Interfaces, double layers and charge-transfer kinetics

Charge can accumulate at an interface

An electrode–electrolyte interface can form an electrical double layer. Changing potential can rearrange this interfacial charge without a net electron-transfer reaction across the interface. That charging current is nonfaradaic. Faradaic current instead accompanies interfacial electron transfer and chemical conversion. Both contributions may appear in a transient measurement.

Imagine making a conducting electrode slightly more positive while it contacts an electrolyte. Electronic charge redistributes in the electrode, and the nearby liquid develops compensating negative excess charge. This can involve enrichment of counterions, depletion of like-charged ions and solvent reorientation. Farther away, the liquid can remain approximately electrically neutral. The separated charge occupies a region: a double layer is not a second metal sheet or a row of newly made reaction products.

The first figure shows a deliberately simple charge account: positive excess on the conductor and negative countercharge in the liquid. Ion and charge symbols indicate the account, not atomic positions, layer thickness or an actual concentration profile. Molecular packing, adsorption, electrolyte composition and potential affect the real structure. A single fixed-capacitance description is a local approximation, not a universal property of every interface.

Let q_DL be the change in charge stored on the electrode side relative to the initial state. For a fixed area and a small potential range with constant differential capacitance C_DL, q_DL = C_DL Δη. Capacitance has units farad, with 1 F = 1 C/V. The changing stored charge contributes I_DL = dq_DL/dt = C_DL dη/dt. Here d/dt means change per unit time; a large current can briefly rearrange charge without producing the corresponding amount of a new chemical substance.

Charge, chemical state and electrochemical potentials supplies current, referenced voltage and reaction-charge bookkeeping. Ionic, electronic and mixed transport explains the moving carriers; Electrode and electrolyte materials explains their connected physical routes. The following interface model defines its additional quantities locally. It does not require an impedance-spectrum calculation.

Reactions require kinetics

Overpotential is an electrode potential displaced from its equilibrium value for the reaction and state. The relation between current and overpotential depends on reaction rates and conditions. Transport can limit the supply of reacting species. Distinguish interfacial charge-transfer limitation from concentration polarisation instead of assigning every voltage loss to one resistance.

For one specified reaction, oxidation and reduction can both occur at equilibrium while their rates balance, giving zero net faradaic current. Displacing the interfacial potential can favour one direction over the other. The net current reflects their difference. Surface composition, catalytic sites, adsorbed species, temperature and the local reactant state can therefore affect rate even when the equilibrium tendency has not changed.

Write the local activation overpotential as η = E_interface − E_eq,local, in volts, using a common reference and the local chemical state for the chosen reaction. In the model below, there is no appreciable concentration change, so local and bulk equilibrium values coincide. Once depletion develops, an offset from the bulk equilibrium potential can also contain concentration polarisation; it is not automatically a pure activation overpotential.

Near a specified equilibrium, a smooth reaction-current relation can sometimes be approximated by I_F = η/R_ct. The positive slope resistance R_ct is in ohms. It represents the local net charge-transfer response for this area and state, not a metallic leakage path or a concentration-independent law at arbitrary voltage. The full Butler–Volmer relation is one nonlinear model of forward and reverse reaction rates; its derivation is unnecessary for the bounded linear case here. A larger driving potential can also activate other reactions rather than simply continue the same straight line.

A small potential step divides current between two processes

Consider an original fictional, uniform working interface initially at equilibrium. A suitable reference and a sufficiently nonlimiting counter-electrode isolate its response. The controlled offset ΔV is stepped from zero to +10.0 mV and held. In this stated circuit, ΔV = I R_s + η: R_s = 100 Ω accounts for the uncompensated series ohmic path, while the interfacial voltage η acts across a parallel C_DL = 100 μF and R_ct = 900 Ω. These are total values for one fixed specimen area, not per-area material constants.

The capacitor branch represents changing stored charge and the reaction branch represents net faradaic transfer. Charge balance at the node is I = I_DL + I_F. Thus (ΔV − η)/R_s = C_DL dη/dt + η/R_ct. This equation says that incoming current either changes interfacial storage or crosses by reaction. The drawing is an equivalent response model; its resistor and capacitor symbols are not separate strips of matter and do not depict literal ion trajectories.

Assume the two linear elements and R_s remain constant over this small excursion, with maintained local composition, no other faradaic reaction, no adsorption-related reaction storage, negligible bulk geometric capacitance on the resolved timescale and no evolving film or diffusion limitation. Real pulse rise time, instrument bandwidth and distributed surfaces may violate those assumptions. The ideal t = 0⁺ entry below is a model limit, not a claim that an instrument resolves an infinitely sharp first instant.

For this model the settling time is τ = C_DL R_s R_ct/(R_s + R_ct) = 9.00 ms. The interfacial offset approaches η_∞ = ΔV R_ct/(R_s + R_ct) = 9.00 mV. If exp(−t/τ) denotes the usual exponential decay, η(t) = 9.00[1 − exp(−t/τ)] mV. You may use the table on a first reading; differentiating that expression gives the storage current and is an optional way to recover the charge-balance equation.

