Variable Charge and Electrical Double Layer of MineralâWater

Article pubs.acs.org/Langmuir

Variable Charge and Electrical Double Layer of Mineral−Water Interfaces: Silver Halides versus Metal (Hydr)Oxides Tjisse Hiemstra* Department of Soil Quality, Wageningen University, P.O. Box 47, NL 6700 AA Wageningen, The Netherlands S Supporting Information *

ABSTRACT: Classically, silver (Ag) halides have been used to understand thermodynamic principles of the charging process and the corresponding development of the electrical double layer (EDL). A mechanistic approach to the processes on the molecular level has not yet been carried out using advanced surface complexation modeling (SCM) as applied to metal (hydr)oxide interfaces. Ag halides and metal (hydr)oxides behave quite differently in some respect. The location of charge in the interface of Ag halides is not a priori obvious. For AgI(s), SCM indicates the separation of interfacial charge in which the smaller silver ions are apparently farther away from the surface than iodide. This charge separation can be understood from the surface structure of the relevant crystal faces. Charge separation with positive charge above the surface is due to monodentate surface complex formation of Ag+ ions binding to I sites located at the surface. Negative surface charge is due to the desorption of Ag+ ions out of the lattice. These processes can be described with the charge distribution (CD) model. The MO/DFT optimized geometry of the complex is used to estimate the value of the CD. SCM reveals the EDL structure of AgI(s), having two Stern layers in series. The inner Stern layer has a very low capacitance (C1 = 0.15 ± 0.01 F/m2) in comparison to that of metal (hydr)oxides, and this can be attributed to the strong orientation of the (primary) water molecules on the local electrostatic field of the Ag+ and I− ions of the surface (relative dielectric constant εr ≈ 6). Depending on the extent of water ordering, mineral surfaces may in principle develop a second Stern layer. The corresponding capacitance (C2) will depend on the degree of water ordering that may decrease in the series AgI (C2 = 0.57 F/m2), goethite (C2 = 0.74 F/m2), and rutile (C2 = ∞), as discussed. The charging principles of AgI minerals iodargyrite and miersite may also be applied to minerals with the same surface structure (e.g., sphalerite and würtzite (ZnS)).

1. INTRODUCTION

(SCM) that has been specially developed for metal (hydr)oxides. Metal (hydr)oxides and Ag halides react quite differently in some respects. Metal (hydr)oxide surfaces are much easier to charge. The low capacity of AgI for charge compared to metal (hydr)oxides has been explained qualitatively by Hiemstra and van Riemsdijk,5 suggesting that it is due to the presence of a layer of primary water of hydration bound to the Ag+ and I− ions on the mineral surface. This water is supposed to be strongly oriented in the local field of the surface ions, leading to electrical saturation of the medium and a correspondingly low dielectric constant and low electrostatic capacitance.6 This molecular picture is supported by recent MD simulations,7 using AgCl(s) as one of the simplest Ag halides. Recently, surface complexation models have been tested for an extensive data set of acid−base titration curves collected for goethite10 and rutile8 in various electrolyte solutions. The charging behavior of rutile (TiO2) has been analyzed by Ridley

One of the most eye-catching properties of colloids is the existence of surface charge created by excess ad(de)sorption of ions (Γi) at the surface that is compensated for by the charge of counter ions and co-ions in the nearby solution. An electrical double layer (EDL) develops that is experienced by colloid and interface chemists studying a variety of colloid and adsorption phenomena. In 1910 and 1913, Gouy1 and Chapman2 developed a theory that is known as the diffuse double layer (DDL) theory. It describes the diffuse distribution of counter ions and co-ions as a function of the distance from a planar charged surface. The model was extended by Stern3 to explain the relationship between the charge and potential of a mercury electrode. According to Stern, counter ions and co-ions remain at some minimum distance of approach from the surface because ions have a finite size. Conceptually, the surface charge is separated from the charge in the diffuse double layer by a Stern or Helmholtz layer, free of charge. Later, the model was refined by Grahame,4 who introduced chemisorption of electrolyte ions. These concepts are used in surface complexation modeling © 2012 American Chemical Society

Received: September 6, 2012 Revised: October 5, 2012 Published: October 11, 2012 15614

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et al.8 using an extensive high-quality data set of acid−base titrations in various electrolytes. The data for rutile are best described using the inner-sphere and outer-sphere complexation of electrolyte cations in combination with a basic Stern layer (BS) model comprising a DDL and a single Stern layer. For goethite, a data set was collected using acid−base titrations in different combinations of cations and anions (Li+, Na+, K+, Cs+, and chloride and nitrate solutions) that were scaled relative to the charge of a stock suspension for internal consistency.6,9 Interpretation with SCM showed that the data could be described well only if the BS model was extended with a second Stern layer, to be called an extended Stern layer (ES) model.10 This difference may point to another physical structure of the double layer that might be related to the alignment of interfacial water molecules. To improve our understanding of the EDL structure, it is of interest to analyze the charging and double-layer properties of other minerals such as AgI(s) using modeling along the same lines as has been done for metal (hydr)oxides. Classically, AgI as a model colloid has been considered from the thermodynamic point of view. In a recent effort, Sugimoto and coworkers11,12 have introduced some surface structural aspects into the thermodynamic approach, but a firm structural basis of the charging process at the mineral−solution interface is not yet available. A mechanistic surface structural approach is essential. The initial modeling, as described in section 2.1, resulted in an unexpected separation of the interfacial charge of the potentialdetermining ions of AgI(s) that cannot be understood without a structural picture of the mineral surface and its charging reactions. Therefore, the development of such a microscopic picture will be the objective of the present study in order to gain insight into the differences and similarities in the EDL of mineral−water interfaces in general. Silver iodide (AgI) is a classical model colloid that has been used to study adsorption phenomena and charge development in relation to the solution composition. Other Ag halogen minerals such as AgBr and AgCl11 may also develop variable charge. The solubility of AgI(s) is the lowest (log KAgI = −16.1 < log KAgBr = −12.3 < log KAgCl = −9.8). The charging of AgI (s) has been examined since the 30 ties of the last century.13 New interest in the behavior of Ag halides may develop because the use of Ag nanoparticles in consumer products is the most rapidly growing application of nanotechnology.14 Ag nanoparticles may transform into Ag halides in specific cases with high halide concentrations, in particular, Cl−. In addition, insight into the charging process of silver halides may be of interest in understanding the surface chemistry of other minerals such as sphalerite and würtzite (ZnS) that have a similar structures to miersite and iodargyrite (AgI), respectively. Because of the better solubility of Ag+ compared to that of halide ions, a neutral Ag halide crystal will be negatively charged when in contact with electrolyte. The negative particle charge can be diminished by the addition of extra Ag+ ions, which may readsorb depending on the solution concentration. At a certain Ag+ concentration, equal numbers of Ag+ and halide ions are present at the particle (i.e., the colloid becomes uncharged). Under this solution condition, the silver halide particle is in its point of zero excess adsorption or point of zero charge (PZC) in the presence of indifferent electrolyte ions. The starting point of our analysis will be an extensive data set for the charging of AgI(s)15,16 as a function of the silver/iodide concentration in solution and the ionic strength. As far as we are aware, these data have not yet been modeled in a successful

manner with a structure-based surface complexation model. The analysis will be done with the charge distribution (CD) approach17 in which the charge is allowed to be distributed at the interface.

