1-pK and 2-pK Protonation Models in the ... - ACS Publications

J. Phys. Chem. B 2001, 105, 9755-9771

9755

1-pK and 2-pK Protonation Models in the Theoretical Description of Simple Ion Adsorption at the Oxide/Electrolyte Interface: A Comparative Study of the Behavior of the Surface Charge, the Individual Isotherms of Ions, and the Accompanying Electrokinetic Effects Wojciech Piasecki and Władysław Rudzin´ ski* Laboratory for Theoretical Problems of Adsorption, Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Ul. Niezapominajek, 30-239 Krako´ w, Poland

Robert Charmas Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie Skłodowska UniVersity, Pl. Marii Curie Skłodowskiej 3, 20-031 Lublin, Poland ReceiVed: April 9, 2001; In Final Form: June 30, 2001

A number of adsorption models have been considered corresponding to various combinations of the 1-pK and 2-pK charging models, with various models of the oxide/electrolyte interface structure. The corresponding theoretical descriptions have been developed, along with the relations between the equilibrium constants, suggested by the appearance of the common intersection point. The equations developed for various adsorption models have been applied then to analyze the behavior of the surface charge, individual isotherms of ions, and electrokinetic data reported by Sprycha for the anatase/NaCl solution system. No decisive proof has been deduced in favor of either the 1-pK or 2-pK charging mechanism, but certain important conclusions have been drawn concerning the applicability of various adsorption models. It was shown that to arrive at a reasonable theoretical description, the assumed model of surface charging must necessarily be combined with a certain model of the oxide/electrolyte interface structure.

Introduction The models of surface complexation are the most commonly applied ones to represent ion adsorption at the oxide/electrolyte interfaces. A lot of such models have been employed. Their construction can differ by assuming different charging mechanisms of surface sites and the distribution of ions in the interfacial region. As far as the charging mechanism is concerned, the two models called the 1-pK and 2-pK charging models are most frequently used. Both these models have some common assumptions, of which the most important are (1) the colloidal particle is represented by a planar surface, (2) electrostatic properties are calculated by assuming that the ion species are point charges, and (3) the equilibria for the formation of complexes between the surface sites and the solution species can be described in exactly the same way as for those composed solely of solution species. The main difference between them lies in the definition of the amphoteric products of the surface hydrolysis uniformly (homogeneously) distributed over the planar surface. The 2-pK model developed by Yates et al.,1 Chan et al.,2 and Davis et al.3,4 assumes that the products of surface hydrolysis can be in one of three protonation states (zero, one, or two protons per site). A group of scientists at the Wageningen Agricultural University (The Netherlands)5-8 have proposed a somewhat different charging model of the surface oxygens. They invoke Pauling’s principle9 of local neutralization of charge to show that the protonated surface oxygens will, in * To whom correspondence should be addressed at the Maria Curie Skłodowska University. Phone: (48) (81) 5375633. Fax: (48) (81) 5375685. E-mail: [email protected].

general, have fractional values of charge and that practically only one protonation step will be observed. That hypothesis is commonly called the 1-pK charging model. Moreover, they argue that various surface oxygens may have different fractional values of charge depending on their local coordination to the cations. That model is known in the literature as the multisite complexation (MUSIC) model of the oxide electrolyte interface.8 For almost the whole past decade, the 2-pK and 1-pK models have been considered as competitive ones corresponding to different real physical situations. The discussion between the two groups of scientists, one preferring the 2-pK model, and the other group preferring the 1-pK model, focused on both the reliability of the assumed physical situations and the advantages of the related theoretical descriptions. The breakthrough came in 1997 with the publication of theoretical papers by Borkovec.10,11 He has shown that the 2-pK and 1-pK approaches represent simply two approximate solutions to a many-body problem, which were obtained by rigorous treatment of the adsorption of ions at the oxide/electrolyte interface. The physical pictures behind these two approaches should not be treated as fully realistic, as was done in the past, but merely as tools to arrive at approximate, simple solutions. Thus, while using the term 1-pK or 2-pK model in our forthcoming considerations, we will have in mind rather two different 1-pK and 2-pK approaches. Theoretically, the 2-pK approach should be the more accurate one, but Borkovec’s model calculations suggested that for the majority of oxide/electrolyte systems the 1-pK approach should be almost as good as the 2-pK one. The reported analyses of the experimental titration isotherms seemed to provide firm support for Borkovec’s suggestions.12

10.1021/jp011299q CCC: $20.00 © 2001 American Chemical Society Published on Web 09/14/2001

9756 J. Phys. Chem. B, Vol. 105, No. 40, 2001 However, while comparing the 1-pK and 2-pK models, the authors considered above all the behavior of the surface charge isotherms. Meanwhile, it has been known for a long time that this observable is not sensitive to the choice of a particular approach (model) to represent its behavior. The same titration isotherm can be equally well represented (fitted) by many theoretical expressions corresponding to various approaches. This situation raised even serious doubts concerning the applicability of the surface complexation models. Sposito,13 for instance, wrote “... the surface complexation models are, in a sense, too successful, i.e., several different models can represent the same set of adsorption data equally well with corresponding chemical parameters in the model taking quite different values”. The surface charge isotherm is a composite one, so there will occur several compensating effects. The differences between various adsorption models (approaches) are much more clearly demonstrated in the behavior of other observables as has been shown, for instance, in the recent papers published by Rudzin´ski and co-workers.14-16 So, fitting simultaneously the experimental data obtained from many experimental sources (techniques) should create a better chance to draw certain conclusions about the applicability of various theoretical approaches. However, not many papers of that kind have been published so far. In the vast majority of the published papers titration isotherms are still employed as the sole experimental source of information. Thus, despite the essential progress made by Borkovec’s works for our better understanding of these adsorption systems at present, we see a necessity of further fundamental studies. For instance, Borkovec’s predictions should be fairly accurate at small surface charges, i.e., in the vicinity of the point of zero charge (PZC). Also, the presence of the ions of the basic electrolyte has not explicitly been considered. Their presence is very essential in all the theoretical descriptions based on the surface complexation models. The above-mentioned uncertainties do not invalidate Borkovec’s fundamental results, but suggest additional investigations to be carried out. Even if 1-pK and 2-pK approaches might equally well account for the fundamental features of these adsorption systems, there remains still the problem of a proper use of these two theoretical tools. Namely, both the 1-pK and 2-pK approaches can result in a variety of theoretical descriptions obtained by assuming various mechanistic models of the structure of the solid/ electrolyte interface, which is shown in the present paper. We also show that one may arrive at different theoretical descriptions by choosing the way in which certain fundamental functions are evaluated. The purpose of the present paper is to discuss a few theoretical descriptions which one may have to consider for a given adsorption system. Some of them are next analyzed in more detail by studying their ability to represent a well-defined experimental systemsthe anatase/NaCl solution interface. That analysis leads to some conclusions and recommendations concerning the use of the surface complexation models. THEORY 1. Theoretical Description Based on the 2-pK Charging Model and the Triple-Layer Model of the Oxide/Electrolyte Interface. Below we briefly stress the assumption of the popular 2-pK surface protonation model combined with a three-layer electrostatic model3,4 of the oxide/electrolyte interface. The schematic picture of the triple-layer model (TLM) is shown in Figure 1.

Piasecki et al.

Figure 1. Diagrammatic presentation of the 2-pK model with three charged planes (TLM): ψ0, δ0, the surface potential and the surface charge density in the 0-plane; ψβ, δβ, the potential and the charge of the β-plane formed by the adsorbed ions (cations C+ and anions A-) of the basic electrolyte; ψd, δd, the potential and the charge diffuse layer; c1, c2 the electrical capacitances, assumed to be constant in the regions between the planes.

According to this complexation model, the potentialdetermining ions H+, the cations C+, and the anions A- of the basic electrolyte form the following surface complexes: SOH0, SOH2+ , SO-C+, and SOH2+A- , where S stands for the surface metal atom. The concentrations of these complexes on the surface are denoted by [SOH0], [SOH2+], [SO-C+], and [SOH2+A-], respectively. [SO-] is the surface concentration of the free sites (unoccupied surface oxygens). According to the mass action law, the following equations and the related equilibrium constants describe the reactions occurring at the oxide/electrolyte interface: Ka1int

SOH2+ 798 SOH0 + H+ Kalint ) Ka2int

{

}

{

}

[SOH0]aH eψ exp - kT0 (1a) [SOH2+]

SOH0 798 SO- + H+ Ka2int )

[SO-]aH eψ exp - kT0 [SOH0]

(1b)

*KCint

SOH0 + C+ 798 SO-C+ + H+ [SO-C+]aH e(ψ0 - ψβ) *KCint ) exp (1c) kT [SOH0]aC

{

}

{

}

*KAint

SOH2+A- 798 SOH0 + H+ + A*KAint )

[SOH0]aHaA e(ψ0 - ψβ) (1d) + - exp kT [SOH2 A ]

where aH is the proton activity in the equilibrium bulk solution and aA and aC are the activities of the anion and cation in the bulk phase. In Figure 1 the quantities c1 and c2 are the electrical capacitances for the capacitors connected in series, assuming that the capacitances are constant within the regions between the planes. As can be deduced from Figure 1, the following relations hold between the charges and the potentials within the compact layer and the diffuse layer:

