- Email: [email protected]
Effective adsorption energy distribution function as a new mean-field characteristic of surface heterogeneity in adsorption systems with lateral interactions
Journal of Colloid and Interface Science 311 (2007) 622–627 www.elsevier.com/locate/jcis
Letter to the Editor
Effective adsorption energy distribution function as a new mean-field characteristic of surface heterogeneity in adsorption systems with lateral interactions Piotr Zarzycki a,b,1 a Department of Chemistry, University of California at Berkeley, Berkeley, CA 94720, USA b Institute of Catalysis and Surface Chemistry, PAS, Krakow, Poland
Received 23 January 2007; accepted 11 March 2007 Available online 20 April 2007
Abstract Recent papers [W. Piasecki, Langmuir 22 (2006) 6761; P. Zarzycki, Langmuir 22 (2006) 11234; P. Zarzycki, J. Colloid Interface Sci. 306 (2007) 328] showed that the distinction between surface heterogeneity and effects of electrostatic interactions becomes problematic for metal oxide/electrolyte systems. This observation suggests that both mentioned factors can be combined into one (“effective” energy). Hence, the effective adsorption energy distribution function (EAEDF) with the Langmuir isotherm can be used in the framework of integral adsorption isotherm (IAI) instead of the classical adsorption energy distribution function (AEDF) with the Fowler–Guggenheim isotherm. The histogram method was used to obtain the effective distributions from the grand canonical Monte Carlo simulations of H+ ion adsorption. The new approach (IAI/EAEDF) gives a much better description of the heterogeneous metal oxide/electrolyte interface than IAI/AEDF. Moreover, reduced quantities are introduced in order to compare different metal oxide/electrolyte systems in a convenient way. © 2007 Elsevier Inc. All rights reserved. Keywords: Mean-field approximation; Dynamic surface heterogeneity; Monte Carlo simulation; Effective adsorption energy distribution function
1. Introduction Surface heterogeneity is a well-established concept in surface science, and it is related to the lack of geometric and energetic uniformity of actual solid surfaces. The term “surface heterogeneity” always refers to a specified adsorption system (adsorbing particles/adsorbent) [1]. The analytical theories of adsorption on heterogeneous solid surfaces are based on the concept of the adsorption energy distribution function (AEDF, χ ). AEDF is defined as differential distribution of the surface sites among various adsorption energies [2]: χ(ε) = dN(ε)/dε. Hence, the number of adsorption centers whose energy lies between ε and ε + dε is represented by dN(ε) = χ(ε) dε. From the physical point of view, χ should
be a discrete function, that is, only a certain number of classes of sites can be distinguished. Moreover, these classes of sites are occupied at the same pressure as the adsorbing gas (so the surface exhibits a patchwise topography) [3]. However, at a certain level of molecular insight each site can be considered as distinct from the others (due to the molecular vicinity of each site and dynamic processes occurring during the adsorption). For this reason (and for the sake of mathematical simplicity) AEDF is often assumed to be a continuous function [3]. The total extent of the covered surface (overall adsorption isotherm θ ) is obtained by integration of the local adsorption isotherm ϑ (describing the adsorption on the surface sites of the adsorption energy ε) weighted by AEDF (χ ) [2,4]: θ (p, T ) = ϑ(p, T , ε)χ(ε, T ) dε, (1) Ω
E-mail addresses: [email protected], [email protected]. 1 Present address: Department of Chemistry, University of California at Berkeley, Berkeley, CA 94720-1460, USA. 0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2007.03.018
where p is the pressure (theories were initially developed for a gas/solid interface), T is the temperature, and Ω is the adsorption energy domain. Equation (1) is known as the integral
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
adsorption isotherm (IAI) and it can be used in a few cases, depending on which functions (θ, ϑ, χ ) are known. If both integral functions (χ, ϑ) are known, Eq. (1) is just the weighted average of adsorption taking place on the surface elements characterized by ε (distributed according to χ ). If the local isotherm (ϑ) and the overall isotherm (θ ) are known, Eq. (1) becomes the Fredholm integral equation of the first kind. In other words, this is a task of finding AEDF for a particular adsorption system [5]. If the overall isotherm (θ ) and AEDF (χ ) are known, there is the problem of the unknown kernel (i.e., ϑ ) [3,6,7]. The IAI approach has many drawbacks. The most important is that the IAI solution is a mean-field type, so the local correlations cannot be considered [8,9]. In other words, AEDF does not describe the surface topography and topology, so the real nature of the surface heterogeneity (surface position dependence of adsorption energy) is treated in the average (meanfield) way [7–10]. Nevertheless, IAI is still one of the most popular methodologies for describing adsorption on heterogeneous surfaces, especially if the simple analytical model for fitting experimental data is required [3,6,11]. IAI was developed for a gas/solid interface, neglecting lateral interactions; however, in almost all adsorption systems adsorbed particles interact with each other in the ad-layer. The simplest theoretical way of including lateral interactions is based on the Bragg–Williams mean-field approach [12], which led to the Fowler–Guggenheim isotherm [13]. It should be emphasized that the term “mean-field” is used in two different contexts: more generally in reference to the average character of IAI as the nonuniform potential energy field characterized by AEDF and more specifically in reference to the lateral interactions (Bragg–Williams mean-field approach). One of the interesting adsorption systems is the metal oxide (including silica)/electrolyte interface. Due to the adsorption of hydrogen ions and electrolyte ions on the charged metal oxide surface, a characteristic charge distribution is formed, known as an electrical double layer (EDL) [14]. The analytical modeling of the considered EDL is based on the surface complexation model [13]. The concept of energetic heterogeneity was introduced into EDL modeling by van Riemsdijk and Koopal et al. [15–18] and then developed by Rudzinski et al. [19], Barrow et al. [23], and Contescu et al. [24]. Recently, Piasecki published a paper in which he discussed the difficulty in separating the electrostatic and energetic heterogeneity factors in the analytical modeling of the metal oxide surface protonation [25]. According to Piasecki [25], the electrostatic interactions can mimic the surface heterogeneity. A similar conclusion can be drown from the recent grand canonical Monte Carlo simulations of proton uptake/release [10] as well as the electrolyte ion adsorption on heterogeneous metal oxide surfaces [9]. The simulation of different topologies/topographies has shown that the lateral repulsive interactions mask effectively almost all surface properties (occurring inside the screening range). For example, if you consider the multipatch surface (each patch characterized by different adsorption energy), the simulation curve (for a system without interactions or with short-range ones) shows the characteristic steps–behavior. However, the long-range interactions of a range
