Computation of Electrode Potentials and Alignment of Electronic Energy Levels Jun Cheng, Marialore Sulpizi, Michiel Sprik ([email protected]) University of Cambridge Contents 1. Reversible and ideally polarizable electrodes 2. Absolute electrode potentials and workfunctions 3. Alignment of vertical and adiabatic energy levels Key source textbooks: F Liquids, Solutions, and Interfaces, W .R. Fawcett (Oxford University Press). BF Electrochemical Methods, A. J .Bard and L R. Faulkner (Wiley) B Fundamentals of Electrochemistry, V. S. Bagotsky (Wiley) G Physical Electrochemistry, E. Gileadi (Wiley). 1.1: Pt(111)/water interface at potential of zero charge -6 Three kind of levels just outside vacuum 0 Electronic levels that are -4 electrode potentials CBM -2 Electronic levels that are -2 not electrode potentials Electrode potentials that -4 0 Fermi are not electronic levels level − -6 2 OH/OH OH/H O potential vs SHE [V] 2 -8 4 energy relative to vacuum [eV] VBM -10 6 Pt (111) water All experimental data 1.2: Pt(111)/water interface at potential of zero charge -6 More electronic levels just outside vacuum 0 IP: Vertical ionization -4 potential of the OH− ion. CBM -2 EA: Vertical electron -2 affinity of OH• radical. -EA -4 0 Fermi λ level − -6 2 OH/OH potential vs SHE [V] λ -8 4 -IP energy relative to vacuum [eV] VBM -10 6 Pt (111) water All experimental data 1.3: The two modes of an electrochemical cell load source I U U anode e e cathode cathode e e anode I I K K K K e e e e Cl Cl ½H2 H ½ 2 H ½H2 Cl ½Cl2 Oxidation Reduction Reduction Oxidation Generating electricity Driving reactions U<Urev U>Urev 1.4: The reversible potential is the potential at open circuit Cl2 g H2 g o 2Cl aq 2H (aq) U Electrode reactions in equilibrium o 'G k T p H2 pCl2 U B ln 2e 2e 2 2 >@H >Cl @ Nernst relation K K Nature of electrodes irrelevant e e ½H2 H ½Cl2 Cl No current U = Urev 1.5: Thermodynamics of electrolytic solutions Chemical potential of species i in phase α ∂Gα α = μi α ∂ni Separation in a standard chemical potentials and activity α α,◦ α μi = μi + kBT ln ai (1) α with activity related to concentration ρi of species i in phase as ρα aα = γα i i i c◦ Due to the neutrality condition (for a 1:1 electrolyte) α − α ρ+ ρ− =0 α α Only the mean activity coefficient γ+γ− can be measured 1.6: Thermodynamics of finite conducting phases Neutrality condition relaxed. Electrochemical potential of species i in phase α ∂G˜α ˜α = (2) μi α ∂ni Separation in a chemical and electric component α α α μ˜i = μi + qiφ (3) • α μi is the chemical potential of species i in phase α • qi is the charge of species i • φα is the inner (or Galvani) potential of phase α Excess charge in conductors accumulates at the boundaries Inner potentials account for the charge inbalance at the boundaries only α Electrostatic solvation interactions are included in μi 1.7: Cell potential as inner potential difference of electrodes Chlorine reduction (slide 1.3) − − Cl2 +2e → 2Cl can be carried out at the cathode of a cell Pt , H2 | H2SO4, H2O || H2O, NaCl | Cl2, Pt with H2 oxidation at the anode + − H2 → 2H +2e Cell potential = potential cathode - potential anode U = φPt − φPt 1.8: Overpotential and electrochemical equilibrium Electrochemical reaction free energy at the cathode S Pt S Pt,◦ S Pt Δ ˜ =2˜ − − 2˜ − =2 − − − 2 − 2 − Gc μCl μe μCl2 μCl μCl2 μe e0 φ φ Similarly for the electrochemical reaction free energy at the anode ˜ S Pt − S − Pt,◦ S − Pt ΔGa =2˜μH+ +2˜μe μH2 =2μH+ μH2 +2μe +2e0 φ φ The salt bridge eliminates the electrical potential difference between S and S φS = φS Adding we obtain the total electrochemical reaction free energy S S Pt Pt Δ ˜ =Δ˜ +Δ˜ =2 − +2 + − − +2 − G Gc Ga μCl μH μCl2 μH2 e0 φ φ = −2e0Urev +2eU The overpotential is the driving force for the electrochemical cell reaction η = U − Urev 1 S S = − − +2 + − − Urev μCl μH μCl2 μH2 2e0 ΔG˜ =2e0η 1.9: Open circuit voltage of cells under redox control Cl2 g H2 g o 2Cl aq 2H (aq) U Electrode reactions in equilibrium o 'G k T p H2 pCl2 U B ln 2e 2e 2 2 >@H >Cl @ Nernst relation K K Nature of electrodes irrelevant e e ½H2 H ½Cl2 Cl No current 1.10: Electro-active species is adsorbed on surface Example: Volmer reaction H aq e m o H U Step in Hydrogen Evolution Reaction Nature of electrode matters. K K e ½H2 H H *H Redox reference Electrosorption 1.11: Open circuit voltage without redox control Electrode acts as a capacitor Q U M pzc C(Q) U Mpzc v Hfermi Point of zero charge for metals Nature of electrode matters. K e K ½H2 H Redox reference Double layer 1.12: Non-polarizable and polarizable interfaces Non-polarizable n • Potential remains constant over a wide U current window • Rest potential Urest equal to reversible Urest potential Urev of redox couple 0 j o • Urest determined by bulk