Ion Adsorption at the Oxide/Electrolyte Interface

ψd )

[

{ } { } { }

δ0 c1

(2a)

K0aH exp -

eψ0 θ0 ) kT θ-

(8a)

-δd c2

(2b)

K+aH2 exp -

2eψ0 θ+ ) kT θ-

(8b)

(2c)

KCaC exp -

eψβ θC ) kT θ-

(8c)

ψ0 - ψβ ) ψ β - ψd )

J. Phys. Chem. B, Vol. 105, No. 40, 2001 9757

(

-δd δd 2kT + ln +1 1/2 |z|e (8 kTI) 80rkTI 0 r 2

)] 1/2

where r is the relative permittivity of the solvent (for water at T ) 25 °C, r ) 78.25), 0 is the permittivity of free space (0 ) 8.854 × 10-12 F/m), and I is the ionic strength of the solution (ions/m3). Equation 2c follows the theory of the diffuse electric double layer. Introducing the fractional surface coverages θi by the individual surface complexes (i ) 0, +, C, A) and free sites (i ) -)

{

KAaH2aA exp -

θi )

θ0 ) [SOH ]/Ns θ+ ) [SOH2 ]/Ns θA ) [SOH2+A-]/Ns θC ) [SO-C+]/Ns -

θ- ) [SO ]/Ns ) 1 -

∑iθi

(i ) 0, +, A, C)

(3)

Kifi 1+

∑iKifi

{

f0 ) exp -

[SO-C+] + [SOH2+A-]

(4)

H+

The equations neglect the share of the free ions in the plane of charge δ0, as well as the share of free ions C+ and Ain the plane β, but eqs 1 point to the presence of such ions at the surface in the compact area of the electric double layer. Within that compact layer, the electroneutrality condition ∑iδi ) 0 (i ) 0, β, d) must be fulfilled, so according to eqs 4 and 5, the value of the diffuse layer charge δd is given by

δd ) B(θ- - θ+)

(6)

For our further purposes we consider the following equivalent set of reactions leading directly to the formation of all the surface complexes onto the free sites SO-

SO- + H+ T SOH0

(7a)

SO- + 2H+ T SOH2+

(7b)

SO- + C+ T SO-C+

(7c)

SO- + 2H+ + A- T SOH2+A-

(7d)

with the corresponding equilibrium constants K0 ) 1/Ka2int, K+ ) 1/Ka1intKa2int, KC ) *KCint/Ka2int, KA ) 1/Ka2int*KAint, respectively, defined by considering the constants of reactions 1a-d. After certain rearrangements, we obtain the following set of equations corresponding to reactions 7:

eψ0 - 2.3(pH) kT

{

fC ) aC exp -

where e is the elementary charge and Ns is the surface density (sites/m2). The charge coming from the specifically adsorbed ions of the basic electrolyte in the β plane, δβ, is given by

(5)

(9)

}

(10a)

f+ ) f02

the surface charge density, δ0, can be expressed as follows:

δβ ) B(θC - θA)

(i ) 0, +, A, C)

where fi (i ) 0, +, A, C) are the following functions of the activities of protons and salt ions:

Ns ) [SO-] + [SOH0] + [SOH2+] +

δ0 ) B(θ + + θA - θ- - θC) B ) eNs

(8d)

The set of nonlinear eqs 8 must be numerically solved to calculate next the individual isotherms of adsorption of ions θi (i ) 0, +, A, C). For that purpose the set of eqs 8 is transformed into the Langmuir-like form

+

0

}

e(2ψ0 - ψβ) θA ) kT θ-

{

fA ) aA exp -

(10b)

eψ0 eδ0 + kT kTc1

}

(10c)

eψ0 eδ0 - 4.6(pH) kT kTc1

}

(10d)

From the nonlinear set of eqs 9, the following equation is obtained

K+f+ + KAfA - KCfC - 1

δ0 ) B

1+

∑iKifi

(i ) 0, +, A, C)

(11)

the solution of which allows δ0 to be calculated for each value of pH. The nonlinear eq 11 with respect to δ0 can be easily solved using an iteration method. Having calculated δ0 values, one can calculate the individual isotherms of ions from eqs 9. To achieve that goal, we have to know also the relation between the surface potential ψ0 and the pH of the bulk solution. At the beginning of electrochemical studies, it was common to apply the Nernst equation, which predicts a potential change equal to 2.303kT/e V for a pH unit. Later on Bousse and co-workers17,18 proposed the following function

2.303(PZC - pH) )

( )

eψ0 eψ0 + sinh-1 kT βkT

(12)

which reduces to the Nernst-like linear form5 in the vicinity of the PZC

ψ0 )

β 2.3kT (PZC - pH) β+1 e

(13)

and becomes the potential charge corresponding to the Nernstian equation, when β/(β + 1) ≈ 1. The nondimensional quantity β has the following meaning:19

9758 J. Phys. Chem. B, Vol. 105, No. 40, 2001

β)

Piasecki et al.

( )

2e2Ns Ka2int cDLkT K int a1

1/2

(14)

where cDL is the linearized double-layer capacitance, which can be theoretically calculated (depending on the salt concentration in the solution) in the way outlined by Bousse et al.17,20 Equation 12 was obtained17 by applying the TLM but without considering the adsorption of electrolyte ions. It was yet assumed that the difference between the equilibrium constant values Ka1int and Ka2int is high enough (Ka2int/Ka1int , 1) and the diffuse charge δd is small enough; i.e., the properties of the electric double layer are determined by the Stern layer. The latter is the case when the electrolyte concentration is equal to or higher than 0.01 mol/dm3. We used eq 12 to calculate heat effects accompanying ion adsorption14,16 and to compare this approximation with other approaches. Applying the Nernstian potential ψ0(pH) in eq 13 or Bousse’s expression for ψ0(pH) in eqs 12 and 14 is not a necessary condition to calculate the function δ0(pH). It can also be done without applying a priori any theoretical expression for ψ0(pH). Let us remark for that purpose that eq 2 can be written as follows:

ψ0 )

[

(

)]

δ0 δd 2kT -δd δd - + + ln +1 1/2 c1 c2 |z|e (8 kTI) 80rkTI 0 r 2

1/2

(15)

Now, from eqs 10 and 11, ψ0 is expressed as a function of δ0 and pH. Having found the function ψ0(pH,δ0), one can express δd defined in eq 6 solely as a function of pH and δ0; i.e., δd ) δd (pH,δ0). After inserting that function into eq 15, one arrives at the equation F(pH,δ0) ) 0, which is next solved numerically to calculate the function δ0(pH). Having calculated δ0(pH), we can express ψ0(pH,δ0) solely as a function of ψ0(pH), and next calculate easily all the θi from eq 10. We call this version of calculation the A POSTERIORI calculations of ψ0(pH), whereas the version based on applying Bousse’s expression 12 further will be called the BOUSSE version of calculations. The experimental titration curves corresponding to different concentrations of basic electrolyte usually have a common intersection point (CIP) in the PZC. That feature has led us to study the consequences of treating it in a formal mathematical way. A point of zero charge is determined by the condition δ0(pH)PZC) ) 0. In most systems the PZC value does not practically depend on salt concentration in the bulk solution (i.e., a CIP occurs at pH ) PZC). Except for the region of very low salt concentrations, one can assume that aC ) aA ) a. Thus, the independence of the PZC of salt concentration can formally be expressed as ∂(PZC)/∂a ) 0. For the triple-layer model (2-pK), the above condition was applied in our previous theoretical works21,22 to arrive at the relations between the value of the PZC and the surface equilibrium constants:

PZC ) /2(pKa1 + pKa2 ) and 1

int

int

PZC ) 1/2(p*KCint + p*KAint) (16) where

pKaiint ) -(log Kaiint) and p*Kiint ) -(log *Kiint) (i ) C, A) (17)

Figure 2. Diagrammatic presentation of the 1-pK model with three charged planes (TLM): ψ0, δ0, the surface potential and the surface charge density in the 0-plane; ψ1, δ1, the potential and the charge coming from the adsorbed ions (cations C+ and anions A-) of the basic electrolyte; ψd, δd, the diffuse layer potential and its charge; c1, c2, the electrical capacitances, constant within the regions between the planes.

