modelling.



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XPS with microjets further allowed 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 (BESi2p) is decreasing linearly with increasing hydrated ion radius resulting in the series BESi2p (Cs+) > BESi2p (K+) > BESi2p (Na+) > BESi2p (Li+). The BE shift ∆BESi2p directly corresponds to the change ∆ϕ0e (with e = elementary charge). As ϕ0 corresponds to 0 mV at the point of zero charge, BESi2p at the point of zero charge can be taken as the direct reference point. With this 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 to 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 Based on 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


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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 GCL 28). 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 1,8 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 experiments

46

revealed that Gdm+ is planar, of triangular shape, and

extremely weakly hydrated.



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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 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 by

43

from in situ photoelectron spectroscopic and potentiometric titration data) permit a characterisation 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.

THEORY


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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 modelled 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



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charge) into the excess charge QSL included in the SL and the excess charge QGCL included in the GCL 50:

QT = QSL + QGCL

(1)

If we normalize on the area of the charged surface, we obtain charge densities: σ 0 = ( σ 0 − σ ζ ) + σ ζ = − σ OHP + σ ζ

(2)

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: Ε SL =

(Φ0 − ζ )

(3)

dSL

where dSL = thickness of the SL. From the Coulomb law follows for the homogeneous field within the SL: Ε SL =

σ0 ε 0 ε r,SL

(4)


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where ε0 is the electric field constant and εr,SL is the dielectric constant within the SL. Simple rearrangement gives: dSL = ε 0 ε r,SL

(Φ0 − ζ ) σ0

(5)

If we compare this equation with the specific capacitance CPPC of an ideal plate-parallel capacitor (CPPC = ε0 εr 1/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:

CSL =

σ0 Φ0 − ζ

(6)

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

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



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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) or disodiumtetraborate decahydrate (p.a., Merck, Darmstadt, Germany) or dipotassium tetraborate tetrahydrate (puriss., Acros Organics, Geel, Belgium) in MilliQ water (18 MΩcm, Merck MilliPore, Darmstadt, Germany). Guanidinium borate buffer was prepared by neutralizing diguanidinium carbonate with boric acid 55:

+ 4 2 + 2 + 2 + ↑ +

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 MilliQ 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 MilliQ water to 100 mL. From this stock solution (c(Gdm+) = 80 mmol L-1) buffers with lower concentration of Gdm+ were prepared by dilution with MilliQ 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.


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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) 60 min, water 60 min, and electrolyte 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 counter ion 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 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 counter ion concentration of 10 mmol L-1.

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



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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 A 56 and data evaluation by moment analysis (determination of the (statistical) central moments

57)

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 3-5 repeated runs. The resulting electropherograms are shown in Figures S1-3 (supporting information). Typical electropherograms are shown in Figures 2a-d. Drift of the baseline, seen in several electropherograms, stems from a drift in the intensity of the D2-lamp employed for


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- 19 -

absorbance detection. This drift can be easily corrected and is not inferring 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: µ ep =

LD L T L L − D T U t mig U t0

(7)

with LD = capillary length to detector, LT = total capillary length, U = applied voltage, tmig = migration time of analyte zone, t0 = migration time of neutral marker. In all instances, 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 L1)

|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.



- 20 -

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| 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 ∆µep 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 kV1 (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 centre 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

Following

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 2 N A ∑ z i2 c i

(10)

ε r ε 0k T

where: zi = charge number (valence) of ith component, ci = molar concentration of ith component. The dimensionless ionic drag coefficients are dependent on the type of the counterion (and the temperature). We obtained 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 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



- 24 -

With the analytic approximation given in Equations (8) to (10), we calculated µep dependent on mcounter, κa, and ζ (refer to Figures 4, 5a-d, and S4). For all counterions there is a distinct minimum of the function |µep| = ƒ(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 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 = ƒ(ζ) (at fixed mcounter and κa). Figure 4 shows the lines representing µep = ƒ(κa) (ζ = -100 V) with varied mcounter (representing electrolytes containing Li+, Na+, K+, or Cs+ as counterion). 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 Smoluchowski equation might introduce an error of more than 50% decrease with regard to the correct value. Highest negative deviation is obtained with the counterion having the lowest limiting ion conductance, i.e. Li+.


