Electrochemical Polarization & Electrode Kinetics

Electro-Chemical Polarization: Core Idea

• Polarization = departure of an electrode’s potential from its equilibrium (open-circuit) value once current flows.

  • Represents the total hindrance to charge transfer at the electrode–electrolyte interface.

  • Sources: slow kinetics (activation), mass-transport limits (concentration), electrical resistance (ohmic).

• Equilibrium vs. non-equilibrium:

  • At zero current ⇒ potential reflects pure thermodynamics.

  • Under finite current ⇒ potential shifts; the shift is the polarization (measured as an overpotential).

Types of Polarization

• Activation Polarization

  • Originates in sluggish electron-transfer or chemical steps.

  • Requires an extra driving force (overpotential) to surmount $\Delta G^{\ddagger}$.

• Concentration (Diffusion) Polarization

  • Reactant arrival / product removal can be slower than reaction rate.

  • Produces concentration gradients at the interface → potential shift.

• Resistance (Ohmic) Polarization

  • Ordinary IR drop through solution or surface films (oxides, salts).

  • Independent of reaction kinetics; proportional to current and total resistance.

Consequences & Quantification

• Lowers battery voltage, raises required electro-plating potential, accelerates corrosion inefficiency, etc.

• Overpotential (also called over-voltage):

  • $\eta = E - E_0$

  • $E$ = actual electrode potential under current.

  • $E_0$ = equilibrium (open-circuit, rest, corrosion) potential.

  • Magnitude of $\eta$ directly measures the degree of polarization.

Anodic vs. Cathodic Polarization

• Either electrode may polarize.

  • Anodic polarization: potential shifts positive ⇒ electrode behaves “more anodic.”

  • Cathodic polarization: potential shifts negative ⇒ electrode behaves “more cathodic.”

• Schematic implications (Fig 1 in transcript):

  • Anode: $E \rightarrow$ more +; example $Zn \rightarrow Zn^{2+}+2e^-$.

  • Cathode: $E \rightarrow$ more –; example $2H^+ + 2e^- \rightarrow H_2$.

Visual Examples of Polarization

Cathodic—Hydrogen Evolution

• Reaction steps:

  1. $2H^+ + 2e^- \rightarrow 2H_{ads}$

  2. $2H<em>{ads} \rightarrow H</em>2(g)$

• If electron supply exceeds proton reduction (slow kinetics) ⇒ electrons pile up → activation polarization (potential more –).

• If $H^+$ diffusion is slow ⇒ concentration polarization with identical negative shift.

Anodic—Iron Dissolution

• Reaction: $Fe \rightarrow Fe^{2+} + 2e^-$

• Slow Fe oxidation ⇒ electrons leave faster than Fe atoms → electron deficit → activation polarization (potential more +).

• Slow diffusion of $Fe^{2+}$ away ⇒ accumulation of positive ions → concentration polarization (also more +).

Ohmic Polarization & IR Drop

• Physical resistance between working and reference electrodes yields $IR_{soln}$ drop (Fig 6).

• Mitigation: Luggin–Haber capillary; keep tip within ≈2 diameters of working electrode.

• $IR_{soln}$ trivial in high-conductivity aqueous media; substantial in organic media, certain soils.

• If film resistance (oxide, hydroxide, salt) present, film viewed as part of overall system “metal / film / solution.”

Electrode Kinetics: Activation-Controlled Reactions

Absolute Reaction-Rate Theory

• Reaction proceeds along a coordinate; reactants cross a free-energy barrier $\Delta G^{\ddagger}$ forming an activated complex $[AB]^\ddagger$.

• Rate constant:

  • $k = \frac{k_B T}{h} e^{-\Delta G^{\ddagger}/RT}$

• Derivation outline (pages 12–15):

  1. Passage frequency of complex $= k<em>B T/h$ (because vibrational energy $h\nu \approx k</em>B T$ at the barrier peak).

  2. Concentration of complexes via equilibrium constant $K^\ddagger = e^{-\Delta G^{\ddagger}/RT}$.

  3. Multiplication gives above $k$ expression.

  • Higher $\Delta G^{\ddagger}$ ⇒ lower $k$ ⇒ slower reaction.

Exchange Current Density & Non-Corroding Electrodes

• Metal $Z$ in its own ion solution: $Z(s) \rightleftharpoons Z^{n+} + ne^-$.

• At $E_0$, forward (oxidation) and reverse (reduction) rates are equal.

  • Current densities equal in magnitude, opposite in sign.

  • $i<em>c = -i</em>a = i_0$ (exchange current density).

  • $i_0$ is not measurable directly—system must be perturbed (polarized).

• Example couples cited: $Cu/Cu^{2+}, Pb/Pb^{2+}, Fe^{3+}/Fe^{2+}, 2H^+/H_2$.

Butler–Volmer Equation & Polarization Curves

• Upon shifting potential $E$ from $E_0$, energy barrier changes:

  • $\Delta G^{\ddagger}<em>{forward} = \Delta G^{\ddagger}</em>0 - \alpha nF (E-E_0)$

  • $\alpha$ (0 < $\alpha$ < 1) = symmetry factor (often ≈ 0.5).

