Charge Regulation in the Electrical Double Layer: Ion Adsorption and

Invited Feature Article pubs.acs.org/Langmuir

Charge Regulation in the Electrical Double Layer: Ion Adsorption and Surface Interactions Gregor Trefalt,† Sven Holger Behrens,‡ and Michal Borkovec*,† †

Department of Inorganic and Analytical Chemistry, University of Geneva, Sciences II, 30 Quai Ernest-Ansermet, 1205 Geneva, Switzerland ‡ School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100, United States

ABSTRACT: Charge regulation in the electrical double layer has important implications for ion adsorption, interparticle forces, colloidal stability, and deposition phenomena. Although charge regulation generally receives little attention, its consequences can be major, especially when considering interactions between unequally charged surfaces. The present article discusses common approaches to quantify such phenomena, especially within classical Poisson−Boltzmann theory, and pinpoints numerous situations where a consideration of charge regulation is essential. For the interpretation of interaction energy profiles, we advocate the use of the constant regulation approximation, which summarizes the surface properties in terms of two quantities, namely, the diffuse layer potential and the regulation parameter. This description also captures some pronounced regulation effects observed in the presence of multivalent ions.

1. INTRODUCTION Interfaces involving fluid phases induce the segregation of ionic charges. Most frequently discussed examples involve water as one of the phases (i.e., solid−water, air−water, and oil− water),1,2 but similar charge segregation may occur equally in organic solvents (e.g., hydrocarbons and ionic liquids). The distinctive phenomenon is the formation of a diffuse layer in the fluid phase, which may also occur on both sides of the interface (e.g., semiconductors and membranes).3,4 The diffuse layer is characterized by a monotonically decaying ionic profile with a typical thickness of 1−100 nm. The diffuse layer charge is neutralized by charges that are localized near the interface, typically within 1 nm or less. This charge segregation induces an electrical potential profile across the interface, and its value at the plane of origin of the diffuse layer is referred to as the diffuse layer potential (Figure 1). The charges localized near the interface are referred to as the inner or compact layer, and they may originate from the charging of ionizable surface groups, charge misbalance due to ionic substitution within a crystal lattice, or the specific adsorption of ions from solution.2 The latter mechanism may also lead to a substantial accumulation of charge at inert substrates, such as, noble metals, hydrocarbons, and air−water interfaces.5−8 Charged interfaces can be characterized by their electric surface potential, surface charge density, or the number of adsorbed ions. A wide range of experimental techniques have been developed to probe these quantities. Electrokinetic © 2015 American Chemical Society

techniques, which include electrophoresis or streaming potential analysis, give access to an electric surface potential, which is referred to as the ζ potential.1,2,9 Even though the precise interpretation of this quantity remains difficult, its value is often comparable to the diffuse layer potential. Numerous methods can be used to quantify the number of ions adsorbed at the interface, and they may include batch depletion analysis, potentiometric titrations, reflectivity, surface tension measurements, or, recently, direct imaging with the atomic force microscope (AFM).1,2,10−13 However, the detailed ionic composition of the adsorbed layers and their charge density profiles may not be easily disentangled. Nevertheless, it is well established that the adsorption of ions modifies the surface charge and the diffuse layer potential and that these quantities may strongly vary with solution composition, especially the pH or ionic strength.10,11 These parameters thus regulate the electrical properties of the interface, and some researchers refer to “charge regulation” in this context.11,14 More frequently, the term “charge regulation” is used in the context of overlapping double layers. For example, this situation occurs in lamellar surfactant liquid crystals, charge-stabilized colloidal crystals, or between two charged aggregating colloidal particles.1,2,10 One normally discusses interactions in such Received: September 28, 2015 Revised: November 24, 2015 Published: November 24, 2015 380

DOI: 10.1021/acs.langmuir.5b03611 Langmuir 2016, 32, 380−400

Langmuir

Invited Feature Article

The appreciation of this interrelation further allows a unified treatment of these effects, but such an approach has been rarely implemented so far.15−17 An important step within this effort was the development of the constant regulation (CR) approximation,26,27 which substantially simplifies the analysis of experimental interaction profiles. A proper description of charge regulation is essential to quantifying interaction forces between surfaces and modifications of the surface charge upon variations of the solution composition. However, this aspect is poorly appreciated, and this article aims at closing this gap. We provide a conceptual framework to discuss charge regulation effects and emphasize the advantages of the CR approximation. The present article mainly focuses on DLVO theory and the underlying PB model. We refer the reader to literature concerning more sophisticated approaches, such as the primitive and explicit solvent models.28−33 Applications of similar concepts to polyelectrolytes, proteins, or biological membranes are not discussed here either, and these topics are covered elsewhere.12,14,25,34,35

2. POISSON−BOLTZMANN (PB) AND DEBYE−HÜ CKEL (DH) THEORY The electrical double layer is normally described with the Poisson−Boltzmann (PB) model.1 This model assumes that the surface charge is neutralized by a diffuse layer, which is modeled in a mean-field fashion, thereby neglecting ion−ion correlations (Figure 1). This approximation is excellent for monovalent electrolytes at low surface potentials and/or sufficiently far away from charged surfaces. The PB theory was extended by various authors to include the finite ionic size, dielectric saturation, and formation of diffuse layers on both sides of the interface (e.g., semiconductors).3,24,36−38 The next level of approximation is the primitive model, which treats all electrostatic interactions between ions rigorously. This model describes the ions as charged hard spheres, but the solvent is still treated in a “primitive” fashion, namely, as a dielectric continuum.28−33 Exact solutions of the primitive model are accessible only with computer simulations, but approximate results can be obtained numerically from appropriate theories, such as integral equations or density functionals. A more detailed description can be achieved by treating the solvent explicitly, whereby one also typically relies on computer simulations. The present article focuses on the classical PB picture, and we refer the reader to the literature concerning more detailed discussions of the extended PB models, primitive models, and explicit solvent models.24,28−33,36−38 In the case of planar geometry, the electric surface potential ψ(x) as a function of the normal distance x from the surface satisfies the Poisson equation

Figure 1. Comparison of two popular models of the electric double layer, namely, the diffuse layer model (left) and the basic Stern model (right). (a) Pictorial representation, (b) charge density profile, and (c) electric potential profile. The diffuse layer model assumes a negligible ionic size, and the basic Stern model considers a finite ionic size by introducing a plane of closest approach.

systems on the basis of the classical theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO), which assumes that these interactions are dominated by van der Waals and electric double-layer forces. The latter are normally described with Poisson−Boltzmann (PB) theory.1,2 When two diffuse layers are brought into close proximity to each other, the overlap of the diffuse layers modifies their profile. Under such conditions, one may surmise that the charge of the inner layer remains constant. However, such constant charge (CC) conditions seem to be infrequent, and the charge in the inner layer is typically regulated by the extent of the diffuse layer overlap. Several researchers have discussed the situation in which the diffuse layer potential remains constant upon approach, thereby referring to constant potential (CP) conditions. Such conditions actually reflect a special case of charge regulation, which may be realistic sometimes. In general, however, the surface charge as well as the diffuse layer potential will be regulated upon approach for a given solution composition, and one refers to these conditions as charge regulation. Such charge-regulation effects have now been clearly identified by direct force measurements with the surface forces apparatus (SFA), AFM, or thin film pressure balance studies.15−20 Overlapping double layers play an equally important role in lamellar liquid crystals, the gating of nanopores, and chargestabilized colloidal suspensions.21−24 We have discussed two types of “charge regulation” phenomena, namely, the variation of the surface charge (or diffuse layer potential) with the separation distance for overlapping double layers or with the solution composition for an isolated double layer. Both phenomena originate from the interplay between the charging of the diffuse layer and of the inner layer. This fact was first clearly established by means of calculations of surface forces coupled to ionization models.25

d2ψ ρ =− ε0ε dx 2

(1)

where ρ is the charge density per unit volume, ε0 is the dielectric permittivity of vacuum, and ε is the dielectric constant of the solvent. The charge density of the diffuse layer is given by the respective ionic profiles, namely,

ρ = q ∑ zici e−ziβqψ i

(2)

where ci denotes the bulk number concentration of ions of type i of charge zi expressed in units of the elementary charge q, and 381


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must be exercised in approximating asymmetric electrolytes with symmetric ones because the PB theory may yield widely different results in these situations.39 The charge density and electric potential profiles are depicted in Figure 1, while the charge−potential relations for the diffuse layer are shown in Figure 2. The latter function

the concentration profile is described by the Boltzman factor where the electric energy ziqψ of the respective ion is weighted by β = 1/(kT), which is the inverse thermal energy, with k denoting the Boltzmann constant and T denoting the absolute temperature. Index i runs over all different ion types in solution. A combination of eqs 1 and 2 leads to the PB equation q d2ψ =− ε0ε dx 2

∑ zicie−z βqψ i

(PB) (3)

i

Because the PB equation often cannot be solved analytically, one considers the linearized Debye−Hückel (DH) equation d2ψ = κ 2ψ dx 2 −1

where κ κ2 =

(DH)

(4)

is the Debye length defined by the relation

2βq2I ε0ε

Figure 2. Schematic representation of different charge regulation mechanisms by means of the surface charge density versus the diffuse layer potential ψD. The equilibrium point is determined by the crossing point (open circles) of the charging curve of the diffuse layer σD (solid lines in color) and of the inner layer σI (dashed lines in gray). (a) Modification of the diffuse layer by decreasing ionic strength or surface separation distance. (b) Modification of the inner layer due to decreasing pH or electrode potential.

(5)

where I is the ionic strength given by I=

1 2

∑ zi 2ci

(6)

i

DH and PB models are equivalent provided the surface potentials are sufficiently low. The Debye length κ−1, which typically is 1−100 nm, is a measure of the extent of the diffuse layer. The simplest situation to consider is an isolated planar charged surface immersed in an electrolyte solution. Although a general solution of the PB equation for arbitrary electrolytes has not yet been found, the analytical expression for the charge− potential relationship reads39 σD = ±[2kTε0ε ∑ ci(e−ziβqψD − 1)]1/2 i

always passes through the origin, and the linear region near the origin reflects the DH relation given in eq 8. The PB relation increases more steeply than the DH case. For fixed diffuse layer charge, the magnitude of the diffuse layer potential increases with decreasing ionic strength because the thickness of the diffuse layer increases, which leads to a decrease in the doublelayer capacitance as suggested by eq 8.

