Introduction to Electrochemistry

The subject of electrochemistry

Electrochemistry studies the conversion between chemical and electrical energy and the interfaces at which this conversion takes place. An electrochemical cell is a system in which at least two electrodes are connected by an ionic conductor (electrolyte). Electrons are carried through metallic conductors in the external circuit, ions within the electrolyte; the two kinds of transport meet at the electrode/electrolyte interface through a charge-transfer (redox) reaction. In a galvanic cell the reaction proceeds spontaneously and produces work (cell, fuel cell); in an electrolytic cell an externally applied voltage drives the reaction (electrolysis, charging).

The electrode/electrolyte interface and the double layer

When a metal is immersed in an electrolyte, an excess of charge forms on its surface and a layer of oppositely charged ions and dipoles forms on the solution side: the electrical double layer. The Helmholtz model treats this structure as two parallel planes of charge (a plate capacitor); the Gouy–Chapman model states that the charge on the solution side forms a diffuse cloud because of thermal motion, and the Stern (and Grahame) model describes the combination of the two — inner and outer Helmholtz planes plus the diffuse layer. The double-layer capacitance (C_dl) is typically 10–40 µF/cm² and shows up in every electrochemical measurement as a non-Faradaic 'charging current'. This capacitance determines the radius of the semicircle in EIS, the background current in CV, and the current spike in the first microseconds of chronoamperometry.

Faradaic and non-Faradaic processes

The current crossing the interface splits into two components: the Faradaic current (i_F), which obeys Faraday's law and is proportional to the charge-transfer reaction, and the capacitive current (i_C = C_dl·dE/dt) arising from charging/discharging of the double layer. Faraday's law links the charge passed to the amount of substance converted: Q = n·F·N. This relationship is the basis of every quantitative electrochemical measurement, from battery capacity to coulometric analysis, from electroplating to corrosion rate.

Why is a three-electrode cell necessary?

The voltage measured in a two-electrode cell contains the overpotentials of both electrodes together with the drop across the electrolyte resistance; the behaviour of a single interface therefore cannot be isolated. In the three-electrode arrangement the working electrode (WE) provides the interface under study, the reference electrode (RE) a fixed and known potential point, and the counter electrode (CE) the path through which the current closes. The potentiostat controls the potential between WE and RE while passing the current between WE and CE; since practically no current flows through the RE, its potential remains stable. A reference electrode close to the cell and of low impedance, a Luggin capillary positioned close to the WE surface and a CE area larger than the WE are the basic rules of a well-designed cell.

Electrolyte, conductivity and supporting electrolyte

The electrolyte carries charge through the migration of ions. Adding a 'supporting electrolyte' at 50–100 times the analyte concentration suppresses transport by migration under the electric field and reduces mass transport to diffusion; at the same time it lowers the solution resistance and thus the ohmic drop. Water, organic solvents (acetonitrile, carbonates), ionic liquids and solid electrolytes offer different potential windows and conductivities; in battery and fuel-cell work the choice of electrolyte sets the upper limit of the cell voltage.

Electrochemical potential and equilibrium

When an electrode reaction O + ne⁻ ⇌ R is at equilibrium, the electrochemical potential of the electrode equals that of the species in solution. The Nernst equation is derived from this equilibrium condition: E = E° + (RT/nF)·ln(a_O/a_R). Since RT/F = 25.69 mV at 25 °C, every tenfold change in the concentration ratio shifts the potential by 59.2 mV in a one-electron reaction. The standard potential E° is the value when all activities equal 1; when concentrations are used instead of activities in practice, a 'formal potential' E°′ is defined, which deviates from E° depending on the ionic strength, pH and complexation of the medium.

Reference electrodes

An absolute single-electrode potential cannot be measured; every potential is reported against a reference. The standard hydrogen electrode (SHE) is 0 V by definition but impractical in the laboratory. Ag/AgCl (3 M KCl ≈ +0.210 V; saturated KCl ≈ +0.197 V vs SHE) and the saturated calomel electrode (SCE, +0.241 V) are standard in aqueous media. In non-aqueous solvents IUPAC recommends the ferrocene/ferrocenium (Fc/Fc⁺) couple as internal reference. In battery research potentials are usually given against Li/Li⁺ (−3.04 V vs SHE) or Na/Na⁺. The temperature coefficient of the reference electrode, the liquid-junction potential and contamination of the filling solution are sources of systematic error; the reference used and the conversion must be stated explicitly in every publication.

Pourbaix diagrams and pH

In reactions involving protons, the Nernst equation contains a pH term; for m protons/n electrons the slope is −(m/n)·59.2 mV/pH. Potential–pH (Pourbaix) diagrams show thermodynamically under which conditions a metal is stable, passive or prone to corrosion. In corrosion engineering the cathodic-protection potential, and in water electrolysis and fuel cells the thermodynamic cell voltage of 1.23 V, are read directly from these diagrams.