2. RESULTS 2.1. Initial Modeling. It is not obvious where the interfacial charge of AgI is located. This is what is experienced when using surface complexation modeling (SCM) in the first approach. Therefore, the charge distribution (CD) concept has been applied to trace it. In the initial modeling, a neutral site S (3 nm−2) was defined that could adsorb Ag+ or I− ions. The charge of the constituting ions (Ag+ and I−) was allowed to be distributed in the compact part of the interface (i.e., the inner Stern layer). Additionally, electrolyte ions were allowed to adsorb in the 1 plane of the electrostatic model. Because the head end of the DDL does not necessarily coincide with the inner Helmholtz plane, a second Stern layer was defined, and the capacitance values of both Stern layers of the extended Stern (ES) model were fitted. The ES model with two Stern layers will reduce to the basic Stern (BS) model with one Stern layer if the capacitance of the second Stern layer (C2) is very high. Our initial modeling shows that the excess adsorption (ΓAg − ΓI) data of AgI(s)15 (Figure 1) can be understood only if

Figure 1. Excess adsorption, ΓAg − ΓI, expressed in mC/m2 as a function of the negative logarithm of the Ag+ activity in solution (pAg) for AgI(s) colloids at different KNO3 electrolyte concentrations (25 °C).15 The lines have been calculated with SCM using the parameters in Table 2. Note that 10 mC/m2 ≈ 0.1 μmol/m2.

negative charge is created on the surface and positive charge is created in the 1 plane. If the formation of negative charge is interpreted as I− adsorption (ΓI) and the formation of positive charge is interpreted as the adsorption of Ag+ ions (ΓAg), it is, without a precise surface structural picture, difficult to understand why the smaller Ag+ ion (r ≈ 66 pm) would reside farther from the surface than the larger I− ion (r ≈ 216 pm). The analysis of the surface structure of the AgI may elucidate this. 2.2. Surface Structure of AgI. Insight into the surface structure of AgI(s) requires information about the crystal structure and its morphology or habit. At room temperature, AgI(s) can be present in two polymorphs, (i.e., a crystal structure with cubic (γ-AgI) or hexagonal (β-AgI) packing).11 Cubic γ-AgI(s) can be found in 15615


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Table 1. Overview of the Structural Properties of Ag Halides component

β-AgI

γ-AgI

AgBr

AgCl

mineral name ion arrangement structure as CN lattice parameters crystal faces crystal shape

iodargyrite hexagonal würtzite α-ZnS 4 a = 0.495 nm c = 0.752 nm 100 edge, 001 planar prisms, rods

miersite cubic sphalerite β-ZnS zinc blende β-ZnS 4 a = 0.650 nm

bromargyrite fcc halite NaCl 6 a = 0.577 nm

chlorargyrite fcc halite NaCl 6 a = 0.555 nm

111, 110 dodecahedrons, tetrahedra

100 (111) cubes

100 (111) cubes

nature as the mineral miersite. The hexagonal β-AgI(s) mineral is known as iodargyrite. Both crystal structures differ from chlorargyrite AgCl(s) and bromargyrite AgBr(s). The latter minerals have a structure similar to that of halite NaCl(s) with a coordination number of CN = 6, whereas CN = 4 in AgI(s) (Table 1). The main morphologies of various macroscopic AgI minerals are shown in Figure 2. Iodargyrite will develop hexagonal

regular spherical shape with some large hexagonal crystals23 that could fit with the habits in Figure 2a,c. When miersite particles approach spherical morphology, the 110 face and its equivalents are dominant and the 111-type faces are minor (Figure 2c). In the case of equal distances for the opposing crystal faces, the calculated contribution of the 110-type faces is 74%. For hexagonal crystals of iodargyrite (Figure 2a), the contribution of the dominant edge faces is 67% at equal distances for all opposing surfaces. In further treatment, the focus will be only on the dominant crystal faces of both minerals (i.e., the 100 face of iodargyrite and the 110 face of miersite). For AgI(s), the surface structure of these faces is the same. There is only a slight difference in the site density. As will be shown, these faces can explain the charging behavior of AgI. The structure of the minor faces will be discussed in the Supporting Information (Figure S1). Figure 3 shows the mineral structure of iodargyrite in a crosssection with the surface structure of the 100 (edge) face on top.

Figure 2. Morphology of iodargyrite (β-AgI), a hexagonal prism (a) with 100 faces at the edge and 001/00-1 faces on the planar side. Morphology of miersite (γ-AgI) with (b) a tetrahedron terminated by the 111 face and its equivalents and (c) a dodecahedron with dominant 110 type of faces beside a contribution of 111 faces.

crystals (Figure 2a) with two main faces (i.e., the 100 face and its equivalents at the edges and the 001/00-1 face on the planar sides). For miersite, the mineral shape of macroscopic crystals can be a tetrahedron with a 111 face and its equivalents (Figure 2b). However, it may also develop predominantly 110-type faces, resulting in a dodecahedron (Figure 2c) approaching a sphere. Kolkmeijer and van Hengel18 found that silver iodide will precipitate from solution in the cubic form (γ-AgI) if Ag+ ions are present in excess and in hexagonal form (β-AgI) if I− ions are present in excess. The latter condition is typical for the preparation of a AgI sol,15 suggesting the presence of iodargyrite. However, miersite may coexist and may be stable19 or not20 with respect to iodargyrite. Moreover, the twinning of both minerals is possible because of the same surface structure of the planar (00-1/001) face of iodargyrite and the 111/-1-1-1 face of miersite. If such twinning occurs, then these faces may become less important in AgI(s) preparations that have both minerals. A AgI sol, freshly prepared in excess I− (pI 5, pAg 11) at 25 °C has spherical nanoparticles with a diameter of approximately d ≈ 10−20 nm.11 This is equivalent to a surface area of ∼100− 50 m2/g. Aging freshly formed sols and precipitates at 80 °C for several days leads to a much lower surface area, typically 1−5 m2/g,15,16,21,22 and the corresponding diameter of the particles is large (i.e., d ≈ 1000−200 nm). The particles have a rather