(PB)

3. IONIC ADSORPTION The adsorption of ions to surfaces has been described on different levels of approximation, including mean-field models, site binding models, and primitive-level models.2,11,40 The DH and PB theories predict an accumulation or depletion of ions near surfaces, but these approaches do not reflect the experimentally observed ionic specificities. To include ionspecific effects, one postulates an attractive interaction between the ions and the substrate, which leads to an adsorption equilibrium between the ions in solution and the ions adsorbed to the surface.11,12,40 The situation is best exemplified with the amply documented case of proton adsorption, which is also referred to as surface ionization. Let us first focus on a water− solid interface decorated with weakly basic primary amine groups with the respective adsorption equilibrium

(7)

where the ± signs refer to positive and negative surface potentials, respectively. This equation defines two fundamental quantities. First, the surface charge σD compensates for the charge of the diffuse layer at the interface. Second, the diffuse layer potential ψD corresponds to the electric potential at the origin of the diffuse layer. Within the DH approximation, the charge−potential relation simplifies to a simple proportionality relation, namely, σD = C DψD

(DH)

(8)

where CD = εε0κ is the diffuse layer capacitance. The diffuse layer acts just like a plate capacitor, whereby the distance between the plates is given by the Debye length κ−1. One relevant special case, where analytical solutions of the PB equation are possible, is the symmetric z:z electrolyte. This situation is clearly important for monovalent 1:1 electrolytes. Symmetric electrolytes containing multivalent ions normally have low solubility, aside from some 2:2 electrolytes (e.g., CuSO4). In these situations, the charge potential relationship, eq 7, simplifies to the Grahame relation ⎛ zβqψD ⎞ 2ε εκ σD = 0 sinh⎜ ⎟ zβ q ⎝ 2 ⎠

R−NH+3 ⇌ R−NH 2 + H+

(10)

where R denotes an anchor group. The corresponding massaction law reads2 K=

ΓR − NH2c H(0)+ ΓR−NH3+

(11)

where K is the equilibrium constant, ΓR−NH2 and ΓR−NH3+ are the surface concentrations of the deprotonated and protonated groups, and c(0) H+ is the concentration of protons near the interface. For ionization reactions, one uses the pK value defined as pK = −log K. The proton concentration at the

(PB) (9)

Asymmetric electrolytes containing multivalent ions can be very soluble, particularly z:1 and 1:z electrolytes involving monovalent ions (e.g., K4Fe(CN)6, LaCl3). However, care 382


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with ξ = −1/2, whereby the fractional charge follows from the valence bond principle.11 Under acidic conditions, the interface becomes positive, whereas under basic conditions, it is negative. The surface is neutral for pH = pK. More detailed ionization models of water−oxide interfaces have been developed by introducing the specific adsorption of other types of ions, by considering additional planes of nearest approach for these ions, and by the distribution of the ionic charge within these planes.40,41 In these models, the sum of the charges on the different adsorption planes corresponds to the inner-layer charge. All of these models thus result in a more complex charging curve of the inner layer, which may further vary with the ionic strength. The latter aspect is in contrast to the simple description presented here, where the inner-layer charge is identical to that of the bare surface and is independent of the ionic strength. The diffuse layer potential is determined by the condition in which the charge of the inner layer compensates for the charge accumulated within the diffuse layer, leading to

interface can be related to its concentration in solution by the Boltzmann relation

c H(0)+ = c H+e−βqψ0

(12)

where ψ0 is the surface potential and cH+ is the molar concentration of protons in the bulk, which can be also expressed as pH = −log cH+. Here, one again approximates the interactions between the adsorbed protons in a mean-field fashion and neglects lateral correlations in the adsorbed layer.12 The simple diffuse layer model assumes that the surface potential is identical to the diffuse layer potential, namely, ψ0 = ψD (Figure 1a). The surface charge density of the inner layer can then be expressed as σI = q ΓR−NH3+

(13)

Because the surface concentration of the ionizable sites remains constant at Γ(0) = ΓR−NH2 + ΓR−NH3+, one can combine eqs 11−13 to express the inner-layer charge as a function of the diffuse layer potential ψD, namely, σI = q Γ(0)

σD(ψD) = σI(ψD)

c H+ c H+ + Ke βqψD

The solution of this equation can be represented graphically as the crossing point between the charge−potential relationships of the diffuse and inner layers (Figure 2). For decreasing ionic strength, the charge of the diffuse layer decreases, the crossing point moves downward, and therefore the surface charge decreases. A similar situation will be discussed for two interacting surfaces in section 3. For decreasing solution pH, the charging curve of the inner layer shifts toward higher potentials, the crossing point moves upward, and the surface charge increases. One may encounter similar situations with metal or semiconductor electrodes whose electric potential can be adjusted with a potentiostat with respect to a reference electrode. Within the simple diffuse layer model, the electrode potential is identical to the diffuse layer potential ψD, whereas in the basic Stern model it corresponds to the surface potential ψ0. However, this approximation seems not to be fully appropriate, and surface ionization models with an additional plane within the solid have been proposed to treat such situations.6,42 Model Results in Monovalent Electrolytes. When the charging curve of the inner layer is known, the proton binding isotherm can be constructed. Representative results in monovalent salt solutions are shown in Figures 3 and 4. Focus first on Figure 3, which illustrates the charging of an interface bearing amine groups in a monovalent salt solution (eq 10). The left column illustrates the pH dependence, and the right one conveys the dependence on the ionic strength. Figure 3a shows the charging curves of the diffuse and inner layers for this particular situation, whereby the crossing points correspond to the equilibrium situations. The resulting surface charge density is shown in Figure 3b, and the diffuse layer potential is shown in Figure 3c. The interface becomes fully ionized in an acidic solution, and it is neutral in a basic one. The diffuse layer potential follows a similar pH dependence, but the potential increases with decreasing ionic strength even when the interface is fully charged. The ionic strength dependence of the diffuse layer potential is therefore inverted with respect to that of the surface charge. With decreasing concentration, the charge density decreases while the surface potential increases. The inversion of the ionic strength dependence of the surface charge density and double-layer potential can be understood as follows. When the ionic strength is increased, the interactions

(14)

This sigmoidal charge−potential relationship of the inner layer is depicted in Figure 2. This relationship can be generalized to surfaces containing ionizable groups of different charge, and the result reads ⎛ ⎞ c H+ ⎜ ⎟ σI = q ∑ Γ(0) + ξ j ⎜ j ⎟ βqψ0 + + K e c ⎝ H ⎠ j j

(15)

where index j runs over all types of sites, Kj is the equilibrium constant of site type j, Γ(0) is the corresponding site density, j and ξj is the respective charge of the deprotonated site in units of elementary charge. We suppress this index when only one type of site is present. The effect of the finite ionic size can be approximately introduced by considering the Stern capacitance defined as CS = σI/(ψ0 − ψD), where ψ0 is the surface potential and ψD is the diffuse layer potential, leading to the so-called basic Stern model (Figure 1b). The respective charging curve of the inner layer, namely, σI versus ψD, can be constructed parametrically through the ψ0 dependence of both quantities. The basic Stern model reduces to the diffuse layer model in the limit of CS → ∞. Other approaches to introduce finite ionic size involve the modification of the PB equation37,38 or consider the ions to be charged hard spheres within the primitive model.33 The ionization model introduced above can also be used to describe surfaces bearing weakly acidic ionizable groups, for example, carboxylic groups, that dissociate according to the equilibrium R−COOH ⇌ R−COO− + H+

(16)

Thereby, we have ξ = −1 (eq 15). The interface is neutral in an acidic solution, and it becomes negatively charged in a basic one. A similar model can be used to model an amphoteric oxide surface, for example, iron oxide (hematite, α-Fe2O3). Hiemstra and van Riemsdijk have shown that this situation can be described with the equilibrium11 Fe−OH 2+1/2 ⇌ Fe−OH−1/2 + H+

(18)

(17) 383


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Figure 4. Charging of an iron oxide interface according to surface reaction 17 described with the basic Stern model. The parameters are ionization constant pK = 8.7, site density Γ(0) = 6.15 nm−2, and Stern capacitance CS = 1.6 F/m2. (a) Charging curves of the diffuse and inner layers, whereby the crossing points (open circles) indicate the equilibrium situation. (b) Surface charge density and (c) diffuse layer potential. The left column shows the pH dependence, and the right one shows the ionic strength dependence. The dashed line in (c, left) shows the Nernst equation (eq 19). The closed symbols indicate conditions used in figures discussed in sections 3 and 4.

Figure 3. Charging of an interface with amine groups ionizing according to reaction 10 as described by the diffuse layer model. The parameters are ionization constant pK = 10 and site density Γ(0) = 0.15 nm−2. (a) Charging curves of the diffuse and inner layers, whereby the crossing points (open circles) indicate equilibrium situations. (b) Equilibrium surface charge density and (c) equilibrium diffuse layer potential. The left column shows the pH dependence, and the right one shows the ionic strength. The dependence of the potential resulting from the Grahame equation (eq 9) is also indicated (c, right). The closed symbols indicate conditions used in figures discussed in sections 3 and 4.

crosses the ordinate at a finite slope whereas for the amine surface the neutral surface is approached only asymptotically. This feature leads to the well-defined crossing point in the charging curves for the iron oxide, but this crossing point is absent for the amine surface. For this reason, the diffuse layer potential of the iron oxide interface is well approximated by the Nernst potential2,27

between the ionizable sites on the surface are screened; therefore, the surface can adsorb an increasing number of protons, thus leading to a higher change density. The diffuse layer charge not only grows by the same amount as the inner layer charge, due to electroneutrality, but also charge accumulates closer to the surface and thus contributes more effectively to reducing the surface potential. Increased electrostatic screening therefore increases the surface charge but lowers the surface potential, which leads to the observed inversion. An analogous situation for an amphoteric iron oxide surface is shown in Figure 4. Under acidic conditions, the surface is positively charged, whereas under basic conditions the surface acquires a negative charge. The surface is neutral at pH 8.7, which leads to the characteristic crossing point, which becomes apparent when the surface charge or the diffuse layer potential is plotted versus pH for different ionic strengths (Figure 4b,c, left). The inverse dependence of the surface charge and diffuse layer potential on the ionic strength can again be observed. The main difference between the situations shown in Figures 3 and 4 is that the charging curve of the inner layer for the iron oxide

