Electron transfer theory and electrocatalysis. Marc Koper Leiden University (NL)

Transcription

1 Electron transfer theory and electrocatalysis Marc Koper Leiden University (NL)

2 Outline Molecular theory of redox reactions Marcus theory ion transfer proton transfer bond breaking role of (metal) catalyst Thermodynamic theory of multiple proton-electron transfer Sabatier principle density functional theory scaling relations role of catalyst and solution ph

3 Butler-Volmer rate law Redox reaction: Red Ox + e - Oxidation rate constant: k = k s exp[αf(e-e 0 )/RT] What is k s? What is α?

4 Outer-Sphere Electron Transfer Bond Breaking Electron Transfer Ox n+ R X e - e - Red (n-1)+ R X - Ion Transfer Dissociative Adsorption - Electron Transfer X 2 transition state Ox n+ δe - Red (n-δ)+ X- X- X δ Xδ

5 Arrhenius law Rate processes are rare events k = Aexp( E / k T) act B Arrhenius law implies equilibrium between normal reactants and active reactants, i.e. the activated complex. Van t Hoff (1884) Arrhenius (1889)

6 The Potential Energy Surface (PES) Activated complex is the saddle point ( col ) of the potential energy surface = transition state reaction coordinate transition state reactant product reaction coordinate 1

7 Transition-State Theory - Assumptions 1. Every time the system reaches the transition state, it unavoidably progresses in the direction of the product state. There is no recrossing of the barrier. 2. The energy distribution among the reactant molecules follows the Maxwell-Boltzmann distribution. Hence the concentration of activated complexes can be calculated from equilibrium theory. 3. Separability of the motion through the transition state from the non-reactive motions. 4. Motion over the barrier is classical - no quantum effects accounted for. Eyring (1935), Evans & Polanyi (1935)

8 Transition-State Theory - Rate Expression k k T Z = B exp( E / k T) b B h Z R h, Planck s constant Z, partition function of the non-reactive modes at the transition state Z R, partition function of the reactant(s) E b, energy difference between activated complex and reactant state ( barrier height )

9 TST rate for escape from 1-D well energy ω E b reaction coordinate Z = 1 1 ZR = 1 exp( hω / 2πk T) ω k = Eb k BT 2π exp( / ) B 2πk BT hω

10 Where is the quantum mechanics? 1. The PES is calculated from quantummechanical principles, on the basis of the Born- Oppenheimer approximation. The electronic motion is considered to be adiabatic. 2. Nuclear motion is treated classically. Nuclear tunneling effects are included in so-called Quantum TS Theory or Quantum Kramers Theory.

11 Reactions in solution energy Coupling to non-reactive solvent modes leads to Brownian motion on PES transition state ω b E b ω r E 0 reactants products reaction coordinate

12 Kramers theory Motion on PES is described by Smoluchowski equation: 2 p( x, t) = D p ( x, t ) + k TD du ( x ) p( x, t) 2 B t x dx x With p(x,t) = probability density of finding the system at reaction coordinate x at time t D k T B = ζ ζ, friction coefficient Solvent friction related to solvent fluctuations by fluctuation-dissipation theorem

13 Kramers rate expression ω b ω r k = Eb kbt = κktst κ = ζ 2π exp( / ), ω b ζ valid for strong friction, the overdamped limit. For moderate friction, Kramers derived: κ = 1/ 2 2 ζ 2 ζ + ωb 4 2 ω b H.A.Kramers, Physica 7 (1940) 284 Brownian Motion in a Field of Force and the Diffusion Model of Chemical Reactions

14 Kramers rate for cusp-shaped barrier Ε b 1/ 2 2 ω r π E b k = exp( Eb / kbt ) 2πζ kbt

15 Electron transfer reactions at electrodes Ox n+ Red (n-1)+ e - Electron is transferred from a continuum of levels on the electrode. Effectively, an electron is transferred from the Fermi level. By changing the electrode potential, the metal levels shift with respect to the electron energy level on the donor or acceptor ion.