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- 25 -

The calculated values can be compared to the experimental values obtained by capillary electrophoresis (refer to Tables S1a-r). Figures 5a-d show 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

8

that calculates ζ from the measured electrophoretic

mobility (refer to Tables S1a-r) via Equations (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 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:

µ eo = −

ε 0 εr ζ η

(11)

We regard this comparison as a possibility to confirm the validity of the employed approximate analytic approach, because nanoparticles and capillary 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



Page 26 of 65

- 26 -

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-dependant. Wang and Pilon

53

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 Saville 62,63 and by Mangelsdorf and White

64

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 to draw the conclusion 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


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- 27 -

to observe a dependence of ζ 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 Figures 5a-d, at this low ionic strength the estimation of ζ is associated with the highest imprecision due to the nonlinearity of the function µ = ƒ(ζ) under conditions of a strong impact of the relaxation effect. For Li+ as counterion, we even observe a distinct maximum of the function µ = ƒ(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 PBE 30 gives access to the electrokinetic charge 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 =

Qeff ε ε kT   eζ   eζ 4 = r 0 κ 2 sinh  tanh + 2 e 4πa  2 kT  κa  4 kT 

  

(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:       2εr ε0 κkT  eζ  1  2 + 1 sinh  σζ =   1+   ( κa)2 e κa  2 kT   2 eζ   cosh       4 kT   

   eζ  8 ln cosh  4 kT      eζ   sinh2     2 kT  


1/2

         

(13)


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- 28 -

The results of both equations include significant errors for κa < 0.5. As in all our cases κa >> 0.5 (refer to Figures 5a-d) 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: σς =

 eζ 2 εr ε0 κ k T sinh e  2kT

  eζ   = 8 n εr ε0 k T sinh    2k T 

(14)

where: n = number density of buffer ions. Equation (14) is the special case for a 1:1electrolyte. The resulting values for σζ (nanoparticles and planar limiting case) are given in Table 3. Results obtained via Equation (12) do not deviate from those obtained via Equation (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 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).


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- 29 -

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 series

9

(or lyotropic sequence

33)

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 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 (upwards) shift due to aggregation effects under conditions of higher ionic strength.



- 30 -

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 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 (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 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 fused-silica


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- 31 -

capillary/buffer (Figure 7 and Table 3). This result 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 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 to model the specific capacitance of the SL with that of an ideal parallel-plate capacitance (with no charges between the plates):



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- 32 -

Φ0 − ζ =

dSL σ0 εr,SL ε0

(15)

This equation is equivalent to that derived for the planar limiting case (refer to Equation (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. Sverjensky

54

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 Equations (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.


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- 33 -

Those data obtained for SNP7 (Ludox SM, T = 25 °C, pH = 9-10, Ι = 50 mmol L-1 or 100 mmol L-1) with different counterions are given in Table 4 (this work and data taken from 43 or 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 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 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 Equation (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



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- 34 -

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 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


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- 35 -

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 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 based on 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 distance 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:



Page 36 of 65

- 36 -

2

FOC(SL)

d   σ0 − σς  1 + SL  a   = σ0

(16)

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

σ0 − σς

(17)

σ0

Results for SNP7 in an electrolyte of Ι = 50 mmol L-1 and pH = 9-10 at T = 25 °C (Equation (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 CSL) 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 increasing 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 Equation (2) and Figure 1):


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- 37 -

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 regression 74). 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 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



- 38 -

measurement (c = 12 mol L-1). Dewan et al. 77 performed molecular dynamics simulations on aqueous alkali chloride solutions near a charged surface (modelling a silica/aqueous electrolyte interface). They found 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

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 workers 14,15,18,19,24

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 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


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account for a fully hydrated silica surface and a purely non-specific ion-solid interaction maintained at higher ionic strength (expected thickness of a monomolecular layer of water on a silica surface = 0.2-0.3 nm 43,78). 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. 2224

This increase in σ0 is accompanied with 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

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 approximation introduced by Ohshima

6

8

adequately describes the mobility-dependent relaxation effect,

which must not be neglected under the conditions of our measurements. We obtain reliable



- 40 -

values for ζ and σζ by modelling the electrophoretic mobility as a function of the sphere radius (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+.

ACKNOWLEDGEMENTS Financial support from the Deutsche Forschungsgemeinschaft (DFG PY 6/11-1) is gratefully acknowledged. AJ thanks the Iraqi Ministry of Higher Education and Scientific


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- 41 -

Research (MoHESR) for providing him with a PhD scholarship (via University of Mosul, Iraq).

SUPPORTING INFORMATION 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, illustration of the iterative scheme employed for the determination of ζ.