ψN =

kT ln 10 (pK − pH) q

(19)

with kT ln 10/q ≃ 59 mV, namely, ψD ≃ ψN, as shown in Figure 4c.Conversely, this approximation is poor for the amine surface. Experimental Results in Monovalent Electrolytes. Let us now illustrate the capability of these models to rationalize experimental data. Figure 5 (top row) shows the charging behavior of carboxylated latex particles.10 The surface charge density shown in Figure 5a has been measured with potentiometric titration, and one observes that the magnitude of the surface charge increases with increasing pH and ionic strength. The diffuse layer potential can be approximated by the surface potential obtained from electrophoretic mobility, which 384


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salt ions.45 Although site densities are not accessible from such experimental charging curves as a result of missing plateaus, they can be evaluated from crystallographic considerations.11 The diffuse layer potentials obtained from surface ionization models compare relatively well with the experimentally determined ζ potentials for the respective oxide particles.44,46 However, such comparisons are not entirely straightforward, especially because of uncertainties in the position of the shear plane, deviations from spherical shape, and particle porosity. The charging of the silica interface could also be predicted with computer simulations based on the primitive model in a satisfactory fashion.33,47 Overall, the ionization of water−oxide interfaces in monovalent electrolyte solutions is rather well understood quantitatively, and we refer to the literature concerning the details.12,33,40,41,44 Multivalent Electrolytes. Let us now investigate the role of multivalent ions on the charging of water−solid interfaces and discuss their influence in the framework of the surface ionization models. We focus on the amine and iron oxide model surfaces first. The model can be simply adapted by employing the charge−potential relationship of the diffuse layer given by eq 7 rather than the Grahame equation (eq 9). The positively charged amine surface is illustrated in Figure 6a,b

Figure 5. Dependence of experimental surface ionization data on solution pH. Carboxyl latex particles with ionization reaction 16 interpreted with the diffuse layer model10 with pK = 4.9 and Γ(0) = 0.60 nm−2. (a) Surface charge density obtained by potentiometric titration and (b) electrokinetic surface potentials by electrophoresis. Calculations compare the diffuse layer potential (dashed lines) to the ζ potential (full lines), which is obtained by shifting the shear plane outward by 0.25 nm. The ζ potentials were calculated from the electrophoretic mobilities with the standard electrokinectic model.10 Oxide particles were interpreted with the basic Stern model. (c) Ludox silica particles11 were interpreted with surface reaction 20 and pK = 7.5, Γ(0) = 8.0 nm−2, and CS = 2.9 F/m2. (d) Iron oxide (hematite) particles43 were interpreted with surface reaction 17 and pK = 8.7, Γ(0) = 6.15 nm−2, and CS = 1.6 F/m2.

can be converted to the ζ potential and is shown in Figure 5b. This conversion is normally carried out with the standard electrokinetic model.1 The ζ potential also increases in magnitude with increasing pH and indeed shows an inverse trend with ionic strength. The model calculations based on the dissociation reaction given in eq 16 rely on the diffuse layer model, and they represent the experimental data rather well. The parameters used were pK = 4.9 and a site density of Γ(0)= 0.6 nm−2. Although the electrophoretic data can also be reasonably well interpreted with the diffuse layer potential, a better representation can be obtained by introducing a shift in the shear plane of 0.25 nm (Figure 5b). The charging behavior of two different oxide surfaces is illustrated in Figure 5 (bottom row). Potentiometric titration data are compared to calculations with the basic Stern model for silica11 and hematite.43 The silica surface is described by the dissociation reaction Si−OH ⇌ Si−O− + H+

Figure 6. Ionization of surfaces calculated with the diffuse layer model in the presence of multivalent ions. Amine surface described with the diffuse layer model and reaction 10 at ionic strengths of (a) 100 and (b) 1.0 mM. Iron oxide surface described with the basic Stern model and reaction 17 at ionic strengths of (c) 100 and (d) 1.0 mM. The amine surface is modeled with pK = 10.0 and Γ(0) = 0.15 nm−2, and the iron oxide surface, with pK = 8.7, Γ(0) = 6.15 nm−2, and CS = 1.6 F/m2.

(top row). For low surface potentials, the diffuse layer model predicts charging curves that are independent of the type of ions, and they collapse at the same ionic strength. This situation is characteristic of the DH model and occurs at high ionic strength as illustrated in Figure 6a. When the surface potential is high, the ionic valence becomes important. This situation is encountered at low ionic strengths and is shown in Figure 6b. When the multivalent ions are co-ions as originating from dissolved z:1 salts, the surface is screened only by the monovalent counterions and the resulting surface charge

(20)

whereby pK = 7.5, Γ(0) = 8 nm−2, and CS = 2.9 F m−2 are the appropriate parameters.11,44 The charging data of iron oxide (hematite, α-Fe2O3) can be interpreted with the reaction given in eq 17, whereb parameters pK = 8.7, Γ(0) = 6.15 nm−2, and CS = 1.6 Fm−2 were used.11 The basic Stern model predicts a somewhat stronger ionic strength dependence than observed experimentally. This effect can be more substantial for other oxides and can be modeled by including the specific binding of 385


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equation (eq 9) as shown in Figure 3c (right). With increasing valence, the potential decreases. For trivalent and tetravalent ions, the diffuse layer potential reverses its sign. This decrease in the surface potential and subsequent charge reversal can be interpreted by the progressive adsorption of the multivalent counterions. Such charge reversal was experimentally documented by electrophoresis or particle aggregation experiments for positively charged iron oxide particles with phosphate or polycarboxylate ions51−53 and for negatively charged particles with metal cations54−56 or multivalent aliphatic amines.57 Figure 7b illustrates the effect of the surface charge density for the adsorption of La3+ ions on silica as reported by James et al.9 Figure 5c may serve to recall that silica is weakly charged at low pH whereas it becomes strongly charged at high pH. At low pH, the ζ potential shown in Figure 7b is hardly influenced by the presence of La3+, but the surface charge increases with increasing La3+ concentration at intermediate pH values. Above pH 5.5, this increase is so pronounced that the surface undergoes a charge reversal. At higher pH, the situation is further complicated by the hydrolysis of La3+, namely, the formation of La(OH)2+ and La(OH)2+ complexes and the surface precipitation of La(OH)3. The latter process probably induces the second charge reversal at high pH. James et al.9 further stress the importance of the hydrolysis of metal ions and surface precipitation for other types of ions, including Co2+, Al3+, and Fe3+. Two charge reversal points with increasing pH were equally reported for rutile in the presence of divalent alkali earth cations.46,49 Therefore, the first reversal point originates from the proton desorption, and the second one, from the adsorption of the divalent metal ions. Similar trends were observed for calcium silicate hydrate and for hydrophobic substrates, albeit featuring only one charge reversal point.8,47 Hydrophobic monovalent ions, surfactants in particular, may also induce a charge reversal.51,58,59 Numerous experimental and modeling studies of the adsorption of multivalent ions to oxide surfaces are available, especially for divalent ones.60,61 The employed surface complexation models are capable of rationalizing the adsorption isotherms even in mixed systems in substantial detail.40,61 However, there is only a limited number of modeling studies attempting to quantify the ζ potential in the presence of multivalent ions. Rahnemaie et al.52 have successfully predicted the dependence of the ζ potential with pH and the amount of added phosphate for an iron hydroxide (goethite, FeOOH). Another modeling study was able to rationalize the existence of two charge reversal points for rutile (TiO2) in the presence of divalent metal ions, albeit with mixed success.40 In spite of these encouraging results, it remains unclear to us to which extent such surface complexation models are able to capture the dependence of diffuse layer potentials on solution composition in the presence of multivalent ions. The approach based on the primitive model seems more encouraging. The charge reversal induced by La3+ ions of sulfate latex particles55 and by Ca2+ and Mg2+ of silica33,47 could be explained with this model based on ion−ion correlation effects without assuming any specific interaction between these ions and the surface. However, for similar latex systems it was not possible to explain the charge reversal by La3+ without assuming specific interactions.56

density is lower than in the monovalent case. For multivalent counterions originating from 1:z salts, the charge density increases because these ions screen the surface charge more effectively. An analogous difference between monovalent and divalent cations was observed experimentally for negatively charged silica particles48 and could also be rationalized within the primitive model.33 The iron oxide surface shown in Figure 6c,d (bottom row) exhibits more complex behavior resulting from its charge reversal. For low surface potentials, the charging curves at the same ionic strength are almost independent of the types of ions present (Figure 6c). At higher surface potentials, the ionic valence becomes relevant (Figure 6d). In this case, the nature of the anions is important at low pH, where the surface is positively charged, and the nature of the cations is significant at high pH, where the surface is negatively charged. The magnitude of the surface charge density increases with increasing valence of the counterions. Such asymmetric charging curves have been reported for various oxides in the presence of divalent cations as well as divalent anions.41,49 The simple PB model presented here assumes that the multivalent ions are indifferent, meaning that they do not adsorb specifically. This assumption is probably realistic for the multivalent co-ions but hardly so for multivalent counterions. In this case, the surface potential is probably further reduced by specific ion adsorption, ion−ion correlations, or both. These effects can be included in mean-field surface complexation models of the charging behavior in the presence of divalent ions.40,41 In such systems, ion-specific effects in the charging behavior could be rationalized by the primitive model by assuming different ionic size but excluding any additional ion− surface interactions.33 An important aspect is the variation of the surface charge density through the adsorption of multivalent counterions. This process may even induce a charge reversal and will strongly modify the respective diffuse layer potentials. Figure 7 reports the ζ potential determined from the electrophoretic mobility in two cases. The dependence of the ζ potential of amidine particles on the ionic concentrations of Cl−, SO4−, Fe(CN)63−, and Fe(CN)64− is shown in Figure 7a.50 For monovalent ions, one observes the decrease in the diffuse layer potential with increasing concentration, as expected from the Grahame

Figure 7. Influence of multivalent ions on the ζ potential of colloidal particles. (a) Dependence on the anion concentration for amidine latex particles at pH 4.0 in the presence of Cl−, SO4−, Fe(CN)63−, and Fe(CN)64− and K+ as the cation.50 (b) Dependence on pH for silica particles in the presence of different concentrations of La3+ ions in 1.0 mM KNO3 electrolyte.9 The ζ potentials were calculated from the electrophoretic mobility with the standard electrokinetic model. The lines serve to guide the eye only.

