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A lattice-gas model for specific ion adsorption at liquid | liquid interfaces
Journal of Electroanalytical Chemistry 500 (2001) 491– 497 www.elsevier.nl/locate/jelechem
A lattice-gas model for specific ion adsorption at liquid liquid interfaces Stefan Frank, Wolfgang Schmickler * Abteilung Elektrochemie, Uni6ersita¨t Ulm, D-89069 Ulm, Germany Received 22 August 2000; received in revised form 6 October 2000; accepted 9 October 2000 Dedicated to Professor Roger Parsons on the occasion of his retirement from the position of the Editor-in-Chief of the Journal of Electroanalytical Chemistry and in recognition of his great contribution to electrochemistry
Abstract A lattice-gas model is used to investigate the specific adsorption of ions at the interface between two immiscible electrolyte solutions. From Monte Carlo simulations, the profiles of particle densities and of the electrostatic potential are obtained. Specific adsorption is shown to affect the potential distribution markedly. In some cases an overshoot of the potential can be observed, an effect that is well known from specific adsorption at metal electrodes. This redistribution of charge and potential can increase the interfacial capacity, shift the potential of zero charge, and influence the rate of electron-transfer reactions. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Liquid liquid interfaces; Specific adsorption; Lattice-gas model; Electron transfer
1. Introduction The capacity of liquid liquid interfaces generally depends on the nature of the ions (see, e.g. Refs. [1,2]). In the absence of adsorption phenomena, this effect has been related to differences in the interaction energies of the ions with the two solvents: ions with a smaller energy of transfer can penetrate deeper into the mixed solvent region at the interface, thus increasing the overlap of space charge regions and hence the capacity [3]. Furthermore, the capacity can be enhanced by other processes that accumulate charge at the interface, in particular by specific adsorption. In contrast to interfaces with electronic conductors, adsorption at liquid liquid interfaces can be caused by ion pairing across the phase boundary. In recent communications [3 – 6], these ideas have been developed within the framework of a lattice-gas model, both by analytic calculations and by Monte Carlo simulations.
* Corresponding author. Tel.: +49-731-5025402; fax: + 49-7315025409. E-mail address: [email protected] (W. Schmickler).
The complementary case of ‘true’ specific adsorption may also occur at liquid liquid interfaces, when a molecule or ion with a hydrophobic and a hydrophilic part is located at the interface with each end in the preferred solvent. This phenomenon has been observed by Fermı´n et al. [7,8], who discuss its implications on the potential distribution and on rates of heterogeneous electron transfer. We believe that this type of adsorption merits further theoretical study, and have therefore Table 1 Interaction constants between the various particles a
S1 S2 K+ 1 At1 K+ 2 A− 2 Dn− 1
S1
S2
K+ 1
A− 1
K+ 2
A− 2
Dn− 1
−w w −u −u u u −r
w −w u u −u −u −s b/3s
−u u t t t t t
−u u t t t t t
u −u t t t t t
u −u t t t t t
r −s b/3s t t t t t
c
c
a In all simulations we have taken u = 3 kT, w =0.47 kT, t= 10 kT, r, s were varied. b For 1st to 3rd neighboring S2. c For 4th to 6th neighboring S2.
0022-0728/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 7 2 8 ( 0 0 ) 0 0 4 3 8 - 1
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S. Frank, W. Schmickler / Journal of Electroanalytical Chemistry 500 (2001) 491–497
investigated this process within the lattice-gas model, contrasting it to interfacial ion pairing. Following the experimental study by Fermı´n et al., we focus our attention on the distribution of the charges and the electrostatic potential, and the resulting effects on electron-transfer rates.