- 42 -

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FIGURE CAPTIONS Figure 1. Schematics of 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. Figure 2. Cumulated superpositions of electropherograms obtained for SNP22(B1) in 4-5 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 0.1 psi (6.89 mbar) 6 s, data rate 16 Hz, moving average 50 points, absorbance detection 214 nm. 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 2a-d and S1. 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). 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: (■)


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SNP7(B2), () SNP12(B2) and () SNP22(B2) (error bars representing standard deviation for five consecutive measurements, for experimental parameters refer to Tables 1 and S1a-p and Figures 2a-d and S1, * = single value). 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 2a-d and S1. 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 2a-d and S1.



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Figure 1. Schematics of 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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Absorbance/mAU

20

(a) SNP22

15

EOF

10

5

0 0

2

4

6

8

10

t/min

20

(b) Absorbance/mAU

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


15

SNP22

EOF 10

5

0 0

5

10

15

t/min


20


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14

(c)

Absorbance/mAU

12 10 8

EOF SNP22

6 4 2 0 0

5

10

15

20

25

30

t/min

14

(d)

12 Absorbance/mAU

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 54 of 65

10 SNP22

EOF 8 6 4 2 0 0

10

20

30

40

t/min Figure 2. Cumulated superpositions of electropherograms obtained for SNP22(B1) in 4-5 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 0.1 psi (6.89 mbar) 6 s, data rate 16 Hz, moving average 50 points, absorbance detection 214 nm.


Page 55 of 65

- 55 -

34

+

Li + Na + K + Gdm

30

-9

2

-1

-1

µep/(-10 m V s )

32

28

26

(a)

24 20

40

60

80

100

120

-1

c(cation)/(mmol L ) +

Li (µeo) + Na (µeo) + K (µeo) + Gdm (µeo) + Li (abs µep SNP12/B2) Na+ (abs µep SNP12/B2) + K (abs µep SNP12/B2) + Gdm (abs µep SNP12/B2)

80

(b) 70 60 50

-9

2

-1

-1

|µ|/(10 m V s )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40 30 20 20

40

60

80

100

120

-1

c(cation)/(mmol L )

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 2a-d and S1.



- 56 -

8

6

-8

2

-1 -1

µep/(-10 m V s )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 56 of 65

Li

+

+ Na + K + Cs

4

0

1

2

3

lg(κa)

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).


Page 57 of 65

- 57 -

- 100 mV - 95 mV - 90 mV - 85 mV - 80 mV - 75 mV - 70 mV - 65 mV - 60 mV - 55 mV - 50 mV - 45 mV - 40 mV - 35 mV - 30 mV

(a) 4 )

1 s 1 V 2 m 8 0 1 / p µe

(

3

2 0

5

10

15

20

25

κa


(b) 4 )

1 s 1 V 2 m 8 0 1 / p µe

3

(

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 0

5

10

15 κa


20

25


- 58 -

(c)


4 )

1 -s 1 V 2 m 8 0 1 / p µe

3

(

*

2 0

5

10

15

20

25

κa - 100 mV - 95 mV - 90 mV - 85 mV - 80 mV - 75 mV - 70 mV - 65 mV - 60 mV - 55 mV - 50 mV - 45 mV - 40 mV - 35 mV - 30 mV

(d) 4 )

1 -s 1 V 2 m 8 0 1 / p µe

3

(

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 58 of 65

*

2 0

5

10

15

20

25

κa 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 standard deviation for five consecutive measurements, for experimental parameters refer to Tables 1 and S1a-p and Figures 2a-d and S1, * = single value).


Page 59 of 65

- 59 -

+

µeo(Li ) + µeo(Na ) + µeo(K ) + µeo(Gdm ) + µep(Li ) + µep(Na ) + µep(K ) + µep(Gdm )

90 80

|ζ|/mV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


70 60 50 40 20

40

60

80

100

120

-1

c(cation)/(mmol L )

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 2a-d and S1.



- 60 -

50 40 -2

|σζ|/(mC m )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 60 of 65

+

µeo(Li ) + µeo(Na ) + µeo(K ) + µeo(Gdm ) + µep(Li ) + µep(Na ) + µep(K ) + µep(Gdm )

30 20 10 0 20

40

60

80

100

120

-1

c(cation)/(mmol L )

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 2a-d and S1.