4. DOUBLE-LAYER INTERACTIONS IN THE SYMMETRIC GEOMETRY Numerous relevant situations involve two identical (or very similar) interfaces interacting across an electrolyte solution, and 386


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distance h, the surface interaction energy of the diffuse layer follows by integration

in this case, we refer to the symmetric geometry. The recent availability of reliable techniques to measure forces involving colloidal particles and surfaces enables us to investigate such interactions directly. The surface forces apparatus (SFA) measures forces between two bent mica sheets.62,63 Forces between colloidal particles and a planar substrate can be probed with total internal reflection microscopy64,65 and the colloidal probe technique.66,67 The latter technique uses the AFM, whereby a colloidal particle is attached to the AFM cantilever. Interactions between two colloidal particles can also be measured with the colloidal probe technique68,69 but equally with optical tweezers and video microscopy. 70,71 The interactions between two surfactant films can be probed with the pressure balance.20,72 The Derjaguin approximation relates the force F between two objects investigated with the surface interaction energy W, according to1,2 F = 2πR eff W

WD(h) =

ψeff =

(23) −1

where A is the amplitude, q is the range of the interaction, and h is the surface separation distance. The same dependence was used previously and was mostly empirically based.50,73,74 The van der Waals interaction can be approximated by the nonretarded expression H WV = − (24) 12πh2

i

⎛ zqβψD ⎞ 4 tanh⎜ ⎟ zqβ ⎝ 4 ⎠

WD = 2ε0εκψD2

(28)

e−κh 1 + (1 − 2p)e−κh

(29)

where one must set p = 1 in the CC case and p = 0 in the CP case. The interaction energy becomes independent of boundary conditions at longer distances and reduces to eq 27 with ψeff = ψD. Because of the dependence of the interaction energies on the square of the diffuse layer potential, the sign of the potential cannot be determined from experimental energy profiles in the symmetric geometry. However, this sign can be found by electrophoresis or from the chemical composition of the surface. The above expression was elegantly generalized by Carnie and Chan.26 Their DH treatment implies that the charge− potential relationship remains a proportionality relation, although the prefactor now depends on the separation distance

where H is the Hamaker constant. This constant is usually positive, which leads to an attractive interaction.2 These interactions may be further weakened by surface roughness and retardation effects. Double-layer interactions can be obtained by solving the PB model for two charged interfaces for the potential profile, but in many situations analytical solutions remain unknown. Nevertheless, the electric potential profile ψ(x) between charged plates can be obtained numerically. When this profile is known, the pressure between the plates can be obtained from the relation1,2,39 Π = kT ∑ ci(e

(27)

where ψD is the diffuse layer potential of the isolated surface. This dependence given in eq 27 reflects the exponential concentration profile in the diffuse layer. For weakly charged interfaces, one has ψeff = ψD, whereas for highly charged ones, the effective potential saturates at ψeff → 4/(zqβ). This value corresponds about to 100 mV for a monovalent electrolyte at room temperature. One should note, however, that the saturation values for asymmetric electrolytes may differ substantially from those for symmetric electrolytes.39,75 The effective potential introduced in eq 28 reflects the precise asymptotics of the DH theory in terms of the more fundamental parameters of the PB theory. Similar reasoning probably applies to the diffuse layer potential in the PB theory, which is also an effective quantity that describes the correct asymptotics of the PB theory in terms of the more fundamental primitive model. Analytical solutions applicable to all separation distances can be obtained from DH theory, but at shorter distances, boundary conditions become important. As surfaces approach, the charge density may remain unchanged irrespective of the separation distance, which is referred to as the CC boundary condition. When the diffuse layer potential remains unchanged upon approach, one refers to the CP boundary condition. These conditions lead to the surface interaction energy1,2

where WV and WD represent the DLVO contributions from van der Waals and diffuse layer interactions. Sometimes, a shortrange non-DLVO contribution is necessary to fit experimental data. In these situations, we assume an exponential form

ε ε ⎛ dψ ⎞ 2 − 1) − 0 ⎜ ⎟ 2 ⎝ dx ⎠

(26)

where for symmetric z:z electrolytes the effective potential is given by

(22)

−ziβqψ

Π(h′) dh′

WD = 2ε0εκψeff 2e−κh

where Reff is the effective radius. For two interacting spheres of radius R, one has Reff = R/2, whereas when one of these spheres interacts with a planar substrate, Reff = R. The latter relation also applies to two perpendicularly crossed cylinders of radius R as used in the SFA. For this reason, many researchers report normalized forces F/Reff. In the present article, we report surface interaction energies W throughout. Classical DLVO theory postulates that the free energy of two interacting interfaces is dominated by two contributions, namely,1,2 W = WV + WD + WE 

WE = Ae−qh

∞

When the surfaces are far apart, this interaction energy can be obtained by means of the superposition approximation. The result reads1,2

(21)

DLVO

∫h

⎛ κh ⎞ σD = ε0εκ tanh⎜ ⎟ψD ⎝2⎠

(25)

(30)

where quantities σD and ψD now refer to the interfaces at a given separation distance h. The charge−potential relationship of the inner layer, however, is a nonlinear function σI(ψD). The

which can be evaluated at any position x in between the plates. From the dependence of the pressure on the separation 387


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idea is to linearize this function around its value for the isolated surface as σI(ψD) = σD(iso) − C I(ψD − ψD(iso))

σ(iso) D

(31)

ψ(iso) D

where and refer to the isolated surface and the inner layer capacitance CI is defined as CI = −∂σI/∂ψD. The approximation introduced in eq 31 has the advantage that the DH equation can be solved for two plates analytically. The resulting interaction energy has exactly the form of eq 29, but parameter p may assume arbitrary values and can be expressed as26,27 p=

CD CI + CD

(32)

where CD is the diffuse layer capacitance of the isolated surface given by eq 8. This approach can be generalized to the nonlinear PB situation27 and is referred to as the CR approximation. Therefore, the diffuse layer capacitance must be evaluated from CD = ∂σD/∂ψD whereby σD is given by eq 7. This approximation is exact at large distances because it assumes that the inner capacitance is given by its value for the isolated surface. The CC and CP conditions can be recovered by setting the regulation parameter to p = 1 and 0, respectively. The regulation parameter may also become negative (p < 0), leading to the so-called sub-CP regime.76 This regime could possibly be realized for sigmoidal adsorption isotherms featuring a critical point, but currently it remains unclear whether such behavior occurs in practice or not. The basic Stern model discussed above does not lead to such sub-CP behavior, provided CS > 0. CC conditions always lead to the largest interaction energies, whereas charge regulation effects will lower those. Clearly, the energy will be decreased when the system has the possibility to relax the surface charge density. The regulation parameter probably also represents an effective quantity, which describes the asymptotic charge regulation characteristics of the interface at large distances. Model Results in Monovalent Electrolytes. The general situation invoking the actual binding isotherm and the PB model must be treated numerically. The diffuse layer potential and the surface charge density again follow from eq 18. In the PB situation, the charge density also decreases with distance at a given potential. The resulting dependencies of interaction free energies, surface charge, and diffuse layer potential on the separation distance are illustrated in Figure 8 at an ionic strength of 10 mM. Two model systems are compared, namely, two interacting amine surfaces at pH 10.0 (left column) and two interacting iron oxide surfaces at pH 11.0 (right column). Under these conditions, the amine surface is weakly positively charged and the iron oxide surface is highly negatively charged (Figures 3 and 4). The interaction energy profiles are shown in Figure 8a. These profiles decrease approximately exponentially, whereby the range of the interaction is given by the Debye length, which is 3.0 nm here. The surface ionization model predicts an interaction energy profile between those for the CC and CP conditions. The corresponding charging curves of the inner and diffuse layer are shown in Figure 8b, whereby the crossing points define the equilibrium condition. One can see that the magnitude of the surface charge density decreases with decreasing separation distance and the magnitude of the diffuse layer potential increases. These trends are confirmed by actual calculations (Figure 8c,d). At contact, the surfaces become

Figure 8. Interaction between charged surfaces across monovalent electrolyte solution of an ionic strength of 10 mM in the symmetric geometry. (a) Surface free energy versus the separation distance, (b) surface charge densities of the diffuse and inner layers versus the diffuse layer potential, (c) surface charge density, and (d) diffuse layer potential versus the separation distance. Left column shows the situation for an amine surface obeying the surface reaction (reaction 10) at pH 10.0 with ψD = 26 mV and p = 0.59. The parameters of the diffuse layer model used are pK = 10.0 and Γ(0) = 0.15 nm−2. The right column shows the situation of an iron oxide surface with the surface reaction (reaction 17) at pH 11.0 with ψD = 104 mV and p = 0.39. The parameters of the basic Stern model used are pK = 8.7, Γ(0) = 6.15 nm−2, and CS = 1.6 F/m2. Full regulation calculations based on surface ionization models (solid line) are compared to the constant regulation (CR) approximation (dashed line). Results for constant charge (CC) and constant potential (CP) conditions delimit the gray region. The insets show schemes of the respective systems.

electrically neutral. The diffuse layer potential attains the Nernst value ψN = 136 mV for the iron oxide surface (eq 19). However, this potential diverges upon approach for the amine surface. This feature is related to the functional form of the charging curve of the inner layer. For the iron oxide surface, this curve cuts the abscissa with a finite slope at ψD = ψN, whereas for the amine surface the abscissa is approached asymptotically. Under CC conditions, the surface charge obviously remains 388


