Modeling of the Coadsorption of Chloride and Hydrogen Ions on Copper Electrode Surface

In the following, we present a Langmuir-like model for the coadsorption of two species on a common surface, where additional interaction parameters are introduced, by which the mutual influence of the two species on their adsorption is taken into account in a phenomenological manner. The kinetics for the coadsorption of two species, A and B, is modeled by the following coupled equations for the temporal evolution of the surface densities

where , Γ_i with i = {A, B} is the surface density of the i-th ion species, Γ⁰_i is the maximal surface density in the absence of the other species, c_i is the ion density in the solution, and k^a_i and k^d_i are the adsorption and desorption coefficients, respectively. Further, θ^free_i is the free surface for the adsorption of the corresponding ion species normalized by the total substrate surface, which is expressed as

where the last term on the rhs of the equations represents the hindering of the adsorption of one species by the adsorbed molecules of the other species. The corresponding factors α and β determine the degree of the hindering. The values of these factors are related to the special kind of the interaction of the two species on the surface. In the following, we consider different cases of this interaction.

(A) Within a simple mechanistic view, the adsorbed species can be considered as rigid bodies covering a species-dependent area a_i on the surface. As the only interaction between the adsorbed molecules, area exclusion between the molecules leads to the following equation

where N_i are the total numbers of the adsorbed species on the total surface A. In the second equation, the relations Γ_i = N_i/A and a_i = 1/Γ⁰_i were used. The same expression results for species B, i. e. θ^free_A = θ_B^free. Comparison with Eqs. 3 and 4 shows that it corresponds to the limiting case α = β = 1.

In general, the interaction between the molecules is more complex. Within an atomistic view, different species may have different preferred adsorption sites on the substrate surface (for example fcc or hcp hollow sites, bridge sites, or on top of substrate atoms).

(B) As a more complex example, one may imagine that the surface area N_Ba_B, covered by adsorbed species B, is not completely forbidden for adsorption of species A. Only within a certain surface part αN_Ba_B (α < 1) species A cannot be adsorbed. Analogously, only the surface part βN_Aa_A (β < 1) is considered to be unavailable for adsorption of species B. Then, the normalized free surfaces for the two species are given by

from which Eqs. 3 and 4 follow.

(C) A further example, which demonstrates the meaning of the coefficients α and β, is a system where each species has its own set of adsorption sites (Γ⁰_A and Γ⁰_B). These sites could be disjunct, e.g. species A on hollow sites and species B on top sites. However, we consider here a more general case where the two species share also some common sites with surface density Γ⁰_AB = Γ_A⁰∩Γ⁰_B. Assuming species to be evenly distributed over common and non-common sites, the surface density of free sites for each species reads

where in the second equations the quantities α = Γ⁰_AB/Γ_A⁰ and β = Γ⁰_AB/Γ_B⁰ have been introduced. Using the relations θ^free_i = Γ_i^free/Γ⁰_i, we obtain again Equations 3 and 4.

(D) For certain systems, replacement processes of an adsorbed species by the other species can be relevant. For instance, adsorbed chloride ions on a copper surface could be replaced by hydrogen ions in solution (compare the results of DFT calculations below). For a system of species A and B, where species B in solution are able to replace adatoms of species A, but not vice versa, those replacement processes are taken into account by the following kinetic equations

where k^r is the kinetic coefficient for the replacement of species A by species B. With the simple approximation 5, i. e. θ^free = 1 − Γ_A/Γ⁰_A − Γ_B/Γ⁰_B, Eqs. 10 and 11 can be rewritten in a form similar to Eqs. 1 and 2

By comparing with Eqs. 1 to 4, one finds that θ^free_A = θ^free and θ^free_B = θ^free + k^rΓ_A/(k^a_BΓ_B⁰). Thus, in the case of an additional replacement mechanism, the kinetic equations can be reformulated in a manner as in Eqs. 3 and 4 where the parameters α and β take the values α = 1 and β = 1 − k^rΓ⁰_A/(k_B^aΓ⁰_B).

