An Introduction to Charge Transfer Resistance

Charge transfer resistance (R_ct) is a major form of resistance within batteries that contributes to a reduction in efficiency. R_ct is related to the activation barrier of electrochemical reactions. In lithium-ion batteries (LIBs) this is particularly impacted the electrode/electrolyte interface. Theoretical energy and power densities are limited by charge transfer resistance which can lead to overvoltages, especially in harsh conditions.

What is the Lithium-ion Charge Transfer Process?

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When considering the resistance to lithium-ion charge transfer, it is important to consider the process as a whole. The process of lithium-ion charge transfer varies for different battery systems. The biggest distinction is the type of electrolyte used. Here both liquid and solid electrolyte systems are discussed:

Liquid Electrolyte Lithium-ion Charge Transfer Process

The lithium-ion (Li⁺) charge transfer process starts when Li⁺ becomes solvated in the electrolyte and an electron (e⁻) is extracted through the cathode. Next the solvated Li⁺ travels through the battery chamber before undergoing de-solvation by the solid electrolyte interphase (SEI). Li⁺ diffuse through the SEI, before receiving an e⁻ from the electrode at the electrode and SEI interface.

The stages of the lithium-ion charge transfer process that are important are intercalation/de-intercalation, transport through layers and solvation/de-solvation as highlighted below:

Solvated lithium ions migrate through the electrolyte during charging and discharging, driven by the applied electric potential.

Solid Electrolyte Lithium-ion Charge Transfer Process

The mechanism of Li⁺ ion transport is dependent on the type of solid electrolyte. There is no solvation step during transfer due to the lack of solvent electrolyte.

For ceramic electrolytes: Li⁺ is transported through mobile point defects in the crystal. The ions move via the vacancy, interstitial or interstitial-substitutional exchange mechanisms. The Li⁺ transport mechanism depends on three factors: carrier type, diffusion pathway, and diffusion type. Carrier types and concentrations are decided by point defects in the ceramic crystal structures, which directly affect ionic conductivity. Li⁺ ions interact with each other, and their surroundings during transport, both of which will affect ionic conductivity significantly.

Resistance from solid electrolyte grain boundaries is generally much higher than the intergrain resistance. The structure and composition of grain boundaries can be really different from the bulk crystal. When the grain boundaries are more resistive than the bulk crystal, "granular" pathways dominate. This is where Li⁺ ions migrate from grain to grain by crossing the grain boundary.

For solid polymer electrolytes: Li⁺ is transported via the segment motion, ion hopping or vehicle mechanism (mass diffusion). Polymer electrolytes have the advantage of good flexibility which facilitates good contavt between the electrolyte and electrode. Polymers are also easy to process, and have low manufacturing costs. The ideal solid polymer electrolytes have high ionic conductivity, near‐unity Li‐ion transference number, wide electrochemical stability window, and high mechanical strength.

Measuring Charge Transfer Resistance

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Charge transfer resistance is a macroscopic parameter typically measured using electrochemical impedance spectroscopy (EIS). EIS measurements characterize the resistance of different processes lithium-ions must undergo in an electrochemical cell. This is done through the fitting of the impedance spectra to an equivalent circuit.

Impedance measurement, often visualized using a Nyquist plot, helps assess how a system resists and stores electrical energy. The plot is two-dimensional, with the X-axis representing the real part of impedance (Zreal), which includes solution resistance (Rs), charge transfer resistance (Rct), and Warburg impedance (W) related to diffusion. The Y-axis shows the imaginary part of impedance (Zimag), reflecting energy storage effects such as capacitance or inductance. Together, these values provide insight into both the resistive and reactive behavior of materials or electrochemical systems.

Arrhenius Equation

The resistance resulting from the charge transfer process experienced by lithium-ions is related to its diffusivity through battery components and de-solvation. The charge transfer resistance (R_CT) follows the relationship based on the thermally activated process represented by the Arrhenius equation:

A Ea

Proportionality coefficient / Frequency factor Activation energy

R T

Gas constant Temperature

As shown in the Arrhenius equation, activation energy is an important indicator of the diffusivity of lithium-ions. The diffusion coefficient decays exponentially with the increase of the energy barrier. In other words, as the amount of energy it takes to undergo a diffusion process increases the likelihood of diffusion occurring decreases more and more rapidly. The diffusion process is defined by lithium-ions crossing the interface between the electrolyte and the electrodes. Therefore the activation energy is defined by the energy required to complete that crossing. The activation energy can be obtained from the slope of a log(1/R_CT) versus the inverse of the temperature plot.