623
comparable to or higher than the patch size made the isotherms smooth [10]. A similar shielding effect of electrostatic interactions is observed if micro/macroscopic correlations between adsorption energies are considered [9]. Moreover, the adsorption isotherm for a highly heterogeneous system neglecting electrostatics and the isotherm for a homogeneous system including lateral repulsive interactions could be indistinguishable (for certain values of interaction/heterogeneity parameters) [26]. These observations suggest that both surface heterogeneity and lateral interactions affect cooperatively the shape of the adsorption isotherm and surface charge density, and to the same extent they can be treated as a single factor. A paper by Prelot et al. [27] exemplified the usefulness of such a combined factor in the analytical interpretation of experimental data. Summing up, these observations allow one to introduce the “effective” adsorption energy distribution function (EAEDF), which also includes the effects of the lateral interactions treated in a meanfield fashion. In this paper, a simple simulation/theoretical study of system behavior is presented by using the concept of EAEDF. Here, both systems, neglecting interactions (EAEDF = AEDF) and including the long-range repulsive interactions (EAEDF more heterogeneous than AEDF), are investigated. In the case of the system with attractive interactions (not considered here), EAEDF will be more homogeneous than static AEDF. Of course, in the limit of the surface coverage → 0 we should observe AEDF = EAEDF. 2. Theory, simulations, and results A new theoretical approach is presented here for a specific case: protonation/deprotonation (based on the 1-pK model [28]) of the energetically heterogeneous metal oxide. According to the 1-pK model the proton uptake/release is governed by the reaction +1/2
≡SOH−1/2 + H+ → ≡SOH2 and +1/2
K ads =
[≡SOH2
]
[≡SOH−1/2 ][H+ ]surf
=
1 , K
(2)
where K ads and K are adsorption and desorption equilibrium constants, respectively. In the case of localized adsorption, the system can be considered as an Ising type (lattice gas), where spin-variable (si ) takes binary values (1, occupied site, or 0, empty site). As stated before, the actual adsorbents are energetically heterogeneous and the most common quantitative characteristics of this phenomenon are given by the AEDF function. For actual sorbents, AEDF possesses a complex shape; however, it is often approximated by the analytical continuous functions. In many actual cases, the quasi-Gaussian function is considered as the best approximation [3,6]. In the case discussed here, the quasiGaussian distribution function (for the considered interface denoted by Γ ) can be written in the form [9,26] (pK = − log(K)
624
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
and pK ∼ ε):
εs 1 2 e2 e2 e−κrij a + + , (5) 2πε0 εw rij 2πε0 εs κεs rij3
of the e.d.l. thickness, 1/κ = 0.43 nm), and e is the elementary charge. This potential formula is valid if εs < εw , a > 0 (a = 0 for metal surface), and κa < 1 (which is the case studied here) [29]. The details of the applied simulation scheme can be found in Refs. [9,10,26,30–33]. There are several ways to include the lateral interaction in the analytical models, but the most popular is the meanfield approach (MFA) [12], leading to the Fowler–Guggenheim isotherm [13] for homogeneous surfaces. The average lateral interactions are given by the interaction parameter ω (in previous papers referred as ¯ ) [26,31], which is defined as ω = W (r ic ) (where c stands for central position on computation i lattice). In the classical MFA the lateral interactions are proportional to the surface coverage [13]. Both adsorption energy (pK) and interaction energy (W, ω) are expressed in units of kB T (kB is Boltzmann constant and T is temperature). The adsorption energies are attributed randomly to each surface site (random topography) using the formula (inverted cumulative distribution function) [10] n mz ∗ , x = pK + 2 arctanh (2ran − 1) tanh (6) m 2n
where qi and qj are the charges of complexes i and j , a is the depth of the charge buried inside the solid (a = 0.25 nm), ε0 is the permittivity of free space, εs is the dielectric constant of the interfacial layer (εs = 3), εw is the dielectric constant of bulk phase (εw = 80), κ is the Debye parameter (inversion
where x and ran are random numbers of the quasi-Gaussian distribution and of a uniform distribution (∈(0; 1)), respectively. In order to get EAEDF and to confirm the validity of Eq. (6) the histogram method was used for pKi (AEDF) and pKi − n j W (rij )sj (EAED).
Γ (pK) =
∗ exp( m 1 n (pK − pK)) , m ∗ 2 F [1 + exp( m n (pK − pK))]
(3)
γ (pK)
where pKmax=pK ∗ +z
F= pKmin
=pK ∗ −z
mz γ (pK) dpK = n tanh , 2n
(4)
where m is the heterogeneity parameter (correlated with the width of the distribution function), pK ∗ is an average value (the position of the maximum), n = 1/ ln(10), F is the normalization constant, and z describes the energetic domain cutoff (see Fig. 1A). The adsorbed ions interact with one another, and the interaction potential function is given by the Borkovec potential [29], W (rij ) = qi qj
Fig. 1. Adsorption energy distribution function (AEDF) (A,B,C), AEDF (B), and EAEDF (C) obtained from simulation using histogram method. The quasi-Gaussian function fitting to “empirical” EAEDF is shown in Part C.
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
According to the IAI approach the adsorption isotherm is given by the definite convoluted integral: ∗ pK +z
θ (pH) =
ϑ(pH, θ)Γ (pK) dpK,
pK ∗ −z
where ϑ(pH, θ) =
exp( n1 (pK − pH − nωθ )) 1 + exp( n1 (pK − pH − nωθ ))
.