Generally, the interactions between the absorbents and with the substrate are very complex. The different scenarios listed above, represent limiting cases and are not mutually exclusive. One example corresponding to case (A) would be the adsorption of inert gases. Strongly different repelling interactions between adsorbed species could lead to a situation considered in case (B). For case (C), it is important that the two species occupy different surface sites of the substrate as for example top and hollow sites. As an example of the replacement scenario (D), we consider below the chlorine-hydrogen-system.

The equilibrium surface densities of the two species in the coadsorbed state are given by the stationary limit of Eqs. 1 and 2. By inserting into those equations formulas 3 and 4 for the normalized free surfaces θ^free_A and θ^free_B, one finds the equilibrium densities

where the abbreviations κ_i = k^a_i/k^d_i have been introduced. If one of the two species is absent, these formulas reduce to the common Langmuir isotherm for a single species. For example for c_B = 0, one obtains Γ^eq_A = Γ_A⁰/(1 + 1/(κ_Ac_A)).

An interesting question is the value of the surface densities in the limit of very high bulk concentration of one species, e.g. for c_A → ∞. In this limiting case, one finds for the saturation surface densities

Thus, we get the remarkable prediction that the saturation surface density Γ^S_A of species A is generally less than Γ⁰_A and depends on the concentration of species B in the solution. Species A reach full coverage Γ^S_A = Γ_A⁰ only if coefficient β = 1, or for the trivial case c_B = 0.

In the following, we show how the derived expressions for the coadsorption of two species can be reformulated as the common Langmuir model for the adsorption of one species only. The kinetic equation for the adsorption of a single species, say A, reads

We have labeled the quantities in the last equation with a tilde to distinguish them from those introduced above. The corresponding equilibrium surface density can be expressed in the following way

where . Our result for Γ^eq_A in the case of the coadsorption of two species, Eq. 14, can be rewritten in a form similar to Eq. 19. By using the expression 16 for the saturation density Γ^S_A, one finds

Comparison of Equations 19 and 20 reveals how the parameters in the common Langmuir model for the description of the adsorption isotherm of only one species are affected by the presence of a second species. The corresponding parameters are rescaled in the following way

Equations 21 and 22 show how the concentration c_B of species B in the bulk solution influences the adsorption of species A.

The temperature dependence of the adsorption and desorption rate constants are commonly written as an Arrhenius equation

where k₀ is a prefactor and E_a the activation energy. The adsorption or desorption of charged species (e. g. H⁺ and Cl⁻ ) represent a charge transfer process.^18,19 Thus, the activation energy will have a contribution due to the bias potential E

where eΔz = e(z^b − z) is the charge transfer between the barrier state with charge ez^b and the corresponding adsorption or desorption state with charge ez^a or ez^d, respectively. This leads to the following potential dependence of the adsorption and desorption rate constants of species i

Using these expressions, we finally find the potential dependence of the characteristic variables κ_i in Equations 14 and 15 for the equilibrium coverages θ^eq_i

where K_i = (k^a_0i/k^d_0i)exp ((E_0i^d − E^a_0i)/(k_BT)) and B_i = e(z^d_i − z^a_i)/(k_BT) have been introduced. Here e(z^d_i − z_i^a) is the charge transfer due to ion adsorption. A similar exponential dependence of the adsorption and desorption rate constants on the bias potential has been proposed in Refs. 9–11,18–20. In the following, we will apply the proposed model equations for the description of the coadsorption of two ion species in the presence of a bias potential to the Cl⁻-H⁺-system.

The adsorption of halides on different metal surfaces has been extensively analyzed in the literature within the framework of the spin polarized density functional theory (DFT). These works concern mainly the adsorption of one species in vacuum (e.g. adsorption on Cu(111)).^21,22,24 Only a few works consider the coadsorption of hydrogen and halides. Coadsorption of hydrogen and chloride ions on Pt(111) has been analyzed for example in Ref. 25. To our knowledge, coadsorption of hydrogen and chlorine on copper has not yet studied by DFT. In order to gain insight into the tendencies of the adsorption for this system, we have performed spin polarized DFT calculations of the Cl-H-coadsorption on the Cu(111) surface. The calculations were done by means of the Vienna ab initio simulation package (VASP),^26,27 using the PBE generalized gradient approximation (GGA) for the exchange-correlation functional²⁸ and the PAW method.^29,30 The wave functions were expanded in plane waves up to a kinetic energy cutoff of 400 eV. The size of the super cell was 3 × 3 or 6 × 6 depending on the chlorine coverage. The Brillouin zone was sampled by 4 × 4 × 1 (2 × 2 × 1) k-points using the Monkhorst-Pack scheme³¹ for small (large) super cell size. Periodic boundary conditions were applied for all calculations with a gap between the surfaces of 25 Å. Dispersion corrections were included by the standard D2 Grimme parametrization.³²