Langmuir


constant, but the diffuse layer potential increases in magnitude. For CP conditions, the diffuse layer potential remains constant, but the magnitude of the surface charge decreases. The calculations based on the surface ionization model are compared to the CR approximation in Figure 8. The CR approximation replaces the charging curve of the inner layer with a straight line such that it matches the surface charge density, diffuse layer potential, and inner capacitance of the isolated surface (eq 31 and Figure 8b). Although this approximation is exact at large separation distances, one finds that the CR approximation remains very good even down to small distances.27 For example, the CR approximation remains excellent down to contact for the iron oxide surface. In this case, the actual charging curve of the inner layer is basically a straight line in the entire relevant region and is thus well approximated by eq 31. For the amine surface, however, the CR approximation becomes inaccurate at smaller separation distances. The reason is that this approximation represents the actual charging curve of the inner layer at low charge densities poorly. Nevertheless, the CR approximation remains reasonable because even at contact the error in the actual surface energy remains below 25%. The CR approximation has been tested against actual surface ionization models in various situations, and the corresponding results are mostly excellent.27,77 Figure 8 further illustrates the relative importance of charge regulation effects for different surface charge densities. The gray region, which is limited by the CC and CP conditions, is substantially larger for the amine surface than for iron oxide, meaning that charge regulation effects are more important for the former. The relevant quantity is the magnitude of the surface charge density. For weakly charged surfaces, regulation effects are more relevant, and for the highly charged ones, less so. In practical terms, however, when a surface is very weakly charged, the double-layer interaction is negligible. Therefore, charge regulation effects are most relevant for moderately charged surfaces, but they become less important for highly charged ones. The essential point to retain is that the CR approximation represents a much simpler alternative to an explicit ionization model for the calculation of interaction energies. In the former case, the surfaces can be characterized by two parameters, namely, the diffuse layer potential (or the corresponding surface charge) and the regulation parameter (or the inner capacitance). These quantities refer to the isolated surface, which leads to a further simplification. We report the diffuse layer potential ψD and the regulation parameter p in the following text. As will be shown, these two quantities suffice to describe double-layer interactions in a satisfactory way. A variation of the surface charge densities between approaching charged interfaces was also found on the basis of the primitive model.33 In many situations, one knows the variation of the diffuse layer potential with the solution properties, such as pH and ionic strength. This quantity can often be approximated by the ζ potential as already discussed in section 2. However, little information is available on the dependence of the regulation parameter on the solution composition. On the basis of the surface ionization models discussed above, the capacitances can be calculated, and the regulation parameter can be evaluated.27 Some representative results for the amine and iron oxide surfaces are shown in Figure 9. One observes that the regulation parameter increases with increasing ionic strength.

Figure 9. Regulation parameter as a function of ionic strength (left) and pH (right). The figure compares (a) amine and (b) iron oxide surfaces. The parameters of the diffuse layer model describing the amine surface with the surface reaction (reaction 10) are pK = 10 and Γ(0) = 0.15 nm−2, and those of the basic Stern model describing the iron oxide surface with reaction 17 are pK = 8.7, Γ(0) = 6.15 nm−2, and CS = 1.6 F/m2.

When one considers its dependence on the solution pH, one finds a minimum near the pK value of the ionizable surface group. These trends can be readily understood. Because the diffuse layer capacitance increases with increasing ionic strength, eq 32 suggests that CC conditions prevail at high salt levels. Near the pK value of the ionizable surface group, the inner capacitance becomes large, leading to a small regulation parameter. This effect is particularly pronounced for the iron oxide surface near the charge neutralization point and at low ionic strengths. In this situation, the system is expected to follow CP conditions closely. Experimental Results in Monovalent Electrolytes. Let us now illustrate the importance of charge regulation effects with experimental surface interaction energy profiles. Figure 10 summarizes such profiles measured in various systems and geometries in monovalent electrolyte solutions. The effect of charge regulation is important in all situations. Consider first the measurements carried out with the SFA between two curved mica sheets in a NaCl solution shown in Figure 10a.15 The data can be well rationalized with the DLVO theory and the CR approximation, leading to regulation parameter p = 0.50. A fit of comparable quality of this energy profile was obtained with a surface ionization model15 similar to the one discussed here. Other authors have also analyzed energy profiles with surface ionization models in a similar fashion.16,17 The difficulty of this approach is that such models contain numerous adjustable parameters that cannot be determined reliably from a single energy profile. Such parameter crosscorrelations were clearly illustrated recently.17 Optimally, an entirely series of force curves recorded under different solution conditions should be analyzed simultaneously. The respective analysis with the CR approximation is substantially simpler because one must determine two parameters only, namely, the diffuse layer potential ψD and the regulation parameter p. 389


Langmuir


because of possible surface heterogeneities, this effect can be minimized by probing several particle pairs. Such measurements involving pairs of similar particles are compared with DLVO calculations in Figure 10c,d. One observes that charge regulation effects are again substantial and that the energy profiles can be well rationalized with the CR approximation. For the silica particles, one finds a regulation parameter of p = 0.56, and for the sulfate particles, p = 0.41. Although CC conditions might be anticipated for sulfate latex particles because of the complete dissociation of surface groups, one observes substantial charge regulation that is probably due to shifts in the equilibrium of specifically adsorbed salt ions. The results shown also include the effect of a short-ranged nonDLVO exponential interaction that is repulsive for silica and attractive for latex (eq 23). Such non-DLVO contributions can be substantial at shorter distances. The lowest row summarizes the resulting diffuse layer potentials for different ionic strengths, and the results for silica particles79 are shown in Figure 10e and those for sulfate latex particles,57 in Figure 10f. The dependence of the diffuse layer potential on the ionic strength can be well rationalized with the Grahame equation (eq 9). When one considers the ionic strength dependence of the regulation parameter, one finds that the fitted values remain relatively constant. This observation is surprising in light of the model calculations shown in Figure 9. These models suggest that this parameter should be increasing with increasing ionic strength, but this trend could not be confirmed experimentally so far. Possibly, this phenomenon could be explained by the specific adsorption of salt ions or by surface charge heterogeneities. The latter effects could also be responsible for the fact that the ζ potentials, which were measured by electrophoresis for the same sulfate particles, are larger in magnitude.57 Nevertheless, we suspect that diffuse layer potentials obtained from force measurements are more reliable than those from electrokinetic experiments, although the required experiments are more laborious and technically challenging for sub-micrometer-sized particles. Multivalent Electrolytes. The presence of multivalent ions in solution may induce distinctive features in the interaction energy profiles. Representative results of model calculations for the amine model surface are shown in Figure 11 at an ionic strength of 10 mM. At pH 11, the surface is only weakly

Figure 10. Comparison of experimental interaction energies (points) with DLVO calculations in the symmetric geometry and monovalent electrolytes invoking the constant regulation (CR) approximation (solid lines). The gray region is delimited with profiles corresponding to CC and CP conditions. Effects of additional non-DLVO exponential attraction are also illustrated (dashed line). (a) Interaction between two mica surfaces15 in 5.25 mM NaCl solution of pH 5.25 with ψD = −27 mV and p = 0.50 (b) between a silica particle and an oxidized silicon wafer78 in a 10 mM KCl solution of pH 5.5 with ψD = −19 mV and p = 0.51, (c) for surfaces of two similar silica particles79 across a 3.0 mM KCl solution of pH 4.0 with ψD = −25 mV and p = 0.56, and (d) for surfaces of two similar sulfate latex surfaces in 5.0 mM KCl solution of pH 4.0 with ψD = −58 mV and p = 0.41.57 (e) Diffuse layer potential as a function of ionic strength for the respective (e) silica79 and (f) sulfate latex surfaces.57 The comparison with the Grahame equation (solid lines) uses charge densities of (e) −2.9 and −19 mC/m2. Hamaker constants used are (a) 9.0 × 10−21, (b) 2.2 × 10−21, (c) 3.3 × 10−22, and (d) 3.5 × 10−21 J, together with shifts of the van der Waals plane by (b) 1.05 and (c) 0.85 nm. The non-DLVO short-range exponential interaction is characterized with decay lengths and amplitudes of (c) 0.30 nm and +0.15 mN/m and (d) 0.32 nm and −64 mN/m.57,74,79 This interaction is neglected in (a, b). The experimental data points at smaller distances in (b, d) are unreliable because of mechanical instabilities. The schemes summarize the experimental geometries used.

Figure 10b shows the surface interaction energy between a silica particle and a planar silica substrate measured with an AFM.78 The energy profile can again be described with DLVO theory and the CR approximation. One obtains a similar regulation parameter p = 0.51. The general difficulty with the sphere-plane geometry is that one cannot easily ascertain that the system investigated is really symmetric. This difficulty can be remedied by measuring the forces between two similar colloidal particles originating from the same batch. Although perfect symmetry cannot be fully assured

Figure 11. Interaction energies versus separation distance of two amine model surfaces in the presence of multivalent ions at an ionic strength of 10 mM in the symmetric geometry. The surface obeys surface reaction 10 and is modeled by the diffuse layer model and parameters pK = 10 and Γ(0) = 0.15 nm−2. Effects of charge regulation are indicated through CC and CP conditions. (a) Weakly charged surface at pH 11.0 and (b) highly charged surface at pH 6.0. The dashed lines in (b) are the DH asymptotes. 390


Langmuir


charged. Under these conditions, the interactions are weak, and they are almost independent of the type of electrolytes (Figure 11a). This situation is characteristic of the DH limit, where the energy profiles depend only on the ionic strength (eq 29). At pH 6.0, the surface is highly charged, and the type of electrolyte affects the interaction profile strongly. The familiar exponential energy profile is valid over a wide range of separation distances only in the case of monovalent electrolyte. One observes marked deviations in asymmetric electrolytes, and these deviations become more pronounced with increasing valence. The deviations are most striking for multivalent co-ions, whereby the resulting energy profile becomes strongly nonexponential and features a sigmoidal shape in the semilogarithmic plot shown. The reason for this behavior is that the interaction is determined by the sole presence of counterions between the surfaces, which leads to an energy profile following approximately WD ∝1/h at intermediate distances.2,75 Moreover, the DH asymptote sets in only at large distances. Deviations can also be observed when the multivalent ions are the counterions, whereby the interactions are strongly reduced by screening. In this case, the DH asymptote sets it earlier. The latter scenario involving multivalent counterions is more hypothetical, however, because such multivalent counterions will adsorb to an oppositely charge substrate strongly. Therefore, they will reduce the diffuse layer potential and the interaction energy will decrease, rather resembling the DH situation shown in Figure 11a. Figure 11 further illustrates the importance of charge regulation for different surface charge densities. Although these effects are obvious for the weakly charged system shown in Figure 11a, they are weaker in the more highly charged system shown in Figure 11b. The same trend was also evident in Figure 8. When the system is very weakly charged, doublelayer interactions become negligible and therefore regulation effects become unimportant. Therefore, one should bear in mind that charge regulation effects are most relevant at intermediate charge densities whereas they become less important for highly charged surfaces. Experimental interaction energy profiles in the symmetric geometry and in the presence of multivalent ions are shown in Figure 12. Such energy profiles can be well described by DLVO theory, except at short distances. We focus first on situations where the multivalent ions represent the counterions. In this case, the charges of the multivalent ion and of the substrate have opposite signs. Charge regulation effects are important for negatively charged sulfate latex particles in the presence of trivalent La3+ ions (Figure 12a).80 A similar situation is encountered when cationic multivalent aliphatic amines and sulfate particles are used (Figure 12b).57 As a rule, the regulation parameters tend to be smaller in the presence of multivalent counterions, typically p < 0.2. These values are surprising because surface ionization models for indifferent multivalent ions rather predict values close to CC conditions. The experimentally observed pronounced charge regulation is possibly caused by the stronger adsorption of these ions and respective shifts of the adsorption equilibrium upon approach. At shorter distances, one observes the presence of an additional short-range non-DLVO attraction, which can be quantified with an additional exponential attraction with a range of 1.0 nm (eq 23). When the fitting of the energy profiles yields small regulation parameters, meaning that CP conditions become a good approximation, the error in the inner-layer capacitance becomes substantial, and one may not even be able to find a

