Reaction Kinetics

How Butler-Volmer, Tafel, and Marcus-Hush-Chidsey equations describe electrochemical reaction rates in batteries.

Reaction kinetics describe how quickly lithium ions can intercalate or deintercalate at the electrode-electrolyte interface. While open-circuit potentials determine the direction of reactions, kinetics determine the rate—making them fundamental to understanding battery power output, efficiency, and the physics behind rate-dependent capacity loss.

What Are Reaction Kinetics?

Reaction kinetics describe the rates of electrochemical reactions that occur at the interface between the electrodes and the electrolyte. These reactions involve:

• Transfer of lithium ions across the interface
• Simultaneous transfer of electrons through the external circuit

As a result, they are strongly influenced by the concentrations and potentials in both the electrodes and the electrolyte. The rate at which these reactions occur directly impacts a battery’s:

• Power output
• Efficiency
• Charge/discharge behavior

Overpotential: The Driving Force

When the battery is operating, the electrodes are no longer at their OCPs, and there is a driving force for the reactions. This driving force is defined as the reaction overpotential ($\eta$):

$\eta = \phi_s - \phi_e - U$

where $\phi_s$ is the electrode potential, $\phi_e$ is the electrolyte potential, and $U$ is the OCP.

The Butler-Volmer Equation

The most commonly used equation to describe reaction kinetics in batteries is the Butler-Volmer equation. It provides a detailed, non-linear relationship between the overpotential and the current density (reaction rate) at the electrode surface:

$i = i_0 \left[ \exp\left(\frac{\alpha_a F \eta}{RT}\right) - \exp\left(-\frac{\alpha_c F \eta}{RT}\right) \right]$

where:

• $i$ is the net current density
• $i_0$ is the exchange current density (reaction rate at equilibrium)
• $\eta$ is the overpotential
• $\alpha_a$ and $\alpha_c$ are the anodic and cathodic charge transfer coefficients
• $F$ is the Faraday constant
• $R$ is the universal gas constant
• $T$ is the temperature

If $\alpha_a = \alpha_c = 0.5$, the two exponentials can be combined into a hyperbolic sine function, which is commonly found in battery models.

Behavior at Different Overpotentials

Overpotential	Behavior
Small	Both terms contribute, leading to a symmetrical response around equilibrium (reversible reaction)
Large	One term dominates (depending on anodic or cathodic), simplifying to the Tafel equation

The Tafel Equation

At higher overpotentials, the Butler-Volmer equation simplifies to the Tafel equation:

$i = i_0 \exp\left(\frac{\alpha_a F \eta}{RT}\right)$

The Tafel equation is most often used to model irreversible reactions, where only one direction of the reaction (e.g., growth of a degradation product) is significant. In this regime, when the overpotential is negative, the current decreases exponentially and becomes negligible, reflecting that the reaction essentially only proceeds in one direction.

Advanced Kinetics: Marcus-Hush-Chidsey

While Butler-Volmer kinetics work well in many situations, they are based on phenomenological assumptions that break down under certain conditions. Marcus-Hush-Chidsey (MHC) theory provides a more fundamental description of electron transfer kinetics rooted in quantum mechanics.

Why Go Beyond Butler-Volmer?

Butler-Volmer assumes that current increases exponentially without limit as overpotential increases. However, experiments show that at very high overpotentials, the reaction rate can saturate or even decrease—behavior that Butler-Volmer cannot capture.

Kinetic Model	Physical Basis	High Overpotential Behavior
Butler-Volmer	Phenomenological	Exponential growth
Marcus-Hush-Chidsey	Quantum mechanical electron transfer	Saturation (inverted region)

The Marcus Theory Foundation

Marcus theory describes electron transfer as requiring reorganization of the solvent and molecular structure. The key parameter is the reorganization energy ($\lambda$), which represents the energy needed to rearrange the environment for electron transfer to occur.

The rate depends on the alignment between the electronic energy levels of the electrode and the redox species. At moderate overpotentials, increasing the driving force speeds up the reaction. But beyond a critical point (the “inverted region”), further increases in overpotential can actually slow the reaction.

Marcus-Hush-Chidsey for Electrodes

MHC theory extends Marcus theory to metal electrodes by integrating over the continuum of electronic states in the metal. The resulting current density is:

$i = i_0 \int_{-\infty}^{\infty} \frac{\exp\left[-\frac{(\lambda - x)^2}{4\lambda k_B T}\right]}{1 + \exp(x - \eta^*)} \, dx$

where $\eta^* = F\eta / k_B T$ is the dimensionless overpotential and $\lambda$ is the reorganization energy.

Unlike Butler-Volmer, MHC kinetics are inherently asymmetric and predict current saturation at high overpotentials—matching experimental observations in many systems.

When to Use MHC Kinetics

MHC kinetics are particularly relevant for:

• Fast charging conditions where high overpotentials are reached
• Certain electrode materials (e.g., some intercalation compounds) where asymmetric kinetics are observed
• Fundamental studies requiring accurate kinetic descriptions

For most practical battery modeling, Butler-Volmer remains the standard choice due to its simplicity and adequate accuracy. MHC kinetics are typically reserved for detailed mechanistic studies or when experimental data shows clear deviations from Butler-Volmer predictions.

Application in Physics-Based Models

In physics-based models, reaction kinetics play a critical role in describing the interfacial behavior between the electrodes and the electrolyte. These kinetics strongly couple the equations for concentrations and potentials in the electrodes and electrolyte.

Equation	Application
Butler-Volmer	Reversible reaction of lithium intercalation into active material particles
Tafel	Irreversible degradation reactions, such as SEI growth

Reaction kinetics are the bridge between the thermodynamics of equilibrium (OCPs) and the dynamic behavior of a battery under use. They are a key factor in the trade-off between current and capacity, which is explored in Battery Capacity.

Related Topics

• Open-Circuit Voltage—the equilibrium potentials that drive reactions
• Battery Capacity—how kinetics affect usable capacity at different C-rates
• Internal Resistance—the macroscopic effect of reaction overpotentials
• SEI Growth—a degradation reaction described by Tafel kinetics