As an important issue of the coadsorption, we analyzed whether the replacement of adsorbed chlorine atoms by hydrogen atoms is energetically favorable. To this end, the energy difference between the two states in Figs. 1a and 1b, where one Cl atom is replaced by H, was calculated. The initial Cl coverage was . Here, the coverage is defined as the number of adsorbed species divided by the number of copper atoms in the topmost (111) surface layer. For the considered atom configurations, we found that the state, where hydrogen replaced chlorine, is energetically unfavorable by 0.6 eV. To be closer to the real electrochemical system, we repeated the calculations for the same set up with initial coverage , but with explicit consideration of the presence of water molecules. In this case, the replacement of chlorine by hydrogen was found to be energetically favorable with an energy gain of 0.9 eV. In order to quantify the stability of coadsorbed states on Cu(111), we calculated a mean adsorption energy for different surface coverages of chlorine and hydrogen. The corresponding adsorption energy was defined by

where E_Cl/H/Slab is the total energy of the whole system and E_Slab is the total energy of the copper slab without adsorbate. E_H2 and E_Cl2 are the energies of the isolated dimers, and N_H and N_Cl are the numbers of adsorbed atoms on the surface. Fig. 2 shows examples of the most stable configurations of coadsorbed states for different coverages with . The preferred adsorption sites of both chlorine and hydrogen are fcc and hcp hollow sites on the surface. The calculated adsorption energies vary between −1 eV for up to −0.5 eV for for . Configurations with were found to be unstable. Obviously, a repelling interaction between the adsorbed atoms leads to a weakening of the adsorption strength. According to a Bader charge analysis,³³ adsorbed chlorine as well as hydrogen are negatively charged. The calculated adsorption energies of individual species have been found to be −0.23 eV and −1.79 eV for H and Cl, respectively. These energy values are in good agreement with previous DFT calculations.^23,24 The mean adsorption energy of the coadsorbed state at a coverage of 2/9 (cf. diagram in Fig. 2) results practically as average of the individual adsorption energies.

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On the basis of our coadsorption model introduced above, we have shown that the adsorption of the ions can be described by the standard Langmuir model with rescaled parameters. In this section, we will apply the proposed reduced model to the adsorption of chloride ions on copper in a highly acidic solution. As mentioned in the Introduction, this problem is of great importance for the electrochemical deposition of copper interconnects within the framework of the damascence technique. The adsorption of Cl and H on copper has been investigated in many studies, both experimentally and theoretically, see e.g.^35–39 To our knowledge, there are only few experiments concerning the quantitative determination of the Cl coverage on copper. Here, we employ experimental findings obtained by Horanyi et al., using the radio-active method.¹⁸ As shown in Fig. 3a, the surface density of chloride on copper depends on the bias potential as well as on the chloride concentration in the bulk solution. In this work, the potential is referred to the standard hydrogen electrode (SHE). In the literature, the adsorption of chloride ions in an acidic solution is often described by the common Langmuir model for a single species without explicit consideration of the presence of hydrogen ions. The corresponding equilibrium surface density is given by formula 19 with A → Cl

This equation reveals the linear dependence of 1/Γ^eq_Cl on 1/c_Cl. The measurement data in Fig. 3a roughly show such a linear dependence as can be seen from the plots in Fig. 3b with the bias potential as curve parameter. By fitting the data in Fig. 3b to a linear function , the saturated surface density has been determined. As a remarkable result, we find from Fig. 3b how the value of changes with the bias potential. The symbols in Fig. 4 show as a function of the electrode potential. It can be seen that the saturation surface density decreases with decreasing potential. From our general formula 21 for the coadsorption of two species, with A → Cl and B → H, we find