Figure 12. Comparison of experimental surface interaction energies in the presence of multivalent ions (points) with DLVO calculations invoking the constant regulation (CR) approximation (solid lines) in the symmetric geometry. The gray region is delimited with profiles corresponding to CC and CP conditions. Effects of additional nonDLVO exponential attraction are also illustrated (dashed lines). (a) Sulfate latex surfaces in a 0.2 mM LaCl3 solution of pH 4.0, leading to ψD = −13 mV and p = 0.19.80 (b) Sulfate latex surfaces in a 0.14 mM solution of linear triethylenetetramine (N4) at pH 4.0 with ψD = −20.1 mV and p = 0.15.57 (c) Amidine latex surfaces across a 0.5 mM LaCl3 solution of pH 4.0 leading to ψD = +64 mV and p = 0.31.80 (d) Silica particle surfaces in a 0.5 mM K4Fe(CN)6 solution of pH 10.0 leading to ψD = −50 mV and p = 0.66.75 Attraction at high salt concentrations and at the charge neutralization point for (e) sulfate latex57 at pH 4.0 and (f) silica74 at pH 5.5. Hamaker constants used are (a−c) 3.5 × 10−21, (d) 3.3 × 10−22, (e) 3.5 × 10−21, and (f) 6.0 × 10−21 J, together with a shift of the van der Waals plane for (d) 0.85 and (f) 1.1 nm. The non-DLVO short-range exponential profile is characterized by a decay length and an amplitude of (a) 1.0 nm and −0.16 mN/m, (b) 1.0 nm and −0.22 mN/m, (d) 0.30 nm and +0.15 mN/m, (e) 1.0 nm and −0.80 mN/m, and (f) 1.0 nm and −0.19 mN/ m. In (c), this interaction is negligible.

sensible estimate for this quantity. One may encounter similar difficulties when the profiles are close to the CC case. The second row of Figure 12 shows energy profiles in situations where the multivalent ions are co-ions.75 In this case, the charge of the multivalent ion and of the substrate has the same sign. This situation is exemplified by two oppositely charged systems, namely, amidine latex particles in the presence of La3+ ions and silica particles in the presence of Fe(CN)64− ions. In both cases, one observes a nonexponential repulsive energy profile. The exponential DH behavior sets in at only relatively long distances. This characteristic dependence was discussed in Figure 11b and is caused by the exclusion of the multivalent ions from in between the surfaces. Effects of charge regulation are relatively minor under these circumstances. 391


Langmuir


Figure 13. Interactions between an amine and iron oxide surface at an ionic strength of 10 mM in the asymmetric geometry. Full regulation results obtained from the respective surface ionization models (full line) are compared to the CR approximation (dashed line). The CC and CP conditions are shown for comparison. The amine surface is always positively charged. (a) Situation at pH 10.0, where the iron oxide surface is negatively charged. In this situation, one has ψD = +26 mV and p = 0.59 for the amine surface and ψD = −63 mV and p = 0.24 for the iron oxide surface. (b) Situation at pH 8.7, where the iron oxide surface is neutral at large separations. Here, the surface parameters are ψD = +58 mV and p = 0.66 for the amine surface and ψD = 0 mV and p = 0.14 for the iron oxide surface. (c) Situation at pH 8.0, where the iron oxide surface is positively charged but undergoes charge reversal upon approach. Here, one has ψD = +69 mV and p = 0.81 for the amine surface and ψD = +35 mV and p = 0.17 for the iron oxide surface. For the amine surface, the diffuse layer model with reaction 10 was used with parameters pK = 10 and Γ(0) = 0.15 nm−2, and for the iron oxide surface, the basic Stern model was used with reaction 17 and with pK = 8.7, Γ(0) = 6.15 nm−2, and CS = 1.6 F/m2. The insets show schemes of the respective systems.

fluctuations are related to charge regulation effects because they will be larger when the surface can regulate its charge more easily. This type of interaction was initially proposed by Kirkwood and Shumaker for proteins and attracted renewed interest recently.35,82,85 These phenomena may also occur simultaneously, and they could be nonadditive.86

On the basis of the examples shown in Figure 12a,b, we have already remarked on the presence of an additional short-ranged non-DLVO attraction. This additional interaction can be visualized more clearly at the charge reversal point. Figure 12e,f compares energy profiles in two systems, namely, sulfate latex particles in the presence of tetravalent oligomeric amine cations57 and silica particles in the presence of different trivalent metal cations.74 In monovalent salt solutions, the energy profile follows the van der Waals interaction. At the charge neutralization point, however, the energy profile is more attractive. This energy profile can be well fitted with an additional exponential law given in eq 23, with a range of 1.0 nm and an amplitude of −0.80 mN/m for the sulfate latex particles and −0.19 mN/m for the silica particles. Similar nonDLVO attraction with a range of 1−3 nm has been observed in other systems containing multivalent counterions.50,74 The origin of this non-DLVO attraction is currently not quite clear to us, but it could be related to ion−ion correlation effects, charge heterogeneities, or charge fluctuations.30,31,81,82 A consideration of ion−ion correlations within the primitive model leads to an additional attraction, although the theoretically predicted range is in the subnanometer regime.31,32,83 Surface charge heterogeneities may also induce additional attractive interactions, as has been documented theoretically and experimentally.81,84 Spontaneous charge fluctuations of the surface charge density may also induce charge heterogeneities, which also lead to attractions. Such

5. DOUBLE-LAYER INTERACTIONS IN THE ASYMMETRIC GEOMETRY Let us now address interaction energies in the asymmetric geometry, meaning that the two interacting surfaces are different. Under such circumstances, regulation effects are of major importance. At large separation distances, the superposition approximation suggests the following interaction energy between two differently charged interfaces1 (1) (2) −κh WD = 2ε0εκψeff ψeff e

(33)

(2) where ψ(1) eff and ψeff are the effective potentials of the left and right surfaces, which are obtained from analogous expressions as given in eq 28. In contrast to the symmetric geometry, however, the above relation permits us to determine the sign of the diffuse layer potential of one surface, provided that the sign of the other one is known. However, eq 33 is valid only at long distances, whereas at shorter distances the actual interaction energy may even differ in sign. For low surface charge densities, the analytical

392


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The second row of Figure 13 illustrates the interactions between an amine and iron oxide surface at pH 8.7. At infinite separation, the iron oxide surface is neutral and the amine surface is again positively charged. This situation corresponds to the charged-neutral case. These two surfaces attract by double-layer interactions. The reason for this attraction is that at shorter distances the iron oxide surface acquires a negative charge through the deprotonation of surface groups whereas the surface charge of the amine surface increases even further through its protonation. The diffuse layer potential of the amine surface decreases to below the value of the isolated surface. At contact, the surface charge densities of both surfaces again compensate for each other, and the two diffuse layer potentials become equal. This feature may lead to different signs of the diffuse layer potential and of the surface charge density at shorter distances. This behavior critically depends on the regulation parameter of the iron oxide surface, which is p = 0.14. This value is below 1/2; therefore, the surfaces attract. If the surfaces satisfy the CC conditions, then the interaction is repulsive, whereas in the CP case one obtains an even stronger attraction. The third row of Figure 13 exemplifies the situation of the amine and iron oxide surfaces at pH 8.0. At infinite separation, both surfaces are positively charged, but the magnitude of the surface charge density of the iron oxide surface is lower than that of the amine surface. Because the sign of the surface charge of both surfaces is the same, they repel at larger distances. However, this repulsion switches to an attraction as the iron oxide surface undergoes a charge reversal upon approach. The iron oxide surface becomes neutral at a distance of 2.8 nm, and at shorter distances, it acquires a negative charge. In this situation, the surfaces are now oppositely charged, which induces an attraction. This interplay between repulsion at longer separation distances and attraction due to charge reversal at shorter distances generates the characteristic maximum in the interaction energy profile. Again, the nature of this transition critically depends on the value of the regulation parameter. For CC conditions, the surfaces always repel, but for CP conditions, the attraction is even stronger at shorter distances. These examples further illustrate that the CR approximation also provides an excellent description of the actual interaction profiles in the asymmetric geometry, basically down to contact. Although the actual distance dependence is more complex than in the symmetric geometry, the CR approximation can again be used with confidence. The reliability of the CR approximation in the asymmetric situation has also been confirmed in other situations.77 Experimental Results in Monovalent Electrolytes. These features can be illustrated with experimental energy profiles in the asymmetric geometry in monovalent electrolyte solutions displayed in Figure 14. The top row shows surface energies measured with the colloidal probe between two different types of particles, namely, sulfate and amidine latex.19 Figure 14a illustrates the situation at pH 4.0, where the sulfate particles are negatively charged and the amidine particles are positively charged. Under these conditions, the surfaces attract and the energy profiles can be well quantified with DLVO theory. However, charge regulation effects are important. This situation is similar to the case shown in Figure 13a. The same system illustrates the charged-neutral case at pH 5.6 (Figure 14b). In this situation, the amidine surface becomes neutral, probably because of the presence of residual sulfate

expressions from DH theory within the CR approximation are known.26 Because these expressions are relatively complicated, let us illustrate the essential behavior with a simpler situation, namely, where one of the isolated surfaces is neutral. We refer to this situation as the charged-neutral case. At long separation distances, only the charged surface forms a diffuse layer. When the surfaces approach, this diffuse layer is squeezed by the neutral surface, which leads to double-layer interactions. When the neutral surface remains neutral upon approach (CC conditions), the interaction will be repulsive because of the ionic confinement between the plates. When the neutral surface is able to easily regulate its surface charge (CP conditions), it will acquire a surface charge of sign opposite to that of the charged surface, and the interaction can become attractive. This qualitative picture can be confirmed by considering the actual interaction energy obtained from DH theory26 WD = 2ε0εκ[ψD(1)]2