Application of eq 15 allows only two of the four constants for the surface reaction equilibria to be treated as independent parameters. This has a very desired effect of reducing the number of best-fit parameters found by computer by adjusting the theoretical expressions to the experimental data. 2. Theoretical Description Based on the 1-pK Charging Model and the Three-Plane Model of the Oxide/Electrolyte Interface. The 1-pK model was proposed by Bolt and Van Riemsdijk,5 and its general reviews and applications were given by Hiemstra and Van Riemsdijk.6,7 The schematic picture of this model for an oxide surface, on which the proton surface complexes SOH(1/2)- and SOH2(1/2)+ exist is shown in Figure 2. In their works based on the 1-pK model, Hiemstra and Van Riemsdijk6,7 placed cations and anions of the basic electrolyte in the diffuse layer, but assumed that they form ionic pairs with surface proton complexes. While accepting the three-layer model, cations and anions should be placed in layer “1” as shown in Figure 2. We will consider that model of the interface first. Protons, along with the anions and cations of the basic electrolyte, form the surface complexes SOH2(1/2)+, SOH2(1/2)+A-, and SOH(1/2)-C+ on the sites SOH(1/2)-. According to the mass action law 1/1KHint

SOH(1/2)- + H+ 98 SOH2(1/2)+ 1

KHint )

{

}

[SOH(1/2)-]aH eψ0 (18a) (1/2)+ exp - kT [SOH2 ]

1K int C

SOH(1/2)- + C+ 798 SOH(1/2)-C+ eψ [SOH(1/2)-C+] 1 KCint ) exp kT1 (18b) (1/2)[SOH ]aC

{ }

1K int A

SOH2(1/2)+ + A- 798 SOH2(1/2)+AKAint )

1

{

}

[SOH2(1/2)+A-] eψ exp - kT1 (18c) [SOH2(1/2)+]aA

The equivalent reactions leading to the formation of these surface complexes on free sites (SOH(1/2)- in this model) have



the following equilibrium constants: K+ ) 1/1KHint, KC ) 1KCint, and KA ) 1KAint/1KHint for adsorption of protons, cations, and anions, respectively. The last equilibrium constant is obtained from the combination of reactions 18a and 18c, which leads us to the following reaction:

{

fA ) aA exp -

KA

SOH(1/2)- + H+ + A- 798 SOH2(1/2)+AKA )

{

-

}

[SOH2 A ] e(ψ0 - ψ1) exp (18d) (1/2)kT [SOH ]aHaA (1/2)+

where ψ0 is the surface potential and ψ1 is the mean potential at the plane of specifically adsorbed counterions. As can be deduced from Figure 2, relations 2 still hold between the charges and the potentials within the compact layer and the diffuse layer. Using the notation

θ+ ) [SOH2(1/2)+]/Ns θA ) [SOH2(1/2)+A-]/Ns θC ) [SOH(1/2)-C+]/Ns θ- ) [SOH(1/2)-]/Ns ) 1 -

∑iθi

(i ) +, A, C)

(19)

{

fC ) aC exp -

eψ0 eδ0 + kT kTc1

}

eδ0 - 2.3(pH) kTc1

(25b)

}

(25c)

Taking into account eq 20, the nonlinear equation system 24 can be transformed further into the one nonlinear equation

1 K+f+ + KAfA - KCfC - 1 (i ) +, A, C) (26) δ0 ) B 2 1 + iKifi

∑

Expression 12 for ψ0(pH) was developed by Bousse et al. by assuming the 2-pK charging mechanism, so it cannot be applied here any longer. Thus, we must accept the A POSTERIORI way in our calculations. The details of the corresponding calculations are given in the Appendix. While applying the Rudzin´ski-Charmas criterion21,22 for a CIP to exist (in the same way as in section 1) to the 1-pK model for pH ) PZC (δ0 ) 0 and ψ0 ) 0), from eq 26 we arrive at the following form of the function U:

U ) K+H + KAaAH - KCaC - 1 ) 0

(27)

the surface charge δ0 can be expressed in the following form:

H ) 10-PZC

(28)

δ0 ) 1/2B(θ+ + θA - θ- - θC)

Expressing formally the independence of the PZC of salt concentration

Ns ) [SOH(1/2)-] + [SOH2(1/2)+] + [SOH(1/2)-C+] + [SOH2(1/2)+A-]

(20)

where

The charge of the specifically adsorbed ions of the basic electrolyte in their plane is given by

δ1 ) B(θC - θA)

(21)

For the whole compact layer, the electroneutrality condition ∑iδi ) 0 (i ) 0, 1, d) must be fulfilled, so

δd ) /2B(θ- + θA - θ+ - θC) 1

{ }

}

{ }

eδ0 θA ) kTc1 θ-

Kifi 1+

∑iKifi

(23b)

(23c)

(i ) +, A, C)

(24)

where fi (i ) +, A, C) are the following functions of proton and salt activities:

{

f+ ) exp -

eψ0 - 2.3(pH) kT

H ) 1KHint and

}

1

KCint

KAint

)1

(31)

or

We may rewrite equation system 23 to the following Langmuirlike form

θi )

(30)

After the set of eqs 27 and 30 are solved, the following relations are obtained:

(23a)

eψ0 θC eδ0 ) KCaC exp + kT kTc1 θKAaHaA exp -

∂U ) K AH - K C ) 0 ∂a

1

eψ0 θ+ K+aH exp ) kT θ-

(29)

and considering that ∂U/∂(PZC) * 0, we arrive at the following equation:

(22)

Now, we write the conditions for the reaction equilibria 18a, 18b, and 18d in the following way:

{

∂(PZC) ∂U/∂a )0 )∂a ∂U/∂(PZC)

(25a)

PZC ) p1KHint and p1KCint ) p1KAint

(32)

p1Kiint ) -log 1Kiint (i ) H, C, A)

(33)

where

Thus, for the 1-pK model, eq 32 reduces the number of unknown equilibrium constants determined from fitting suitable experimental data from three to one. This feature has not been fully employed by the authors using the 1-pK model in their analysis of experimental data. Reducing the number of free (best-fit) parameters appears to be extremely important in further generalizations of the 1-pK model, introducing a large number of a priori unknown parameters. 3. Developing Relations between the Equilibrium Constants for Generalization of the 1-pK Model Called MUSIC. Generalization of the 1-pK model, taking into account various


Piasecki et al.

types of reactive oxygen groups present on metal oxides, called MUSIC, was proposed by Hiemstra et al.8 The multisite complexation model was based on the principle of local neutralization of charge in ionic crystals as proposed by Pauling.9 In ionic structures the charge of a cation is neutralized by the opposite charges of surrounding anions. Obviously, the charge of an anion is compensated in the same way. If the degree of neutralization of the charge of an anion is expressed per bond, one is led to the definition of a formal bond valence (V) as the charge (Z) of a cation divided by its coordination number (CN). The overall chemical formula for a metal oxide (or hydroxide) can be written as MeO(CN)/n (or Me(OH)(CN)/n), where n is the coordination number of the anion. For example, for alumina hydroxide Z ) 3, CN ) 6, V ) 1/2, n ) 2, and the chemical formula is Al(OH)6/2. On the surface of an ionic crystal, not all of the charge of anions is neutralized by cations of the solid, due to the existence of broken bonds at the interface. When a metal oxide or hydroxide is in solution, the protonation of its surface can neutralize the charge of anions. It can be represented by the two reactions

Men-OnV-2 + H + T Men-OHnV-1

Kn,1

Men-OHnV-1 + H + T Men-OH2nV

Kn,2 (34b)

(34a)

where Kn,1 and Kn,2 denote the affinity constants for the proton adsorption reactions on the two different types of surface groups. The MUSIC model also combines the crystal morphology with different kinds of protonation of surface oxygens. When different crystal planes are developed on an oxide surface, fractions of different coordinated surface species on each of them should be taken into account. The amount of reactive surface oxygens having different coordination numbers depends on the kind of crystal planes exposed to the surface. In their paper, Hiemstra and Van Riemsdijk6 gave examples of different oxides, and the way of treating them using their MUSIC model. For example, for goethite three types of surface oxygens are assumed to be present: singly, doubly, and triply coordinated ones. The doubly coordinated oxygens are considered as inert, so the following proton adsorption reactions occur:

Fe-OH(1/2)- + H+ T Fe-OH2(1/2)+

K1,2

(35a)

Fe3-O(1/2)- + H+ T Fe3-OH(1/2)+

K3,1

(35b)

At the same time, three important types of crystal planes have been reported for geothite: the (100), (010), and (001) faces. The (100) plane has singly, doubly, and triply coordinated oxygens present in equal amounts, but the other planes have only singly and doubly coordinated surface oxygens. Because of great similarity of the last two faces, they may be treated as one face. The authors6 report that the fraction of the (100) face equals half of the total surface area. In this way, they calculate how many surface groups of each kind follow reactions 35. According to their paper,6 TiO2 (rutile) has singly and doubly coordinated surface groups, which are able to react with protons in the following way:

Ti-OH(1/3)- + H+ T Ti-OH2(2/3)+

K1,2

(36a)

Ti2-O(2/3)- + H+ T Ti2-OH(1/3)+

K2,1

(36b)

Adsorption of protons at the triply coordinated surface oxygens is very low.23 When one assumes that the material in the suspension is large crystals, the rutile should have three crystal faces (as dominant ones): (110), (101), and (100) faces in the ratio 3:1:1, respectively.23 Following the crystallographic structure, for these crystal faces the amounts of singly and doubly coordinated ions should be equal. The above considerations can be generalized by writing eqs 34 in a slightly different form:

SOz- + H+ T SOH1-z

(37a)