Page 61 of 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


- 61 -

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+

c/

µep

µep

µep

µep

µep

µeo

(mmol L-1)

(SNP7/B2)/

(SNP12/B1)/

(SNP12/B2)/

(SNP22/B1)/

(SNP22/B2)/

(B2)/

(mm2 kV-1 s-1)

(mm2 kV-1 s-1)

(mm2 kV-1 s-1)

(mm2 kV-1 s-1)

(mm2 kV-1 s-1)

(mm2 kV-1 s-1)

20

-32.96 (±0.03)

-33.18 (±0.10)

-33.13 (±0.04)

-33.51 (±0.07)

-35.16 (±0.03)

69.05 (±0.06)b

40

-30.89 (±0.03)

-31.93 (±0.01)

-32.24 (±0.02)

-33.33 (±0.03)

-34.88 (±0.02)

58.73 (±0.05)b

60

-29.94 (±0.04)

-31.55 (±0.05)

-31.72 (±0.03)

-34.16 (±0.07)

-34.47 (±0.05)

52.21 (±0.04)b

80

-29.04 (±0.03)

-30.91 (±0.01)

-30.93 (±0.02)

-33.56 (±0.03)

-33.54 (±0.05)

47.66 (±0.13)b

100

-28.17 (±0.07)

-30.14 (±0.01)

-30.15 (±0.05)

-32.64 (±0.08)

-32.63 (±0.05)

44.36 (±0.05)b

120

-27.54 (±0.06)

-29.55 (±0.10)

-29.54 (±0.08)

-31.84 (±0.04)

-31.81 (±0.08)

41.53 (±0.04)b

20

-32.63 (±0.02)

-30.71 (±0.13)

-32.79 (±0.03)

-34.20 (±0.11)

-34.72 (±0.08)

63.98 (±0.04)c

40

-30.53 (±0.03)

-31.93 (±0.01)

-31.82 (±0.02)

-34.01 (±0.06)

-34.33 (±0.02)

53.67 (±0.38)c

60

-29.39 (±0.01)

-30.39 (±0.03)

-31.00 (±0.01)

-32.53 (±0.04)

-33.58 (±0.04)

47.58 (±0.06)c

80

-28.23 (±0.01)

-29.74 (±0.01)

-29.92 (±0.02)

-32.00 (±0.05)

-32.36 (±0.04)

42.78 (±0.04)c

100

-27.21 (±0.04)

-28.44 (±0.08)

-28.92 (±0.03)

-30.96 (±0.05)

-31.15 (±0.04)

39.03 (±0.05)c

120

----------a

-27.23 (±0.04)

----------a

----------a

----------a

36.36 (±0.04)c

20

-32.49 (±0.02)

-32.36 (±0.03)

-32.22 (±0.01)

-34.80 (±0.13)

-33.67 (±0.05)

59.87 (±0.02)d

40

-29.89 (±0.08)

-31.15 (±0.02)

-30.93 (±0.05)

-34.22 (±0.06)

-32.19 (±0.03)

51.86 (±0.13)d

60

-28.72 (±0.11)

----------a

-29.78 (±0.08)

-33.28 (±0.19)

-31.35 (±0.02)

45.35 (±0.08)d

80

-27.25 (±n.d.)

----------a

-28.79 (±0.19)

-32.53 (±0.01)

-30.46 (±0.21)

40.34 (±0.24)d

100

----------a

----------a

----------a

----------a

----------a

37.41 (±0.13)d

120

----------a

----------a

----------a

----------a

----------a

34.77 (±0.04)d

20

----------a

----------a

-30.35 (±0.02)

----------a

-32.04 (±0.02)

55.78 (±0.02)e

40

-27.47 (±0.03)

----------a

-28.71 (±0.05)

----------a

-30.72 (±0.11)

45.95 (±0.11)e

60

-25.50 (±0.18)

----------a

-26.90 (±0.15)

----------a

-28.70 (±0.19)

38.94 (±0.04)e

80

-23.90 (±0.05)

----------a

-25.36 (±0.30)

----------a

-26.88 (±n.d.)

