Correlating Electrode Tortuosity and McMullin Number with Battery Rate Performance

1. Introduction

Improving cell energy density motivates the development of thicker electrodes. However, as electrode thickness increases, liquid-phase lithium-ion transport through the porous coating becomes progressively more restricted — increasing internal resistance, reducing active material utilization, and degrading both cycle performance and rate capability^[1].

Electrode tortuosity quantifies the degree of curvature of the ionic transport pathway through the porous electrode structure, it is another important parameter related to the transmission characteristics besides porosity^[2]. Beyond porosity (\(\varepsilon\)), tortuosity (\(\tau\)) is the critical microstructural variable governing ionic transport: two electrodes with identical porosity but different pore connectivity and geometry can have dramatically different tortuosity values and therefore very different rate performance. The MacMullin number (\(N_m = \tau / \varepsilon\)), combines both parameters into a single experimentally accessible metric that characterizes the effective ionic permeability of an electrode coating and predicts rate capability without requiring extended cell cycling.

Researchers conventionally evaluate electrode rate and cycle performance by assembling coin cells or pouch cells — but the long testing cycles required for this approach limit R\&D throughput. Electrode tortuosity testing via symmetric cell EIS provides a rapid, non-cycling alternative: a single EIS measurement can rank electrodes by their ionic transport efficiency in hours rather than days, enabling systematic screening of electrode formulation, compaction density, coating thickness, and manufacturing process variables.

2. Test Conditions and Methods

2.1 Test Equipment

• Symmetric cell assembly and EIS testing: The IEST EIC1400 Multi-Channel Ion Conductivity Test System as shown in Figure 1, integrating four-channel symmetric cell assembly fixtures, electrochemical impedance test, fitting software, and a high-purity argon atmosphere — enabling multi-channel rapid EIS measurement for electrode tortuosity determination. Frequency range: 0.1–1,000 Hz.
• Half-cell rate testing: 2032 coin cells with graphite anodes (three thickness variants) vs. lithium metal, cycled at 0.1C, 0.2C, and 0.5C to measure rate-dependent capacity and capacity retention.

Figure 1. IEST EIC1400 Multi-Channel Ion Conductivity Testing System — integrating symmetric cell assembly, argon atmosphere, and EIS measurement for electrode tortuosity and MacMullin number determination.

2.2 Electrode Preparation

Graphite slurry was coated at three doctor-blade gap settings to produce electrodes nominally 100 µm, 200 µm, and 400 µm thick (wet gap control). All electrodes were dried and handled under controlled conditions to ensure comparable porosity and binder distribution across the three thickness variants — isolating thickness as the primary variable.

2.3 Testing Protocol

• Electrode tortuosity measurement: Stack electrode–separator–electrode in each of the four channels → seal chamber, evacuate and backfill with high-purity argon → inject electrolyte quantitatively into each channel → equilibrate 10 min → measure EIS spectrum → extract R_ion from Nyquist plot → compute MacMullin number using Equation (2).
• Rate performance test: Charge and discharge at 0.1C, 0.2C, and 0.5C for each electrode thickness variant; record capacity and capacity retention at each rate.

2.4 McMullin Number Calculation Method

The ionic resistance \(R_{ion}\) of the electrode coating is related to tortuosity \(\tau\), porosity \(\varepsilon\), electrolyte conductivity \(\sigma\), electrode thickness \(d\), and electrode area \(A\) by Equation (1):

\[
\tau = \frac{R_{ion} \cdot A \cdot \varepsilon \cdot \sigma}{d} \tag{1}
\]

Since direct measurement of electrode porosity \(\varepsilon\) requires complex and time-consuming characterization, the MacMullin number (also written McMullin number) \(N_m = \tau / \varepsilon\) is used instead as the standard metric for electrode tortuosity assessment. \(N_m\) is calculated directly from EIS data using Equation (2), requiring only \(R_{ion}\), electrode geometry, and electrolyte conductivity:

\[
N_m = \frac{\tau}{\varepsilon} = \frac{R_{ion} \cdot A \cdot \sigma}{d} \tag{2}
\]

The impedance of the symmetric cell was tested using the electrochemical workstation, and the EIS obtained is shown in Figure 2. Which shows a characteristic two-segment response — a high-frequency line segment and a low-frequency line segment — reflecting the ionic transport signature in the absence of electrochemical reactions. The ionic resistance \(R_{ion}\) is extracted by extending the low-frequency line segment until it intersects the real axis, then subtracting the high-frequency real-axis intercept: \(R_{ion} = \Delta Z_{Re} / 3\). Substituting into Equation (2) yields the MacMullin number, which can be analyzed as the degree of the zigzagging of the electrode.