(7)
The local isotherm (ϑ ) is given by the Fowler–Guggenheim isotherm, where the lateral interactions are represented by the MFA term ωθ . As mentioned, we combined the surface heterogeneity effect (i.e., AEDF) and the lateral interactions (using MFA) into one factor, called effective AEDF. In this study, we determined the shape of EAEDF from the GCMC simulations using histogram methods (see Fig. 1C). The EAEDF obtained from the simulation scheme (histograms) were the bell-shaped functions, so we can expect that incorporation of lateral interactions in AEDF does not change the class of the function. The peak position of EAEDF can be easily predicted using MFA (pK ∗eff = pK ∗ − nωθ ) [9,26] and only the “effective” heterogeneity parameter (meff ) has to be estimated. Concluding, the analytical form of EAEDF (if AEDF is given by (7)) can be written as Γ MFA (pK, θ ) eff
exp( m n(θ) (pK ∗ − nωθ − pK)) meff (θ ) = MFA . F (θ ) [1 + exp( meff (θ) (pK ∗ − nωθ − pK))]2
(8)
n
By using the concept of EAEDF Eq. (7) can be rewritten to the form pK ∗ −nωθ+z
θ (pH) =
ϑ(pH)Γ MFA (pK, θ ) dpK,
pK ∗ −nωθ−z
where ϑ(pH) =
exp( n1 (pK − pH)) 1 + exp( n1 (pK − pH))
.
(9)
625
The local isotherm in Eq. (9) is given by the Langmuir isotherm (i.e., neglecting lateral interactions). Instead of fitting the analytical EAEDF (i.e., meff , Eq. (8)) we use the “experimental” histograms obtained during the GCMC course (Γ MFA → Γ simulation ). In order to compare different metal oxide/electrolyte systems it is convenient to introduce the reduced pH units (τ ) and the reduced surface charge density σ0∗ by the following definitions: σ0 τ = pH − pzc, σ0∗ = (10) . eNS The various metal oxide/electrolyte interfaces differ in the observed pzc (point of zero charge) and the surface site density (whose magnitude affects σ0 ). The reduced pH unit (τ ) unifies the x axis, while reducing the surface charge density σ0∗ unifies the y axis for all types of the metal oxide/electrolyte interface. The surface charge density (for 1-pK model/considering only H+ ions uptake/release) in the reduced quantities is simply the charge balance of the interface: 1 σ0∗ (τ ) = θH (τ ) − . (11) 2 The quantity σ0∗ describes the acid/base properties of an average surface site (formal “effective” charge of the surface). Due to division by NS , reduced surface charge density becomes the intensive thermodynamic parameter, so the comparison is more reliable than σ0 (Fig. 2). In Fig. 3A the surface charge density curves obtained from grand canonical Monte Carlo simulations are presented for systems including and neglecting lateral interactions. In order to mimic the bulk phase, periodic boundary conditions are applied in two dimensions, and in order to obtain good statistics the simulations are averaged over 10 system replicas. The surface is represented by the lattice: 100 × 100 (lattice knots), which corresponds to surface site density of 1 site/nm2 . In Fig. 3B the average lateral interactions calculated in a static-MFA manner (ωθ ) are compared with those obtained in a dynamic-MFA version (ω(θ)) from Monte Carlo simulation (following one of the system replicas). Averaging over all replicas gives exactly the same result as static MFA.
Fig. 2. Evolution of adsorption energy distribution function (AEDF) to effective adsorption energy distribution functions (EAEDF) (shielding/amplification of the “effective” surface heterogeneity due to the lateral interaction). In Part B: evolution of AEDF for the system considered here.
626
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
Fig. 3. Part A: reduced surface charge density (σ0∗ , charge balance) obtained from simulation. Part B: comparison the average lateral interactions in a dynamic-MFA (ω(θ)) and static-MFA (ω · θ ). Part C: comparison of “experimental data” (obtained in simulation) and IAI/AEDF and IAI/EAEDF theory (numerical integration). The pzc is equal 10 (system neglecting interactions) and 5.29 (system including interactions). The average interaction parameter ω is equal to 24.36 (static-MFA).
In Fig. 3C a comparison of the theory (IAI approach) and the “computer experiment” (grand canonical Monte Carlo simulation) is shown. The IAI approach with EADEF and the Langmuir adsorption isotherm gives better agreement than IAI with AEDF and the Fowler–Guggenheim adsorption isotherm. The improvement became more visible in strongly acidic (high negative τ , positively charged surface) and strongly alkaline (high positive τ , negatively charged surface) solutions. Near pzc (τ = 0) both analytical approaches gives a relatively good approximation (Fig. 3C). 3. Discussion and summary In this article, the concept of “effective” adsorption energy distribution function is introduced as a new way to consider surface heterogeneity in combination with the lateral interactions effect. The evolution of AEDF due to lateral interactions becomes important only in the case of long-range interactions, that is, when the energetic effect of adsorption depends on both changing the state of a particular surface site (occupied/empty) and the interactions with the rest of the adsorbed particles. The presented results have shown that EAEDF belongs to the same function class as the static AEDF (in this case, quasi-Gaussian functions). It was proved by fitting quasi-Gaussian (with meanfield shifting of peak position) to histograms obtained from computer simulations (Fig. 1C). In the EAEDF approach the interaction factor is transferred from the local adsorption isotherm to the energy adsorption dis-
tribution function. A few advantages of this approach can be seen immediately, for example, the CA approximation used to solve IAI still contains the information of interactions, while in the classical approach (CA based on AEDF) it does not. Moreover, the analytical approach (IAI) based on EAEDF/Langmuir gives a better approximation than that based on AEDF/Fowler– Guggenheim (see Fig. 3C). The concept of EAEDF can be introduced into strictly analytical modeling [20–22] by defining the particular form of function Γ (pK, θ, ψ0 (θ )). EAEDF is a more general concept than only AEDF-incorporated lateral interactions in a mean-field fashion. The dynamic character of AEDF originates from many sources: adsorbate-induced reconstruction/relaxation of the surface (concept introduced by Somorjai [34,35] and widely studied theoretically by Cerofolini et al. [36–38]) and the change of energetic/electronic state of the surface or even a change of the surface character (e.g., from hydrophilic to hydrophobic). However, in this letter only the evolution of AEDF due to lateral interactions is considered. Note that the dynamic character of AEDF does not mean time dependence. The term “dynamic” is used to emphasize the AEDF dependence of the surface coverage (in contrast with classical IAI formalism where dχ/dθ = 0). Even though the presented approach considers only the equilibrium properties, the time evolution of AEDF can be treated in a similar way. In addition to the discussion of AEDF, we introduced definitions of the reduced surface charge density σ0∗ and parameter τ (reduced pH scale for all metal oxides and related materials).