With t and τ in the same units, I_DL(t) = 100 exp(−t/τ) μA, I_F(t) = 10.0[1 − exp(−t/τ)] μA and I(t) = 10.0 + 90.0 exp(−t/τ) μA. Initially all the resolved model current charges the double layer; η and net reaction current start at zero. Later the storage current falls while the reaction current rises. A decaying total trace therefore need not mean a decreasing reaction rate.

At the model steady limit, I_DL is zero but I_F = I = 10.0 μA. The series path drops I R_s = 1.00 mV and the interface retains 9.00 mV, closing the supplied 10.0 mV account. Using the whole 10.0 mV as η would instead assign 11.1 μA to the 900 Ω reaction branch. A terminal or remote-reference voltage is not automatically the local reaction driving potential.

An area under total current is not automatically reaction charge

Current is a rate. Charge over an interval is the area under its current-versus-time curve, q = ∫I dt. Over 0 to T, the model charge divides into Δq_DL = C_DL η(T) and q_F = ∫I_F dt. The first remains as changed interfacial storage at the endpoint; the second is the net charge attributed to the specified reaction. Their sum equals the external charge q_ext.

For T = 27.0 ms = 3τ, exp(−3) ≈ 0.049787. The stored change is (100 × 10⁻⁶ F)(8.551916 × 10⁻³ V) = 0.855192 μC. Integrating the rising reaction current gives q_F = I_∞{T − τ[1 − exp(−T/τ)]}, with I_∞ = 10.0 μA. Substitution gives q_F = 0.184481 μC. The total is 1.039672 μC. One μC is 10⁻⁶ C and one ms is 10⁻³ s; mixing those prefixes would obscure the account.

If the sole counted faradaic reaction transfers two electrons per reaction event, its net amount of reaction progress is q_F/(2F), using the rounded Faraday constant F = 96,500 C/mol. The result is about 9.559 × 10⁻¹³ mol, or 0.9559 pmol. Dividing the total external charge by 2F would instead give 5.3869 pmol, about 5.64 times too large. Even the correct q_F split does not identify a particular product if several faradaic reactions are possible; chemical selectivity needs its own evidence.

In this maintained-state approximation, storage approaches the finite 0.900 μC value, while a continuing net reaction can keep transferring charge. A real finite reactant inventory or changing interphase eventually changes the model. The short response is not a complete battery capacity or an indefinitely sustainable reaction claim.

A blank helps only with an explicit comparison model

A nonreacting control can test whether a charging response is present. But simply subtracting its current from a reacting trace does not always recover reaction current. Removing a reaction changes how the interfacial potential develops, even if the two specimens have the same C_DL and series resistance.

Take a hypothetical ideal blocking control with the same 100 Ω, 100 μF and +10.0 mV step but no faradaic branch. Its storage time is R_s C_DL = 10.0 ms and its current is I_blank = 100 exp[−t/(10.0 ms)] μA. At 27.0 ms it carries 6.72055 μA. The reacting total carries 14.48084 μA, so their difference is 7.76028 μA. Yet the reacting specimen’s actual model I_F is 9.50213 μA. The control does not have the same interfacial voltage trajectory or storage current as the reacting specimen.

The blank is still informative: it demonstrates a possible charging contribution and challenges an all-reaction interpretation. A justified decomposition can use independently checked resistance/capacitance, a measured or modelled interfacial potential, time-resolved charge balance and reaction-specific chemical evidence. Altering electrolyte or suppressing a reaction can also change capacitance, area or adsorption, so even equal C_DL in this numerical control is a supplied condition, not a general experimental fact.

Change the cause while keeping the comparison interpretable

Reaction at a surface and delivery to that surface are different steps. A consuming reaction can lower local reactant concentration; diffusion, migration or convection then affect supply. The local equilibrium potential and the charge-transfer rate can both change with that concentration. A current plateau therefore is not by itself a measurement of one intrinsic R_ct, and a long decaying transient is not by itself proof of double-layer charging.

The table gives a separate original fictional steady comparison, not values from the small-step circuit. Charging has settled within the observation, and reference, ohmic correction, bulk composition, temperature, area and surface state are matched. Only external liquid supply is deliberately changed. The listed displacement is E_interface − E_eq,bulk; because local composition can differ, it is not asserted to equal the activation η used in the fixed-composition circuit.

At the larger displacement the maintained current rises from 20.0 to 50.0 μA when supply is improved, a factor of 2.50. Under the stated checks this is evidence that transport contributes to the limitation; it is not evidence that the material’s intrinsic reaction sites became 2.50 times faster. The unchanged 8.00 μA at the smaller displacement does not prove an exactly pure charge-transfer regime or an absence of smaller unresolved transport effects.

A useful materials comparison keeps the reaction, reference, potential window, exposed-area convention, surface preparation, temperature, local composition, timing and measurement resolution with the result. If accessible area changes, total R_ct and C_DL can change without a new intrinsic rate per site. If a film grows, the same circuit may no longer apply. Porous electrodes, microstructure and phase changes and Operating stability, degradation and material-to-system evidence continue those separate architecture and history questions.