SOHz- + H+ T SOH21-z

(37b)

where z is an absolute value of the charge (0 < z e 1) of the surface complex of the SOz- or SOHz- type. The equilibrium constants for both these reactions are labeled with one symbol, 1/1KHint, because we take into account only one of the two surface reactions 37. The first of these reactions is taken into consideration when the absolute value of the charge (z) of the SOz- complex is smaller than 1. The SOH1-z complex formed then has a positive charge, and it is rather impossible for the second proton to be attached to the SOH1-z complex according to reaction 37b. Later on the SOz- complexes are assumed to be so-called “free surface sites” (with designation θ- for their coverage). The remaining SOH1-z complexes are designated by θ+. The case when the system is described only by the second reaction (eq 37b) will take place when the absolute value of the charge (z) of the SOz- complex in eq 37a is greater than 1. In this case the SOH1-z complex is formed immediately according to reaction 37a, and its charge is smaller than zero. Thus, while modeling the surface charging, it can be assumed that the complex SOHz- is already there, and can be treated as a “free adsorption site” (with designation θ- for its coverage). Adsorption of protons takes place on these free sites according to reaction 34b, and the formed SOH21-z complexes are designated by θ+ for their surface coverage. As we mentioned before, for both protonation reactions we will use the same symbol, 1/KHint, for appropriate equilibrium constants. However, it may happen that two kinds of coordinated oxygens may appear on an oxide surface simultaneously. Then, we use the symbol 1/KH(1)int for reaction 37a and the symbol 1/1KH(2)int for reaction 37b. Now let us remark that the modeling of surface complexation in the case of reaction 37b is actually the same, provided that SOHz2- complexes are treated in the same way as the “free” surface sites SOz1-. We, therefore, introduce the following notation including also the adsorption of the ions of basic electrolyte. For reaction 37a

θ- ) θ-(1) ) [SOz1-]/Ns(1) θ+ ) θ+(1) ) [SOH1-z1]/Ns(1) θC ) θC(1) ) [SOz1-C+]/Ns(1) θA ) θA(1) ) [SOH1-z1A-]/Ns(1) (38a) whereas for reaction 34b

θ- ) θ-(2) ) [SOz2-]/Ns(2) θ+ ) θ+(2) ) [SOH1-z2]/Ns(2) θC ) θC(2) ) [SOz2-C+]/Ns(2) θA ) θA(2) ) [SOH1-z2A-]/Ns(2) (38b)



where Ns(1) and Ns(2) are the surface densities (in appropriate units) of the two kinds of coordinated oxygens, respectively. Now let us consider first the simple case when only one kind of surface oxygen is present (i ) 1 or i ) 2), due to either reaction 37a or reaction 37b. The surface charge, δ0, is then given by the expression

(

δ0(i) ) ziBi

)

1 - zi 1 - zi θ + θ - θC(i) - θ-(i) zi +(i) zi A(i) Bi ) eNs(i) (39)

The charge of the specifically adsorbed ions of the basic electrolyte is then given by eq 21, whereas the diffuse charge δd(i) is given by

(

δd(i) ) ziBi θ-(i) -

)

1 - zi 1 - zi θ+(i) θ + θA(i) zi zi C(i)

(40)

The nonlinear equation system 24 can be transformed into one nonlinear equation for δ0:

1 - zi 1 - zi K+(i)f+(i) + KA(i)fA(i) - KC(i)fC(i) - 1 zi zi

δ0(i) ) ziBi

1+

∑jKj(i)fj(i)

(j ) +, C, A) (41)

This nonlinear equation can be easily solved numerically to give δ0 as a function of pH. Having calculated this function, we evaluate the individual adsorption isotherms of all ions from eqs 24. Now we study the consequences of the appearing CIP by saying that, at pH ) PZC, δ0 ) 0 and ψ0 ) 0, no matter what the salt concentration is. Then, according to eq 41, the function Ui takes the following form:

U(i) )

1 - zi 1 - zi K+(i)H + KA(i)aAH - KC(i)aC - 1 ) 0 zi zi (42)

Except for very low electrolyte concentrations, one can assume that aA ) aC ) a. Thus, the independence of the CIP of the electrolyte concentration can, formally, be expressed as follows:

∂U(i) ∂a

)

1 - zi KA(i)H - KC(i) ) 0 zi

(43)

After the set of eqs 42 and 43 are solved, the following relations are obtained

H)

zi 1 K int and 1 - zi H(i)

1

KC(i)int

KA(i)int

1

)1

(44)

or in another form

PZC ) p1KH(i)int - log

zi and p1KC(i)int ) p1KA(i)int 1 - zi (45)

Thus, when only one kind of oxygen is present on a surface, eqs 45a,b reduce the number of independent equilibrium constants from three to one. It is also interesting to note that in the case of the 1-pK model and the considered case when only one kind of surface oxygen is present, the equilibrium constants

p1KC(i)int and p1KA(i)int are always equal to each other, whether the surface protonation has the character described by eq 37a or by eq 37b. Assuming that the oxide particles in the investigated suspension may still expose various crystal planes toward solution, and that the surface oxygens of these planes may still possess different degrees of ionization, brings one to a variety of complicated situations when a theoretical description is attempted. This complexity has not yet receiVed the sufficient attention of the researchers applying the 1-pK MUSIC approach. Below we are going to discuss briefly some of the most probable situations, and their theoretical descriptions. When many kinds of surface oxygens are present at the oxide/ electrolyte interface, the developed equations will depend on the surface topography. Basically, three topographies are then possible: (1) the “patchwise” topography when the suspension is carrying crystals having various crystal planes, and only one kind of surface oxygen is present on one crystal plane, (2) the “mixed” topography, when various (basically two types) oxygens appear on every crystal plane, and form microscopically fairly homogeneous domains, and (3) the “mediate” topography, when only one kind of surface oxygen appears on certain crystal planes, whereas other crystal planes may contain both kinds of surface oxygens. We should finally realize that one may face in a given system more than one kind of surface oxygen reacting in the way outlined in eq 37a or 37b. Thus, the 1-pK model introduces a number of a priori unknown equilibrium constants. Also, still other parameters appearsthe proportions of the various crystal planes and oxygen densities on these planes. Thus, certain simplifications are necessary to make the 1-pK model applicable. Hiemstra and Van Riemsdijk6 assume that if the oxide crystals are sufficiently large, the relative contribution to the whole crystal surface from various surface planes can be estimated from the crystallographic data. They also assume a priori that certain equilibrium constants should be equal. Drawing formal consequences of the existence of a CIP is one of the best ways to decrease the number of a priori unknown parameters, because it does not involve any physical assumption. As we have shown above, describing adsorption in terms of the 1-pK model may involve a lot of physical situations. While drawing the formal consequences of the existence of a CIP, we consider here only some of them for illustration. Let the first one be the situation when an oxide crystal possesses two crystal planes, the relative contributions of which are equal to x1 and x2 ) 1 - x1, respectively. Next, we assume that the protonation on the first crystal plane is described by eq 37a, whereas on the second plane it is described by eq 37b. We also assume that the crystal planes are sufficiently large that they may have their “own” diffuse charges δd(1) and δd(2). Then, for a given pH value we calculate δ0(1) and δ0(2) by assuming a certain set of parameters for i ) 1 and i ) 2: 1KH(i)int, 1K int 1 int L R L R C(i) , KA(i) , xi, Ns(i), c1(i) , and c1(i) (where c1(i) and c1(i) are the electrical capacitances for the left and right branches of the titration curve, respectively). Not all of these parameters are independent. This is because instead of two values of Ns(i) and two values of xi we can accept the average surface density, Ns ) ∑ixiNs(i) (i ) 1, 2), and one value of xi (∑jxj ) 1, for j ) 1, 2). At pH ) PZC

δ0Σ(pH)PZC) ) x1δ0(1)(pH)PZC) + x2δ0(2)(pH)PZC) ) 0 (46a)


Piasecki et al.

and

∂δ0Σ(pH)PZC) )0 ∂a

(46b)

Let us remark, however, that in general

δ0(1)(pH)PZC) * 0 and δ0(2)(pH)PZC) * 0 (47) because PZC is a macroscopic property of the system. So, the derivative in eq 46b will have to be calculated from a set of equations. The model considered so far could be called “ideally patchwise”. Now, let us consider the model which we may call “ideally mixed”. This could be the case when the crystal has only one crystal plane with two kinds of surface oxygens, the protonation of which is described by eqs 37a and 37b, respectively. Let further x1 and x2 ) 1 - x1 denote the fractions of the two kinds of surface oxygens. So, now, we have the situation that c1 and c2 as well as ψ0, δ0, δd, etc. are common for these two kinds of surface oxygens. The conditions for a CIP to exist are still given by eqs 46a,b, but in eqs 25 ψ0 and δ0 must be equal to zero. The condition 46a takes takes the following explicit form:

1 - zi 1 - zi K+(i)H + K Ha - KC(i)a - 1 zi zi A(i)

2

U)

xiziBi ∑ i)1

1 + K+(i)H + KA(i)Ha + KC(i)a

)0

(48a)

As for the derivative ∂δ0∑(pH)PZC)/∂a ) 0 2 KA(i) - K+(i)KC(i) ∂U )0 ) xiBi ∂a i)1 (1 + K H + K Ha + K a)2 +(i) A(i) C(i) (48b)

∑

Equation 48b can be transformed into a equation with a different form of the denominator: 2 1 ∂U ) xiBi1 ∂a i)1 K

∑

+(i)

1

KA(i) - 1KC(i)

(1 + K+(i)H + KA(i)Ha + KC(i)a)

2

)0 (48c)

Thus, it can be seen that one may consider many more complicated models. This is because an ideal crystal may have more than two crystal planes exposed to solution, and each of these crystal planes may have one or two kinds of surface oxygens. Generally, comparison of the 1-pK and 2-pK charging models based on the studies of experimental data will not be an easy problem. It will require possibly precise knowledge of the crystallographic structure of the investigated oxide sample. Next, it would be necessary to have at one’s disposal possibly diverse information obtained from various independent experiments, not only the δ0-pH dependence but also individual adsorption isotherms of the ions measured radiometrically, electrokinetic curves, etc. 4. Behavior of Electrokinetic Effects as an Additional Verification of the Applied Approach. The electrokinetic effects accompanying formation of the electric double layer may be a source of important information. The ζ-potential measurements are carried out frequently, and their theoretical interpretation is apparently simple.