16 Outer-Sphere Electron Transfer Ox n+ + e - Red (n-1)+ Ox n+ Red (n-1)+ e - Sequence of events: 1. The reactant moves close to the electrode surface, but does not adsorb (outer Helmholtz plane, say) 2. The solvent assumes a suitable intermediate non-equilibrium configuration (the transition state) 3. The electron is exchanged radiationless 4. The system (solvent) relaxes to its new equilibrium configuration

17 The Marcus Potential Energy Surface free energy V Red transition state n-1 n V Ox+e 1. Minima at q = n-1 (Red) and q = n (Ox + e - ) because these are the equilibrium solvent configurations. 2. Deviations from equilibrium are assumed to be harmonic; V Red and V Ox+e are parabolic in q. 3. V Ox+e can be shifted up and down by changing the electrode potential. generalized solvent coordinate q

18 The solvent reorganization energy λ free energy Red λ Ox q λ is the difference in energy between a non-equilibrium Ox species with a Red solvation shell and an Ox species with its proper equilibrium solvation shell, taking into account only the slow modes of solvation (i.e. the electronic polarization is always equilibrated) λ = e 2 0 2a 1 ε opt 1 ε s = kj/mol

19 Movie of electron transfer Cl - Cl 0 Cl 0 + e - Cl - C.Hartnig, M.T.M.Koper, J.Am.Chem.Soc. 125 (2003) 9840

20 Inner-sphere and outer-sphere modes Ox n+ Red (n-1)+ e - Apart from the reorganization of the solvent (outer sphere), ligands or a strongly bound first layer of solvent molecules also reorganizes (inner sphere). These modes are usually treated separately.

21 Activation energy V Red (q) = λ[q-(n-1)] 2 V Ox+e (q) = λ[q-n] 2 - η Transition state: V Red (q * ) = V Ox+e (q * ) Activation energy: G act = (λ η) 2 4λ λ = solvent reorganization energy η = reaction free energy = e 0 (φ φ 0 ) (φ φ 0 ) = overpotential

22 Transfer coefficient Butler-Volmer definition α = - d G/dη Marcus theory predicts: α = 1 2 η 2λ α is the amount of charge transferred to the transition state (Hush) α is potential dependent, leading to a non-linear Tafel plot α is independent of temperature

23 Nonlinear solvent reorganization Cl-O Cl-H Effective radius gets smaller with higher charge; Increasing dielectric saturation with higher charge Orientation of water depends on charge: strongest change in electrostriction from 0 to -1 C.Hartnig, M.T.M.Koper, J.Chem.Phys. 115 (2001) 8540

24 Potential dependence of transfer coefficient Electrochemical ET to a ferrocene tethered to a gold electrode via a alkane thiol chain. C.E.D.Chidsey, Science 251 (1991) 919

25 What happened to the prefactor? k = ν exp( G / k T) n b B Marcus suggested the reaction to be adiabatic and ν n to be a collision frequency of the two reactants. However, it is more likely to be a typical frequency of the solvent motion or the inner-sphere vibration, whichever is dominant. ν n ω in = 2 π

26 Non-adiabatic ET Electron tunneling at the TS is rate determining in case of weak electronic coupling (small orbital overlap H ab ) k v el = v = el exp( G 2H h 2 ab 1/ 2 (Calculated by application of Fermi s Golden Rule for electronic transitions.) b π λtotk B / T k B T ) V.G.Levich, Adv.Electrochem.Electrochem.Eng 4 (1966) 249

27 Evidence for adiabatic ET Adiabatic ET: reaction rate depends only on solvent fluctuations, not on electron tunneling probability. There should be no metal dependence of the reaction rate. [Ru(NH 3 ) 6 ] 2+/3+ couple metal k (cm s -1 ) Pt 1.2 Pd 1.0 Au 1.0 Cu 1.2 Ag 1.2 Pt/Tl ad 1.3 T.Iwasita, W.Schmickler, J.W.Schultze, Ber.Bunsenges.Phys.Chem. 89 (1985) 138