2. The lattice-gas model We have adapted our lattice gas model used in former simulations [3,6] to the problem at hand. Here we shall summarize its basic features and focus on new elements. Simulations were performed on a simple cubic lattice with dimensions 100× 16 ×16, the long extension being the z direction. Periodic boundary conditions apply in the x and y directions. As in our previous studies the system consists of two immiscible solvents S1 and S2, − ions K+ 1 and A1 preferentially solvated in solvent 1, − but poorly solvable in solvent 2, and ions K+ 2 and A2 preferentially dissolved in solvent 2. In addition, there are anions Dn1 − (n= 1 or 2) preferentially dissolved in solvent 1, which can be specifically adsorbed at the interface. Each lattice point is occupied by one particle, which interacts with its nearest neighbor through chemical forces. These interactions are summarized in a 7× 7 matrix (see Table 1). In order to reduce the number of parameters many interaction constants were chosen to have the same absolute values. Solvent particles of the same type attract each other with a constant − w (w\ 0), whereas unlike solvent particles repel each other with w. Ions A− and K+ have an interaction constant − u (u\0) with their preferred solvent and a constant u with the other solvent. The cross-interaction t between ions has been set to a high value in order to avoid pair formation between ions; in particular, there is no ion-pair formation across the phase boundary. Generally, these interaction constants are independent of the particle’s other nearest neighbors, with one exception: the interaction of the adsorbing anion Dn1 − with solvent S2 depends on the number of neighbors of this type: for the first three neighbors of type S2 the interaction is attractive with a constant (−s, s \0), for subsequent neighbors of this type it is repulsive, with a characteristic constant of (3s). The interaction of these ions with S1 is always repulsive (−r, r \0), independent of other neighbors. This rule ensures that for this particle the energetically most favorable sites are at the interface. In all simulations, we have chosen values of the interaction parameters u, r and s which result in the same energy of transfer from the preferred to the unpreferred solvent for all ions. In addition to the nearest-neighbor interactions each ion experiences the average electrostatic potential gen-
erated by the other ions, which is calculated by solving the Poisson –Boltzmann equation subject to the boundary conditions (− ) = 0 and d/dy(− ) = 0; this approach is tantamount to using a mean-field approximation for the Coulomb interaction. For this purpose we have to specify the lattice constant, which we take as 4 A, , and the dielectric constants of the two solvents. Solvent 1 with a dielectric constant of m1 =80 is meant to represent water, solvent 2 with m1 =10 represents an organic liquid. Since the composition of our system varies rapidly near the interface, we use an effective dielectric constant m(z)= f1(z)m1 + f2(z)m2 in this region, where f1(z) and f2(z) are the mole fractions of the aqueous and the organic phase, respectively.
3. Monte Carlo simulations Initially, the region z \0 was filled with solvent − molecules S2 and equal numbers N of ions K+ 2 and A2 . Conversely, the region z 50 was filled with solvent S1, − K+ 1 A1 as base electrolyte and a lower concentration of − n− (K+ ) and of 1 n D1 . In total, the number of anions A cations K+ was N, the number of adsorbing anions 2 Dn1 − was M, and the number of the common positive counterions K+ 1 was N+nM. The system was equilibrated by place exchanges between neighboring particles using the Metropolis [9] Monte Carlo algorithm. Subsequently, the distributions of the particles and the electrostatic potential were obtained by performing of the order of 106 Monte Carlo steps per particle. To polarize the interface between the two solutions, both phases were charged by adding equal numbers of oppositely charged excess ions; for example, to charge solution 1 positively, ions K+ were placed into the 1 region zB 0 and ions A− placed into the region z \0. 2 The number of adsorbing ions Dn1 − was not changed. After equilibration, the bulk solutions were found to be electroneutral, whereas the excess charges formed two oppositely charged space-charge regions at the interface, as expected. By comparing the single particle energy of adsorbed Dn1 − with Dn1 − in the bulk region, we obtained the adsorption energy Ead, which can be split into a chemical part E chem and an electrostatic part ad E elad. From the profile of the electrostatic potential, the total potential drop D =(− ) −( ) is obtained, D being positive for positively charged solution 1. The differential capacity C= d|/dD can be calculated by numerical differentiation. To investigate the effect of specific ion adsorption at the liquid liquid interface, we have performed several sequences of simulations varying the parameters that govern the adsorption energy Ead, and keeping all other parameters constant. Each sequence consisted of simulations with various charge densities and hence various
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493
4. Results
4.1. Distribution of charge and potential
Fig. 1. Distribution of the charge density and the electrostatic potential for uncharged phases. Thick solid line: no specific adsorption; thin solid line: adsorbing ion with charge −1 and interaction parameters r =2 and s =4; dotted line: adsorbing ion with charge −2 and interaction parameters r =1 and s =5.
Table 2 Shift Dpzc of the potential of zero charge (pzc) and chemical part of the adsorption energy (E chem ad ) at the pzc for various values of interaction parameters r and s, and ion charge −n r/kT, s/kT
2, 4
2, 5
n
1
2
1
2
Dpzc/mV E chem ad /kT
6 −2.6
10 −2.7
12 −5.5
18 −5.7
In the absence of the adsorbing ion Dn1 − the uncharged system shows zero average charge density for all lattice coordinates z and thus no potential drop across the interface. In the presence of Dn1 − , adsorption causes a sharp negative peak in the charge density at the interface, which is balanced by a positive space charge region in solution 1 (see Fig. 1) — by definition, there is no excess charge in solution 2. The electrostatic potential drops across the dipolar charge distribution in solution 1, while the charge and the potential distributions in solution 2 are practically unaffected. The bulk of solution 1 is then at a positive potential D compared to solution 2. This potential difference D is equal to the shift Dpzc in the potential of zero charge (pzc) caused by the specific adsorption of the ions at the interface, since in our model both bulk phases have the same potential in the absence of specific adsorption. As expected, this shift is larger, the greater is the difference s− r in the interaction constants, which govern the strength of the adsorption, and the higher the charge on the ion (see Table 2). The positive shift of the potential in solution 1 persists when the interface is polarized. Fig. 2(a) and (b) show the situation when there is a positive excess charge in solution 1. In this case specific adsorption increases both the charge stored at the interface and the concomitant potential drop. At potentials below the pzc the adsorbed charge has the opposite sign as the surface charge in solution 2. The sum of these charges, which has to be compensated by the space charge in solution 1, is smaller than in the absence of specific adsorption; it can even be negative for sufficiently strong adsorption and not too high surface charges. In the latter case, the potential distribution in solution 1 shows an inverted region (see dashed line in Fig. 2(d)). Such an overshoot of the potential has not been observed in simulations for ion pairing [6]. An ion pair is a neutral species, and hence has a weaker effect on the potential profile. For the same reason it has only a small effect on the position of the pzc.