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
The reduced units allow one to analyze the various metal oxide/electrolyte systems in a convenient way. Acknowledgments The author thanks the Foundation for Polish Science for awarding the Maria Sklodowska-Curie Fellowship (Foreign Fellowships for Young Ph.D.’s) and Kris Szornel (Lawrence Berkeley National Laboratory) for helpful discussions. References [1] D.H. Everett, Langmuir 9 (1993) 2586–2592. [2] R. Sips, J. Chem. Phys. 16 (1948) 490–495. [3] W. Rudzinski, D.H. Everett, Adsorption of Gases on Heterogeneous Surface, Academic Press, New York, 1991. [4] R. Sips, J. Chem. Phys. 18 (1950) 1024–1026. [5] G.A. Cerofolini, Surf. Sci. 24 (1971) 391–403. [6] W. Rudzinski, W.A. Steele, G. Zgrablich, Equilibria and Dynamics, Elsevier, London, 1997. [7] M. Borówko, in: J. Tóth (Ed.), Adsorption: Theory, Modeling, and Analysis, Dekker, New York, 2002. [8] A. Patrykiejew, M. Borówko, in: M. Borówko (Ed.), Computational Methods in Surface and Colloid Science, Dekker, New York, 2000. [9] P. Zarzycki, J. Colloid Interface Sci. 306 (2007) 328–336. [10] P. Zarzycki, Langmuir 22 (2006) 11234–11240. [11] M. Jaroniec, R. Madey, Physical Adsorption on Heterogeneous Surface, Elsevier, Amsterdam, 1988. [12] W.L. Bragg, E.J. Williams, Proc. R. Soc. London A 145 (1934) 699. [13] R.H. Fowler, E.A. Guggenheim, Statistical Thermodynamics, Cambridge Univ. Press, London, 1949. [14] W. Stumm, J.J. Morgan, Aquatic Chemistry: Chemical Equilibria and Rates in Natural Waters, Wiley, New York, 1996. [15] W.H. van Riemsdijk, G.H. Bolt, L.K. Koopal, J. Blaakmeer, J. Colloid Interface Sci. 109 (1986) 219–228.
627
[16] W.H. van Riemsdijk, J.C.M. de Wit, L.K. Koopal, G.H. Bolt, J. Colloid Interface Sci. 116 (1987) 511–522. [17] L.K. Koopal, W.H. van Riemdsijk, C.M. de Wit, M.F. Benedetti, J. Colloid Interface Sci. 166 (1994) 51–60. [18] A.W.M. Gibb, L.K. Koopal, J. Colloid Interface Sci. 134 (1990) 122– 138. [19] W. Rudzinski, R. Charmas, S. Partyka, Langmuir 7 (1991) 354–362. [20] W. Rudzinski, R. Charmas, S. Partyka, F. Thomas, J.Y. Bottero, Langmuir 8 (1992) 1154–1164. [21] W. Rudzinski, R. Charmas, S. Partyka, J.Y. Bottero, Langmuir 9 (1993) 2641–2651. [22] W. Rudzinski, R. Charmas, W. Piasecki, F. Thomas, F. Villieras, B. Prelot, J.M. Cases, Langmuir 14 (1998) 5210–5225. [23] N.J. Barrow, G.W. Brümmer, R. Strauss, Langmuir 9 (1993) 2606– 2611. [24] C. Contescu, J. Jageillo, J.A. Schwarz, Langmuir 9 (1993) 1754–1765. [25] W. Piasecki, Langmuir 22 (2006) 6761–6763. [26] P. Szabelski, P. Zarzycki, R. Charmas, Langmuir 20 (2004) 997–1002. [27] P. Prélot, R. Charmas, P. Zarzycki, F. Thomas, F. Villiéras, W. Piasecki, W. Rudzinski, J. Phys. Chem. B 106 (2002) 13280–13286. [28] G.H. Bolt, W.H. Riemsdijk, in: G.H. Bolt (Ed.), Soil Chemistry, Elsevier, Amsterdam, 1982. [29] M. Borkovec, Langmuir 13 (1997) 2608–2613. [30] P. Zarzycki, P. Szabelski, R. Charmas, J. Comput. Chem. 25 (2004) 704– 711. [31] P. Zarzycki, P. Szabelski, R. Charmas, J. Comput. Chem. 26 (2005) 1079– 1088. [32] P. Zarzycki, P. Szabelski, R. Charmas, Appl. Surf. Sci. 252 (2005) 752– 758. [33] P. Zarzycki, J. Colloid Interface Sci. 297 (2006) 204–214. [34] G.A. Somorjai, Langmuir 7 (1991) 3176–3182. [35] G.A. Somorjai, Chem. Rev. 96 (1996) 1223–1235. [36] G.F. Cerofolini, Surf. Sci. 336 (1996) 597–606. [37] G.F. Cerofolini, G. Boara, S. Agosteo, A. Foglio Para, Langmuir 13 (1997) 913–918. [38] M.O. Vlad, G.F. Cerofolini, P. Oefner, J. Ross, J. Phys. Chem. B 109 (2005) 21241–21257.