28 Evidence for non-adiabatic ET Non-adiabatic ET: reaction rate depends explicitly on the strength of the electronic coupling, i.e. the electron tunneling probability. A distance dependence of the reaction rate is expected. H ( R) = H 0 exp( βr) AB AB Ferrocene couple attached to Au through alkyl thiol tail. Smalley et al. J.Phys.Chem.99 (1995) 13141

29 Electron transfer rate to cytochrome c Rate limited by reorganization 6 e - 4 H O O O O O O H O O O O O O ln(k app /s -1 ) 2 0 S S Au S S S Au Electron tunneling k = e -d/β S -2 Au-HS-(CH 2 ) n -COOH n(thiol length) β = 1.09 per methylene M.T.de Groot, T.H.Evers, M.Merkx, M.T.M.Koper, Langmuir 23 (2007) 729

30 Zusman s theory Dielectric friction is related to longitudinal relaxation time: ζ ω τ dielec = 2 s L Together with Kramers overdamped rate expression for a cusp-shaped barrier, this gives: k 1/ 2 1 λ = exp( λ / 4 kbt ) τ L 16π kbt L.D.Zusman, Chem.Phys. 49 (1980) 295

31 Evidence for slow solvent dynamics Plots of log k vs. -log t L may indicate some solvent dynamical effects for high-friction ( slow ) solvents. Fe + /Fe and Co + /Co with different ligands in different solvents M.J.Weaver, Chem.Rev. 92 (1992) 463

32 Electrochemical ion transfer Ox n+ Red (n-δ)+ δe - e.g. halide adsorption I - I (1-δ)- + δe - or metal deposition Zn e - Zn ad or Zn(Hg) The ion has to penetrate the electric double layer and lose part of its solvation shell. In the adsorbed state, the ion may be only partially (dis)charged.

33 Anderson-Newns Hamiltonian describes the exchange of an electron between an isolated orbital (the adsorbate) and a continuum of levels (metal). Electronic part: H elec = ε a n a + Σ k ε κ n k + Σ k [V ak c a+ c k + V ka c k+ c a ] Solvent part: H solv = λq 2 + 2λ(z - n a )q ε k metal ε adsorbate ε a

34 The electronic interaction parameter electronic energy ε = 2πΣ k V ak 2 δ(ε ε k ) metal solution ε F εa density of states describes the broadening of the adsorbate energy level due to electron exchange.

35 Schmickler s model for ion transfer 1. distance dependent reorganization energy λ, taken from molecular dynamics simulations. 2. distance dependent electronic interaction, taken from quantum chemical calculations (x) = 0 exp(-x/l) W.Schmickler, Chem.Phys.Lett. 237 (1996) 152

36 Solvation energy vs. distance Energy of the chloride-solvent and chlorine-solvent interaction as a function of the distance from the electrode surface. ion atom E.Spohr, Chem.Phys.Lett. 207 (1993) 214 C.Hartnig, M.T.M.Koper, J.Phys.Chem.B 108 (2004) 3824

37 Thallium reduction on Hg Tl + + δe - Tl (1-δ)+ + (1-δ)e - Tl (1-δ)+ Tl(Hg) -1 (a) 0.5 (b) energy/ev x/a Tl q q saddle point local minimum x/a Transfer coefficient: fraction of the double-layer potential traversed at the transition state. Tl +

38 Lead reduction on Hg: two mechanisms Pb 2+ + e - Pb + Pb + + δe - Pb (1-δ)+ + (1-δ)e - Pb (1-δ)+ Pb(Hg) Pb 2+ + δe - Pb (2-δ)+ + (2-δ)e - Pb (2-δ)+ Pb(Hg) M.T.M.Koper, W.Schmickler, J.Electroanal.Chem. 450 (1998) 83