The equality of the potentials ζ ) ψd, at least for low values of surface potentials and low concentrations of electrolytes in the bulk phase, has been commonly assumed in the literature. At higher values of the potential and higher concentrations, the viscosity close to the surface increases due to the increase of the surface concentration. Subsequently, the boundary plane of the mobile layer moves more deeply into the solution, and the anticipated value |ζ| becomes lower than the value |ψd|. The relationship ψd(Z) was obtained from the accurate solution of the Poisson equation and has the following form as discussed by Blesa and Kallay et al.:24-26

ψd(Z) )

2kT 1 + gd exp{-κZ} ln e 1 - gd exp{-κZ}

(49a)

where

gd )

exp{eψd/2kT} - 1 exp{eψd/2kT} + 1

(49b)

κ is the Debye-Huckel reciprocal thickness of the double layer

κ)

( ) 2e2I r0kT

1/2

(49c)

One should remember that, depending on the assumed model, we will have to accept different definitions of the charge density in the d-plane, i.e., eq 6 for the 2-pK model, eq 40 for the 1-pK model with the assumption that only one kind of coordination of the surface oxygens is present (i ) 1 or i ) 2), and the general equation

δd )

∑ixiδd(i)

(50)

for the 1-pK model depending on the topography of the surface. Another situation is when the crystal planes are not sufficiently large and they do not have their “own” diffuse charges δd(i). In such a case one should calculate the “average” δd from the electroneutrality condition of the whole double layer. Discussion A Numerical Quantitative Analysis of the Experimental Data Based on the 1-pK and 2-pK Protonation Models. Comparison of the 1-pK and 2-pK approaches that could lead to reliable conclusions in favor of applying one of them should fulfill two obvious conditions. (1) The studied oxide surface should represent a possibly simple structure that would allow for reducing the number of uncertainties which may appear. (2) The experimental data should carry some possibly rich information coming from various independent sources. Despite a large body of experimental data that has been published so far, it is not easy to find a data collection that would meet the two criteria mentioned above. After studying a number of such data sets, we have decided to choose for our analysis the data for the system NaCl/anatase, published by Sprycha.27,28 These data contain (1) titration curves measured for various concentrations of the basic electrolyte NaCl, (2) the individual adsorption isotherms of Na+ and Cl- ions measured by using radiometric methods, and (3) electrokinetic measurements of the ζ-potential. We only chose the data set for concentration of an inert electrolyte equal to 0.01 mol/dm3 because only in this case were



Figure 4. Curie-Wulff scheme of the anatase crystal. The proportions of all edges and planes are in agreement with those in the picture of the natural crystal of anatase.29

Figure 3. Diagrammatic presentation of the 1-pK model with two charged planes (BSM): ψ0, δ0, the surface potential and the surface charge density in the 0-plane; ψd, δd, the diffuse layer potential and its charge; cBS, the electrical capacitance of the rigid layer, constant within the region between the planes.

all three kinds of experimental data reported. The ζ-potential was determined by means of electrophoretic mobility measurements. Sprycha did not explain, in detail, how the electrokinetic potential was determined from the data obtained in these measurements.27,28 Probably he used the Smoluchowski formula. Then, there was also another important reason for choosing anatase connected with its crystallographic properties. While anticipating the results of our discussion, we may say that the probability of exposing one crystallographic plane is so dominate here that the contribution to adsorption from other crystallographic planes can be neglected. At the same time we have two kinds of surface oxygens on that dominant crystallographic plane. So, it is possibly the simplest example of physical systems to which MUSIC approach should be applied. Now we explain certain principles of our investigation, based on comparison of the theoretical predictions offered by various approaches with the monitored behavior of various experimental data. For us, the basis for our conclusions was the comparison with the experimental data which are directly measured. These are the δ0(pH), θC(pH), and θA(pH) data. The quantitative interpretation of electrokinetic data involves making a number of additional assumptions, such as the one concerning the position Z of the shear plane. Therefore, while looking for a certain set of parameters (equilibrium constants, capacitances, etc.), we accepted the parameter sets leading to the simultaneous best agreement between the theoretically and the experimentally determined titration isotherms, δ0(pH), and the radiometrically monitored individual isotherms of cation and anion adsorption, θC(pH) and θA(pH). After the best set of parameters was accepted, the corresponding ζ(pH) function was calculated and next compared with the data deduced from the electrokinetic measurements. The last comparison, however, was treated as an important suggestion concerning the applicability of the assumed approaches. Anatase has a tetragonal I41/amd structure with a ) 3.785 Å and c ) 9.914 Å.29 The structure of the anatase surface has been the subject of an advanced theoretical study published by Zio´łkowski.30 His theoretical considerations suggest that the most probable faces of anatase real crystals are the (001) and (101) ones. Zio´łkowski included a photo of a natural anatase crystal about 3 mm in length. Figures 4 and 5

Figure 5. Structure of the surface of the (101) face of anatase, shown in two different ways: (A) the top view, in which the big empty circles denote surface oxygens located in the plane of the paper, the black big circles are the oxygens located 1.6 Å below that plane, the footballlike big circles are the oxygens located only 0.8 Å below the plane, and the small circles denote the titania cations; (B) the side view on the (101) surface, in which the big empty circles denote oxygen ions and the small shadowed circles are the titania ions.

show the schematic picture of that natural anatase sample. That picture keeps the right proportions of all the crystal edges and planes. Simple calculations show that the contribution of the (001) faces to the total crystal surface is less than 3%. This would suggest that the surface of anatase should, to a very good approximation, be considered as exhibiting the one crystal plane (101) toward solution. Ludwig and Schindler31 studying the adsorption of heavy metal ions at the anatase/electrolyte interface have accepted such an assumption. The structure of the surface face (101) of anatase is shown in Figure 5. One can see easily the appearance of singly and doubly coordinated surface oxygens, in equal proportions. Such a model of the anatase surface has been accepted in the abovementioned work by Ludwig and Schindler.31 Anatase represents, thus, possibly the simplest example of an adsorption system to which the MUSIC approach can be applied. We have only two kinds of surface oxygens, and a fully mixed surface topography, with the common values of δ0 and ψ0 to be assumed when calculating θi. As the first step to compare the 1-pK and 2-pK approaches, we assumed that all the surface oxygens are identical. Sprycha, who analyzed his experimental data, accepted such an assumption. He used his well-known graphical method28,29 to determine the physical constants appearing in the adsorption equations corresponding to the 2-pK TLM. They are collected in Table 1.


Piasecki et al.

TABLE 1: Values the the Parameters Found for the Anatase/NaCl System When Described in Terms of the 2-pK TLM, and Applying Various Versions of Calculationsa Sprycha’s TLM parameters

BOUSSE TLM

A POSTERIORI TLM

PKint a1 ) 4.0 pKa2int ) 8.0* p*KCint ) 5.8 p*KAint ) 6.2* c1L ) 1.1 F/m2 c1R ) 1.2 F/m2

pKa1int ) 3.2 pKa2int ) 8.8* p1KCint ) 5.55 p1KAint ) 6.45* c1L ) 1.0 F/m2 c1R ) 1.1 F/m2

pKa1int ) 3.2 pKa2int ) 8.8* p1KCint ) 5.55 p1KAint ) 6.45* c1L ) 0.95 F/m2 c1R ) 1.05 F/m2

a The first column is the set of values found by Sprycha applying his graphical method.27,28 The second column is the set of parameters found by us when applying Bousse’s expression 12 for ψ0(pH), whereas the third column is the set of values found by us when applying the A POSTERIORI version of calculations. The asterisk denotes the values which were found by applying the Rudzin´ski-Charmas criterion. The fitted experimental data were measured at 25 °C, and the NaCl concentration equals 0.01 mol/dm3. The Ns value was accepted to be 12 sites/nm2.