4.2. Adsorption beha6ior potential differences between the solutions. The chemical adsorption energy E chem is controlled by the interacad tion constants r and s, whereas the electrostatic part E elad is controlled by the ion charge number − n. For comparison, we have performed a sequence of simulations in the absence of the adsorbing ion. The mean concentrations of the base electrolytes were chosen as 3.9 ×10 − 3 particles per lattice point, corresponding to a 0.1 M solution. The mean concentration of the adsorbing salt was one order of magnitude lower.
The surface excess Y of the adsorbing ions is determined both by the chemical driving force, which is specified through the chemical interactions r and s, and by the electrostatic interactions. Fig. 3 shows Y as a function of the potential drop D for several system parameters. A negative potential drop drives the ions towards the interface, but even for small and intermediate positive potentials there is still a sizable amount of adsorbed ions. The diamonds and circles in the figure
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correspond to ions with the same chemical interactions but different charge numbers. Over most of the range investigated the monovalent ion has a greater surface excess than the divalent ion; obviously, the larger adsorbate –adsorbate repulsion keeps the excess of the ions with the larger charge smaller. Only large negative
potentials can overcome the mutual repulsion of the adsorbates, and then the excess of the divalent ion becomes larger. Other things being equal, a smaller chemical driving force (squares in Fig. 3) always entails a smaller surface excess. Note that it is not possible to express the surface excess in terms of a coverage: with
Fig. 2. Distribution of the charge density and the electrostatic potential at the interface. Data are shown for the absence of adsorbing ion (thick solid lines); adsorbing ion with charge − 1 and interaction parameters r= 2 and s =4 (thin solid lines); adsorbing ion with charge − 2 and interaction parameters r =1 and s =5 (dotted lines). The surface charge density of solution 1 is 0.78 mC/cm2 in (a) and (b), and − 0.78 mC/cm2 in (c) and (d).
Fig. 3. Dependence of the surface excess Y of Dn1 − , in particles per site, on the potential drop D. Interaction parameters r= 1 and s= 5, ion charge − 1 (diamonds) or − 2 (circles); interaction parameters r =2 and s= 4, ion charge − 2 (squares) and ion charge −1 (crosses).
S. Frank, W. Schmickler / Journal of Electroanalytical Chemistry 500 (2001) 491–497
495
Fig. 4. Surface charge density | versus total potential drop D for absence of adsorbing ion (filled squares); interaction parameters r =2 and s= 4, ion charge − 1 (crosses) or −2 (squares); interaction parameters r =1 and s =5, ion charge − 1 (diamonds) or −2 (circles).
Fig. 5. Differential capacity C versus total potential drop D for absence of adsorbing ion (solid line), and for two cases of specific adsorption: interaction parameters r= 2 and s= 4, ion charge −2 (dotted); interaction parameters r =1 and s= 5, ion charge − 2 (dashed).
increasing potential drop the interface becomes rougher, and hence the number of potential adsorbate sites larger.
4.3. Interfacial capacity The potential-dependence of Y is reflected in the dependence of the charge density | on the potential drop D (Fig. 4), where differences to simulations without specific adsorption arise from the density of the
specifically adsorbed charge. As expected, these differences are particularly marked for negative potential drops. As a result, the differential capacity (Fig. 5) is steeper at negative potentials. At positive potentials, there are no significant differences — the deviations between the curves are within the errors incurred by numerical differentiation. The minimum of the capacity curves, which corresponds to the potential of zero charge, is shifted to positive values as described above.