Figure 3. Cross section of AgI showing the mineral structure of iodargyrite, β-AgI(s) with internal rows of tetra-coordinated Ag+ (small spheres) and I− (large spheres) ions, and at the top the termination of the 100 face with triply coordinated surface groups that lack one bond, resulting in ≡Ag+0.25 and ≡I−0.25 sites (Ns = 2.90 nm−2 each). Note that one bond is not visible in this projection. The excess adsorption of Ag+ results in a positive surface. These adsorbed Ag+ ions are located above the 100 face, as indicated. Negative charge can be created by the desorption of Ag+ ions, which are removed from the 100 face as suggested from the interpretation of the charging data (Figure 1) with the CD model. The dominant 110 face of miersite has the same type of structure as the edge (100) face of iodargyrite but with a slightly higher site density (Ns = 3.37 nm−2).

In the interior of the mineral, all ions are 4-fold coordinated, but note that in the 2D representation in Figure 3 only three Ag−I bonds are visible. For equal charge distribution of Ag+ (z = +1) in the coordination sphere, the 4-fold coordination (CN = 4) results in a Pauling bond valence of ν ≡ z/CN = 0.25 vu. 15616


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On the surface, Ag+ and I− ions are present at lower coordination. For the 100 face in Figure 3, all ions on the surface are triply coordinated, although in the 2D representation only two Ag−I bonds are visible. The surface groups can be represented as ≡Ag+0.25 and ≡I−0.25. The net charge of the surface group has been calculated with charge bookkeeping using the sum of valences, defined as ∑ν − zion, in which ν is the bond valence and zion is the formal valence of the ion in the surface group (i.e., for ≡Ag, zion = +1 vu; for ≡I, zion = −1 vu). At the 100 face of iodargyrite (Figure 3), both types of surface groups (≡Ag+0.25 and ≡I−0.25) are present in equal numbers and the surface is neutral. The 100 face can be positively charged by adsorbing Ag+ ions. The Ag+ ion will protrude from the surface (Figure 3). If adsorbed ions can be considered to be an extension of the crystal structure,24 then the Ag+ ion will coordinate to one ≡I−0.25 group. When the Ag+ concentration in solution is lower than the value in the PZC, the surface will be negatively charged (negative Ag+ excess adsorption). In principle, negative charge can be created on the 100 face by the adsorption of I− ions that have to protrude from the surface. Another possibility is to desorb Ag+ ions from the surface. In that case, negative charge is created on the surface whereas in the case of I− adsorption the charge will be at approximately the same position where the Ag+ ions reside when bound in excess. If adsorbed Ag+ and I− ions both protrude from the surface, then no significant charge separation is expected. However, if the particle charge is due to only the adsorption and desorption of Ag+ ions, then positive charge is created above the surface and negative charge is created on the surface, respectively. 2.3. Surface Complexation Modeling. On the basis of the above analysis of the surface structure, our initial surface complexation modeling can be improved. Below, a consistent set of charging reactions will be defined using the ≡ I−0.25 and ≡ Ag+0.25 sites of the dominant faces as representative of AgI(s). For clarity, we will refer only to the 100 face of iodargyrite (βAgI(s)) because the dominant faces of both minerals have the same surface structure, as mentioned above. The site density is set at the mean value (Ns = 3.1 nm−2) of both dominant faces. In the modeling, the formation of aqueous AgIn1‑n (n = 0−3) complexes has been included because these may contribute to the ionic strength, and the activities of Ag+(aq) and I−(aq) have been linked by the AgI solubility product as documented in the Supporting Information (Table S1). 2.3.1. Reactions. The triply coordinated ≡Ag+0.25 and ≡I−0.25 sites of the 100 face of iodargyrite (Figure 3) may release or accept Ag+, respectively, leading to the definitions ≡Ag +0.25 ⇔ ≡

−0.75

+Ag +(aq)

KdAg

≡I−0.25 + Ag +(aq) ⇔ ≡ I−0.25 − Ag +1

Table 2. Reactivity of the 100 Face of Iodargyrite and the 110 Face of Miersite (Figure 3)a

K aI

1 2 3 4 5

Δz0

Δz1

−1 0 0 0 0

0 +1 −1 −1 +1

Δz2

log K

0 0 0 0 0

−4.87 ± 0.15 +4.02 ± 0.04 ≪b −1.02 ± 0.05 −2.16 ± 0.12

The charge distribution coefficients and formation constants for surface species at the AgI solid solution interface are given. The capacitance of the inner and outer Stern layers are respectively C1 = 0.16 ± 0.01 F/m2 and C2 = 0.47 ± 0.03 F/m2. The site densities of reference sites ≡I−0.25 and ≡Ag+0.25 are each 3.1 nm−2. R2 = 0.997 and n = 125 data points. bThe affinity is too low to be revealed by the model.

The charge of the adsorbed I− ion is allocated to the 1 plane (i.e., Δz1 = −1, Table 2). Note that if potential-determining ions are also located in the 1 plane then surface charge (σ0) and excess adsorption (Γ) are no longer identical. The collected data refer to excess adsorption Γ, and this is shown in the figures. The excess adsorption Γ has been expressed in mC/m2 rather than μmol/m2. To account for interaction with electrolyte ions, we may also define ≡Ag +0.25 + NO3−(aq) ⇔ ≡ Ag +0.25···NO3−

K NO3 (4)

−0.25

≡I

+

−0.25

+ K (aq) ⇔ ≡ I

···K

+

KK

(5)