(2p2 − 1)e−2κh 1 + (2p1 − 1)(2p2 − 1)e−2κh

(34)

ψ(1) D

where and p1 are the diffuse layer potentials of the left (charged) interface and p2 is the regulation parameter of the neutral surface with ψ(2) D = 0. All of these quantities refer to isolated surfaces. At long separation distances, the interaction energy behaves as WD ∝ (2p2 − 1)e−2κh. Therefore, the energy profile is repulsive for p2 > 1/2 whereas it becomes attractive for p2 < 1/2. Another unusual feature of this double-layer interaction is that the energy profile decays with half the Debye length. The superposition approximation given in eq 33 incorrectly suggests that the interaction vanishes. In fact, that approximation represents only the large distance term, which indeed vanishes in the charged-neutral case. Model Results in Monovalent Electrolytes. Let us exemplify the nature of double-layer interactions in the asymmetric geometry with model calculations first. Figure 13 shows the dependence on surface separation distance for various quantities characterizing interacting amine and iron oxide surfaces at an ionic strength of 10 mM for three different pH values shown in the three rows. Leftmost column 1 displays the interaction energy. Columns 2 and 3 show the diffuse layer potential and surface charge density of the amine surface, and columns 4 and 5 show the corresponding quantities for the iron oxide surface, respectively. The first row of Figure 13 illustrates the situation at pH 10.0. At infinite separation, the two surfaces are oppositely charged, and the interactions are attractive at longer distances. At shorter distances, however, the CR and CP conditions remain attractive, while the CC condition leads to repulsion. For the amine surface, the surface charge density increases upon approach because of protonation of the surface amine groups. At the same time, the diffuse layer potential decreases. For the iron oxide surface, the surface charge density decreases because of deprotonation, and the diffuse layer potential increases at first but decreases close to contact. The degrees of protonation of the respective surfaces adjust such that the charge densities of the two surfaces precisely compensate for each other and the two diffuse layer potentials become the same. Obviously, the diffuse layer potential remains constant in the CP case, but in the CC case, the charge density remains constant. Moreover, profiles of the diffuse layer potential and of the surface charge density for the CR conditions do not necessarily lie in between the CC and CP conditions. The latter features differ from those for the symmetric geometry. 393


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boundary conditions. Figure 14d shows the energy profile between a silica surface and a self-assembled monolayer (SAM) on gold whose potential was adjusted by means of a potentiostat.6 The surface functional groups are OH groups originating from aliphatic alcohols incorporated into the SAM. At long separation distances, the surfaces are oppositely charged. Because the surfaces regulate their charge only weakly, the energy profile becomes repulsive at shorter separation distances. The weak regulation is probably caused by the charge segregation within the SAM layer. Naively, one would expect that a metal surface connected to a potentiostat should follow CP conditions. However, this assumption is obviously not satisfied in the present system and equally not in others.42 The present SAM system also illustrates that inert surfaces without ionizable groups may accumulate substantial charges by means of ion adsorption from solution. Multivalent Electrolytes. Charge regulation is equally important in systems containing multivalent ions in the asymmetric geometry.80,87−89 These ions adsorb strongly to an oppositely charged surface; therefore, they reduce the surface charge and may induce a charge reversal. Close to this charge reversal point, one of the surfaces will be weakly charged or even neutral, and in such circumstances, charge regulation effects are expected to be substantial. Figure 15 summarizes typical energy profiles measured with the colloidal probe technique and provides a comparison with

Figure 14. Comparison of experimental interaction energy profiles in the asymmetric geometry in monovalent salt solutions (points) with calculations based on DLVO theory. The gray region is delimited with profiles corresponding to CC and CP conditions. Interaction between amidine and sulfate particles compared to the constant regulation (CR) approximation (solid lines) at (a) pH 4.0 and an ionic strength of 0.2 mM with ψD = +70 mV for amidine and −45 mV for sulfate and (b) pH 5.6 and an ionic strength of 0.2 mM with ψD = 0 mV for amidine and −54 mV for sulfate. The regulation parameters are 0.41 for amidine and 0.33 for sulfate.19 (c) Interaction between silica and mica surfaces coated with poly(2-vinylpyridine) (PVP) at pH 6.5 and an ionic strength of 0.1 mM with ψD = −95 mV for silica and −22 mV for the PVP surface and CP conditions (p = 0).18 (d) Interaction between silica and a SAM-coated gold electrode under potentiostatic control at pH 8.0 and an ionic strength of 1.2 mM with ψD = −46 mV and p = 0.6 for silica and ψD = +13 mV and p = 1 for gold.6 Hamaker constants used are (a, b) 4.5 × 10−21 and (c) 1.2 × 10−20 J, whereas in (d) the van der Waals interaction is negligible.

groups from the polymerization initiator.19 Nevertheless, the surfaces attract because the amidine surface can regulate its charge as a result of the corresponding regulation parameter of p = 0.41. This situation resembles the one discussed in Figure 13b. In this particular system, the corresponding interaction profiles in the two symmetric geometries (i.e., pairs of two amidine particles or two sulfate particles) could be equally realized, and the respective energy profiles could be quantified with DLVO theory with the same parameters as in the asymmetric geometry.19 When only interaction profiles in the asymmetric geometry are available, the respective parameter determination becomes nontrivial. Within the CR approximation, one needs two parameters to describe each surface, namely, the diffuse layer potential and the regulation parameter, which leads to four unknown parameters in the asymmetric geometry. In our view, the most reliable strategy is to determine two of these parameters from the interaction profiles in the symmetric geometry and then obtain the remaining ones in the asymmetric geometry. Figure 14c shows the measurement of an energy profile at pH 6.5 between a silica surface and mica coated with poly(vinylpyridine) (PVP).18 Under these conditions, both surfaces are negatively charged, and the interaction profile can be explained by strong charge regulation, which is close to CP

Figure 15. Comparison of experimental interaction energy profile data in asymmetric geometry in salt solutions containing multivalent ions (points), with calculations based on DLVO theory invoking the constant regulation (CR) approximation (solid lines). The gray region is delimited with profiles corresponding to CC and CP conditions. Interactions between amidine and sulfate latex particles at pH 4.0 (a) in a 0.2 mM LaCl3 solution with ψD = +50 mV and p = 0.41 for amidine and ψD = −12 mV and p = 0.19 for sulfate and (b) in a 0.1 mM K4Fe(CN)6 solution with ψD = −7.0 mV and p = 0.12 for amidine and ψD = −72 mV and p = 0.33 for sulfate.80 Interactions between a silica particle and an amine-functionalized silica surface (c) in 0.1 mM CoC12H30N8Cl3 solution88 at pH 6.5 with ψD = +1.1 mV and p = 0.11 for silica and ψD = +12 mV and p = 0.80 for the amine surface and (d) in a 0.05 mM K4Fe(CN)6 solution87 of pH 6.0 with ψD = −50 mV and p = 0.50 for silica and ψD = −11 mV and p = 0.15 for the amine surface. The Hamaker constants used are (a, b) 3.5 × 10−21 and (c, d) 5.0 × 10−22 J. 394


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particle deposition, suspension rheology, and membrane permeability.10,21−24,44,91,92 However, the role of charge regulation has not been investigated to any comparable depth as for the situations discussed in sections 4−6. Nevertheless, these aspects are relevant discussed below. Salt-Free Systems. Charging processes in salt-free systems are influenced by ion adsorption and charge regulation effects. A typical system features a lamellar liquid crystal made of charged bilayers containing ionic surfactants. Small ions in such a system neutralize the bilayer charge; therefore, the aqueous phase is charged because it contains only counterions. These charged bilayers repel through double-layer forces and generate an osmotic pressure that can be measured experimentally. When the separation distance between the bilayers is decreased, the pressure will increase. The situation is analogous to the interaction between two charged surfaces discussed above, and the pressure will depend on the regulation characteristics of the surfactant layers. Under CC conditions, which might be realistic for strongly acidic surfactants, the charge of the bilayer will remain constant, and the pressure will increase strongly with decreasing separation distance. When the charge is easily regulated, as one would expect for fatty acid bilayers, for example, the pressure profile should be less repulsive. However, the quantitative treatment of this situation has to be adapted to the situation without (or with little) added salt, whereby the respective pressure or interaction energy profiles are modified. In particular, the interaction energy no longer decays exponentially but rather as a power law.2,75 A dialyzed suspension of charged nanoparticles in water without added salt will behave similarly.21−23 In this situation, the aqueous phase contains only counterions. A salt-free suspension of charged particles will further generate a substantial osmotic pressure. This pressure will depend on the particle concentration but equally on the regulation characteristics of the surface. A quantitative description of salt-free suspensions is challenging because of the many-body nature of the problem.93 However, one may resort to the so-called cell model whereby one considers a single particle enclosed in a spherical cell that contains the neutralizing counterions.23,94 The interactions within the cell are then treated on the PB or primitive model level, and those between particles are taken into account on the DH level by means of an effective charge that is obtained from the long-distance asymptote. This effective charge depends on particle concentration and will be influenced by charge regulation effects. These effects will therefore also modify the phase behavior of charged colloidal particles, their crystallization kinetics, or their sedimentation processes.21,22,95 Particle Homoaggregation. The aggregation of colloidal particles is directly determined by interaction forces between these constituents; therefore, regulation effects are relevant as well.10,92 When the same (or similar) particles form aggregates, one refers to homoaggregation and one deals with the symmetric geometry. DLVO theory is capable of rationalizing the characteristic trends reasonably well, sometimes even quantitatively. One finds that at low ionic strength or in highly charged systems the aggregation rates are slow, whereas at high ionic strength or in weakly charged systems the aggregation is rapid. The transition between the two regimes is rather sharp and is referred to as the critical coagulation concentration (CCC). The effect of charge regulation on the aggregation of particles was analyzed for different systems, including carboxylated latex and silica.10,44 The aggregation rate decreases with the decreasing regulation parameter, but the difference