39 Electron and ion transfer compared Electron transfer Ion transfer reaction occurs at a distance from the surface due to strongly inner solvation sphere reaction coordinate: solvent and/or inner-sphere reorganization transfer coefficient = 1/2 at equilibrium transfer coefficient is potential dependent transfer coefficient is not temperature dependent reaction involves penetration of the electric double layer reaction coordinate is mainly the distance from the electrode surface activation energy correlates with the ion solvation energy transfer coefficient depends on the structure of the double layer transfer coefficient is temperature dependent

40 Concerted bond breaking and electron transfer R X e.g. methylchloride reduction e - R X - CH 3 Cl + e - CH 3 + Cl - The methylchloride does not adsorb onto the metal electrode J.M.Saveant, J.Am.Chem.Soc. 109 (1987) 6788

41 Hamiltonian for bond breaking ET H = H elec + H solv + H bond-breaking H bond-breaking is modeled by a kind of switching function: H bond-breaking = [1-n a ] V R-X + n a V R+X - where n a is the number operator of the antibonding LUMO orbital of the R-X molecule V(r) V R-X V R+X - r, distance between R and X D e r 0 r M.T.M.Koper, G.A.Voth, Chem.Phys.Lett. 282 (1998) 100

42 PES for BBET products 5 products R. + X r / A reactant 3 saddle q (a) 4 E(q,r) / ev r / A saddle reactant R-X q (b) Small, weak electronic interaction

43 Activation energy of bond breaking G act = (λ+d e-η) 2 4(λ+D e ) transfer coefficient 1 α = 2 η 2(λ+D e ) amount of charge transferred to the antibonding orbital

44 Adsorption of molecules: backdonation the metal donates electronic charge to the antibonding orbital leading to a weakening of the intramolecular bond. 5 4 δe - x / A (q T,r T ) Large, strong electronic interaction leads to chemical bond q

45 Catalysis and electron transfer e.g. Cl 2 + 2e - 2 Cl - or H 2 2 H + + 2e - X 2 transition state x r X- X- X δ- Xδ- M.T.M.Koper, G.A.Voth, J.Chem.Phys. 109 (1998) 1991 E.Santos, M.T.M.Koper, W.Schmickler, Chem.Phys.Lett. 419 (2006) 421 E.Santos, M.T.M.Koper, W.Schmickler, Chem.Phys. 344 (2008) 195

46 PES of catalytic BBET 2 X ads E / ev 2 X x / A X - 5 r / A

47 Proton-coupled electron transfer A + H + + e - ET A - + H + PT CPET PT AH + + e - ET AH CPET = Concerted Proton-Electron Transfer S. Hammes-Schiffer, A.A.Stuchebrukhov, Chem.Rev.110 (2010) 6939 M.T.M.Koper, Phys.Chem.Chem.Phys. 15 (2013) 1399

48 Hamiltonian of PCET j.grimminger, S.Bartenschlager, W.Schmickler, Chem.Phys.Lett. 416 (2005) 316 M.T.M.Koper, Phys.Chem.Chem.Phys. 15 (2013) 1399 ( ) * 2 1 2,2 1,1 * ), ( n n q q q q q q c c V c c V n n c c V c c V n n q q H A p p e e p e p p e e pt pt p p k k A k A k k k k k A A p e β λ λ λ λ λ ε ε ε ε = Electron affinity + solvation energy - Proton affinity = pk a (AH + )

49 PES for coupled and decoupled PCET E A A AH + AH AH q e q p E act = λ + λ + 2λ p e 4 M.T.M.Koper, Phys.Chem.Chem.Phys. 15 (2013) 1399

50 PT reorganization energy λ p HA H + + A - Electrostatically, this is like the creation or annihilation of a dipole. Reorganization energy of the solvent is then the dielectrically slow part of the solvation energy of a dipole, as given by the Kirkwood-Onsager expression: λ p = 1 4πε ε 1 ε 1 s opt 0 2ε s + 1 2ε opt + 1 µ a 3 M.T.M.Koper, Chem.Sci. 4 (2013) 2710

51 Cross reorganization energy (a) λ = 0 (b) λ < 0 (c) λ > 0 H + H + θ A A A H + e - e - e - M.T.M.Koper, Chem.Sci. 4 (2013) 2710