Table 1 also collects the values of the same parameters obtained by us, while applying Bousse’s expression 12 to represent ψ0(pH), and the Rudzin´ski-Charmas criteria21,22 to decrease by two the number of best-fit parameters (the BOUSSE version of calculations). Finally, the third column of Table 1 contains the set of parameter values found by applying the Rudzin´ski-Charmas criteria, and accepting the A POSTERIORI version of calculations, in which ψ0(pH) is the a posteriori result of solving the system of adsorption equations (see the Appendix). In performing the calculations, we assumed that there are two different values of the c1 parameter, depending on the pH value, one for the acidic region (c1L for pH < PZC) and another one for the basic region (c1R for pH > PZC). This is in agreement with the suggestions of Sprycha28 and Blesa and Kallay.24 Figures 6 and 7 show the comparison of the theoretical functions δ0(pH), θC(pH), θA(pH), and ζ(pH), calculated by using the 2-pK TLM and the parameters collected in Table 1, with the experimental values reported by Sprycha. Once again, we emphasize that the parameters collected in Table 1 were found by the simultaneous best fits of δ0(pH), θC(pH), and θA(pH), so the theoretical functions ζ(pH) are solely a posteriori results in our calculations. A certain small improvement between theory and experiment can be observed in Figure 6 when one uses Bousse’s eq 12. Figure 7 shows that using the A POSTERIORI procedure to calculate ψ0(pH) in the 2-pK model leads to absolutely unrealistic predictions of the electrokinetic data. While applying the 1-pK MUSIC approach to represent the anatase/NaCl interface, along with the method of charging the two kinds of surface oxygens, we arrive at two sets of surface complexation reactions. For the singly coordinated surface oxygens we have

SOH

(1/3)-

+

1/1KH(1)int

+ H 798 SOH2 1K

(2/3)+

int C(1)

SOH(1/3)- + C+ 798 SOH(1/3)-C+ int 1K A(1)

SOH2(2/3)+ + A- 798 SOH2(2/3)+A-

(51a) (51b) (51c)

whereas for the doubly coordinated oxygens we have 1/1KH(2)int

SOH(2/3)- + H+ 798 SOH2(1/3)+

(52a)

Figure 6. (A) Comparison of the experimental values of δ0(pH) (•••) reported by Sprycha for the anatase/NaCl system, with the theoretical ones calculated by applying the 2-pK TLM and Sprycha’s parameters collected in Table 1 (---), and next by using our parameters in Table 1, obtained by applying Bousse’s eq 12 (s). (B) Comparison of the corresponding ζ(pH) functions with their experimental values (•••) reported by Sprycha. (C) Comparison of the θC(pH) and θA(pH) functions with the experimental ones. 1K int C(2)

SOH(2/3)- + C+ 798 SOH(2/3)-C+ 1K int A(2)

SOH2(1/3)+ + A- 798 SOH2(1/3)+A-

(52b) (52c)

While carrying out the calculations of the surface charge δ0, ζ-potential, and individual adsorption isotherms of ions θi, we always took the two eqs 48a,b into account, to decrease by two the number of best-fit parameters. Looking to eq 48c, which is just eq 48b written in a different form, we can see that one possible solution of eqs 48a,b is when

KC(1)int ) 1KA(1)int and

1

1

KC(2)int ) 1KA(2)int

(53)

because then eq 48c can be solved for one pair of values of KH(1)int and 1KH(2)int. It is interesting to note that the relations 53 were accepted by Hiemstra and Van Riemsdijk6,7 in their papers based on the 1-pK approach. However, in trying to solve the Rudzin´ski-Charmas equation system 48a,c, one discovers its intriguing property. Namely, for a fixed value of the parameters 1KC(1)int and 1KC(2)int, one obtains infinite pairs of values of KH(1)int and 1KH(2)int which solve the equation system 48a. The only restriction is that these pairs of values must fulfill the following condition:

(pKH(1)int + p1KH(2)int)/2 ) PZC

(54)



Figure 7. Comparison of the theoretical functions calculated by using the 2-pK TLM and the BOUSSE procedure, along with the corresponding parameters collected in Table 1 (s) with the theoretical functions calculated by using the A POSTERIORI procedure and the parameters from Table 1 (---). Also, the experimental data (•••) are shown for comparison. Other details are as in Figure 6.

i.e., KH(1)int and 1KH(2)int must always be symmetrically located toward PZC. In the case of the anatase (101) face/NaCl interface, eqs 53 create serious doubts. This is because both Na+ cations and Clanions are probably adsorbed by a largely electrostatic mode on the already existing surface complexes SOH(1/3)-, SOH2(2/3)+ or SOH(2/3)-, SOH2(1/3)+ having different charges, thus attracting Na+ and Cl- ions with different strengths. If Na+ and Cl- ions are adsorbed by purely Coulombic forces, it seems much more reasonable to assume the following relations:

KC(1)int ) 1KA(2)int and

1

KC(2)int ) KA(1)int

1

(55)

While fixing a certain pair of values 1KC(1)int and 1KC(2)int and next solving the Rudzin´ski-Charmas equation system 48a,c, one discovers again the same intriguing property of that equation system. The solution is obtained for an infinite number of pairs of the values 1KH(1)int and 1KH(2)int, which, however, must fulfill condition 54. Thus, in the next step of our investigation we studied the behavior of our observables δ0, θC, θA, and ζ as functions of pH, predicted by the three 1-pK TLM approaches. First, we assumed that all the surface oxygens have identical features. Figure 8 shows the comparison with experiment of the theoretical predictions based on the TLM, and obtained by assuming the 2-pK and 1-pK charging mechanisms. One can see that the combination of the TLM interface model and 1-pK

Figure 8. Comparison of the theoretical functions calculated by using the 2-pK homogeneous TLM and the BOUSSE version of calculations (s) with the theoretical functions calculated by using the 1-pK HOMOGENEOUS TLM (---). The theoretical values are calculated by using the parameters collected in Tables 1 and 2.

charging mechanism leads to unrealistic predictions concerning the electrokinetic effects. This has to be connected with the failure of the combination of TLM, 2-pK, and the A POSTERIORI procedure to fit the electrokinetic data observed in Figure 7. So, what is common for Figures 7 and 8 is the failure in combining the TLM and the A POSTERIORI procedure to predict the features of the electrokinetic data. This conclusion receives further support from the model calculations presented in Figure 9. Next, we assumed the MUSIC model along with the alternative relations p1KC(1)int ) p1KA(1)int and p1KC(2)int ) p1KA(2)int offered by the Rudzin´ski-Charmas criterion. Finally, we studied the second MUSIC alternative relations p1KC(1)int ) p1KA(2)int and p1KC(2)int ) p1KA(1)int also offered by the Rudzin´ski-Charmas criterion. Table 2 collects the parameters, found by fitting the experimental data for δ0, θC, and θA by the theoretical expressions developed for the three 1-pK approaches described above. Figures 8-11 bring several comparisons between the theoretical functions δ0(pH), θC(pH), and θA(pH) and ζ(pH) calculated by applying various approaches, and the experimental data reported by Sprycha. We can see that introducing the MUSIC model does not help to eliminate the failure of the TLM combined with the A POSTERIORI procedure to reproduce the behavior of the observed electrokinetic data. Thus, in the last attempt to eliminate that failure, we decided to check whether the assumption Z * 0 might help to eliminate that failure. All calculations of the related theoretical ζ(pH) functions, presented in Figures 6-11, have been made by assuming that


Figure 9. Comparison of the theoretical functions calculated by using the 2-pK homogeneous TLM and the A POSTERIORI version of calculations (s) with the theoretical functions calculated by using the 1-pK HOMOGENEOUS TLM (---). Other details are as in Figure 6.

TABLE 2: Values of the Parameters Found by Us While Fitting Sprycha’s Experimental Data for δ0(pH), θC(pH), and θA(pH) by the Theoretical Expressions Corresponding to the TLM of the Interface and Various 1-pK Modelsa 1-pK HOMOGENEOUS TLM p1KHint ) 6.0 p1KCint ) p1KAint ) -0.75 c1L ) 0.95 F/m2 c1R ) 1.05 F/m2 c2 ) 0.2 F/m2

MUSIC-1 TLM

MUSIC-2 TLM

pKH(1)int ) 5.5 p1KH(2)int ) 6.5 p1KC(1)int ) p1KA(2)int ) -0.6 p1KC(2)int ) pKA(1)int ) -1.1 c1L ) 1.0 F/m2 c1R ) 1.1 F/m2 c2 ) 0.2 F/m2

pKH(1)int ) 5.5 p1KH(2)int ) 6.5 p1KC(1)int ) pKA(1)int ) -0.6 p1KC(2)int ) p1KA(2)int ) -0.4 c1L ) 1.1 F/m2 c1R ) 1.1 F/m2 c2 ) 0.2 F/m2

a The column HOMOGENEOUS is the set of data obtained by assuming that all the surface oxygens have identical features. The raw MUSIC-1 TLM is for the MUSIC TLM, and the assumptions that int 1 int 1 int p1Kint C(1) ) p KA(2) and that p KC(2) ) pKA(1) are offered by the Rudzin´ski-Charmas criterion. The third column MUSIC-2 TLM is for the MUSIC TLM, and the alternative assumptions that p1Kint C(1) ) p 1 int 1 int Kint ´ skiA(1) and that p KC(2) ) p KA(2) are also offered by the Rudzin Charmas criterion.

the shear plane is located at Z ) 0, so ζ ) ψd. Now, we are going to check how the assumption that Z > 0 may improve the theoretical predictions for ζ(pH), obtained by applying the various adsorption approaches. These model investigations are shown in Figures 12-16. From the above presented model investigations, it follows that taking Z > 0 substantially improves the predicted behavior of the ζ(pH) functions calculated for the 2-pK adsorption model

Piasecki et al.