DLVO theory. In these situations, it is essential to solve the PB equation for an asymmetric electrolyte, possibly even for the appropriate mixture of asymmetric and monovalent electrolytes, and take solution speciation into account. The top row shows results obtained with amidine and sulfate particles.80 When this system is studied in a solution containing 0.2 mM La3+ ions, the surfaces are oppositely charged at large separation distances and the energy profile is attractive as shown in Figure 15a. However, comparison with the calculation shows that the amidine surface regulates the charge substantially. In a solution containing 0.1 mM Fe(CN)64− ions, the amidine surface undergoes a charge reversal due to the adsorption of Fe(CN)64− ions. This situation is illustrated in Figure 15b, where both surfaces are negatively charged at large separation distances. For this reason, one observes repulsion at larger distances, but as the surfaces approach, the interactions become attractive. This attraction is caused by a charge reversal because Fe(CN)64− desorbs upon approach and the amidine surface attains a positive charge. Similar situations were also realized with a silica particle and a planar amino-functionalized silica substrate.87,88 The aminofunctionalized silica surface is positively charged. In a solution of 0.1 mM trivalent cationic cobalt complex [CoC12H30N8]3+, the silica surface becomes neutralized through the adsorption of this complex. The experimental results illustrated in Figure 15c show an attractive profile.88 This attraction is primarily due to charge regulation because the calculation shows that the CC conditions lead to repulsion. In a solution containing 0.05 mM Fe(CN)64−, the aminofunctionalized surface becomes negatively charged through the adsorption of this species.87 The interaction energy profile is shown in Figure 15d. Because both surfaces are negatively charged, they repel at longer distances. Upon approach, the amino-functionalied surface undergoes a charge reversal back to its positive charge, which induces an attraction at shorter distances. Besteman et al.87 interpreted the latter data with an empirical double-exponential energy profile, which in fact resembles the long-distance expansion of the DH solution in the asymmetric geometry.26 However, they did not discuss the role of charge regulation, and it is comforting to see that the PB model combined with the CR approximation quantifies their data perfectly well. These interaction profiles were also qualitatively rationalized with computer simulations on the primitive model level without assuming any specific interactions between the ions and the substrate.90 All of these examples illustrate that double-layer interactions in the asymmetric geometry can be interpreted with DLVO theory in a straightforward fashion, provided charge regulation effects are taken into account. In contrast to the symmetric geometry, charge regulation is of major importance in the asymmetric setting, especially when one of the surfaces is weakly charged or uncharged. Moreover, the simple CR approximation rationalizes the interaction profiles quantitatively. However, additional non-DLVO interactions seem to be of lesser importance in the asymmetric geometry than in the symmetric one.

6. OTHER PHENOMENA Charge regulation effects have been exemplified with ionic adsorption and the interaction between surfaces. Numerous other situations exist where charge regulation can also be of importance. These include the properties of ionizable surfactant aggregates, salt-free colloidal suspensions, particle aggregation, 395


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the classical boundary conditions of CC and CP can be recovered by setting p = 1 or 0, respectively. Especially in the asymmetric geometry, the effects of charge regulation can be huge, and they may even determine the sign of the interaction energies. In such systems, attempts to rationalize DLVO interactions only in terms of the classical CC and CP conditions appear futile to us. Charge regulation is equally important for ion adsorption to isolated interfaces. Although many researchers focus on the underlying adsorption processes, the charge distribution within the interface is addressed less frequently. However, the adsorption of charged species will modify the charging state of an interface and will necessarily induce variations of the diffuse layer potential or even induce a reversal of its sign. A large suite of surface ionization models have been developed in the past,40,41 and they are used to rationalize ion adsorption to interfaces. Precisely the same models can be used to evaluate interaction energies or pressures. Although some authors have attempted to extract the respective parameters entering the surface ionization models from measured interaction profiles, this approach is plagued by the large number of parameters involved and the insensitivity of the energy profiles to some of them.15−17 That approach could probably be improved by extracting these parameters simultaneously from a series of force curves determined under different solution conditions, but such an approach seems to be a daunting task. We instead advocate the use of the simpler CR approximation, which can be applied to individual force profiles and involves the determination of one additional quantity from the interaction profiles, namely, the regulation parameter. Because this parameter can also be evaluated from surface ionization models for an isolated surface, a direct comparison with such model predictions seems to be more straightforward. Currently, however, simple surface ionization models predict rather different trends for the regulation parameter than the ones observed experimentally. To us it is currently not obvious whether one deals with experimental errors, the presence of additional non-DLVO interactions, or shortcomings of current surface ionization models. The applicability of the PB equation in the presence of multivalent ions should be equally reconsidered. Experimental force profiles in such systems can be well described with PB theory in symmetric and asymmetric geometries down to few nanometers by adjusting the diffuse layer potential and the regulation parameter for the respective surfaces.80 Computer simulations of the primitive model suggest that PB theory breaks down in such systems, when the actual surface parameters are being considered.28,30,31,33 We suspect, however, that PB theory represents an excellent approximation at longer distances from the surface, provided that appropriate effective parameters are used. Although this point of view also seems to be supported by some theoretical studies,29,32 more detailed investigations would be needed to address the range of validity of the PB approach in the presence of multivalent ions. In particular, this question could be further clarified by a detailed comparison between experimental force profiles and primitive model calculations.

between CC and CP conditions remains relatively modest. This aspect becomes particularly relevant because the experimental errors in the aggregation rates are often substantial, and the scatter of such data points may extend through the CC and CP conditions. Moreover, the interpretation of particle aggregation rates is further obscured by the influence of surface charge heterogeneities, which may enhance the rate of aggregation substantially.96 Multivalent ions influence aggregation rates strongly. The classical Schulze−Hardy rule states that multivalent counterions of valence z lead to a shift in the CCC described by the empirical relation1

CCC ∝ z −n

(35)

where n ≃ 6 for highly charged surfaces. Experimental CCCs indeed decrease strongly with increasing valence, but the dependence is typically weaker, often rather approaching the DH result for weakly charged surfaces with n ≃ 2. The fact that multivalent counterions may adsorb specifically may complicate the situation further. For multivalent co-ions and highly charged surfaces, one has n ≃ 1 and the dependence is weaker.97 Although the aggregation rates in the presence of multivalent ions must also be influenced by charge regulation effects, little is known on this aspect. Particle Heteroaggregation and Deposition. When two different colloidal particles aggregate in solution, one refers to heteroaggreagtion. Particle deposition on a planar substrate can be viewed as a special case of heteroaggreagtion in which one of the particles is much smaller than the other one.92 In these systems, the asymmetric geometry dictates the relevant interactions, meaning that the two surfaces involved are different. In this case, one of the surfaces could be weakly charged, which is a condition where charge regulation effects will be important. The dependence of the heteroaggregation or deposition rate on the ionic strength was shown to be qualitatively different for different boundary conditions. Under CC conditions, the aggregation rate may strongly decrease with decreasing ionic strength and feature a CCC. Under CP conditions, however, the aggregation rate may increase with decreasing ionic strength slowly. In fact, the CCC may be directly related to the regulation parameter. The precise behavior critically depends on the regulation characteristic of the surfaces in question. Although some experimental results for heteroaggregation and deposition rates exist,98,99 the surfaces involved are often not very well characterized. Although these phenomena are expected to be strongly influenced by charge regulation effects, the available data do not allow us to address these aspects in detail. Interestingly, charge regulation was proposed to represent the principal mechanism responsible for the adsorption of nanoparticles in the so-called halo stabilization.91

7. OUTLOOK This article emphasizes the importance of charge regulation within the electrical double layer. Charge regulation is essential in the interpretation of force or pressure profiles involving interfaces and/or particles. The well-known case of a CP boundary condition used to rationalize such interaction profiles is one special case of charge regulation conditions, and the effects of charge regulation on such interaction profiles can be more complex. We advocate the use of the CR approximation, which uses a single regulation parameter p to quantify these effects. This parameter is a property of the isolated surface, and

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 396


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Biographies

Michal Borkovec is a full professor in the Department of Analytical and Inorganic Chemistry at the University of Geneva, Switzerland. He received his Ph.D. in chemical physics at Columbia University in 1986 and thereafter worked as a lecturer at the Swiss Federal Institute of Technology (ETH) in Zürich. In 1998, he was appointed as an associate professor at Clarkson University and was promoted to full professor in 2000. He moved to Geneva shortly thereafter. His research focus is the physical chemistry of colloids, surfaces, and polymers. He is also interested in applications of these topics in environmental engineering and industrial process control. He is an elected member of the Swiss National Research Council and the Swiss Academy of Engineering Sciences.

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ACKNOWLEDGMENTS We thank Klemen Bohinc, Tjisse Hiemstra, Peter Kralchevsky, Christophe Labbez, and Rudi Podgornik for useful discussions and comments. This research was supported by the University of Geneva, Swiss National Science Foundation through awards 140327 and 159874 and the U.S. National Science Foundation through awards 1134398 and 1160138.

Gregor Trefalt is a senior lecturer in the Department of Analytical and Inorganic Chemistry at the University of Geneva. He obtained his Ph.D. in 2012 at the Jožef Stefan Institute in Ljubljana, Slovenia, in the area of colloidal processing and ceramic materials. Subsequently, he moved to the University of Geneva, where he first worked as a postdoctoral research associate and was then promoted to his current position. His research focuses on colloidal interactions and particle aggregation in aqueous solutions, ionic liquids, and nonpolar media.

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ABBREVIATIONS AFM atomic force microscope CC constant charge CCC critical coagulation concentration CP constant potential CR constant regulation DH Debye−Hückel DLVO Derjaguin, Landau, Verwey, and Overbeek PB Poisson−Boltzmann PVP poly(vinylpyridine) SAM self-assembled monolayer SFA surface forces apparatus REFERENCES

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