The electrolyte ions will be located in the 1 plane, as given in Table 2 by Δz1. 2.3.2. Modeling Results. Using the surface structure of Figure 3 as representative of AgI(s), the data of Bijsterbosch and Lyklema15 (Figure 1) can be described very well (R2 = 0.997). The affinity constants are given in Table 2. In the modeling, I− adsorption as a monodentate complex was allowed (eq 3), but no binding of I− to the ≡Ag+0.25 surface site was found. This suggests that the affinity of I− ions for ≡Ag+0.25 is relatively low, which is in line with a recent MD simulation for the AgCl(s)/0.05 M KCl solution interface.7 Zarzycki and Rosso7 found no chloride innersphere complexation with the AgCl surface despite the very high concentration of Cl− ions in solution. The absence of the complexation of I− ions with ≡Ag+0.25 surface sites in our modeling (eq 3) suggests that the stability of triply coordinated ≡Ag+0.25 is sufficiently high in the interface and no coordination with an I− ion is required. In contrast to I−, Ag+ may form a monodentate surface complex (eq 2). The binding constant is log K = 4 (Table 2). This interaction with a single ≡I−0.25 group is weaker than the binding of silver ions to three triply coordinated I− groups (≡I−0.25), forming ≡Ag+0.25. The affinity of the latter can be estimated from the reverse of the Ag+ desorption reaction (eq 1). The constant for Ag+ binding in a tridentate complex is about 1 log K unit higher than for the monodentate complex formation of Ag+. A relatively high Ag+ concentration is required to obtain zero excess Ag adsorption with a corresponding charge. A relatively high Ag+ concentration suggests a higher solubility of Ag+ ions compared to that of I− ions, and hence it points to a higher stability of I− in the lattice. For this reason, no desorption

(2)

Upon desorption of Ag+ (eq 1), three Ag−I bonds on the surface are broken (Figure 3), leading to a defect with three ≡I−0.25 groups that are represented by ≡−0.75. The charge created by desorption is allocated to the surface plane (i.e., Δz0 = −1, Table 2). The charge of the adsorbed Ag+ ion (eq 2) is placed in the 1 plane in a first approach (Δz1 = +1, Table 2). In principle, the ≡Ag+0.25 site may also bind an I− ion. Therefore, this has been allowed in the modeling by defining ≡Ag +0.25 + I−(aq) ⇔ ≡ Ag +0.25 − I−

equation

a

(1)

K aAg

species ≡−0.75 ≡I−Ag+ ≡Ag−I− ≡Ag···NO3− ≡I···K+

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reaction for I− in the lattice has been defined. Moreover, the introduction of the surface desorption reaction of I− did not improve the quality of the fit, but we note that such a reaction can also not be excluded on the basis of the data. 2.3.3. Surface Speciation. The above parametrization can be used to calculate the surface speciation. In Figure 4, the Ag+ adsorption and desorption is given as a function of the concentration in solution (pAg) for 10−4−1 M KNO3 systems.

suggested by mass titrations. This observation apparently contrasts with the work of Lyklema and Golub22 measuring the excess Ag+ adsorption in the pH range of 4−7.3. The data in this study do not show the pH dependency of the excess adsorption (ΓAg − ΓI) and PZC (Figure S3). The absence of a pH dependency implies that the electrostatic potentials corresponding to excess Ag+ adsorption in the inner compact part of the EDL (ψ0 and ψ1) are not seriously affected by the pH. This is supported by modeling. Hence, any pH dependency of the IEP of AgI(s) must then be related to the acid−base properties of adsorbed water that is present beyond the inner compact part of the EDL. The development of a corresponding theoretical framework for the pH dependency of the electrokinetics is beyond the scope of this article. 2.3.5. Charge Distribution. In the above analysis, a simple approach was followed in the sense that the charge of adsorbed Ag+ (z = 1) in an ≡I−Ag innersphere complex has been fully attributed to the same electrostatic plane where the weakly bound K+ ions also reside. However, in the CD model as used for ion adsorption by metal (hydr)oxides, the charge of the innersphere complexes is distributed in the compact part of the EDL between the surface and the Stern plane and is not fully located in the 1 plane in the case of innersphere complexation.8 The question arises as to whether this can also be done for Ag+ as the innersphere complex at the AgI−water interface. Conceptually, the CD value is related to the geometry of the surface complex, defined in terms of types of ligands and corresponding bond lengths with the metal ion. The geometry can be approached with molecular orbital (MO) calculations. 2.3.5.1. Geometry of the Ag Surface Complex. In solution, solvated silver ions are coordinated to at least four water molecules (CN = 4)27.28 The reported average Ag−O distance varies between about 232−242 pm and depends on the method applied.27 Recently, a unifying interpretation28 of EXAFS and LAXS data suggested that the structure might be asymmetric, having two different Ag−O bond lengths. Using two short (232 pm) and two significantly longer Ag−O bonds (248 pm) describes both data sets well.28 The given distances are equivalent, with bond valences of about 0.3 and 0.2 vu, respectively. The presence of two Ag−O distances is also found for hydrated Ag+ in some clusters optimized with MO/DFT calculations. 29 The suggested asymmetry can be best reproduced using in the calculations the EDF2 functional, as shown in the Supporting Information (Table S3). To estimate the charge distribution of the Ag+ ion bound by ≡I−0.25 as the monodentate innersphere surface complex, the geometry of the surface complex has been optimized with MO/ DFT/EDF2. To mimic the surface, an I4Ag2−AgI cluster (charge = −1 vu) has been defined that represents the mineral and surface structure of the 100 face of iodargyrite (Figure 5). The positions of these ions were fixed. On top of the cluster, an additional Ag+ ion was placed and linked to the ≡I ion. To complete the tetrahedral coordination sphere, −OH2 was attached to both interface Ag+ ions and nine water molecules were added as secondary water (Figure 5). The positions of the adsorbed Ag+ ion and the water molecules were optimized. The modeling results of Figure 5 are summarized in Table 3. 2.3.5.2. Charge Distribution (CD) Values. The geometries derived can be used for a bond valence analysis. The actual bond valence s can be calculated31,32 with

Figure 4. Positive adsorption of Ag+ as a monodentate innersphere complex as a function of the negative logarithm of the Ag+ activity in solution (pAg) for five different background electrolyte concentrations of KNO3. Negative adsorption and charge are created by the desorption of Ag+ from the surface. The lines have been calculated with the parameters in Table 3. At an Ag+ adsorption of ±0.5 μmol/ m2, about 10% of each type of site has accepted or released an Ag+ ion if each type of group has a site density (Ns) of 5.15 μmol/m2 (3.1 nm−2). This indicates that only a few percent of the sites have changed.