Figure 10. Comparison of the theoretical functions calculated by using the 2-pK homogeneous TLM and the A POSTERIORI version of calculations (s) with the theoretical functions calculated by using the 1-pK MUSIC-1 TLM (---). The corresponding parameters used in these calculations are collected in Table 2. Other details are as in Figure 6.

when using Bousse’s relation 12. On the contrary, taking Z > 0 makes worse the theoretical functions ζ(pH) calculated for the 1-pK adsorption models. Generally speaking, the assumption Z > 0 makes worse the theoretical predictions based on the TLM, and using the A POSTERIORI procedure to calculate ψ0(pH). Then our theoretical studies showed that much better behavior of the theoretical ζ(pH) functions is predicted when one combines the 1-pK charging mechanism with the simpler basic Stern model (BSM), shown schematically in Figure 3. In BSM all the equations developed so far reduce to simpler ones, by making c2 ) ∞. So, there is no need to write again the explicit form of the equations corresponding to BSM. However, the parameters estimated by fitting the experimental data will take somewhat different values. We show it first by considering the 1-pK models of adsorption. So, Table 3 being a counterpart of Table 2 collects the values of the parameters obtained by making c2 ) ∞ in all the equations developed for the 1-pK models. In other words, Table 3 collects the parameter values corresponding to the 1-pK BSM. Figure 17 shows that the classical 2-pK TLM approach and ψ0(pH) calculated from Bousse’s relation 12 yield a fit similar to that of the simplest 1-pK BSM approach, both of them assuming that the surface oxygens have identical properties. As far as the electrokinetic data are concerned, the 2-pK TLM (Bousse) approach leads to better predictions in the region of smaller |ζ| values whereas the 1-pK BSM leads to better predictions for the highest |ζ| values.



Figure 13. Effect of Z on the function ζ(pH), calculated for the 2-pK homogeneous TLM, using the A POSTEORI procedure and the parameters collected in Table 1. The solid line is for Z ) 0 Å, the long-dashed line is for Z ) 5 Å, and the short-dashed line is for Z ) 10 Å. Also, the experimental data are shown for comparison.

Figure 11. Comparison of the theoretical functions calculated by using the 1-pK MUSIC-1 TLM (s) and the 1-pK MUSIC-2 TLM (---). Other details are as in Figure 6.

Figure 14. Effect of Z on the function ζ(pH), calculated for the 1-pK homogeneous TLM and the parameters collected in Table 2. The solid line is for Z ) 0 Å, the long-dashed line is for Z ) 5 Å, and the shortdashed line is for Z ) 10 Å. Also, the experimental data are shown for comparison.

Figure 12. Effect of Z on the function ζ(pH), calculated for the 2-pK homogeneous TLM, using the BOUSSE procedure and the parameters collected in Table 1. The solid line is for Z ) 0 Å, the long-dashed line is for Z ) 5 Å, and the short-dashed line is for Z ) 10 Å. Also, the experimental data are shown for comparison.

Figure 18 shows that introducing the MUSIC concept (energetic heterogeneity) to the BSM of the interface does not improve the agreement between theory and experiment. The 1-pK MUSIC BSM approach yields almost the same agreement as the 1-pK BSM approach assuming energetic equivalence of surface oxygens. Moreover, the 1-pK MUSIC-2 BSM approach cannot fit the individual isotherms of ion adsorption, though

somewhat better agreement is obtained between the theoretically predicted and experimentally monitored electrokinetic data. As the BSM of the interface is necessary to arrive at the reasonable fit of the experimental data by equations corresponding to the 1-pK charging mechanism, we decided to check what will happen when one combines the 2-pK charging mechanism with the BSM interface model. The results are shown in Figure 19. While assuming the 2-pK charging mechanism, and using Bousse’s eq 12 to calculate ψ0(pH), one arrives at an essentially similar agreement with experiment for both the TLM and BSM of the interface. Then, using the A POSTERIORI procedure to calculate ψ0(pH) leads to a much worse prediction of the behavior of the electrokinetic data. Conclusions Below we summarize our observations concerning the comparison of the 2-pK and 1-pK approaches.


Piasecki et al. TABLE 3: Values of the Parameters Found by Us While Fitting the Experimental Data for δ0(pH), θC(pH), and θA(pH) by the Theoretical Expressions Corresponding to the BSM of the Interface, and to Various 1-pK Modelsa 1-pK HOMOGENEOUS BSM p1Kint H ) 6.0 1 int p1Kint C ) p KA ) -0.75

cL1 ) 0.95 F/m2 cR1 ) 1.05 F/m2 c2 ) ∞ a

MUSIC-1 BSM

MUSIC-2 BSM

pKint H(1) ) 5.5 p1Kint H(2) ) 6.5 1 int p1Kint C(1) ) p KA(2) ) -0.6 int p1Kint C(2) ) pKA(1) ) -1.1 cL1 ) 0.95 F/m2 cR1 ) 1.05 F/m2 c2 ) ∞

pKint H(1) ) 5.5 p1Kint H(2) ) 6.5 int p1Kint C(1) ) pKA(1) ) -0.6 1 int p1Kint C(2) ) p KA(2) ) -0.4 cL1 ) 1.1 F/m2 cR1 ) 1.1 F/m2 c2 ) ∞

Other details are the same as in Table 2.

Figure 15. Effect of Z on the function ζ(pH), calculated for the 1-pK MUSIC-1 model and the parameters collected in Table 2. The solid line is for Z ) 0 Å, the long-dashed line is for Z ) 5 Å, and the shortdashed line is for Z ) 10 Å. Also, the experimental data are shown for comparison.

Figure 16. Effect of Z on the function ζ(pH), calculated for the 1-pK MUSIC-2 model and the parameters collected in Table 2. The solid line is for Z ) 0 Å, the long-dashed line is for Z ) 5 Å, and the shortdashed line is for Z ) 10 Å. Also, the experimental data are shown for comparison.

(1) To arrive at a reasonable agreement between theory and experiment when one assumes the 2-pK charging mechanism, the ψ0(pH) function has to be calculated from Bousse’s eq 12. Using the A POSTERIORI procedure to calculate ψ0(pH) leads to a similar fit of the charging isotherm δ0(pH) and of the individual isotherms of ion adsorption θi, but makes worse the agreement between the theoretically predicted and the experimentally observed electrokinetic data. (2) The above conclusions are true when either the TLM or BSM of the interface is assumed. Both of these interface models lead then to a similar agreement between theory and experiment when the 2-pK charging mechanism is assumed. (3) To arrive at a reasonable agreement between theory and experiment, when one assumes the 1-pK charging mechanism, the BSM of the interface has necessarily to be accepted. (4) Introducing the MUSIC concept to the 1-pK model does not affect much the agreement between theory and experiment.

Figure 17. Comparison of the theoretical functions calculated by using the 2-pK homogeneous TLM and the BOUSSE version of calculations (s) with the theoretical functions calculated by using the 1-pK HOMOGENEOUS BSM (---). Also the experimental data (···) are shown for comparison. (A) δ0(pH) functions. (B) ζ(pH) functions. (C) θC(pH) and θA(pH) functions. The theoretical values are calculated by using the corresponding parameters collected in Tables 1 and 2.

(5) Using the best of the 2-pK and 1-pK models leads to a similar agreement between theory and experiment. Of course all the above conclusions should be treated with caution. They have been drawn from the analysis of just one anatase/NaCl system. However, it is to be expected that future analyses of other adsorption systems will remain valid at least for some of the above conclusions. The present analysis does not give significant evidence in favor of one or another approach, but suggests that making an assumption about the charging mechanism must be necessarily



Figure 18. Comparison of the theoretical functions calculated by using the 1-pK MUSIC-1 BSM (s) and the 1-pK MUSIC-2 BSM (---). Other details are as in Figure 17.

accompanied by a suitable assumption concerning the structure of the oxide/electrolyte interface. Our analysis also shows the possible large variety of adsorption models, which have to be considered when the energetic heterogeneity of the surface oxygens is taken into consideration. This is especially true in the case of the 1-pK models. Acknowledgment. We express our thanks to Professor Van Riemsdijk for stimulating discussions during W.R.’s visit to his laboratory at the Wageningen Agricultural University and to Professor Stanisław Pikus from the Department of Crystallography, Maria Curie Skłodowska University, for many helpful discussions. This work has been carried out as a part of PANCNRS joint project.

Figure 19. Comparison of the theoretical functions calculated by using the 2-pK BSM and the BOUSSE procedure, along with the parameters collected in Table 1 (s) and the theoretical functions calculated by using the A POSTERIORI procedure for the 2-pK BSM and the parameters from Table 1 (---). Also, the experimental data (···) are shown for comparison. (A) Comparison of the δ0(pH) values. (B) Comparison of the ζ(pH) functions. (C) Comparison of θC(pH) and θA(pH) values.

from which ψ0 can be calculated as a function of δ0, pH, and a ) aC ) aA. From the two roots of eq A2 only the following one has physical meaning

[

- δ0K0H + Y)

aKC X(δ0 + B)

+ aKAH2X(δ0 - B)

2K+H2(δ0 - B) aKC δ0K0H + + aKAH2X(δ0 - B) X(δ0 + B)

[[

]

]

2

- 4K+H2(δ0 - B)(δ0 + B)

]

1/2

2K+H2(δ0 - B)

(A3) Appendix 1. Calculation of δ0(pH), ψ0(pH), and θi(pH) Functions for the 2-pK TLM Using the A POSTERIORI Procedure. Let us denote

( )

eδ0 X ) exp kTc1

( )

eψ0 and Y ) exp kT

(A1)

Then eq 11 can be rearranged to the following quadratic equation

[

K+H2(δ0 - B)Y2 + δ0K0H +

aKC X(δ0 + B)

]

+

aKAH2X(δ0 - B) Y + (δ0 + B) ) 0 (A2)

or alternatively

kT ψ0 ) - ln e

[[

(

δ0K0H +

[

- δ0K0H +

aKC X(δ0 + B)

+ aKAH2X(δ0 - B)

2K+H2(δ0 - B) aKC

X(δ0 + B)

+ aKAH2X(δ0 - B)

]

2

]

-

- 4K+H2(δ0 - B)(δ0 + B)

2K+H2(δ0 - B)

]

)

1/2

(A4) Using relations A3 and A4 in eq 10, the θi defined in eq 9 and δd defined in eq 6 are expressed now as functions of δ0, pH, and the electrolyte activity a. Next, the obtained δd(δ0,pH,a) function is inserted into eq 15, in which also ψ0 is expressed as a function of δ0, pH, and a through eq A4. Then, eq 15 takes


Piasecki et al.