Letter to the Editor
Effective adsorption energy distribution function as a new mean-field characteristic of surface heterogeneity in adsorption systems with lateral interactions Piotr Zarzycki a,b,1 a Department of Chemistry, University of California at Berkeley, Berkeley, CA 94720, USA b Institute of Catalysis and Surface Chemistry, PAS, Krakow, Poland
Received 23 January 2007; accepted 11 March 2007 Available online 20 April 2007
Abstract Recent papers [W. Piasecki, Langmuir 22 (2006) 6761; P. Zarzycki, Langmuir 22 (2006) 11234; P. Zarzycki, J. Colloid Interface Sci. 306 (2007) 328] showed that the distinction between surface heterogeneity and effects of electrostatic interactions becomes problematic for metal oxide/electrolyte systems. This observation suggests that both mentioned factors can be combined into one (“effective” energy). Hence, the effective adsorption energy distribution function (EAEDF) with the Langmuir isotherm can be used in the framework of integral adsorption isotherm (IAI) instead of the classical adsorption energy distribution function (AEDF) with the Fowler–Guggenheim isotherm. The histogram method was used to obtain the effective distributions from the grand canonical Monte Carlo simulations of H+ ion adsorption. The new approach (IAI/EAEDF) gives a much better description of the heterogeneous metal oxide/electrolyte interface than IAI/AEDF. Moreover, reduced quantities are introduced in order to compare different metal oxide/electrolyte systems in a convenient way. © 2007 Elsevier Inc. All rights reserved. Keywords: Mean-field approximation; Dynamic surface heterogeneity; Monte Carlo simulation; Effective adsorption energy distribution function
1. Introduction Surface heterogeneity is a well-established concept in surface science, and it is related to the lack of geometric and energetic uniformity of actual solid surfaces. The term “surface heterogeneity” always refers to a specified adsorption system (adsorbing particles/adsorbent) [1]. The analytical theories of adsorption on heterogeneous solid surfaces are based on the concept of the adsorption energy distribution function (AEDF, χ ). AEDF is defined as differential distribution of the surface sites among various adsorption energies [2]: χ(ε) = dN(ε)/dε. Hence, the number of adsorption centers whose energy lies between ε and ε + dε is represented by dN(ε) = χ(ε) dε. From the physical point of view, χ should
be a discrete function, that is, only a certain number of classes of sites can be distinguished. Moreover, these classes of sites are occupied at the same pressure as the adsorbing gas (so the surface exhibits a patchwise topography) [3]. However, at a certain level of molecular insight each site can be considered as distinct from the others (due to the molecular vicinity of each site and dynamic processes occurring during the adsorption). For this reason (and for the sake of mathematical simplicity) AEDF is often assumed to be a continuous function [3]. The total extent of the covered surface (overall adsorption isotherm θ ) is obtained by integration of the local adsorption isotherm ϑ (describing the adsorption on the surface sites of the adsorption energy ε) weighted by AEDF (χ ) [2,4]: θ (p, T ) = ϑ(p, T , ε)χ(ε, T ) dε, (1) Ω
E-mail addresses: [email protected], [email protected]. 1 Present address: Department of Chemistry, University of California at Berkeley, Berkeley, CA 94720-1460, USA. 0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2007.03.018
where p is the pressure (theories were initially developed for a gas/solid interface), T is the temperature, and Ω is the adsorption energy domain. Equation (1) is known as the integral
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
adsorption isotherm (IAI) and it can be used in a few cases, depending on which functions (θ, ϑ, χ ) are known. If both integral functions (χ, ϑ) are known, Eq. (1) is just the weighted average of adsorption taking place on the surface elements characterized by ε (distributed according to χ ). If the local isotherm (ϑ) and the overall isotherm (θ ) are known, Eq. (1) becomes the Fredholm integral equation of the first kind. In other words, this is a task of finding AEDF for a particular adsorption system [5]. If the overall isotherm (θ ) and AEDF (χ ) are known, there is the problem of the unknown kernel (i.e., ϑ ) [3,6,7]. The IAI approach has many drawbacks. The most important is that the IAI solution is a mean-field type, so the local correlations cannot be considered [8,9]. In other words, AEDF does not describe the surface topography and topology, so the real nature of the surface heterogeneity (surface position dependence of adsorption energy) is treated in the average (meanfield) way [7–10]. Nevertheless, IAI is still one of the most popular methodologies for describing adsorption on heterogeneous surfaces, especially if the simple analytical model for fitting experimental data is required [3,6,11]. IAI was developed for a gas/solid interface, neglecting lateral interactions; however, in almost all adsorption systems adsorbed particles interact with each other in the ad-layer. The simplest theoretical way of including lateral interactions is based on the Bragg–Williams mean-field approach [12], which led to the Fowler–Guggenheim isotherm [13]. It should be emphasized that the term “mean-field” is used in two different contexts: more generally in reference to the average character of IAI as the nonuniform potential energy field characterized by AEDF and more specifically in reference to the lateral interactions (Bragg–Williams mean-field approach). One of the interesting adsorption systems is the metal oxide (including silica)/electrolyte interface. Due to the adsorption of hydrogen ions and electrolyte ions on the charged metal oxide surface, a characteristic charge distribution is formed, known as an electrical double layer (EDL) [14]. The analytical modeling of the considered EDL is based on the surface complexation model [13]. The concept of energetic heterogeneity was introduced into EDL modeling by van Riemsdijk and Koopal et al. [15–18] and then developed by Rudzinski et al. [19], Barrow et al. [23], and Contescu et al. [24]. Recently, Piasecki published a paper in which he discussed the difficulty in separating the electrostatic and energetic heterogeneity factors in the analytical modeling of the metal oxide surface protonation [25]. According to Piasecki [25], the electrostatic interactions can mimic the surface heterogeneity. A similar conclusion can be drown from the recent grand canonical Monte Carlo simulations of proton uptake/release [10] as well as the electrolyte ion adsorption on heterogeneous metal oxide surfaces [9]. The simulation of different topologies/topographies has shown that the lateral repulsive interactions mask effectively almost all surface properties (occurring inside the screening range). For example, if you consider the multipatch surface (each patch characterized by different adsorption energy), the simulation curve (for a system without interactions or with short-range ones) shows the characteristic steps–behavior. However, the long-range interactions of a range