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4.4. Electron-transfer reactions Species adsorbed from the bulk of one phase may exchange electrons with particles pertaining to the other phase. Such a particular case of interfacial electron transfer has been observed by Fermı´n et al. [7], who have studied the heterogeneous photoreduction of specifically adsorbed anionic porphyrins (meso-tetrakis(4carboxyphenyl)porphyrinato zinc(II), ZnTPPC) by a neutral quencher. They observed rather unusual current –potential curves; we have therefore simulated electron exchange with an adsorbed species in our model. In general, the rate of an electron-transfer reaction depends both on the local concentration of the reactants and on the driving force. For the special case of an electron exchange between spherical species at a liquid liquid interface the rate constant can be written in the form [10]:
& &
k= B dz
red dz%z ox h (z)z i (z%) g(z −z%) +
exp[− sg(z−z%)] exp
1 s
e0[(z) −(z%)] 2kT
n
(1)
with g(z−z%)=max(R, z −z% ). In this equation z ox h (z) and z red (z%) are the distributions of the two reactants, i normalized to unity in the corresponding bulk phases a and b. The factor in square brackets comes from the integration parallel to the interface; R is the sum of the two radii, and s the decay length of the electronic overlap between the reactants, which also gives rise to the first of the exponential terms. The last term contains the driving force; here, a transfer coefficient of 1/2 has been assumed, which is the theoretical value when
the driving force is small compared to the energy of reorganization. B is the pre-exponential factor. In our calculations we have considered electron transfer from a neutral donor in the solution 2 to the adsorbed species: + 1) − Dn1 − + ND(n + N+ 1
(2)
The distribution of the acceptor is obtained directly from our simulations and for the neutral donor we have assumed that its concentration follows that of the solvent molecules 2, and that it vanishes in the bulk of solution 1; this should be a reasonable approximation for a neutral species. Explicit calculations have been performed for n= 2 and interaction parameters r=1 and s= 5; the electronic decay length was set to s=1 A, − 1. The resulting current –potential curves are shown in Fig. 6. Their somewhat unusual shape is the result of three factors: the width of the interface, the interfacial concentration of the reactants, and the driving force. In the region between − 0.1 and about 0.2 V the current shows an increase with the driving force; however, the effective transfer coefficient is quite small, of the order of 0.1, because the surface concentration of the acceptor decreases with increasing potential. At potentials beyond 0.2 V the decrease in the concentration dominates, and the current becomes less. A strong decrease of the current with increasing potential is observed in the region below −0.1 V. In this range, the width of the interface, and hence the reaction zone, becomes thinner as the system approaches the potential of zero charge; in addition, the surface concentration of the adsorbed donor decreases, and these two factors combine to result in a high negative slope.
Fig. 6. Current–potential curves for electron transfer to the adsorbed species. System parameters: r = 1, s = 5, ion charge −2.
S. Frank, W. Schmickler / Journal of Electroanalytical Chemistry 500 (2001) 491–497
Obviously, these curves do not show the simple behavior predicted by the model of Schmickler [10], who predicted that the current – potential curves should be mainly governed by changes in the concentration. However, his work was directed at the much simpler case of comparatively low electrolyte concentrations and the absence of specific adsorption. So our results do not contradict this earlier work.
497
stant is difficult, qualitatively this effect seems to be well established. In our model, the potential dependence of the apparent rate constant is governed by several effects: the variation of the surface concentration of the adsorbed reactant, the change in the driving force, and in the thickness of the interfacial region. This can give rise to rather complicated current –potential curves, which makes a comparison with experimental results difficult.
5. Discussion Qualitatively, the differential capacity behaves quite similarly in the cases of specific ion adsorption and ion pair formation [6], though pair formation has a somewhat stronger effect. Experimental findings [7] agree qualitatively with our results: they show a shift of the pzc and a steeper rise of the capacity negative of the pzc. These effects are somewhat larger in the experimental data than in our simulations, probably because adsorption is stronger and the ionic charge higher. However, the distribution of the potential is rather different from the one we observed for ion-pair formation [6]. Only in the case of specific adsorption have we found an overshoot of the potential at the interface. The question is how this modified potential distribution affects charge transfer reactions at the interface. Fermı´n et al. [7] have studied the heterogeneous photoreduction of a specifically adsorbed anionic porphyrin (ZnTPPC) by a neutral quencher; they observed an increase of the electron-transfer rate constant k with decreasing surface concentration of the porphyrin. Since the concentration of the quencher should be independent of potential, they explained this by an increasing driving force. Though a quantitative determination of the rate con-
.
Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
References [1] Z. Samec, A. Troja´nek, J. Langmeier, J. Electroanal. Chem. 444 (1998) 1. [2] D. Homolka, P. Ha´jkova, V. Marecek, Z. Samec, J. Electroanal. Chem. 159 (1983) 283. [3] T. Huber, O. Pecina, W. Schmickler, J. Electroanal. Chem. 467 (1999) 203. [4] C.M. Pereira, W. Schmickler, F. Silva, M.J. Sousa, J. Electroanal. Chem. 436 (1997) 9. [5] C.M. Pereira, W. Schmickler, A.F. Silva, M.J. Sousa, J. Phys. Lett. 268 (1997) 13. [6] S. Frank, W. Schmickler, J. Electroanal. Chem. 483 (2000) 18. [7] D.J. Fermı´n, Z. Ding, H. Dung Duong, P.-F. Brevet, H.H. Girault, J. Phys. Chem. B 102 (1998) 10334. [8] D.J. Fermı´n, H. Dung Duong, Z. Ding, P.-F. Brevet, H.H. Girault, Phys. Chem. Chem. Phys. 1 (1999) 1461. [9] N. Metropolis, A.W. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [10] W. Schmickler, J. Electroanal. Chem. 428 (1997) 123.
A lattice-gas model for specific ion adsorption at liquid liquid interfaces Stefan Frank, Wolfgang Schmickler * Abteilung Elektrochemie, Uni6ersita¨t Ulm, D-89069 Ulm, Germany Received 22 August 2000; received in revised form 6 October 2000; accepted 9 October 2000 Dedicated to Professor Roger Parsons on the occasion of his retirement from the position of the Editor-in-Chief of the Journal of Electroanalytical Chemistry and in recognition of his great contribution to electrochemistry
Abstract A lattice-gas model is used to investigate the specific adsorption of ions at the interface between two immiscible electrolyte solutions. From Monte Carlo simulations, the profiles of particle densities and of the electrostatic potential are obtained. Specific adsorption is shown to affect the potential distribution markedly. In some cases an overshoot of the potential can be observed, an effect that is well known from specific adsorption at metal electrodes. This redistribution of charge and potential can increase the interfacial capacity, shift the potential of zero charge, and influence the rate of electron-transfer reactions. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Liquid liquid interfaces; Specific adsorption; Lattice-gas model; Electron transfer
1. Introduction The capacity of liquid liquid interfaces generally depends on the nature of the ions (see, e.g. Refs. [1,2]). In the absence of adsorption phenomena, this effect has been related to differences in the interaction energies of the ions with the two solvents: ions with a smaller energy of transfer can penetrate deeper into the mixed solvent region at the interface, thus increasing the overlap of space charge regions and hence the capacity [3]. Furthermore, the capacity can be enhanced by other processes that accumulate charge at the interface, in particular by specific adsorption. In contrast to interfaces with electronic conductors, adsorption at liquid liquid interfaces can be caused by ion pairing across the phase boundary. In recent communications [3 – 6], these ideas have been developed within the framework of a lattice-gas model, both by analytic calculations and by Monte Carlo simulations.
* Corresponding author. Tel.: +49-731-5025402; fax: + 49-7315025409. E-mail address: [email protected] (W. Schmickler).
The complementary case of ‘true’ specific adsorption may also occur at liquid liquid interfaces, when a molecule or ion with a hydrophobic and a hydrophilic part is located at the interface with each end in the preferred solvent. This phenomenon has been observed by Fermı´n et al. [7,8], who discuss its implications on the potential distribution and on rates of heterogeneous electron transfer. We believe that this type of adsorption merits further theoretical study, and have therefore Table 1 Interaction constants between the various particles a
S1 S2 K+ 1 At1 K+ 2 A− 2 Dn− 1
S1
S2
K+ 1
A− 1
K+ 2
A− 2
Dn− 1
−w w −u −u u u −r
w −w u u −u −u −s b/3s
−u u t t t t t
−u u t t t t t
u −u t t t t t
u −u t t t t t
r −s b/3s t t t t t
c
c
a In all simulations we have taken u = 3 kT, w =0.47 kT, t= 10 kT, r, s were varied. b For 1st to 3rd neighboring S2. c For 4th to 6th neighboring S2.
0022-0728/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 7 2 8 ( 0 0 ) 0 0 4 3 8 - 1
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S. Frank, W. Schmickler / Journal of Electroanalytical Chemistry 500 (2001) 491–497
investigated this process within the lattice-gas model, contrasting it to interfacial ion pairing. Following the experimental study by Fermı´n et al., we focus our attention on the distribution of the charges and the electrostatic potential, and the resulting effects on electron-transfer rates.