δ0 )

the form of a nonlinear equation

F(δ0,pH,a) ) 0

1 - zi 1 - zi f+ K+(i)f+(i) + KA(i)fA(i) - a2KC(i) - 1 zi zi fA

(A5)

from which δ0 is calculated as a function of pH and a. Having calculated this function, we can express ψ0 in eq A4 solely as a function of pH. 2. Calculation of δ0(pH), ψ0(pH), and θi(pH) Functions for the 1-pK TLM Using the A POSTERIORI Procedure. By the substitutions given in eq A1, eq 20 is transformed into the following form

1 HK+Y + aHKAX - aKC(Y/X) - 1 δ0 ) B 2 1 + HK+Y + aHKAX + aKC(Y/X)

∑ xiziB

i)1,2

1 + K+(i)f+(i) + KA(i)fA(i) + a2KC(i)

f+

fA (j ) +, C, A) (A10)

Equation A10 is next transformed into the quadratic equation with respect to f+

a2f+2 + a1f+ + a0 ) 0

(A6)

(A11)

which has two roots from which ψ0 is obtained as a function of δ0, pH, and the electrolyte activity a

1 B(aHKAX - 1) - δ0(1 + aHKAX) 2 Y) aKC aKC 1 - B 1HK+ δ0 1 + HK+X 2 HK+X

[(

) (

)]

(A7)

The next mathematical operations are very similar to those described for the 2-pK TLM A POSTERIORI procedure. 3. Calculation of δ0(pH), ψ0(pH), and θi(pH) Functions for the 1-pK MUSIC TLM Using the A POSTERIORI Procedure. The procedure described below applies to the case of anatase, so eq 41 is used to represent the function δ0(ψ0,pH,a), with x1 ) x2 ) 1/2. The topography is the “ideal mixed” one, so ψ0 and δ0 are assumed to be the same for every point of the surface. Thus, δ0 is considered to be given by the following expression:

δ0 ) 1 - zi 1 - zi K+(i)f+(i) + KA(i)fA(i) - KC(i)fC(i) - 1 zi zi

∑ xiziB

1+

i)1,2

∑j

Kj(i)fj(i) (j ) +, C, A) (A8)

Taking into consideration that

{ }

eψ0 K+(1)f+ ) K+(1)aH exp kT eδ0 KA(1)fA ) KA(1)aHa exp kTc1

{

{ } }

eψ0 eδ0 f+ ) a2KC(1) KC(1)fC ) KC(1)a exp + kT kTc1 fA

{ }

eψ0 kT eδ0 KA(2)fA ) KA(2)aHa exp kTc1 K+(2)f+ ) K+(2)aH exp -

{

KC(2)fC ) KC(2)a exp -

{ } }

eψ0 eδ0 f+ + ) a2KC(2) kT kTc1 fA

eq A8 can be written in the following form:

-a1 ( (a12 - 4a2a0)1/2 2a2

f+ ) where

(

a2 ) δ0 K+(2) +

(A9b) (A9c) (A9d) (A9e)

)(

a2KC(2) fA

( (

K+(1) +

)( )(

)

a2KC(1) fA

-

) )

a2KC(2) 1 - z1 a2KC(1) K+(1) x1z1B K+(2) + fA z1 fA x2z2B K+(1) +

[(

a1 ) δ0 K+(1) +

a2KC(1) 1 - z2 a2KC(2) K+(2) (A13a) fA z2 fA

)

a2KC(1)

( [

fA

K+(2) +

(1 + KA(2)fA) +

(

K+(2) +

[(

)

a2KC(2) fA

x1z1B (1 + KA(2)fA)

(

)(

]

(1 + KA(1)fA) -

)

a2KC(1) 1 - z1 K+(1) + z1 fA

a2KC(2) 1 - z1 KA(1)fA - 1 fA z1

x2z2B K+(1) + (A9a)

(A12)

)(

)]

-

)

a2KC(1) 1 - z2 KA(2)fA - 1 + fA z2

(1 + KA(1)fA)

(

)]

a2KC(2) 1 - z2 K+(2) z2 fA

(A13b)

a0 ) δ0(1 + KA(1)fA)(1 + KA(2)fA) 1 - z1 KA(1)fA - 1 x1z1B(1 + KA(2)fA) z1 1 - z2 x2z2B(1 + KA(1)fA) KA(2)fA - 1 (A13c) z2

(

(

)

)

From the definition of f+ it follows that f+ > 0, so from the two roots in eq A12 we choose only the positive one. Knowing f+, we calculate ψ0:

(A9f) ψ0 )

kT aH ln e f+

(A14)

Ion Adsorption at the Oxide/Electrolyte Interface Next, using the iteration method, we solve eq A10 to obtain the value of the surface charge δ0. References and Notes (1) Yates, D. E.; Levine, S.; Hearly, T. W. J. Chem. Soc., Faraday Trans. 1 1974, 70, 1807. (2) Chan, D.; Perram, J. W.; White, L. R.; Hearly, T. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046. (3) Davis, J. A.; James, R. O.; Leckie, J. O. J. Colloid Interface Sci. 1978, 63, 480. (4) Davis, J. A.; Leckie, J. O. J. Colloid Interface Sci. 1978, 67, 90. (5) Bolt, G. H.; Van Riemsdijk, W. H. In Physicochemical Models in Soil Chemistry B, 2nd ed.; Bolt, G. H., Ed.; Elsevier: Amsterdam, 1982. (6) Hiemstra, T.; Van Riemsdijk, W. H. Colloids Surf. 1991, 59, 7. (7) Hiemstra, T.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1996, 179, 488. (8) Hiemstra, T.; Van Riemsdijk, W. H.; Bolt, G. H. J. Colloid Interface Sci. 1989, 133, 91. (9) Pauling, L. The Nature of the Electrostatic Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1967. (10) Borkovec, M. Langmuir 1997, 13, 2608. (11) Borkovec, M.; Daicic, J.; Koper, G. J. M. Proc. Natl. Sci. U.S.A. 1997, 94, 3499. (12) Lutzenkirchen. J. EnViron. Sci. Technol. 1998, 32, 3149. (13) Sposito, G. J. Colloid Interface Sci. 1993, 91, 329. (14) Rudzinski, W.; Charmas, R.; Piasecki, W.; Thomas, F.; Villieras, F.; Prelot, B.; Cases, J. M. Langmuir 1998, 14, 5210.

J. Phys. Chem. B, Vol. 105, No. 40, 2001 9771 (15) Charmas, R. Langmuir 1999, 15, 5635. (16) Rudzinski, W.; Panas, G.; Charmas, R.; Kallay, N.; Preocanin, T.; Piasecki, W. J. Phys. Chem. B 2000, 104, 11912. (17) Bousse, L.; de Rooij, N. F.; Bergveld, P. IEEE Trans. Electron DeVices 1983, 30, 1263. (18) Bousse, L.; Maindl, J. D. In Geochemical Processes Mineral Surfaces; Davis, J. A., Hayes, K. F., Eds.; ACS Symposium Series No. 323; American Chemical Society: Washington, DC, 1986; p 79. (19) Van den Vlekkert, H.; Bousse, L.; de Rooij, N. F. J. Colloid Interface Sci. 1988, 122, 336. (20) Bousse, L.; Bergveld, P. J. Electroanal. Chem. 1983, 152, 25. (21) Rudzinski, W.; Charmas, R.; Partyka, S.; Foissy, A. New J. Chem. 1991, 15, 327. (22) Rudzinski, W.; Charmas, R.; Partyka, S.; Thomas, F.; Bottero, J. Y. Langmuir 1992, 8, 1154. (23) Hiemstra, T.; De Wit, J. C. W.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1989, 133, 105. (24) Blesa, M. A.; Kallay, N. AdV. Colloid Interface Sci. 1988, 28, 111. (25) Kallay, N.; Tomic, M. Langmuir 1988, 4, 559. (26) Tomic, M.; Kallay, N. Langmuir 1988, 4, 565. (27) Sprycha, R. Habilitation Thesis, Maria Curie Skłodowska University, Lublin, Poland, 1986. (28) Sprycha, R. J. Colloid Interface Sci. 1984, 102, 173. (29) Wyckoff, W. G. Crystal Structures; Interscience: New York, 1969. (30) Ziolkowski, J. Surf. Sci. 1989, 209, 536. (31) Ludwig, C.; Schindler, P. J. Colloid Interface Sci. 1995, 169, 284.

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