623
comparable to or higher than the patch size made the isotherms smooth [10]. A similar shielding effect of electrostatic interactions is observed if micro/macroscopic correlations between adsorption energies are considered [9]. Moreover, the adsorption isotherm for a highly heterogeneous system neglecting electrostatics and the isotherm for a homogeneous system including lateral repulsive interactions could be indistinguishable (for certain values of interaction/heterogeneity parameters) [26]. These observations suggest that both surface heterogeneity and lateral interactions affect cooperatively the shape of the adsorption isotherm and surface charge density, and to the same extent they can be treated as a single factor. A paper by Prelot et al. [27] exemplified the usefulness of such a combined factor in the analytical interpretation of experimental data. Summing up, these observations allow one to introduce the “effective” adsorption energy distribution function (EAEDF), which also includes the effects of the lateral interactions treated in a meanfield fashion. In this paper, a simple simulation/theoretical study of system behavior is presented by using the concept of EAEDF. Here, both systems, neglecting interactions (EAEDF = AEDF) and including the long-range repulsive interactions (EAEDF more heterogeneous than AEDF), are investigated. In the case of the system with attractive interactions (not considered here), EAEDF will be more homogeneous than static AEDF. Of course, in the limit of the surface coverage → 0 we should observe AEDF = EAEDF. 2. Theory, simulations, and results A new theoretical approach is presented here for a specific case: protonation/deprotonation (based on the 1-pK model [28]) of the energetically heterogeneous metal oxide. According to the 1-pK model the proton uptake/release is governed by the reaction +1/2
≡SOH−1/2 + H+ → ≡SOH2 and +1/2
K ads =
[≡SOH2
]
[≡SOH−1/2 ][H+ ]surf
=
1 , K
(2)
where K ads and K are adsorption and desorption equilibrium constants, respectively. In the case of localized adsorption, the system can be considered as an Ising type (lattice gas), where spin-variable (si ) takes binary values (1, occupied site, or 0, empty site). As stated before, the actual adsorbents are energetically heterogeneous and the most common quantitative characteristics of this phenomenon are given by the AEDF function. For actual sorbents, AEDF possesses a complex shape; however, it is often approximated by the analytical continuous functions. In many actual cases, the quasi-Gaussian function is considered as the best approximation [3,6]. In the case discussed here, the quasiGaussian distribution function (for the considered interface denoted by Γ ) can be written in the form [9,26] (pK = − log(K)
624
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
and pK ∼ ε):
εs 1 2 e2 e2 e−κrij a + + , (5) 2πε0 εw rij 2πε0 εs κεs rij3
of the e.d.l. thickness, 1/κ = 0.43 nm), and e is the elementary charge. This potential formula is valid if εs < εw , a > 0 (a = 0 for metal surface), and κa < 1 (which is the case studied here) [29]. The details of the applied simulation scheme can be found in Refs. [9,10,26,30–33]. There are several ways to include the lateral interaction in the analytical models, but the most popular is the meanfield approach (MFA) [12], leading to the Fowler–Guggenheim isotherm [13] for homogeneous surfaces. The average lateral interactions are given by the interaction parameter ω (in previous papers referred as ¯ ) [26,31], which is defined as ω = W (r ic ) (where c stands for central position on computation i lattice). In the classical MFA the lateral interactions are proportional to the surface coverage [13]. Both adsorption energy (pK) and interaction energy (W, ω) are expressed in units of kB T (kB is Boltzmann constant and T is temperature). The adsorption energies are attributed randomly to each surface site (random topography) using the formula (inverted cumulative distribution function) [10] n mz ∗ , x = pK + 2 arctanh (2ran − 1) tanh (6) m 2n
where qi and qj are the charges of complexes i and j , a is the depth of the charge buried inside the solid (a = 0.25 nm), ε0 is the permittivity of free space, εs is the dielectric constant of the interfacial layer (εs = 3), εw is the dielectric constant of bulk phase (εw = 80), κ is the Debye parameter (inversion
where x and ran are random numbers of the quasi-Gaussian distribution and of a uniform distribution (∈(0; 1)), respectively. In order to get EAEDF and to confirm the validity of Eq. (6) the histogram method was used for pKi (AEDF) and pKi − n j W (rij )sj (EAED).
Γ (pK) =
∗ exp( m 1 n (pK − pK)) , m ∗ 2 F [1 + exp( m n (pK − pK))]
(3)
γ (pK)
where pKmax=pK ∗ +z
F= pKmin
=pK ∗ −z
mz γ (pK) dpK = n tanh , 2n
(4)
where m is the heterogeneity parameter (correlated with the width of the distribution function), pK ∗ is an average value (the position of the maximum), n = 1/ ln(10), F is the normalization constant, and z describes the energetic domain cutoff (see Fig. 1A). The adsorbed ions interact with one another, and the interaction potential function is given by the Borkovec potential [29], W (rij ) = qi qj
Fig. 1. Adsorption energy distribution function (AEDF) (A,B,C), AEDF (B), and EAEDF (C) obtained from simulation using histogram method. The quasi-Gaussian function fitting to “empirical” EAEDF is shown in Part C.
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
According to the IAI approach the adsorption isotherm is given by the definite convoluted integral: ∗ pK +z
θ (pH) =
ϑ(pH, θ)Γ (pK) dpK,
pK ∗ −z
where ϑ(pH, θ) =
exp( n1 (pK − pH − nωθ )) 1 + exp( n1 (pK − pH − nωθ ))
.
