Supporting Information A Universal Approach to Determine the Free Energy Diagram of an Electrocatalytic Reaction

Transcription

1 Supporting Information A Universal Approach to Determine the Free Energy Diagram of an Electrocatalytic Reaction Kai S. Exner 1,, Iman Sohrabnejad-Eskan 1, Herbert Over 1,* 1 Physical Chemistry Department, Justus-Liebig-University Giessen, Heinrich-Buff-Ring 17, 3539 Giessen, Germany University of Sofia, Faculty of Chemistry and Pharmacy, Department of Physical Chemistry, 1 James Bourchier Avenue, 1164 Sofia, Bulgaria * Corresponding author: herbert.over@phys.chemie.uni-giessen.de Closed Microkinetic Solution according to the Volmer-Heyrovsky Mechanism The two-step Volmer-Heyrovsky mechanism consists of an electrochemical Volmer step that is followed by the electrochemical Heyrovsky step. The reverse reactions of the Volmer and the Heyrovsky step are allowed to proceed. Therefore, we obtain the following reaction equations with rate constants k 1 for the Volmer step, k -1 for the reverse Volmer step, k for the Heyrovsky step and k - for the reverse Heyrovsky step: k 1 m S+ X ± e S X k S + ± 1 m X S X e m S X + X ± e S+ X k (1) () (3) k S X m + S X + X ± e In equations (1) (4), S denotes the active site of the electrocatalyst and X the active species (X = Cl in case of CER or X = H + in case of HER) that is forming the product X = Cl or X = H for CER and HER, respectively. The reaction rate r for product formation is given by the equation: dp( X x ) m r = = ka( X ) a( e ) θ ( S X ) k a( X ) θ ( S) (5) dt The activities of electrons, the species X and the product X are assumed to be constant, a(e ) = a(x) = a(x ) = 1. The coverage θ(s X) of the reaction intermediate (RI) can be obtained as solution of the following differential equation: () =+ (S) (S X) (S X)+ (S) (6) Taking into account the balance of active sites S (S)+ (S X)=1 (7) (4) 1

2 and steady-state conditions, i.e. () translates to: (S X)= =0, the coverage θ(s-x) of the reaction intermediate (8) According to equations (7) and (8), the coverage θ(s) of the electrocatalyst s active site is given by: (S)= (9) Applying equations (5) (9), the reaction rate r can be expressed as: = (10) The rate constants k 1, k -1, k and k - are calculated employing transition state theory and micro reversibility summarized in equations (11) to (14): = # (11) = ( # ) ( ) (1) = ( # ) (13) = # ( ) (14) Here, G 1 # and G # denote the transition state (TS) free energies of the Volmer and the Heyrovsky step, respectively, while G 1 TD indicates the free energy of the reaction intermediate S-X (cf. Figure S1). α 1 and α are the symmetry factors for Volmer and Heyrovsky step. k B, h and e denote Boltzmann constant, Planck constant and the elementary charge, respectively. η indicates the absolute value of the applied overpotential, i.e. the electrode potential U in excess of the corresponding reversible electrode potential U o, namely η = U U o CER = +η CER > 0 or η = (U U o HER) = η HER > 0. The reaction rate r translates into the current density j via Faraday s law: = Γ (15) The number of transferred electrons z is two for the Volmer-Heyrovsky mechanism, while Γ act stands for the number of active sites on the catalyst s surface per cm surface area ( cm - and cm - for RuO (110) and Pt(111), respectively). From equations (10) (15), we obtain the following closed microkinetic solution for the Volmer-Heyrovsky mechanism j(η):

3 ()= ( ) ( ) # # # ( ) # ( ) (16) Equation (16) reveals that the overall current density j as function of the applied overpotential η depends on six parameters: The surface density of active sites Γ act, the TS free energies G # 1 and G #, the symmetry factors α 1 and α and the free energy of the reaction intermediate G TD 1. While the density of active sites can be calculated by the number of active sites per area of the surface unit cell, all other parameters can be adopted from the free energy landscape in Figure S1. Figure S1: Free energy diagram of the Volmer-Heyrovsky mechanism, in which the active species X is forming the product X via the coupled Volmer and Heyrovsky steps, for η = 0 V. While G 1 # and G # denote the transition state (TS) free energies of the Volmer and the Heyrovsky step, α 1 and α indicate the symmetry factors of the Volmer and the Heyrovsky step, respectively. G 1 TD constitutes the free energy of the reaction intermediate S X. i) CER over RuO (110): Based on the evaluation of the experimental Tafel plot (cf. Figure 5), we obtain the following optimized parameters for the CER over RuO (110): G # 1 = 0.77 ev, G # = 0.89 ev, α 1 = 0.69 and α = The ab initio Pourbaix diagram (cf. Figure 6) reveals G TD 1 = 0.34 ev and the surface density of active sites amounts to cm - in case of RuO (110). Figure S depicts the applied overpotential as function of the experimentally measured current density in the form of a Tafel plot as well as the fit function according to the closed microkinetic solution of equation (16) for the CER over RuO (110). It turns out that the closed microkinetic solution 3