Open-circuit potential (OCP) and mixed potential

The OCP measured while no current is drawn is the Nernst potential if a single equilibrium reaction exists. Where two or more reactions proceed simultaneously, as in corrosion, the measured value is a 'mixed potential' (E_corr): the point at which anodic and cathodic currents balance, which is not a thermodynamic equilibrium. In batteries the change of OCP with rest time (relaxation) shows the concentration gradients inside the cell slowly disappearing; GITT is based on this principle.

Overpotential and activation

The deviation from the equilibrium potential (η = E − E_eq) is called the 'overpotential' and drives the reaction in one direction. For the charge-transfer step the Butler–Volmer equation links current density to overpotential: j = j₀·[exp(α_a·F·η/RT) − exp(−α_c·F·η/RT)]. Here j₀ is the exchange current density (the common magnitude of the forward and reverse reaction rates at equilibrium) and α_a, α_c are the anodic and cathodic transfer coefficients (α_a + α_c = 1 for a single-step one-electron reaction). If j₀ is large the electrode behaves 'reversibly' and the potential stays close to Nernst; if j₀ is small the electrode is 'irreversible' and a large overpotential is needed for the same current.

The Tafel limit and the Tafel slope

If |η| is large enough (in practice > ~50–100 mV) one term of Butler–Volmer is neglected and the logarithmic Tafel relation is obtained: η = a + b·log|j|. The Tafel slope b = 2.303·RT/(α·F); for α = 0.5 it is 118 mV/decade at 25 °C. The Tafel slope carries information about the mechanism: in hydrogen evolution, for instance, 120, 40 and 30 mV/decade suggest that the Volmer, Heyrovsky and Tafel steps respectively are rate-determining. Extrapolation of the Tafel lines to E_corr gives the corrosion current (i_corr); this is the classical route to corrosion-rate measurement.

Small overpotential: the linear region and polarisation resistance

When |η| ≪ RT/F the exponentials become linear: j ≈ j₀·(α_a+α_c)·F·η/RT. The charge-transfer resistance defined in this region, R_ct = RT/(nF·j₀), appears in EIS as the diameter of the semicircle. In corrosion its counterpart is the polarisation resistance R_p, linked to the corrosion current by the Stern–Geary relation: i_corr = B/R_p, B = β_a·β_c/[2.303(β_a+β_c)]. The method is fast and non-destructive; the constant B, however, requires the Tafel slopes to be known.

Marcus theory and limits

Butler–Volmer is a phenomenological model; Marcus–Hush theory predicts that the transfer coefficient changes with overpotential and that the current saturates at very large η. In practice the measured current is limited not only by charge transfer but also by mass transport and ohmic drop; for a reliable Tafel analysis these effects must be corrected or shown to be absent.

The three transport mechanisms

Electroactive species reach the electrode surface in three ways: diffusion under a concentration gradient, migration under an electric field and convection through macroscopic movement of the solution. The Nernst–Planck equation combines the three in a single expression. Supporting electrolyte suppresses migration; in unstirred, short experiments natural convection is neglected and transport is described by diffusion alone — most theoretical analyses rest on this 'pure diffusion' assumption.

Fick's laws and the diffusion layer

Fick's first law defines the flux by the gradient (J = −D·∂c/∂x), the second the change of concentration with time (∂c/∂t = D·∂²c/∂x²). If the surface concentration is driven to zero after a potential step, the diffusion layer grows as √(πDt) and the current decays as t^(−1/2) according to the Cottrell equation: i = nFAc√D/√(πt). Typical D values are of the order of 10⁻⁵ cm²/s in aqueous solution and 10⁻⁸–10⁻¹² cm²/s in polymer and solid electrolytes; in battery electrodes solid-state diffusion is usually rate-determining.

Forced convection: Levich and Koutecký–Levich

At a rotating disk electrode (RDE) the fluid dynamics creates a diffusion layer of constant thickness (δ = 1.61·D^(1/3)·ν^(1/6)·ω^(−1/2)); the limiting current is proportional to ω^(1/2) according to the Levich equation. If kinetics and transport are both limiting, the Koutecký–Levich equation 1/i = 1/i_k + 1/i_L is used: a plot of 1/i against ω^(−1/2) gives the kinetic current from the intercept and n and D from the slope. In electrocatalysis the determination of the number of electrons transferred in ORR (2 or 4) relies on this analysis.

From solution diffusion to solid-state diffusion

In intercalation electrodes the chemical diffusion coefficient of species such as Li⁺ within the solid is calculated from GITT, PITT and the low-frequency Warburg region of EIS. All three methods rest on Fick's laws but with different boundary conditions; agreement of the results in order of magnitude is a sign that the model is valid. Not knowing the particle size and geometry is the largest source of uncertainty in these calculations.