(7)
The local isotherm (ϑ ) is given by the Fowler–Guggenheim isotherm, where the lateral interactions are represented by the MFA term ωθ . As mentioned, we combined the surface heterogeneity effect (i.e., AEDF) and the lateral interactions (using MFA) into one factor, called effective AEDF. In this study, we determined the shape of EAEDF from the GCMC simulations using histogram methods (see Fig. 1C). The EAEDF obtained from the simulation scheme (histograms) were the bell-shaped functions, so we can expect that incorporation of lateral interactions in AEDF does not change the class of the function. The peak position of EAEDF can be easily predicted using MFA (pK ∗eff = pK ∗ − nωθ ) [9,26] and only the “effective” heterogeneity parameter (meff ) has to be estimated. Concluding, the analytical form of EAEDF (if AEDF is given by (7)) can be written as Γ MFA (pK, θ ) eff
exp( m n(θ) (pK ∗ − nωθ − pK)) meff (θ ) = MFA . F (θ ) [1 + exp( meff (θ) (pK ∗ − nωθ − pK))]2
(8)
n
By using the concept of EAEDF Eq. (7) can be rewritten to the form pK ∗ −nωθ+z
θ (pH) =
ϑ(pH)Γ MFA (pK, θ ) dpK,
pK ∗ −nωθ−z
where ϑ(pH) =
exp( n1 (pK − pH)) 1 + exp( n1 (pK − pH))
.
(9)
625
The local isotherm in Eq. (9) is given by the Langmuir isotherm (i.e., neglecting lateral interactions). Instead of fitting the analytical EAEDF (i.e., meff , Eq. (8)) we use the “experimental” histograms obtained during the GCMC course (Γ MFA → Γ simulation ). In order to compare different metal oxide/electrolyte systems it is convenient to introduce the reduced pH units (τ ) and the reduced surface charge density σ0∗ by the following definitions: σ0 τ = pH − pzc, σ0∗ = (10) . eNS The various metal oxide/electrolyte interfaces differ in the observed pzc (point of zero charge) and the surface site density (whose magnitude affects σ0 ). The reduced pH unit (τ ) unifies the x axis, while reducing the surface charge density σ0∗ unifies the y axis for all types of the metal oxide/electrolyte interface. The surface charge density (for 1-pK model/considering only H+ ions uptake/release) in the reduced quantities is simply the charge balance of the interface: 1 σ0∗ (τ ) = θH (τ ) − . (11) 2 The quantity σ0∗ describes the acid/base properties of an average surface site (formal “effective” charge of the surface). Due to division by NS , reduced surface charge density becomes the intensive thermodynamic parameter, so the comparison is more reliable than σ0 (Fig. 2). In Fig. 3A the surface charge density curves obtained from grand canonical Monte Carlo simulations are presented for systems including and neglecting lateral interactions. In order to mimic the bulk phase, periodic boundary conditions are applied in two dimensions, and in order to obtain good statistics the simulations are averaged over 10 system replicas. The surface is represented by the lattice: 100 × 100 (lattice knots), which corresponds to surface site density of 1 site/nm2 . In Fig. 3B the average lateral interactions calculated in a static-MFA manner (ωθ ) are compared with those obtained in a dynamic-MFA version (ω(θ)) from Monte Carlo simulation (following one of the system replicas). Averaging over all replicas gives exactly the same result as static MFA.
Fig. 2. Evolution of adsorption energy distribution function (AEDF) to effective adsorption energy distribution functions (EAEDF) (shielding/amplification of the “effective” surface heterogeneity due to the lateral interaction). In Part B: evolution of AEDF for the system considered here.
626
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
Fig. 3. Part A: reduced surface charge density (σ0∗ , charge balance) obtained from simulation. Part B: comparison the average lateral interactions in a dynamic-MFA (ω(θ)) and static-MFA (ω · θ ). Part C: comparison of “experimental data” (obtained in simulation) and IAI/AEDF and IAI/EAEDF theory (numerical integration). The pzc is equal 10 (system neglecting interactions) and 5.29 (system including interactions). The average interaction parameter ω is equal to 24.36 (static-MFA).
In Fig. 3C a comparison of the theory (IAI approach) and the “computer experiment” (grand canonical Monte Carlo simulation) is shown. The IAI approach with EADEF and the Langmuir adsorption isotherm gives better agreement than IAI with AEDF and the Fowler–Guggenheim adsorption isotherm. The improvement became more visible in strongly acidic (high negative τ , positively charged surface) and strongly alkaline (high positive τ , negatively charged surface) solutions. Near pzc (τ = 0) both analytical approaches gives a relatively good approximation (Fig. 3C). 3. Discussion and summary In this article, the concept of “effective” adsorption energy distribution function is introduced as a new way to consider surface heterogeneity in combination with the lateral interactions effect. The evolution of AEDF due to lateral interactions becomes important only in the case of long-range interactions, that is, when the energetic effect of adsorption depends on both changing the state of a particular surface site (occupied/empty) and the interactions with the rest of the adsorbed particles. The presented results have shown that EAEDF belongs to the same function class as the static AEDF (in this case, quasi-Gaussian functions). It was proved by fitting quasi-Gaussian (with meanfield shifting of peak position) to histograms obtained from computer simulations (Fig. 1C). In the EAEDF approach the interaction factor is transferred from the local adsorption isotherm to the energy adsorption dis-
tribution function. A few advantages of this approach can be seen immediately, for example, the CA approximation used to solve IAI still contains the information of interactions, while in the classical approach (CA based on AEDF) it does not. Moreover, the analytical approach (IAI) based on EAEDF/Langmuir gives a better approximation than that based on AEDF/Fowler– Guggenheim (see Fig. 3C). The concept of EAEDF can be introduced into strictly analytical modeling [20–22] by defining the particular form of function Γ (pK, θ, ψ0 (θ )). EAEDF is a more general concept than only AEDF-incorporated lateral interactions in a mean-field fashion. The dynamic character of AEDF originates from many sources: adsorbate-induced reconstruction/relaxation of the surface (concept introduced by Somorjai [34,35] and widely studied theoretically by Cerofolini et al. [36–38]) and the change of energetic/electronic state of the surface or even a change of the surface character (e.g., from hydrophilic to hydrophobic). However, in this letter only the evolution of AEDF due to lateral interactions is considered. Note that the dynamic character of AEDF does not mean time dependence. The term “dynamic” is used to emphasize the AEDF dependence of the surface coverage (in contrast with classical IAI formalism where dχ/dθ = 0). Even though the presented approach considers only the equilibrium properties, the time evolution of AEDF can be treated in a similar way. In addition to the discussion of AEDF, we introduced definitions of the reduced surface charge density σ0∗ and parameter τ (reduced pH scale for all metal oxides and related materials).