Determination of all unknown parameters in Eq. 30 at once by fitting the potential dependence of the data in Fig. 4 was not unique. However, in all cases we found that the value of α was almost one. With α = 1, Eq. 30 simplifies to

Using the general formula 27, we obtain the following potential dependence of

Since for positive bias potential E, chloride adsorption is more favorable than hydrogen adsorption, we expect B_H > 0. Then, the saturation surface density as a function of the bias potential represents a sigmoid function with for E → +∞ and for E → −∞. By introducing a critical bias potential E_c via the equation (1 − β)K_Hc_H = exp (B_HE_c), the chloride saturation density can be expressed as

Fit of the data points in Fig. 4 to Eq. 33 yields the solid curve in Fig. 4. The corresponding parameter values are Γ⁰_Cl = 3.21 × 10^{− 10} mol/cm², B_H = 6.66 V^{− 1} and E_c = −0.226 V.

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In the linear relationship between 1/Γ^eq_Cl and 1/c_Cl, we have introduced the parameter (compare Eq. 29). The fit of this parameter is presented in Fig. 5a as a function of . Together with the derived dependence of the saturation surface density on the bias potential in Fig. 4, it is possible to derive the value of as a function of the potential. The corresponding data points are shown in Fig. 5b. According to the general formula 22 derived above and considering the fit result α = 1, the ratio is given by

By inserting the potential dependence introduced in Eq. 27, we find

Because of a considerable scatter of the data points in Fig. 5b, a general fit of all unknown parameters in Eq. 35 is unreliable. Thus, in a rough manner, we approximated the prefactor in front of K_Clexp ( − B_ClE) by one. This corresponds to the limiting case β = 0 in Eq. 35 and yields a lower estimate of the parameter B_Cl. As a consequence the dependence of on the hydrogen bulk concentration cancels. The fit results read then B_Cl = −1.4 V^{− 1} and K_Cl = 125 m³/mol. The corresponding value of the charge transfer for chloride adsorption is eΔz = 0.036 e. For comparison, Foresti et al. studied the adsorption of chloride ions on the Ag(111) surface and found a partial charge transfer of 0.07 e (see Tab. 1 of Ref. 34). As a tendency, with increasing value of β, our fit suggests a weaker exponential potential dependence of the parameter since then the magnitude |B_Cl| decreases.

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In summary, the above analysis of the measurements of the adsorbed chloride surface density by Horanyi et al.¹⁸ provides us with the following basic data to predict the chloride equilibrium coverage. The potential dependence of the saturated surface concentration is given by Eq. 33 and that of the ratio of the adsorption and desorption rate by . The corresponding parameter values K_Cl, B_Cl, Δz, Γ⁰_Cl, B_H, and E_c are listed in Table I. With these data, the chloride equilibrium surface density was calculated according to model Equation 29 as a function of the electrode potential for different chloride concentrations in the bulk solution. As can be seen in Fig. 6a, the simulation results agree very well with the measurements. We calculated the equilibrium surface densities also for bulk concentrations higher than in the experiments. For c_Cl = 1 mM and 10 mM, the equilibrium surface density Γ^eq_Cl is very close to the saturation value . Increasing the chloride bulk concentration beyond 1 mM practically has no influence on the equilibrium surface density of chloride in the studied potential regime.

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For comparison, we also studied a simplified adsorption model without consideration of a potential dependence of the saturation value of the surface density , similar to approaches in Refs. 9–11,18–20. Because of our approximation β = 0 in Eq. 35, this corresponds to the limiting case of complete absence of adsorbed hydrogen. Fit of the measurements in Fig. 6a for a special bulk concentration, say c = 0.1 mM, yields the parameters mol/cm² and B_Cl = 5.5 V^{− 1}. The corresponding charge transfer during the adsorption process becomes 0.14 e. With these parameter values, the surface density of adsorbed chloride can be calculated for other bulk concentrations. As shown in Fig. 6b, there is a considerable disagreement between simulation and measurements. This clearly demonstrates the importance of including the potential dependence of the saturation surface density in the model description of chloride adsorption in an acidic solution.