Worked examples

Try the reasoning

1. Is every measured current faradaic?

   Show the answer and reasoning

   No.

   Interfacial charging also produces current.

2. Why state overpotential relative to equilibrium?

   Show the answer and reasoning

   It describes departure from a defined equilibrium potential.

   A bare applied voltage can include other cell contributions.

3. In the same ideal model, double C_DL to 200 μF while keeping ΔV, R_s and R_ct fixed. Predict the new time constant, initial and steady total currents, and total current at 9.00 ms. Does a larger capacitance prove a faster reaction?

   Show the answer and reasoning

   τ becomes 18.0 ms; the initial limit remains 100 μA and the steady current remains 10.0 μA. At 9.00 ms the total is about 64.588 μA. Faster reaction is not established.

   The initial and steady limits are set by the stated resistances and step. Doubling C_DL slows the interfacial voltage rise: I = 10 + 90 exp(−9/18). It increases finite storage, not this model’s R_ct or its steady net reaction current.

4. Instead keep C_DL = 100 μF and R_s = 100 Ω, but supply a verified local R_ct = 400 Ω at the same area/state and +10.0 mV step. Find steady current, steady η and τ. Can that establish a universal superior material?

   Show the answer and reasoning

   20.0 μA, 8.00 mV and 8.00 ms. It establishes the supplied local response, not universal superiority.

   I_∞ = 0.010/(100 + 400) A; η_∞ = I_∞ × 400 Ω; τ = 100 × 10⁻⁶ × 100 × 400/500 s. Intrinsic site activity, area, selectivity, stability and other conditions still need their own comparison.

5. At the original model’s steady 10.0 μA, an observer divides the full +10.0 mV step by 900 Ω and reports 11.1 μA of reaction. Identify the missing term and the correct local driving potential.

   Show the answer and reasoning

   The series drop is 1.00 mV, leaving η = 9.00 mV and I_F = 10.0 μA.

   The measured offset includes I R_s. Subtract it using the stated sign convention before assigning the local interfacial response. This correction is not a claim that all possible voltage losses are ohmic.

6. For the original model at 54.0 ms, η is 8.977691 mV and integrated external charge is 1.347992 μC. How much is stored and how much is net reaction charge? Is the small instantaneous storage current at 54.0 ms a reason to ignore all earlier charging?

   Show the answer and reasoning

   Stored change is about 0.897769 μC; net reaction charge is about 0.450223 μC. Earlier charging cannot be ignored.

   Use C_DL η for the change retained at the endpoint, then subtract it from the external integral. A small final storage rate does not make the accumulated stored charge zero.

7. Apply −10.0 mV from the same initial equilibrium, assuming the same symmetric linear model remains valid. What happens to the signs of η, storage current and net reaction current? Is a negative current an unphysical negative amount of substance?

   Show the answer and reasoning

   All three signed quantities reverse. A negative net reaction current represents the opposite direction under the chosen convention, not a negative physical particle count.

   The model equations are linear about this equilibrium, so changing the step sign changes the response sign. Real asymmetric reactions or a different potential window need not obey this mirror model.

8. In the separate matched steady table at +50.0 mV relative to bulk equilibrium, improved supply raises current from 20.0 to 50.0 μA. A report calls this a 2.50-fold increase in intrinsic catalyst activity. Is that supported?

   Show the answer and reasoning

   No. The ratio is 2.50, but the controlled change supports a transport contribution under the stated matched conditions.

   Supply changes the reacting surface’s chemical environment. Local equilibrium and local rates can both respond. Keep the controlled bulk-referenced offset distinct from a proven activation overpotential and do not relabel a supply change as a site-activity measurement.

9. A charge-balance analysis reliably separates 0.200 μC of faradaic charge, but two reduction reactions may carry it. Can dividing that charge by the desired reaction’s electron count establish the desired product amount?

   Show the answer and reasoning

   No. The partition between the faradaic reactions is still unknown.

   Removing nonfaradaic storage does not establish chemical selectivity. Product-specific measurements or another justified reaction partition are needed; the total faradaic charge alone cannot supply it.

10. At a verified equilibrium for one reversible reaction, the measured net faradaic current is zero. Does that prove that no forward or reverse reaction event occurs?

    Show the answer and reasoning

    No. Opposing rates can be nonzero and equal.

    Net current is their signed difference. Distinguish equilibrium balance from an ideally blocked interface, where the considered charge-transfer pathway is absent in the model.

What to carry forward

• Separate charging, reaction and transport.
• Interpret kinetics within a defined interfacial state.
• A transient can combine a falling storage current with a rising net reaction current.
• State the potential reference and distinguish the local activation driving potential from ohmic and concentration contributions.
• A stored-charge change and a net reaction-charge integral have different physical meanings.
• A blank or a supply change becomes discriminating evidence only with the comparison’s state and response assumptions.