In the PZC, a considerable amount of positive charge is present above the surface in the electrostatic 1 plane because of monodentate complex formation of Ag+ with ≡I−0.25 (Figure 4). This implies that at the PZC the surface is correspondingly quite negatively charged. This is rather unusual and very different from what is assumed in classical models.16,25 It implies that in the PZC, the potential of the surface will be negative, whereas usually a zero potential is assumed for the PZC. In addition, the Ag+ ion can be considered to be the single potential-determining ion in the present charging mechanism of the AgI(s) surface. The charge of Ag+ adsorbed above the surface in the 1 plane contributes to particle charge over a large range of concentration (Figure 4). When the net charge of the particles (0 plane plus 1 plane) is positive, increasing the ionic strength will increase the adsorption. The reason is the better screening of the positive field by the electrolyte ions. The positive potential in the 1 plane, which is repulsive for adsorbed Ag+ ions, will decrease, allowing more Ag+ adsorption. Above the PZC, the opposite will occur. At a high pAg (i.e., very low Ag+ concentrations), there is no significant adsorption of Ag+ at the 1 plane (Figure 4). This mechanism will make the overall charging strongly asymmetric, especially at high ionic strength. 2.3.4. Excess Adsorption and IEP. Recently, it has been suggested that the IEP of Ag halide particles is affected by the pH.26 Only in an acidic environment (pH 3) do IEP and PZC coincide, which has been attributed to the dissociation of adsorbed water, releasing protons into the solution, as

s = e−(R − R 0)/ B 15618

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modeling, we will discuss first the possibility of AgNO30 ion pair formation on the surface. 2.3.5.3. AgNO30 Ion Pair Formation. As found with our modeling (Table 2), the log K value for the ion pair formation of K+ with ≡I−0.25 on the surface is very small (log K ≈ −2.2) whereas the fitted log K for NO3− ion pair formation is much larger (log K ≈ −1.0). This might be an indication that part of the NO3− interaction is due to the formation of an adsorbed AgNO3 ion pair. Ion pair formation is also found in solution for Ag+(aq) + NO3−(aq) ⇔ AgNO30(aq) with log KAgNO3 = −0.29. On the surface, the interaction of nitrate with adsorbed Ag+ ions can be defined as ≡I−0.25 + Ag +(aq) + NO−3 (aq) ⇔ ≡ I−0.25AgNO30 K AgNO

(7)

3

The formation of this complex will be discussed in the next paragraph. 2.3.5.4. Final CD Modeling. Table 4 contains information about the final modeling in which the CD approach is applied

Figure 5. Structure of a hydrated I4Ag2−AgI−Ag cluster (z = −1) optimized with MO/DFT/EDF2/6-31+G** with pseudopotentials for Ag and I. The ions of the mineral template are fixed (triangles). The surface groups are represented by triply coordinated (a) Ag+and (c) I− ions. The hydrated Ag+ ion on top (a) is attached to the triply coordinated surface group ≡ I−0.25(c). The silver ions (a and b) are 4fold coordinated, primarily interacting with three and one −OH2 ligand, respectively.

Table 4. Reactivity of AgI Represented by the 100 Face of Iodargyrite and the 110 Face of Miersite (Figure 3)a

Table 3. Bond Distancesa in the Coordination Sphere of a Singly Coordinated Ag+ Ion of a Hydrated AgI Cluster Calculated with MO/DFT Using the EDF2 Functional and 6-31+G** Basis Set with LACVP Pseudopotentials for Ag and I bond

bond length (pm) b

Ag(a)−O3 I(c)−Ag(a)c parameter R0 n0 n1

231, 234, 242 286 value 188.8 0.26g 0.74g

species

equation

Δz0

Δz1

Δz2

log K

≡−0.75 ≡I−Ag+ ≡I−AgNO3 ≡Ag···NO3− ≡I···K+

1 2 7 4 5

−1 0.26 0.26 0 0

0 +0.74 −0.26 −1 +1

0 0 0 0 0

−4.79 ± 0.20 +3.64 ± 0.04 +3.27 ± 0.04 −1.52 ± 0.14 −2.06 ± 0.17

a

The charge distribution coefficients and formation constants for surface species at the AgI solid solution interface are given. The capacitance of the inner and outer Stern layers are respectively C1 = 0.15 ± 0.01 and C2 = 0.57 ± 0.04 F/m2. The site densities of ≡I−0.25 and −≡Ag+0.25 are each 3.1 nm−2. R2 = 0.998 and n = 125 data points.

reference value 232−248d 278−282e reference value

using the above-derived CD value and allowing the formation of an adsorbed AgNO3 ion pair. The ion pair formation equation (eq 7) has a fitted log K value (3.27) that is slightly lower than for the primary Ag+ binding to ≡I−0.25 (3.64). From the difference, the affinity of NO3− for adsorbed Ag+ can be derived, resulting in log K = −0.37 ± 0.06. This value is very similar to the ion pair formation reaction in solution (log K = −0.29). With the introduction of AgNO3 ion pair formation on the surface, the ion pair formation of NO3− with the ≡Ag+0.25 surface group decreases significantly in the CD approach from log K = −0.71 to −1.53 ± 0.14. The latter value is more on the order found for the ion pair formation of K+ with ≡I−0.25 (log K = −2.06 ± 0.17) and F− ions with ≡Ag+0.25 (log K = −2.40 ± 0.17). The latter value has been found by fitting the charging of AgI(s) in KF solutions using the data of Lyklema and Overbeek16 (Figure S4, Table S4). The introduction of charge distribution and the interaction of nitrate with adsorbed silver does not improve the (already high) quality of the fit, but it may rationalize the fitting parameters. The capacitances of the Stern layers found with the CD approach are C1 = 0.15 ± 0.01 F/m2 and C2 = 0.57 ± 0.04 F/ m2. The capacitance values are very similar to the values found with the point charge approach (Table 3), implying that the obtained capacitance values are not strongly model-dependent. 2.3.6. Reactivity of AgCl(s) and AgBr(s). As given in Table 1, the mineral structure of AgCl(s) and AgBr(s) differs from that of AgI(s). The coordination number is larger (CN = 6), in line

184f 0.25h 0.75h

a In picometers. bAg(a)−O3 refer to Ag−O bonds of H2O with Ag+ as defined in Figure 5. cI(c)−Ag(a) refers to the surface Ag−I bond as defined in Figure 5. dIn solution with CN = 4.28 eEquivalent bonds in solid iodargyrite. fReference length R0 (eq 6) in picometers for Ag−O bonds in minerals.30 gCalculated ionic charge distribution coefficients for the 0 and 1 planes. hIonic charge distribution in the case of a Pauling distribution.