Figure 2. Nyquist plot of symmetric cell EIS for electrode tortuosity measurement. Ionic resistance \(R_{ion}\) is extracted as \(\Delta Z_{Re} / 3\) from the intercept difference between high-frequency and extrapolated low-frequency line segments.

3. Results Analysis

3.1 MacMullin Number Results vs. Electrode Thickness

EIS fitting of the three graphite electrode variants yielded the following MacMullin numbers (Figure 3a–d):

Table 1. Electrode thickness effect on MacMullin number and rate capability

Electrode Thickness	MacMullin Number (Nm)	Rate Capability Rank	Interpretation
100 µm	4.61	Best	Lowest electrode tortuosity; most direct ionic transport pathways; highest capacity retention at all test rates
200 µm	6.15	Intermediate	Moderate electrode tortuosity; increased transport distance reduces high-rate utilization
400 µm	6.61	Worst	Highest electrode tortuosity; greatest deviation from straight-path transport; most degraded rate performance

The trend is clear: increasing electrode thickness increases the MacMullin number, indicating more tortuous ion pathways and reduced effective ionic conductivity.

Figure 3. Symmetric cell EIS Nyquist plots and MacMullin number (McMullin number) results: (a) 100 µm, Nm = 4.61; (b) 200 µm, Nm = 6.15; (c) 400 µm, Nm = 6.61; (d) MacMullin number comparison — confirming that electrode tortuosity increases monotonically with coating thickness.

The trend is physically intuitive: as electrode thickness increases, the pore network through which lithium ions must transport becomes longer and more complex. Blind pores, narrow pore throats, and tortuous inter-particle channels — each contributing to electrode tortuosity — become statistically more prevalent and more consequential over greater coating depths. The result is that ionic resistance scales faster than linearly with thickness, producing MacMullin numbers that increase from 4.61 (100 µm) to 6.61 (400 µm) — a 43% increase for a 4× thickness increase.

3.1.1 Electrode Pore Structure and Its Relationship to Tortuosity

Electrode pore size and pore connectivity are the microstructural determinants of electrode tortuosity. Porous battery electrodes contain three types of pore features that each contribute to the MacMullin number: (1) through-pores — pores that span the full electrode thickness and provide direct ionic transport pathways; these minimize electrode tortuosity; (2) dead-end (blind) pores — pores connected to the electrode surface on only one side; ions enter but must reverse direction to continue transport, increasing effective path length and tortuosity; (3) constricted pore throats — narrow junctions between pore bodies that restrict ionic flux and increase local resistance, amplifying the MacMullin number disproportionately to their volume fraction. As electrode thickness increases, the probability of encountering blind pores and constricted throats over the transport path increases — explaining why the 400 µm electrode has a significantly higher MacMullin number (6.61) than the 100 µm electrode (4.61), despite comparable porosity.

3.2 Two Methods for Electrode Tortuosity Measurement

Figure 4. Electrochemical methods for electrode tortuosity determination: (a) eRDM Polarization-Interrupt Method — MacMullin number from potential relaxation slope; (b) eSCM Symmetric Cell EIS Method — MacMullin number from ionic resistance fitting on Nyquist plot.^[3]

Two standard electrochemical methods are used to determine electrode tortuosity and McMullin number:

• Polarization-Interrupt Method (eRDM): a fixed DC current is applied to polarize the cell, creating a lithium-ion concentration gradient. When the current is interrupted, the gradient relaxes and the potential decays to zero. A semi-log plot of potential vs. time is fitted to extract the MacMullin number from the relaxation curve slope.
• Symmetric Cell EIS Method (eSCM): the electrode is assembled in a symmetric non-intercalating cell, EIS is measured, and R_ion is extracted from the Nyquist plot to calculate the MacMullin number via Equation (2). This is the method used in this study and implemented in the IEST EIC1400 system.