• Substituting into rate expression + Faraday’s law gives net current density:

  • $i = i<em>0 \left[ e^{\alpha nF (E-E</em>0)/RT} - e^{-(1-\alpha)nF(E-E_0)/RT} \right]$

• Characteristics (Fig 11):

  • At small $\eta$ both terms matter ⇒ curve is exponential but symmetric.

  • At large |$\eta$| one term dominates ⇒ straight-line (Tafel) behavior on semi-log plot.

Nernst Limit and Reversible Systems

• For rapid (reversible) one-electron reactions, $k^0$ large ⇒ FAR term ≫ 1 ⇒ the Butler–Volmer simplifies to Nernst relation: $\frac{[B]}{[A]} = e^{F(E-E^0)/RT}$.

Tafel Equation (High |η| Region)

Anodic branch

• When cathodic term negligible: $i \approx i_0 e^{\alpha nF \eta/RT}$

• Taking log (base 10):

  • $\log i = \log i_0 + \frac{\alpha nF}{2.303 RT} \eta$

  • Slope $b_a$:

  • $b_a = \frac{2.303 RT}{\alpha nF}$ (V per decade).

Cathodic branch

• For large negative $\eta$: $i \approx -i_0 e^{-(1-\alpha)nF \eta/RT}$

• Slope $b_c$:

  • $b_c = -\frac{2.303 RT}{(1-\alpha)nF}$ (negative sign indicates cathodic direction).

• Extrapolating either straight line back to $\eta = 0$ gives $i_0$.

Practical Plotting of Polarization Curves

• Galvanostatic method: impose current → measure resulting $E$; historically produced plots with $\log |i|$ on x-axis.

• Potentiostatic method (modern): step/scan $E$ → record $i$; $E$ is independent variable and plotted on x-axis.

• Figures 12a–c illustrate different orientations; always label axes clearly (note inversion of polarity if plot rotated).

Reversible vs. Irreversible Potentials

• Reversible potential requires:

  1. Metal ions of same species present (unit activity for standard potential).

  2. No interfering foreign ions/films.

• Real corrosion often lacks these conditions → electrode exhibits an irreversible (mixed) potential not predicted by Nernst.

• Example: iron dissolution in acid

  • Global reaction: $Fe + 2H^+ \rightarrow Fe^{2+} + H_2$.

  • Actually composed of two independent partial reactions (Fe oxidation & H^+ reduction) each with own kinetics (Fig 13).

Mixed Potential Theory (Wagner–Traud)

• Any overall electrochemical process = sum of separate anodic & cathodic partial reactions.

• At steady state (no external current):

  • Total cathodic current density = total anodic current density (charge conservation).

  • Common potential reached = corrosion (mixed) potential $E_{corr}$.

• For Fe in acid (example):

  • $i<em>{H^+} + i</em>{Fe^{2+}}^{(red)} = i<em>{H</em>2}^{(ox)} + i_{Fe}^{(ox)}$.

  • Simplifies to equality of net hydrogen evolution and iron dissolution currents.

• Corrosion rate $i_{corr}$:

  • Magnitude of either anodic or cathodic branch at $E_{corr}$.

  • Cannot be measured directly—deduced by extrapolating Tafel lines back to $E_{corr}$.

• Modified Butler–Volmer for corroding metal:

  • $i = i<em>{corr}\left[ e^{\alpha nF (E-E</em>{corr})/RT} - e^{-(1-\alpha)nF(E-E_{corr})/RT} \right]$.

  • $E_{corr}$ lacks thermodynamic meaning; determined strictly by kinetics of all simultaneous partial reactions.

Key Equation Toolbox (quick reference)

• Overpotential: $\eta = E-E_0$

• Activation-controlled rate constant: $k = \frac{k_B T}{h} e^{-\Delta G^{\ddagger}/RT}$

• Exchange current density equality: $|i<em>a| = |i</em>c| = i<em>0$ at $E</em>0$.

• Butler–Volmer (single reaction): $i = i_0\big[e^{\alpha nF\eta/RT} - e^{-(1-\alpha)nF\eta/RT}\big]$.

• Tafel slopes: $b<em>a = \frac{2.303 RT}{\alpha nF},\quad b</em>c = -\frac{2.303 RT}{(1-\alpha)nF}$.

• Mixed potential steady state: $\sum i<em>{cathodic} = \sum i</em>{anodic},\; E = E_{corr}$.

• Corroding Butler–Volmer: $i = i<em>{corr}\big[e^{\alpha nF(E-E</em>{corr})/RT} - e^{-(1-\alpha)nF(E-E_{corr})/RT}\big]$.

Experimental & Practical Notes

• Luggin–Haber capillary reduces solution IR error; tip distance ≈ 2 × outer diameter keeps current distribution unaltered.

• Ohmic drop negligible in high-conductivity aqueous electrolytes but critical in low-conductivity organics or soils.

• Tafel slopes expressed in mV decade⁻¹; sign convention: anodic (+), cathodic (–).

• Always specify temperature, $n$, $\alpha$, solution resistivity, and surface area when reporting kinetic parameters.