2. The lattice-gas model We have adapted our lattice gas model used in former simulations [3,6] to the problem at hand. Here we shall summarize its basic features and focus on new elements. Simulations were performed on a simple cubic lattice with dimensions 100× 16 ×16, the long extension being the z direction. Periodic boundary conditions apply in the x and y directions. As in our previous studies the system consists of two immiscible solvents S1 and S2, − ions K+ 1 and A1 preferentially solvated in solvent 1, − but poorly solvable in solvent 2, and ions K+ 2 and A2 preferentially dissolved in solvent 2. In addition, there are anions Dn1 − (n= 1 or 2) preferentially dissolved in solvent 1, which can be specifically adsorbed at the interface. Each lattice point is occupied by one particle, which interacts with its nearest neighbor through chemical forces. These interactions are summarized in a 7× 7 matrix (see Table 1). In order to reduce the number of parameters many interaction constants were chosen to have the same absolute values. Solvent particles of the same type attract each other with a constant − w (w\ 0), whereas unlike solvent particles repel each other with w. Ions A− and K+ have an interaction constant − u (u\0) with their preferred solvent and a constant u with the other solvent. The cross-interaction t between ions has been set to a high value in order to avoid pair formation between ions; in particular, there is no ion-pair formation across the phase boundary. Generally, these interaction constants are independent of the particle’s other nearest neighbors, with one exception: the interaction of the adsorbing anion Dn1 − with solvent S2 depends on the number of neighbors of this type: for the first three neighbors of type S2 the interaction is attractive with a constant (−s, s \0), for subsequent neighbors of this type it is repulsive, with a characteristic constant of (3s). The interaction of these ions with S1 is always repulsive (−r, r \0), independent of other neighbors. This rule ensures that for this particle the energetically most favorable sites are at the interface. In all simulations, we have chosen values of the interaction parameters u, r and s which result in the same energy of transfer from the preferred to the unpreferred solvent for all ions. In addition to the nearest-neighbor interactions each ion experiences the average electrostatic potential gen-
erated by the other ions, which is calculated by solving the Poisson –Boltzmann equation subject to the boundary conditions (− ) = 0 and d/dy(− ) = 0; this approach is tantamount to using a mean-field approximation for the Coulomb interaction. For this purpose we have to specify the lattice constant, which we take as 4 A, , and the dielectric constants of the two solvents. Solvent 1 with a dielectric constant of m1 =80 is meant to represent water, solvent 2 with m1 =10 represents an organic liquid. Since the composition of our system varies rapidly near the interface, we use an effective dielectric constant m(z)= f1(z)m1 + f2(z)m2 in this region, where f1(z) and f2(z) are the mole fractions of the aqueous and the organic phase, respectively.
3. Monte Carlo simulations Initially, the region z \0 was filled with solvent − molecules S2 and equal numbers N of ions K+ 2 and A2 . Conversely, the region z 50 was filled with solvent S1, − K+ 1 A1 as base electrolyte and a lower concentration of − n− (K+ ) and of 1 n D1 . In total, the number of anions A cations K+ was N, the number of adsorbing anions 2 Dn1 − was M, and the number of the common positive counterions K+ 1 was N+nM. The system was equilibrated by place exchanges between neighboring particles using the Metropolis [9] Monte Carlo algorithm. Subsequently, the distributions of the particles and the electrostatic potential were obtained by performing of the order of 106 Monte Carlo steps per particle. To polarize the interface between the two solutions, both phases were charged by adding equal numbers of oppositely charged excess ions; for example, to charge solution 1 positively, ions K+ were placed into the 1 region zB 0 and ions A− placed into the region z \0. 2 The number of adsorbing ions Dn1 − was not changed. After equilibration, the bulk solutions were found to be electroneutral, whereas the excess charges formed two oppositely charged space-charge regions at the interface, as expected. By comparing the single particle energy of adsorbed Dn1 − with Dn1 − in the bulk region, we obtained the adsorption energy Ead, which can be split into a chemical part E chem and an electrostatic part ad E elad. From the profile of the electrostatic potential, the total potential drop D =(− ) −( ) is obtained, D being positive for positively charged solution 1. The differential capacity C= d|/dD can be calculated by numerical differentiation. To investigate the effect of specific ion adsorption at the liquid liquid interface, we have performed several sequences of simulations varying the parameters that govern the adsorption energy Ead, and keeping all other parameters constant. Each sequence consisted of simulations with various charge densities and hence various
S. Frank, W. Schmickler / Journal of Electroanalytical Chemistry 500 (2001) 491–497
493
4. Results
4.1. Distribution of charge and potential
Fig. 1. Distribution of the charge density and the electrostatic potential for uncharged phases. Thick solid line: no specific adsorption; thin solid line: adsorbing ion with charge −1 and interaction parameters r =2 and s =4; dotted line: adsorbing ion with charge −2 and interaction parameters r =1 and s =5.