P. Zarzycki / Journal of Colloid and Interface Science 311 (2007) 622–627
The reduced units allow one to analyze the various metal oxide/electrolyte systems in a convenient way. Acknowledgments The author thanks the Foundation for Polish Science for awarding the Maria Sklodowska-Curie Fellowship (Foreign Fellowships for Young Ph.D.’s) and Kris Szornel (Lawrence Berkeley National Laboratory) for helpful discussions. References [1] D.H. Everett, Langmuir 9 (1993) 2586–2592. [2] R. Sips, J. Chem. Phys. 16 (1948) 490–495. [3] W. Rudzinski, D.H. Everett, Adsorption of Gases on Heterogeneous Surface, Academic Press, New York, 1991. [4] R. Sips, J. Chem. Phys. 18 (1950) 1024–1026. [5] G.A. Cerofolini, Surf. Sci. 24 (1971) 391–403. [6] W. Rudzinski, W.A. Steele, G. Zgrablich, Equilibria and Dynamics, Elsevier, London, 1997. [7] M. Borówko, in: J. Tóth (Ed.), Adsorption: Theory, Modeling, and Analysis, Dekker, New York, 2002. [8] A. Patrykiejew, M. Borówko, in: M. Borówko (Ed.), Computational Methods in Surface and Colloid Science, Dekker, New York, 2000. [9] P. Zarzycki, J. Colloid Interface Sci. 306 (2007) 328–336. [10] P. Zarzycki, Langmuir 22 (2006) 11234–11240. [11] M. Jaroniec, R. Madey, Physical Adsorption on Heterogeneous Surface, Elsevier, Amsterdam, 1988. [12] W.L. Bragg, E.J. Williams, Proc. R. Soc. London A 145 (1934) 699. [13] R.H. Fowler, E.A. Guggenheim, Statistical Thermodynamics, Cambridge Univ. Press, London, 1949. [14] W. Stumm, J.J. Morgan, Aquatic Chemistry: Chemical Equilibria and Rates in Natural Waters, Wiley, New York, 1996. [15] W.H. van Riemsdijk, G.H. Bolt, L.K. Koopal, J. Blaakmeer, J. Colloid Interface Sci. 109 (1986) 219–228.
627
[16] W.H. van Riemsdijk, J.C.M. de Wit, L.K. Koopal, G.H. Bolt, J. Colloid Interface Sci. 116 (1987) 511–522. [17] L.K. Koopal, W.H. van Riemdsijk, C.M. de Wit, M.F. Benedetti, J. Colloid Interface Sci. 166 (1994) 51–60. [18] A.W.M. Gibb, L.K. Koopal, J. Colloid Interface Sci. 134 (1990) 122– 138. [19] W. Rudzinski, R. Charmas, S. Partyka, Langmuir 7 (1991) 354–362. [20] W. Rudzinski, R. Charmas, S. Partyka, F. Thomas, J.Y. Bottero, Langmuir 8 (1992) 1154–1164. [21] W. Rudzinski, R. Charmas, S. Partyka, J.Y. Bottero, Langmuir 9 (1993) 2641–2651. [22] W. Rudzinski, R. Charmas, W. Piasecki, F. Thomas, F. Villieras, B. Prelot, J.M. Cases, Langmuir 14 (1998) 5210–5225. [23] N.J. Barrow, G.W. Brümmer, R. Strauss, Langmuir 9 (1993) 2606– 2611. [24] C. Contescu, J. Jageillo, J.A. Schwarz, Langmuir 9 (1993) 1754–1765. [25] W. Piasecki, Langmuir 22 (2006) 6761–6763. [26] P. Szabelski, P. Zarzycki, R. Charmas, Langmuir 20 (2004) 997–1002. [27] P. Prélot, R. Charmas, P. Zarzycki, F. Thomas, F. Villiéras, W. Piasecki, W. Rudzinski, J. Phys. Chem. B 106 (2002) 13280–13286. [28] G.H. Bolt, W.H. Riemsdijk, in: G.H. Bolt (Ed.), Soil Chemistry, Elsevier, Amsterdam, 1982. [29] M. Borkovec, Langmuir 13 (1997) 2608–2613. [30] P. Zarzycki, P. Szabelski, R. Charmas, J. Comput. Chem. 25 (2004) 704– 711. [31] P. Zarzycki, P. Szabelski, R. Charmas, J. Comput. Chem. 26 (2005) 1079– 1088. [32] P. Zarzycki, P. Szabelski, R. Charmas, Appl. Surf. Sci. 252 (2005) 752– 758. [33] P. Zarzycki, J. Colloid Interface Sci. 297 (2006) 204–214. [34] G.A. Somorjai, Langmuir 7 (1991) 3176–3182. [35] G.A. Somorjai, Chem. Rev. 96 (1996) 1223–1235. [36] G.F. Cerofolini, Surf. Sci. 336 (1996) 597–606. [37] G.F. Cerofolini, G. Boara, S. Agosteo, A. Foglio Para, Langmuir 13 (1997) 913–918. [38] M.O. Vlad, G.F. Cerofolini, P. Oefner, J. Ross, J. Phys. Chem. B 109 (2005) 21241–21257.