34.46 (±0.15)e

a

not determined, b refer to Table S1b, c refer to Table S1g,

d

refer to Table S1l, e refer to Table S1q



Page 62 of 65

- 62 -

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 Smoluchowski equation.

c/(mmol L-1)

Cation

Li+

Na+

K+

Gdm+

a

SNP7/B2

SNP12/B1

SNP12/B2

SNP22/B1

SNP22/B2

Capillary

ζ/mV

ζ/mV

ζ/mV

ζ/mV

ζ/mV

ζ/mV

20

(-78)

(-83)

(-82)

(-67)

(-76)

-88.7

40

-66.6

-66.4

-67.7

-59.7

-64.2

-75.4

60

-61.0

-61.4

-62.0

-58.7

-59.4

-67.1

80

-56.6

-57.2

-57.3

-55.4

-55.3

-61.2

100

-52.9

-53.7

-53.8

-52.2

-52.2

-57.0

120

-50.4

-51.2

-51.2

-49.8

-49.8

-53.3

20

-72.3

-69.7

-71.8

-66.9

-68.8

-82.2

40

-63.0

-59.9

-63.4

-60.1

-60.9

-68.9

60

-57.7

-56.5

-58.1

-54.0

-56.3

-61.1

80

-53.2

-53.1

-53.6

-51.4

-52.2

-54.9

100

-49.7

-49.0

-50.1

-48.5

-48.8

-50.1

120

-------a

-45.6

-------a

-------a

-------a

-46.7

20

-69.1

-66.9

-66.5

-66.2

-62.8

-76.9

40

-59.4

-59.5

-58.9

-59.2

-54.7

-66.6

60

-54.7

-------a

-53.8

-54.7

-50.9

-58.2

80

-49.9

-------a

-50.1

-51.8

-48.0

-51.8

100

-------a

-------a

-------a

-------a

-------a

-48.0

120

-------a

-------a

-------a

-------a

-------a

-44.7

20

-------a

-------a

-62.2

-------a

-60.0

-71.6

40

-53.8

-------a

-54.2

-------a

-52.3

-59.0

60

-47.6

-------a

-47.9

-------a

-46.3

-50.0

80

-43.0

-------a

-43.5

-------a

-41.9

-44.3

not determined


Page 63 of 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


- 63 -

Table 3. Electrokinetic charge density σζ obtained via Eq. (7) for the nanoparticles investigated compared to that of capillary inner wall (Grahame equation). Cation

Li+

Na+

K+

Gdm+

c/ (mmol L-1)

SNP7/B2

SNP12/B1

SNP12/B2

SNP22/B1

SNP22/B2

Meanb

Capillary

σζ/(C m-2)

σζ/(C m-2)

σζ/(C m-2)

σζ/(C m-2)

σζ/(C m-2)

σζ/(C m-2)

σζ/(C m-2)

rel. dev.c /%

20

(-0.0443)

(-0.0455)

(-0.0446)

(-0.0311)

(-0.0371)

(-0.0396)

-0.0451

(-12.3)