4 in equation (16) matches remarkably well the experimental current density as function of the applied overpotential η CER of CER over RuO (110) in the complete overpotential range. Figure S: Tafel plot of the CER over RuO (110). Black dots indicate the experimentally measured current density, while the fit function according to the model of a Volmer- Heyrovsky mechanism (equation (16)) is posed as green line. It turns out that the closed microkinetic solution is able to describe the current density j of CER in the complete overpotential range of 0.0 V < η CER < 0. V. The free energies in Figure S1 enter the analytical formula (16) only in the denominator. A careful inspection of these exponential terms in the denominator reveals that the first (G # 1 + α e η) and the last term (G # (1 α 1 ) e η) dominate the other two terms (G # TD G 1 + α 1 e η) and (G # 1 G TD 1 (1 α ) e η) as long as G TD 1 >> 0 ev. Actually, G TD 1 = 0.34 ev so that equation (16) simplifies to: = () ( ) # # ( ) (17) G 1 TD does not enter equation (17) so that the current density j as a function of η is independent of G TD 1. In the Tafel regime, the numerator is dominated by the first exponential term, while the second exponential term tends to zero with increasing overpotential η. Therefore, equation (17) can be written in the Tafel regime as: = # ( ) # ( ) (18) 4

5 In equation (18) the denominator is governed by two exponential terms, (G # # 1 + α e η) or (G (1 α 1 ) e η). Figure S3 depicts the change in free energy of both terms as function of the applied overpotential η. It turns out that for small overpotentials, i.e. η < 0.10 V, the term (G # (1 α 1 ) e η) is much larger than (G # 1 + α e η). An applied overpotential of η = 0.15 V is identified as crossover overpotential, since for this overpotential value both terms reveal the same free energy. For η > 0.15 V the term (G # 1 + α e η) prevails over (G # (1 α 1 ) e η). Figure S3: Free energy G # as function of the applied overpotential η for the terms (G # 1 + α e η) and (G # (1 α 1 ) e η). While for η < 0.15 V the term (G # (1 α 1 ) e η) prevails over (G # 1 + α e η), the reverse case is observed for η > 0.15 V. According to (G # (1 α 1 ) e η) >> (G 1 # + α e η) for η < 0.10 V, the term (G 1 # + α e η) is negligible resulting in: = exp # exp( ) (19) Therefore, the logarithm of the current density in the first Tafel regime (0.03 V < η < 0.10 V) is approximated by: log()=log # +( ) =log( )+ According to equation (0) the TS free energy G # 5 (0) and the symmetry factor α of the Heyrovsky step can be determined from the interception with the log(j) axis (logarithm of the exchange current density log (j 0 )) and from the slope of the log (j) vs. η plot (Tafel slope b),

6 respectively. This result, which is based on microkinetic modeling, is in accordance with the framework of the generalized Butler-Volmer equation (cf. Section ) assuming quasiequilibrium of the reaction intermediates preceding the rate-determining reaction step with the reactants. For the second Tafel region, i.e. η > 0.15 V, the term (G 1 # denominator of equation (18) leading to: + α e η) dominates the = exp # exp (1) Consequently, the logarithm of the current density in the second Tafel regime (η > 0.15 V) can be written as: log()=log # + =log( )+ Equation () reveals that the TS free equation G 1 # and the symmetry factor α 1 of the Volmer step can be determined from the interception with the log(j) axis (log (j 0 )) and from the slope of the log (j) vs. η plot (b), respectively, which matches with the analysis based on the Butler- Volmer formalism (cf. Section 3). () ii) HER over Pt(111): The same analysis as for the CER over RuO (110) can be conducted for the HER over Pt(111), since the reaction is assumed to proceed via a Volmer-Heyrovsky mechanism. The number of active sites amounts to cm - for Pt(111) and the surface Pourbaix diagram indicates G 1 TD = 0.35 ev (cf. Figure 14). The transition state free energies G 1 # and G # as well as the symmetry factors α 1 and α are optimized based on the experimental data of Brülle et al. 1 We obtain G 1 # = ev, G # = 0.75 ev, α 1 = und α = Figure S4 depicts the experimentally measured Tafel plot and the fit function according to the closed microkinetic solution in equation (16) for the HER over Pt(111). 6

7 Figure S4: Tafel plot of the HER over Pt(111). Black dots indicate the experimentally measured current density, while the fit function according to the model of a Volmer- Heyrovsky mechanism (equation (16)) is superimposed as green line. The closed microkinetic equation models the current density j of HER in the overpotential range of 0.04 V < η HER < 0.1 V remarkably well. Figure S4 reveals that the closed microkinetic formula of equation (16) describes the Tafel plot for HER over Pt(111) reasonably well for 0.04 V < η HER < 0.1 V. There are systematic deviations between our model function and the experimental data for lower overpotentials, i.e. η HER < 0.04 V. This finding might be ascribed to kinetically hindered bubble formation that limits the reaction rate of HER for small overpotentials. Consequently, the current density is systematically overestimated in the model function for η HER < 0.04 V. (1) Brülle, T.; Schneider, O.; Stimming, U. Z. Phys. Chem. 01, 6,