Table 2 Shift Dpzc of the potential of zero charge (pzc) and chemical part of the adsorption energy (E chem ad ) at the pzc for various values of interaction parameters r and s, and ion charge −n r/kT, s/kT
2, 4
2, 5
n
1
2
1
2
Dpzc/mV E chem ad /kT
6 −2.6
10 −2.7
12 −5.5
18 −5.7
In the absence of the adsorbing ion Dn1 − the uncharged system shows zero average charge density for all lattice coordinates z and thus no potential drop across the interface. In the presence of Dn1 − , adsorption causes a sharp negative peak in the charge density at the interface, which is balanced by a positive space charge region in solution 1 (see Fig. 1) — by definition, there is no excess charge in solution 2. The electrostatic potential drops across the dipolar charge distribution in solution 1, while the charge and the potential distributions in solution 2 are practically unaffected. The bulk of solution 1 is then at a positive potential D compared to solution 2. This potential difference D is equal to the shift Dpzc in the potential of zero charge (pzc) caused by the specific adsorption of the ions at the interface, since in our model both bulk phases have the same potential in the absence of specific adsorption. As expected, this shift is larger, the greater is the difference s− r in the interaction constants, which govern the strength of the adsorption, and the higher the charge on the ion (see Table 2). The positive shift of the potential in solution 1 persists when the interface is polarized. Fig. 2(a) and (b) show the situation when there is a positive excess charge in solution 1. In this case specific adsorption increases both the charge stored at the interface and the concomitant potential drop. At potentials below the pzc the adsorbed charge has the opposite sign as the surface charge in solution 2. The sum of these charges, which has to be compensated by the space charge in solution 1, is smaller than in the absence of specific adsorption; it can even be negative for sufficiently strong adsorption and not too high surface charges. In the latter case, the potential distribution in solution 1 shows an inverted region (see dashed line in Fig. 2(d)). Such an overshoot of the potential has not been observed in simulations for ion pairing [6]. An ion pair is a neutral species, and hence has a weaker effect on the potential profile. For the same reason it has only a small effect on the position of the pzc.
4.2. Adsorption beha6ior potential differences between the solutions. The chemical adsorption energy E chem is controlled by the interacad tion constants r and s, whereas the electrostatic part E elad is controlled by the ion charge number − n. For comparison, we have performed a sequence of simulations in the absence of the adsorbing ion. The mean concentrations of the base electrolytes were chosen as 3.9 ×10 − 3 particles per lattice point, corresponding to a 0.1 M solution. The mean concentration of the adsorbing salt was one order of magnitude lower.
The surface excess Y of the adsorbing ions is determined both by the chemical driving force, which is specified through the chemical interactions r and s, and by the electrostatic interactions. Fig. 3 shows Y as a function of the potential drop D for several system parameters. A negative potential drop drives the ions towards the interface, but even for small and intermediate positive potentials there is still a sizable amount of adsorbed ions. The diamonds and circles in the figure
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correspond to ions with the same chemical interactions but different charge numbers. Over most of the range investigated the monovalent ion has a greater surface excess than the divalent ion; obviously, the larger adsorbate –adsorbate repulsion keeps the excess of the ions with the larger charge smaller. Only large negative
potentials can overcome the mutual repulsion of the adsorbates, and then the excess of the divalent ion becomes larger. Other things being equal, a smaller chemical driving force (squares in Fig. 3) always entails a smaller surface excess. Note that it is not possible to express the surface excess in terms of a coverage: with
Fig. 2. Distribution of the charge density and the electrostatic potential at the interface. Data are shown for the absence of adsorbing ion (thick solid lines); adsorbing ion with charge − 1 and interaction parameters r= 2 and s =4 (thin solid lines); adsorbing ion with charge − 2 and interaction parameters r =1 and s =5 (dotted lines). The surface charge density of solution 1 is 0.78 mC/cm2 in (a) and (b), and − 0.78 mC/cm2 in (c) and (d).
Fig. 3. Dependence of the surface excess Y of Dn1 − , in particles per site, on the potential drop D. Interaction parameters r= 1 and s= 5, ion charge − 1 (diamonds) or − 2 (circles); interaction parameters r =2 and s= 4, ion charge − 2 (squares) and ion charge −1 (crosses).
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Fig. 4. Surface charge density | versus total potential drop D for absence of adsorbing ion (filled squares); interaction parameters r =2 and s= 4, ion charge − 1 (crosses) or −2 (squares); interaction parameters r =1 and s =5, ion charge − 1 (diamonds) or −2 (circles).
Fig. 5. Differential capacity C versus total potential drop D for absence of adsorbing ion (solid line), and for two cases of specific adsorption: interaction parameters r= 2 and s= 4, ion charge −2 (dotted); interaction parameters r =1 and s= 5, ion charge − 2 (dashed).
increasing potential drop the interface becomes rougher, and hence the number of potential adsorbate sites larger.