Butler-Volmer Equation

The Butler-Volmer equation is the foundational model in electrochemical kinetics that describes how current is related to the rate of electron transfer reactions at an electrode–electrolyte interface. In the cases of electrochemical engineering and battery modeling, the macroscopic Butler-Volmer equation is used which considers bulk electrochemical processes (quantities expressed per mole):

i i0 α n

Net current density Exchange current density Charge transfer coefficient Number of electrons transferred

F R T η

Faraday Constant Gas constant Temperature Overpotential (difference from equilibrium potential)

The charge transfer coefficient (α) is often referred to as the symmetry factor because it determines the dependence of the current density on overpotential. It is useful in indicating the direction the redox reaction is going. At α = 0.5 the oxidation and reduction reactions are balanced or symmetrical". When α increases towards 1 then the redox reaction shows greater current in the cathodic region. When α decreases towards 0 then the redox reaction shows greater current in the anodic region.

At small overpotentials (low η), such as in EIS, the Butler-Volmer equation can be linearized:

i i0 n F

Net current density Exchange current density Number of electrons transferred Faraday Constant

R T η

Gas constant Temperature Overpotential (difference from equilibrium potential)

This is a linear relationship between current density and overpotential. This linearization simplifies calculations and analysis. This happens because when overpotential (η) is very small then the arguments of the exponentials are much less than 1. Ohm's law is then used to identify charge transfer resistance relationship as overpotential is voltage.

For large overpotentials, different simplified butler-volmer equations can be used. If the large overpotentials are positive, the anodic simplified butler-volmer (green line) tracks very nicely with the full butler-volmer line. If the large overpotential is negative then the cathodic simplified butler-volmer equation (purple line) tracks well.

Tafel plot

The Tafel plot is a graph of the log of the current (i) vs overpotential η. This differs from the Butler-Volmer where overpotential vs. i/i₀ is plotted. The charge transfer coefficient and exchange current density can be determined from the plot. The anodic and cathodic current profiles have different slopes:

• Anodic Current: (1-α)F/RT
• Cathodic Current: αF/RT

All factors that influence the charge transfer resistance are highlighted by the exchange current density:

F k0 Aeff

Faraday Constant Intrinsic rate constant Effective active surface area

cOx cRed α

Concentration of oxidized species Concentration of reduced species Charge transfer coefficient

The exchange current density represents the current density, which flows in each direction, anodic and cathodic, when the overall current is zero.

Factors Limiting Lithium-ion Charge Transfer Kinetics

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There are a range of factors limiting lithium-ion charge transfer kinetics, specifically connected to the selected battery components.

If we consider the factors that influence the exchange current density, we can see how the charge transfer resistance is directly impacted. Exchange current densities reflect intrinsic rates of electron transfer between the lithium ions and the electrode. Intrinsic rate constant, effective active surface area, the concentration of the redox species and charge transfer coefficient are all contributing factors. The effect on charge transfer is summarized in the table below:

Factor	Description	Effect on Rct
De-solvation	Slower de-solvation of lithium ions in electrolyte	Higher Rct
Surface area/morphology/roughness	More active sites for charge transfer	Lower Rct
Solid electrolyte interphase properties	Thick/resistive SEI	Higher Rct
Electrolyte composition	Poor ion mobility/solvation	Higher Rct
Ion conductivity	Poor conductivity	Higher Rct
Temperature	Lower temperature	Higher Rct
State of charge	Depends on electrode material & SOC
Interface chemistry	Unstable/interrupted contact between electrolyte and electrode	Higher Rct
Mechanical degradation	Cracks or particle isolation/lithium plating	Higher Rct

Electrolyte - Solid vs liquid

Solid Electrolyte: Solid electrolyte does not require the Li+ ions to undergo de-solvation, removing a classic charge transfer resistive process faced by liquid electrolytes. However, solid electrolytes face other challenges such as poor contact due to differences in crystal structure between electrode and electrolyte components. Grain formation and gaps mean that contact between electrodes and electrolyte is less than 100%. This can be resolved by adding buffer layers. Grain boundary conduction is a limiting factor in most cases of ceramic solid state electrolyte charge transport. It increases the activation energy of io transport and reduces the overall ionic conductivity.

Liquid Electrolyte: The ion-transport and de-solvation process is mainly controlled by the interactions between electrolyte components: dipole-diploe interaction between solvents, ion-dipole interaction between the cation/anion and solvent, and ion-ion interactions between cation and anion. These interactions are controlled by the distance between the two components, the charge of the ion, the solvent molecule angle, the dielectric constant and the dipole moment of the solvent. As temperature increases ionic conductivity of solvents increase due to faster thermal movement.

• We must balance the need to disassociate the Li+ from anions via strong solvent dielectric constants which can lead to strong dipole-dipole interactions which can hinder ion transport. Solvent electrolyte considerations must be made.

Links to Internal Resistance

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Charge transfer resistance is one form of internal resistance. Ohmic and mass transport resistance also contribute but each are more prevalent at different times and voltages. Ohm's law (V = IR) describes the relationship between voltage, current and resistance in an electrical circuit. Therefore, it can be used to calculate resistance based on known potential and voltages of a given circuit. The schematic below shows a voltage-time plot during constant current showing the time order of the different resistances:

It is this relationship between voltage, current and resistance that allows us to use experiments which control and monitor voltage and current in order to determine resistance.

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