in which R0 is an element-specific reference length and B is a constant (usually B = 37 pm). The application of eq 6 using the data in Table 3 leads to the ionic bond valence distribution of the adsorbed Ag+ ion, attributing Δn0 = +0.26 vu to the surface group and Δn1 = +0.74 vu to the water ligands. The correction for any changes in the dipole orientation of adsorbed water6 is negligible for this complex, leading to Δz0 = +0.26 and Δz1 = +0.76 vu as estimated CD values. Although two types of ligands (−I− and −OH2) are involved, the charge on Ag+ is almost equally distributed in the coordination sphere (i.e., close to a Pauling distribution). Before applying the CD values in the final 15619


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with a smaller relative size of these halide ligands. The coordination number CN = 6 leads to a Pauling bond valence of 0.167 vu. Two main crystal faces can be found for NaCl-like structures (i.e., the 100 face and its equivalents and the 111 face and its equivalents). For chlorargyrite (AgCl), the usual habit is a cube with 100 faces. Termination on the 100 faces of both minerals gives surface groups with a coordination number of CN = 5, and the groups can be defined as ≡Ag+0.167 and ≡Cl−0.167. Both groups are present in equal amounts, and the surface is neutral. Model results of Sugimoto and Shiba12 suggest for all silver halides (AgCl, AgBr, and AgI) asymmetry in ion adsorption similar to that given in Figures 1 and 4. To our knowledge, no direct experimental charging data for colloidal AgCl(s) or AgBr(s) are available that allow us to reveal by analogy the above-defined AgI charging mechanism. However, some important information comes from recent MD simulations showing that the Cl− ion will not bind to ≡Ag+0.167 on the surface,7 despite the presence of a high Cl− concentration. In the MD simulation, a stoichiometric AgCl crystal (2.5 × 2.5 × 2.5 nm3 with 100 faces) was built in contact with a 0.05 M KCl solution (1 KCl and 1046 H2O molecules). The 0.05 M chloride concentration is far above the point of zero excess adsorption of AgCl (PZC ≈ 5.2); therefore, the surface should be negatively charged. If the Cl− ion in the simulation box on average would bind only at the surface, then the corresponding charge would be −25 mC/ m2, which is a reasonable number (Figure 1). However, the MD simulation shows that Cl− does not adsorb as an innersphere complex, and no negative surface charge is created. How can this discrepancy be understood? The formation of negative charge on the AgCl 100 face at a high Cl− concentration can be due to the desorption of Ag+ (i.e., the same charging mechanism for AgCl(s) as suggested above for AgI(s)). The formation of negative charge by this mechanism starting with a neutral AgCl box is not easily simulated with MD because it requires the release of one Ag+ ion that cannot be present in a small solution box because it would create a very high Ag+ concentration, whereas the equilibrium concentration above the PZC needs to be ≪10−5 M. It illustrates the present limitation of the MD approach for pCl < PZC.

ψ0 =

⎫ θA 2.3RT 2.3RT ⎧ ⎨log log(A+) + + log KA ⎬ F F ⎩ 1 − θA ⎭ (10)

Introducing the PZC with ψ0 = 0 at concentration A0 and a relative loading θA,0 leads to ψ0 =

2.3RT 2.3RT {log(A+) − log(A +0 )} + F F ⎧ ⎫ θA /θA,0 ⎨log ⎬ (1 − θA )/(1 − θA,0) ⎭ ⎩ ⎪

⎪

⎪

⎪

(11)

If the number of sites were infinite, then the mixing entropy θA/(1 − θA) would not vary with loading, being equal to the value in the PZC, θA,0/(1 − θA,o). Then, the last term in eq 11 is zero, resulting in Nernst's law, which relates the surface potential ψ0 to the logarithm of the activity (A+) in solution with a slope of 59.1 mV at 25 °C. The relative deviation from Nernst's law can be expressed in a factor α, defined in ψ = α2.3RT/F log(A+).34 The factor α will depend on the ratio θA/(1 − θA). Because it is extremely difficult to measure subtle differences as a function of loading, the mean slope (α2.3RT/F) will be evaluated as done in the literature.35 3.1.2. Data. 3.1.2.1. Suspensions. The above expression is valid for ion binding to one type of site at the same electrostatic potential ψ. In our model, the ions are present at different sites and electrostatic positions. For this reason, Nernstian behavior will be assessed numerically by making full speciation calculations. According to our modeling, a zero surface potential (ψ0) is reached at pAg = 3.4 ± 0.1 (Figure S2). This point is two units lower than the PZC (pAg 5.45 ± 0.05), given in Figure 4. The AgI(s) suspension behaves nearly Nernstian (as follows from Figure 6). The deviation from Nernst's law varies with electrolyte concentration. The change in the electrostatic surface potential with the logarithm of the Ag+ activity decreases from 58.7 to 56.9 mV/pAg in the range of I = 10−4−1 M. AgI electrodes made from microcrystalline AgI behave as almost Nernstian (59.1 mV/pAg at 25 °C).16,25

3. ELECTRICAL DOUBLE LAYER 3.1. Nernst's Law. 3.1.1. Theory. Nernst's law is not obeyed by mineral surfaces with a limited number of sites for binding the potential-determining ions. This is due to the surface mixing entropy.33 The principle of non-Nernstian behavior can be illustrated for a single charging reaction with the adsorption of potential-determining ion A+ on surface site Sn with charge n: Sn + A+ ⇔ SAn + 1

KA

(8)

with KA =

θA (A+)e−Fψ0/ RT 1 − θA

(9) Figure 6. Mean fractional deviation from Nernst's law (solid lines) for AgI(s) as a function of the ionic strength I in comparison to very recent experimental data collected for single-crystal electrodes of AgBr(s)35 (spheres) and AgCl(s) (triangles25 and squares34). The behavior of classical electrodes with microcrystalline AgI(s) is almost Nernstian.25

in which θA is the mole fraction of sites occupied by A+ (i.e., [SAn+1]/[ST] and [ST] = [Sn] + [SAn+1]). The factor θA/(1 − θA) expresses the ratio between sites filled with A+ and empty sites. Rewriting eq 9 relates the surface potential ψ0 and the concentration of the potential-determining ion: 15620


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original paper on the interpretation of the capacitance of AgI,5 this can be explained by the strong orientation of the water molecules. This water can be considered to be the primary hydration water of the Ag+ and I− ions on the surface at which the water dipole orients. The capacitance C of a macroscopic capacitor is related to the plate distance d according to