activities n Polarizable • Current remains zero over a wide U potential window • Urest Rest potential Urest dependent on 0 j o electrode surface and composition electrolytic solution (no redox control) 1.13: Equivalent circuit representation Modelling the response to small perturbations in potential (steps or harmonic) Cdl Cdl: Double layer capacitance, varying with charge (potential) RF: Interface (Faradaic) resistance, exponentially dependent on potential Rs Rs: Solution (Ohmic) resistance, independent potential RF • RF large: polarizable interface Only alternating current (AC) response • RF small: non-polarizable interface: Sustains direct current (DC) Note: This model is an oversimplification 1.14: Computation of electrode potentials: Outline Should apply to both reversible and ideally polarizable electrodes • We cannot use Nernst law (no electron exchange between metal and solution) • Instead potentials will be formulated in terms of transfer to vacuum Potentials vs SHE of M|S interface separated in absolute potentials ◦ U | (she) = U | (abs) − U + (abs) M S M S H /H2 Absolute potentials can be identified with workfunctions 1. S. Trasatti, Pure & Appl. Chem., 1986, 58, 955–966 2. S. Trasatti, Electrochim. Acta, 1990, 35, 269–271 See also Fawcett(F) 2.1: Inner, outer and surface potentials The inner potential is separated in outer potential ψα and surface potential χα φα = ψα + χα The outer or Volta potential is measured at a point in vacuum just outside the phase boundary Coming from infinity this point is • close enough so that all work againts surface charge has been carried out. • far enough so that image and dipole forces have not yet caught on. The outer potential accounts for the work against surface charge 2.2: Separate potential for work againts surface dipoles Definition of the real potential α α α αi = μi + qiχ (4) relation to the electrochemical potential α α α μ˜i = αi + qiψ α α α μ˜i , αi and ψ can be determined by thermodynamic experiment • α ∞ μ˜i is the work to insert a particle i from in phase α (Eq. 2). α • qiψ is the work to bring a particle i from ∞ to “just outside” phase α. • α αi is therefore the work to insert it from there in phase α The implication is that α α μi , φα and χ cannot be determined by experiment without further “extra-thermodynamical” assumptions 2.3: Workfunctions and chemical potentials Minus the real potential with the gas-phase chemical potential as reference α − α,◦ − g,◦ Wi = αi μi defines the work function to be distinguished from the solvation free energy α,◦ α,◦ − g,◦ ΔsGi = μi μi (5) which compares chemical potentials. The difference is the surface potential α,◦ − α α ΔsGi = (Wi + qiχ ) Rearranging the standard real potential can be written as α,◦ − α g,◦ αi = Wi + μi (6) Example: aqueous proton ◦ − g,◦ αH+ = WH+ +Δf GH+ (7) g,◦ g,◦ where μH+ =Δf GH+ is the formation free energy of the gas-phase proton 1 + − H2(g) → H (g) + e (vac) 2 Note that the gas-phase reference state for the electron is an electron at rest g,◦ ⇒ α,◦ − α μe =0 αe = We (8) 2.4: Interface and contact potentials Special notation for the difference in inner potential accross an interface β α αΔβφ = φ − φ Similarly for the difference in outer potential (contact potential) β α αΔβψ = ψ − ψ D 'E \ Triple interface potential loop E D β α \ \ αΔβφ = αΔβψ + χ − χ (9) Only αΔβψ can be measured F E FD D I E I D 'E I 2.5: Electron exchange between two metals Copper wire welded to a platinum electrode: e\ Pt Electrochemical equilibrium Cu 'Pt \ Cu Pt μ˜e =˜μe e\ Cu Pt Substituting Eq. 3 Cu We We Cu − Cu Pt − Pt μe e0φ = μe e0φ ~Cu ~Pt Pe Pe and Eq. 1 with ae =1 Cu,◦ − Cu Pt,◦ − Pt μe e0φ = μe e0φ We find for the Galvani potential difference Pt,◦ − Cu,◦ e0CuΔ Ptφ = μe μe (10) Standard chemical potentials are instrinsic (but unmeasurable) quantities ⇒ The interface potential at a metal-metal contact is a constant and so is the contact potential (using Eqs. 4,6,8 and 9) Pt,◦ − Cu,◦ − Pt Cu Cu − Pt e0CuΔ Ptψ = μe μe χ + χ = We We which can be measured using a Kelvin probe (see F) 2.6: Two similar metals not in equilibrium Two copper wires connected to two electrodes, Cu and Cu, at different potential Cu Cu − Cu Cu Cu − Cu μ˜e = μe e0φ , μ˜e = μe e0φ Cu Cu Then, because μe = μe we have e0CuΔ Cu φ = CuΔ Cu μ˜e The chemical work for exchanging an electron between electrode terminals equals the electrical work for exchange of a unit negative test charge provided the terminals are made of the same metal 2.7: Electrochemical cell for X•/X− couple load Cu Cu' Cell reaction • 1 − + U X (s)+ H2(g) → X (s)+H (s) 2 M anode Pt cathode Potential vs SHE e e ΔrG UX•/X− (she) = − e0 S S' Assuming standard conditions at anode K K e e • ½H2 H X X Oxidation Reduction Electrochemcial specification of cell .