Figure 5. Critical comparison of eRDM and eSCM electrode tortuosity methods for four electrodes with identical porosity but different pore geometries: eRDM produces identical MacMullin numbers for all four (insensitive to pore structure); eSCM correctly differentiates their electrode tortuosity — demonstrating that the symmetric cell EIS method more accurately reflects actual ionic transport performance.^[3]

Figure 5 demonstrates a key limitation of the eRDM method: for electrodes with identical porosity but different pore connectivity, eRDM produces identical MacMullin numbers regardless of pore geometry — because it is sensitive only to bulk concentration gradients, not to local pore structure. The eSCM method resolves these differences, producing distinct McMullin numbers that reflect actual ionic transport efficiency. The IEST EIC1400 system implements the eSCM method and therefore provides electrode tortuosity data that more faithfully represents real battery rate performance.

3.3 Correlation Between MacMullin Number and Rate Performance

Figure 6 and Table 2 demonstrate the capacity and capacity retention of negative electrode sheets with different thicknesses at different multiplicities. It can be seen that the capacity of each electrode decreases with the increase of multiplicity, but the capacity retention rate is 100 μm > 200 μm > 400 μm, which indicates that the electrode obtained by 100 μm coating has the best multiplicity performance, and the 400 μm electrode has the worst performance. Combined with the data of McMullin number, it can be seen that with the increase of the thickness of the electrode, the curvature of the electrode becomes larger, and the multiplicity performance of the cell becomes worse subsequently.

Figure 6. Rate performance of graphite anodes at three electrode thicknesses: (a) capacity-voltage profiles at 0.1C, 0.2C, 0.5C; (b) rate capacity retention curves — retention order: 100 µm (Nm = 4.61) > 200 µm (Nm = 6.15) > 400 µm (Nm = 6.61), directly correlating with electrode tortuosity.

Table 2. Rate performance at different coating thicknesses (specific capacity, mAh/g)

Rate	100 μm		200 μm		400 μm
	Charge (mAh/g)	Discharge (mAh/g)	Charge (mAh/g)	Discharge (mAh/g)	Charge (mAh/g)	Discharge (mAh/g)
0.1C	259.713	260.862	225.379	227.152	175.783	175.084
0.2C	222.513	224.121	191.696	193.351	139.694	139.111
0.5C	136.860	137.320	98.330	98.448	57.428	57.312

The electrochemical performance test results and the pore curvature test results can completely correspond to each other, which indicates that we can predict the performance of the electrode through the test of the electrode pore curvature, correlate the electrode structure and performance prediction quickly, and accelerate the design and process development of the electrode.

4. Summary

Symmetric cell EIS measurement of electrode tortuosity on graphite anodes of three coating thicknesses confirms that electrode tortuosity increases with electrode thickness — from Nm = 4.61 at 100 µm to Nm = 6.61 at 400 µm — and that this tortuosity increase corresponds directly to degraded rate capability across all test C-rates.

Key findings: electrode tortuosity, quantified as the MacMullin number (\(N_m = \tau / \varepsilon\)), increases from 4.61 to 6.61 as graphite anode thickness increases from 100 \(\mu\)m to 400 \(\mu\)m — a 43\% increase in electrode tortuosity for a 4× increase in thickness, reflecting the greater prevalence of dead-end pores and constricted pore throats in thicker coatings. Rate capacity retention order at all test rates (0.1C, 0.2C, 0.5C) matches the MacMullin number ranking exactly, confirming that symmetric cell EIS tortuosity testing can rank electrode rate performance without cell cycling. Beyond electrode thickness, the same EIC1400 EIS method applies to screening electrode formulation (binder, carbon black), porosity (compaction density), active material morphology, electrolyte type, and separator choice — providing a rapid microstructure-to-performance correlation tool for battery electrode development.

5. References

[1] Sun Weibing et al. A low curvature thick electrode and its preparation method and application. CN115312777A. 2022.

[2] Wang Chenyang, Zhang Anbang, Chang Zenghua, et al. Progress in structure design and preparation of porous electrodes for lithium ion batteries[J]. Materials Engineering, 2022, 50 (1): 67-79.

[3] Tuan-Tu Nguyen, Arnaud Demortière, Benoit Fleutot, Bruno Delobel, Charles Delacourt & Samuel J. Cooper. The electrode tortuosity factor: why the conventional tortuosity factor is not well suited for quantifying transport in porous Li-ion battery electrodes and what to use instead [J]. Electrochem. Soc. 2018,165:A388.

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