40

-0.0468

-0.0444

-0.0457

-0.0363

-0.0401

-0.0416

-0.0482

-13.6

60

-0.0494

-0.0477

-0.0484

-0.0428

-0.0436

-0.0456

-0.0490

-6.9

80

-0.0506

-0.0494

-0.0495

-0.0454

-0.0453

-0.0474

-0.0495

-4.2

100

-0.0512

-0.0504

-0.0504

-0.0467

-0.0467

-0.0486

-0.0500

-2.9

120

-0.0521

-0.0515

-0.0515

-0.0480

-0.0479

-0.0497

-0.0501

-0.7

Meand

-------a

-0.0487

-0.0491

-0.0438

-0.0447

-0.0466

-0.0494

SDd

-------a

0.0028

0.0022

0.0046

0.0030

0.0032

0.0008

RSDe/%

-------a

5.6917

4.5152

10.5676

6.7980

6.7804

1.5862

20

-0.0395

-0.0352

-0.0368

-0.0309

-0.0322

-0.0338

-0.0393

-14.1

40

-0.0434

-0.0385

-0.0417

-0.0366

-0.0373

-0.0385

-0.0418

-7.8

60

-0.0459

-0.0427

-0.0443

-0.0383

-0.0405

-0.0415

-0.0428

-3.2

80

-0.0467

-0.0448

-0.0453

-0.0412

-0.0419

-0.0433

-0.0426

1.6

100

-0.0473

-0.0448

-0.0460

-0.0425

-0.0429

-0.0441

-0.0421

4.6

120

-------a

-0.0446

-------a

-------a

-------a

-0.0446

-0.0423

5.4

Meand

-------a

-0.0431

-0.0443

-0.0397

-0.0407

-0.0424

-0.0423

SDd

-------a

0.0027

0.0019

0.0027

0.0024

0.0025

0.0004

RSDe/%

-------a

6.2876

4.2502

6.7752

6.0040

5.8139

0.9363

20

-0.0371

-0.0333

-0.0329

-0.0354

-0.0283

-0.0325

-0.0352

40

-0.0401

-0.0382

-0.0377

-0.0304

-0.0322

-0.0346

-0.0396

-12.6

60

-0.0427

-------a

-0.0400

-0.0358

-0.0354

-0.0371

-0.0400

-7.3

80

-0.0432

-------a

-0.0416

-0.0390

-0.0377

-0.0394

-0.0394

0.1

100

-------a

-------a

-------a

-------a

-------a

-------a

-0.0399

-------a

120

-------a

-------a

-------a

-------a

-------a

-------a

-0.0399

-------a

Meand

-------a

-0.0382

-0.0398

-0.0351

-0.0351

-0.0370

-0.0398

SDd

-------a

-------a

0.0020

0.0043

0.0028

0.0024

0.0003

RSDe/%

-------a

-------a

4.9299

12.3954

7.8696

6.4907

0.6313

20

-------a

-------a

-0.0301

-------a

-0.0265

-0.0283

-0.0314

-9.9

40

-0.0353

-------a

-0.0338

-------a

-0.0304

-0.0321

-0.0333

-3.6

60

-0.0359

-------a

-0.0346

-------a

-0.0315

-0.0331

-0.0326

1.4

80

-0.0361

-------a

-0.0350

-------a

-0.0320

-0.0335

-0.0322

4.0

Meand

-------a

-------a

-0.0345

-------a

-0.0313

-0.0329

-0.0327

SDd

-------a

-------a

0.0006

-------a

0.0008

0.0007

0.0006

RSDe/%

-------a

-------a

-------a

1.7728 2.6151 2.1735 1.7027 not determined, b arithmetic mean of values determined for SNP12 and SNP22, c rel. dev. = relative deviation in percent ={[σζ(Capillary) – σζ(Mean)]/σζ(Capillary)} (-100), d without value for c = 20 mmol L-1, e RSD = relative standard deviation a


-5.6

0.2 -7.7

-6.8

0.6

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Page 64 of 65

- 64 -

Table 4. Stokes radius rS of electrolyte cation, hydrated cation radius rhyd, calculated thickness of the Stern layer dSL, surface potential Φ 0, 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.

Cation

Li+ 50 mmol L-1

Na+ 50 mmol L-1

K+ 50 mmol L-1

Gdm+ 50 mmol L-1

Na+ 100 mmol L-1

a

rS/

rhyd/

dSL/

dSL/

Φ0 /

ζ/

σ0/

n0/

σζ /

σOHP/

nOHP/

nm

nm

nm

nm

mV

mV

(C m-2)

(nm-2)

(C m-2)

(C m-2)

(nm-2)

0.238 0.382a

0.80b

0.78c

-415d

-63.8e

-0.172f

-1.07

-0.048g

0.124

0.77

72%

0.49

0.184 0.358a

0.74b

0.71c

-385d

-60.4e

-0.173f

-1.08

-0.045g

0.128

0.80

74%

0.53

0.125 0.331a

0.60b

0.57c

-325d

-57.1e

-0.179f

-1.12

-0.041g

0.138

0.86

77%

0.67

0.179 (0.18)h

-----h

-----h

-----h

-50.7e

-----h

-----h

-0.036g

-----h

-----h

-----h

-----h

0.184 0.358a

0.62b

0.60c

-355d

-49.7e

-0.194

-1.21

-0.047g

0.147

0.92

76%

0.64

FOC(SL)

CSL/ (F m-2)

data from 72, b data from 43,67, c this work, calculated with Equation (5), d data from 43,67, chloride, pH = 10, calculated from Si 2p binding energy, e

this work, interpolated data for c = 50 mmol L-1, SNP7, pH = 9.2, refer to Table 2, f data from 43,67, potentiometric titration, chloride, pH = 10, g this work, interpolated data for c = 50 mmol L-1, SNP7, pH = 9.2, refer to Table 3, h no data available (approximation rhyd = rS for Gdm+).


Page 65 of 65

- 65 -

TOC Graphic

8

µep/(-10-8m2V-1s-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Li+ Na+ K+ Cs+

6

ζ = -100 mV +

2

λ0(Li ) = 38.69 S cm mol

-1

+ 2 -1 λ0(Na ) = 50.11 S cm mol

4

+

2

-1

λ0(K ) = 73.50 S cm mol

+ 2 -1 λ0(Cs ) = 79.91 S cm mol

1

10

κa

100

1000


Quantification of Zeta-Potential and Electrokinetic ... - ACS Publications

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