The difference can be due to, for instance, a difference in log K. A decrease in Δlog K by 1 unit leads to a significant switch to Nernstian behavior, as follows from model simulations. 3.1.2.2. Single-Crystal Electrodes. For single-crystal electrodes made from a large AgBr(s)35 or AgCl(s)25,34 crystal, the deviation from Nernst's law is larger. The larger deviations may point to a larger difference in the binding constants (Δlog K), which might be related to another type of mineral structure (Table 1). Our model for AgI(s) suggests a decrease in the Nernstian slope at increasing ionic strength. The data for single crystals of AgBr(s) and AgCl(s) show the opposite trend (Figure 6). Preocanin et al.34 have described this salt dependency with a site-binding model by varying the inner capacitance with the electrolyte level. At low ionic strength, the capacitance is set rather high, stimulating charging. At higher ionic strength, its value is strongly decreased, leading to less charging. This approach might be inconsistent with the charging behavior, but the model has not been verified because of the lack of charging data. 3.2. Capacitances. 3.2.1. EDL Structure. The physical structure of the EDL in terms of the alignment of interfacial water has been characterized with spectroscopy, molecular modeling, and other measurements for a considerable number of aqueous interfaces. Water alignment may differ for materials. In a classical experiment, Pashley and Israelachvili36 have measured the force−distance relationship for two mica plates that are pressed together. At large distances, the repulsive force increases gradually, as is expected for an overlapping diffuse double layer. Once the plates are a small distance apart, the force starts to oscillate. The water molecules and counterions move in steps because of the increased ordering in the neighborhood of the surface, acting as a hard wall. The layer thickness of water ordering on muscovite is equivalent to about three water molecules.6,36 The first two water layers have a thickness of about 0.37 nm, as found by recent atomic force microscopy.37 Beyond the first two water layers, the structuring decays.36 This interpretation agrees with X-ray reflectivity data for muscovite.38 Experimentally, significant water ordering has also been found on a variety of other surfaces such as a charged metallic Ag electrode (111 face)39 and the 021 faces of α-Al2O3 (corundum) and α-Fe2O3 (hematite).40−42 The presence of about three layers of physisorbed water (∼0.8 nm) is also found for a crystal face of goethite (α-FeOOH) with MD simulations43 at very high ionic strength (1.2 M). This physisorbed water was strongly structured in layers, and the diffusion coefficient of the water molecules was strongly reduced. Beyond this distance, only some weak features exist. For other minerals, much less ordering of interfacial water is found. For instance, for the 100 face of quartz (SiO2)38 and the 110 faces of rutile (TiO2)44,45 and cassiterite (SnO2),46 only one distinct layer of water molecules is observed with possibly some weak ordering in a second layer. This difference in ordering may lead to a different structure of the compact part of the EDL. Weak ordering supports the use of a single Stern layer in the SCM (basic Stern model), whereas an extended Stern layer approach with two Stern layers might be appropriate for surfaces with a strong ordering of water. 3.2.2. Inner Stern Layer. The capacitance values that have been derived in our modeling (Table 3) show an interesting feature. The inner capacitance is very low. In our model concept, the low number points to a low dielectric constant in the compact part of the DDL. As suggested previously in our

C=

εrε0 d

(12)

in which the relative and absolute dielectric constants are given by εr and ε0 (8.85 × 10−12 CV−1m−1), respectively. According to our structural model for AgI (Figure 3), the distance d between the surface and 1 plane is d ≈ 0.4 nm. In combination with the fitted capacitance value (C1 = 0.15 F/m2), the relative dielectric constant is rather close to εr ≈ 6, which is a typical number at electrical field saturation.5 Another reason for the low dielectric constant of the inner Stern layer region might be the low relative dielectric constant (εr) of the mineral itself. For β-AgI, εr = 7 has been reported.47 However, it can be shown that for metal oxides no relation exists between the value of the dielectric constant of the solid and the inner Stern region of the interface. The inner Stern layer capacitances of goethite and the edge faces of gibbsite are about 0.9−1.0 F/m2. If the average size of the counterions (e.g., d = 0.35 nm) is a measure of the minimum distance of approach, then the equivalent relative dielectric constant is εr ≈ 406 for goethite and the edge faces of gibbsite. For wellcrystallized rutile (TiO2),8 the capacitance of C1 = 0.64−0.71 F/m2 is equivalent to εr ≈ 35. The first number (εr ≈ 40) is about halfway between the dielectric constant of free water (εr = 78) and the solid (goethite εr = 12, gibbsite εr = 9) However, the value for rutile εr ≈ 35 is much lower than the dielectric constant of the solid (rutile εr = 120) and the free solution phase (water εr = 78). This indicates that the interfacial dielectric constant is not necessarily a property that follows directly from the solid or solution phase. 3.2.3. Outer Stern Layer. Another interesting observation made with the analysis of the charging of AgI(s) is the capacitance of the second Stern layer. The mean fitted number is C2 = 0.57 ± 0.05 F/m2, which is slightly lower than found for goethite (C2 = 0.74 ± 0.10 F/m2).9 As mentioned above, for goethite a relative strong ordering of water is found in MD simulations43 supporting the use of an extended Stern layer model. The low value of C2 for AgI(s) suggests a similar process. Recently, the charging behavior of an extensive, highquality data set for rutile (TiO2) has also been analyzed with the CD model.8 For this mineral, there is no indication of the presence of a second Stern layer. The data for rutile TiO2 are best described using a basic Stern model. This is in line with MD simulations,45 suggesting no structured water beyond a distance of ∼0.4 nm, a typical distance for the Stern layer in a basic Stern model.