4.3. Interfacial capacity The potential-dependence of Y is reflected in the dependence of the charge density | on the potential drop D (Fig. 4), where differences to simulations without specific adsorption arise from the density of the
specifically adsorbed charge. As expected, these differences are particularly marked for negative potential drops. As a result, the differential capacity (Fig. 5) is steeper at negative potentials. At positive potentials, there are no significant differences — the deviations between the curves are within the errors incurred by numerical differentiation. The minimum of the capacity curves, which corresponds to the potential of zero charge, is shifted to positive values as described above.
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4.4. Electron-transfer reactions Species adsorbed from the bulk of one phase may exchange electrons with particles pertaining to the other phase. Such a particular case of interfacial electron transfer has been observed by Fermı´n et al. [7], who have studied the heterogeneous photoreduction of specifically adsorbed anionic porphyrins (meso-tetrakis(4carboxyphenyl)porphyrinato zinc(II), ZnTPPC) by a neutral quencher. They observed rather unusual current –potential curves; we have therefore simulated electron exchange with an adsorbed species in our model. In general, the rate of an electron-transfer reaction depends both on the local concentration of the reactants and on the driving force. For the special case of an electron exchange between spherical species at a liquid liquid interface the rate constant can be written in the form [10]:
& &
k= B dz
red dz%z ox h (z)z i (z%) g(z −z%) +
exp[− sg(z−z%)] exp
1 s
e0[(z) −(z%)] 2kT
n
(1)
with g(z−z%)=max(R, z −z% ). In this equation z ox h (z) and z red (z%) are the distributions of the two reactants, i normalized to unity in the corresponding bulk phases a and b. The factor in square brackets comes from the integration parallel to the interface; R is the sum of the two radii, and s the decay length of the electronic overlap between the reactants, which also gives rise to the first of the exponential terms. The last term contains the driving force; here, a transfer coefficient of 1/2 has been assumed, which is the theoretical value when
the driving force is small compared to the energy of reorganization. B is the pre-exponential factor. In our calculations we have considered electron transfer from a neutral donor in the solution 2 to the adsorbed species: + 1) − Dn1 − + ND(n + N+ 1
(2)
The distribution of the acceptor is obtained directly from our simulations and for the neutral donor we have assumed that its concentration follows that of the solvent molecules 2, and that it vanishes in the bulk of solution 1; this should be a reasonable approximation for a neutral species. Explicit calculations have been performed for n= 2 and interaction parameters r=1 and s= 5; the electronic decay length was set to s=1 A, − 1. The resulting current –potential curves are shown in Fig. 6. Their somewhat unusual shape is the result of three factors: the width of the interface, the interfacial concentration of the reactants, and the driving force. In the region between − 0.1 and about 0.2 V the current shows an increase with the driving force; however, the effective transfer coefficient is quite small, of the order of 0.1, because the surface concentration of the acceptor decreases with increasing potential. At potentials beyond 0.2 V the decrease in the concentration dominates, and the current becomes less. A strong decrease of the current with increasing potential is observed in the region below −0.1 V. In this range, the width of the interface, and hence the reaction zone, becomes thinner as the system approaches the potential of zero charge; in addition, the surface concentration of the adsorbed donor decreases, and these two factors combine to result in a high negative slope.
Fig. 6. Current–potential curves for electron transfer to the adsorbed species. System parameters: r = 1, s = 5, ion charge −2.
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Obviously, these curves do not show the simple behavior predicted by the model of Schmickler [10], who predicted that the current – potential curves should be mainly governed by changes in the concentration. However, his work was directed at the much simpler case of comparatively low electrolyte concentrations and the absence of specific adsorption. So our results do not contradict this earlier work.
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stant is difficult, qualitatively this effect seems to be well established. In our model, the potential dependence of the apparent rate constant is governed by several effects: the variation of the surface concentration of the adsorbed reactant, the change in the driving force, and in the thickness of the interfacial region. This can give rise to rather complicated current –potential curves, which makes a comparison with experimental results difficult.
5. Discussion Qualitatively, the differential capacity behaves quite similarly in the cases of specific ion adsorption and ion pair formation [6], though pair formation has a somewhat stronger effect. Experimental findings [7] agree qualitatively with our results: they show a shift of the pzc and a steeper rise of the capacity negative of the pzc. These effects are somewhat larger in the experimental data than in our simulations, probably because adsorption is stronger and the ionic charge higher. However, the distribution of the potential is rather different from the one we observed for ion-pair formation [6]. Only in the case of specific adsorption have we found an overshoot of the potential at the interface. The question is how this modified potential distribution affects charge transfer reactions at the interface. Fermı´n et al. [7] have studied the heterogeneous photoreduction of a specifically adsorbed anionic porphyrin (ZnTPPC) by a neutral quencher; they observed an increase of the electron-transfer rate constant k with decreasing surface concentration of the porphyrin. Since the concentration of the quencher should be independent of potential, they explained this by an increasing driving force. Though a quantitative determination of the rate con-
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Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
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