4. CONCLUSIONS The above work can be summarized with a number of conclusions. • The surface complexation modeling of excess adsorption data suggests the separation of charge by potentialdetermining ions at the AgI interface. Negative charge is created on the surface, and positive charge is located above the surface at some distance. 15621


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• The observed charging behavior can be understood from the molecular structure of the mineral faces of AgI(s). Positive charge is due to monodentate innersphere complexation by a triply coordinated iodide surface group (≡I−0.25) completing its coordination sphere with Ag+. Negative particle charge can be explained by the depletion of Ag+ from the mineral surface by desorption (e.g., 100 face of iodargyrite). • The excess adsorption data of AgI(s) can be described with the CD model. Interfacial CD has been estimated from the MO/DFT optimized geometry of a hydrated AgI cluster using the EDF2 functional, which most accurately predicts the asymmetric structure of hydrated Ag+ ions in solution. • The Ag+ ion distributes its charge very regularly in the tetrahedral coordination sphere of the monodentate surface complex, resulting in a charge distribution with Δz0 = 0.26 and Δz1 = 0.74 vu, showing charge dominance in the 1 plane, in agreement with modeling. • The electrical double layer on AgI(s) consists of two Stern layers. The inner capacitance value is low, about 0.15 ± 0.01 F/m2, suggesting a low value for the relative dielectric constant of the medium, being close to a number typical for the strong orientation of water in a saturated field, which is expected for the primary hydration water of ions. Secondary hydration water is also rather strongly structured in the AgI−water interface because surface complexation modeling points to the presence of a second Stern layer with a relatively low capacitance of 0.57 ± 0.04 F/m2. It is suggested that in general the absence or presence of a second Stern layer (i.e., the use of a BS or an ES model in SCM) is mineraldependent and related to the number (n) of layers of structured water (n ≈ 1−3). • AgI(s) has a pH-independent excess adsorption of the constituting ions. A pH-dependent IEP will be due to adsorbed water present in the EDL beyond the inner Stern layer.

(2) Chapman, D. L. A contribution to the theory of electrocapillarity. Philos. Mag. 1913, 6, 475−481. (3) Stern, O. Zur theory der electrolytischen doppelschicht. Z. Electrochem. 1924, 30, 508−516. (4) Grahame, D. C. The Electrical Double Layer and the Theory of Electrocapillarity. Chem. Rev. 1947, 41, 441−501. (5) Hiemstra, T.; Van Riemsdijk, W. H. Physical chemical interpretation of primary charging behaviour of metal (hydr)oxides. Colloids Surf. 1991, 59, 7−25. (6) Hiemstra, T.; Van Riemsdijk, W. H. On the relationship between charge distribution, surface hydration and the structure of the interface of metal hydroxides. J. Colloid Interface Sci. 2006, 301, 1−18. (7) Zarzycki, P.; Rosso, K. M. Molecular dynamics simulation of the AgCl/electrolyte interfacial capacity. J. Phys. Chem. C 2010, 114, 10019−10026. (8) Ridley, M. K.; Hiemstra, T.; van Riemsdijk, W. H.; Machesky, M. L. Inner-sphere complexation of cations at the rutile-water interface: a concise surface structural interpretation with the CD and MUSIC model. Geochim. Cosmochim. Acta 2009, 73, 1841−1856. (9) Rahnemaie, R.; Hiemstra, T.; Van Riemsdijk, W. H. A new structural approach for outersphere complexation, tracing the location of electrolyte ions. J. Colloid Interface Sci. 2006, 293, 312−321. (10) Westall, J.; Hohl, H. A comparison of electrostatic models for the oxide/solution interface. Adv. Colloid Interface Sci. 1980, 12, 265− 294. (11) Sugimoto, T.; Shiba, F. A new approach to interfacial energy. 3. Formulation of the absolute value of the solid-liquid interfacial energy and experimental collation to silver halide systems. J. Phys. Chem. B 1999, 103, 3607−3615. (12) Sugimoto, T.; Shiba, F. A new approach to interfacial energy. 4. Effects of adsorption of halide ions and gelatin on solid-liquid interfacial energies of silver halides. J. Phys. Chem. B 1999, 103, 3616− 3625. (13) Overbeek, J. T. G., Electrochemistry of the Double Layer. In Colloid Science I; Kruyt, H. R., Ed.; Elsevier: Amsterdam, 1952; pp 115−190. (14) Anonymous. The Project on Emerging Nanotechnologies, September 6, 2012. (15) Bijsterbosch, B. H.; Lyklema, J. Interfacial electrochemistry of silver-iodide. Adv. Colloid Interface Sci. 1978, 9, 147−251. (16) Lyklema, J.; Overbeek, J. T. G. Electrochemistry of silver iodide. The capacity of the double layer at the silver iodide-water interface. J. Colloid Interface Sci. 1961, 16, 595−608. (17) Hiemstra, T.; Van Riemsdijk, W. H. A surface structural approach to ion adsorption: the charge distribution (CD) model. J. Colloid Interface Sci. 1996a, 179, 488−508. (18) Kolkmeijer, N. H.; Van Hengel, J. W. A. Ü ber das regulare und das hexagonale silberjodid. Z. Krist. 1934, 88, 317−322. (19) Hoshino, S. Structure and dynamics of solid-state ionics. Solid State Ionics 1991, 48, 179−201. (20) Burley, G. Polymorphism of Silver Iodide. Am. Mineral. 1963, 48, 1266−1276. (21) Lyklema, J. Electrical double layer on silver iodide - influence of temperature and application to sol stability. Discuss. Faraday Soc. 1966, 42, 81−90. (22) Lyklema, J.; Golub, T. Electrical double layer on silver iodide and overcharging in the presence of hydrolyzable cations. Croat. Chem. Acta 2007, 80, 303−311. (23) Van den Hul, H. J.; Lyklema, J. Determination of specific surface areas of dispersed materials. Comparison of negative adsorption method with some other methods. J. Am. Chem. Soc. 1968, 90, 3010− 3015. (24) Spadini, L.; Manceau, A.; Schindler, P. W.; Charlet, L. Structure and stability of Cd2+ surface complexes on ferric oxides. 1. Results from EXAFS spectroscopy. J. Colloid Interface Sci. 1994, 168, 73−86. (25) Kallay, N.; Preocanin, T.; Supljika, H. Measurement of surface potential at silver chloride aqueous interface with single-crystal AgCl electrode. J. Colloid Interface Sci. 2008, 327, 384−387.

ASSOCIATED CONTENT

S Supporting Information *

Structure of minor crystal faces of AgI(s). Solution and surface chemistry of KNO3/AgI. Calculated electrostatic interface potentials. pH dependency of the charging of AgI(s).22 Test functional in MO/DFT optimization of hydrated Ag(OH2)4+. Charging of AgI(s) in KF solutions. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +31 317 48 2342. Fax: +31 317 419000. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS Part of this work has been funded by NanoNextNL (FES 5120756-02), which is greatly appreciated. REFERENCES

(1) Gouy, G. Sur la constitution de la charge électrique à la surface d’un électrolyte. J. Phys. (Paris) 1910, 9, 457−468. 15622


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