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Solid-Phase Lithium Concentration Distribution and Average State of Charge Evolution in Blended Cathodes of Lithium-Ion Batteries Under a One-Dimensional Isothermal Model

This study employs a 1D isothermal COMSOL model with an "additional porous electrode" framework to reveal that blended cathodes exhibit non-monotonic kinetic behavior driven by open-circuit potential overlap, identifying intermediate NMC811 ratios as kinetically limited while terminal ratios offer synergistic optimization for electrode design.

Original authors: Yue Zhang, Zenglong Zhang, Qi Zhang, Zihao Wang, Duqiang Xin

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yue Zhang, Zenglong Zhang, Qi Zhang, Zihao Wang, Duqiang Xin

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Mixing Batteries Like a Smoothie

Imagine you are making a smoothie. You have two ingredients:

LCO (Lithium Cobalt Oxide): Think of this as a steady, reliable runner. It's consistent but moves a bit slowly.
NMC811 (A high-nickel material): Think of this as a sprinter. It's fast and can carry a lot of energy, but it's a bit more volatile.

Scientists often try to mix these two materials inside a single battery to get the best of both worlds: the speed of the sprinter and the stability of the runner. The goal is to make a battery that holds more energy and charges faster.

However, the researchers in this paper found that simply mixing them isn't as simple as adding 50% of one and 50% of the other. Sometimes, mixing them creates a traffic jam inside the battery.

The Problem: The "Traffic Jam" in the Middle

The researchers used a computer model (a digital simulation) to watch what happens inside the battery when they mix these two materials in different ratios. They looked at how "lithium" (the energy carriers) moves inside the tiny particles of the battery.

They discovered a surprising "Goldilocks" problem:

Too much LCO (20% mix): The battery behaves mostly like the steady runner. It's safe and stable, but not super fast.
Too much NMC811 (80% mix): The battery behaves mostly like the sprinter. It's fast and holds a lot of energy.
The "Middle" Mix (40% to 60%): This is where things go wrong.

The Analogy: Imagine a highway with two lanes. One lane is for slow cars (LCO) and one for fast sports cars (NMC811).

If the road is mostly slow cars, everyone drives at a steady pace.
If the road is mostly sports cars, everyone speeds along.
But if you have an even split (50/50), the fast cars try to overtake the slow cars, but the lanes are too narrow. They get stuck, brake suddenly, and create a massive traffic jam.

In the battery, this "traffic jam" happens because the two materials have different "open-circuit potential" (a fancy way of saying they want to release energy at different voltages). When mixed in the middle range, they fight over the current. This causes a kinetic mismatch, meaning the lithium gets stuck inside the particles, creating a huge concentration difference between the center and the surface of the particle. This stress can eventually crack the battery particles, just like a traffic jam causes stress on a car engine.

The Solution: Stick to the Ends

The study found that to get the best performance, you should avoid the middle mix entirely.

For safety and long life: Use a mix with 20% NMC811. The slow runner (LCO) dominates, keeping the traffic flowing smoothly.
For maximum speed and energy: Use a mix with 80% NMC811. The sprinter dominates, creating a clear path for fast energy delivery.

The "middle" mixes (40% and 60%) actually performed worse than expected because the internal stress was so high that it limited how much energy the battery could actually deliver.

How They Did It: The "Extra Layer" Trick

Usually, computer models treat a battery electrode as if it's made of just one type of material. To study a mix, the researchers invented a clever trick called the "additional porous electrode" method.

The Analogy: Imagine you are trying to measure the flow of water through a sponge that has two different types of holes in it. Standard models would just guess an "average" hole size.
The researchers' method is like saying, "Okay, we will treat this sponge as if it has two separate layers of holes right on top of each other."

Layer 1: The LCO holes.
Layer 2: The NMC811 holes.
Both layers share the same water (electrolyte) and the same pressure (voltage), but the water moves through them at different speeds based on their own rules.

This allowed them to see exactly how the two materials competed and cooperated without making the computer model too complicated.

The "Stress Test": Is the Model Reliable?

Finally, the researchers wanted to make sure their computer model wasn't just a fluke. They asked: "What if our numbers for how fast lithium moves are slightly wrong?"

They tested the model by changing the speed of lithium movement by 20% (making it faster or slower).

The Result: The model was incredibly stable. Even with these big changes, the results only wiggled a tiny bit in the very first few seconds. After about 100 seconds, the lines on the graph merged perfectly.
The Takeaway: This means their predictions are robust. Even if the real-world materials aren't perfect, the model's advice on how to mix the battery (avoid the middle, stick to the ends) is trustworthy.

Summary

This paper is like a traffic report for battery designers. It tells us that when mixing two different battery materials, don't try to split them 50/50. That middle ground causes a "traffic jam" that stresses the battery. Instead, lean heavily toward one material or the other (20% or 80%) to keep the energy flowing smoothly and safely.

Problem Statement

The integration of blended cathode systems (mixing multiple active materials) offers a pathway to enhance the energy density and kinetics of lithium-ion batteries. However, conventional porous electrode theories, such as the Doyle-Fuller-Newman (DFN) model, typically rely on single-active-component assumptions. These models fail to accurately describe the cooperative electrochemical reactions and internal kinetic heterogeneity that arise when multiple materials coexist within a single electrode. Specifically, there is a lack of systematic theoretical understanding regarding the quantitative relationship between material intrinsic parameters, particle-scale concentration gradients, and electrode-scale macroscopic responses in blended systems. Existing studies often treat blended cathodes as equivalent single materials via volume-averaging, obscuring the independent diffusion kinetics and interfacial reaction characteristics of constituent materials. Furthermore, the optimal blending ratio to balance diffusion uniformity and capacity performance remains unclear, particularly for high-nickel/low-nickel systems like LCO and NMC811.

Methodology

To address these gaps, the authors developed a one-dimensional (1D) isothermal lithium-ion battery model using COMSOL Multiphysics. The core innovation is an "additional porous electrode" modeling strategy. This approach allows for the simultaneous definition of multiple active material phases (specifically LiCoO2 [LCO] and LiNi0.8Mn0.1Co0.1O2 [NMC811]) within the same cathode domain.

Key methodological features include:

Decoupled Parameters: While the distinct material phases share the identical local electrolyte concentration field and solid-phase potential distribution, they maintain completely independent solid-phase lithium diffusion coefficients, interfacial reaction kinetic parameters, and equilibrium potential functions.
Current Superposition: The total volumetric reaction current density is derived through the linear superposition of the reaction contributions from each constituent phase, governed by independent Butler-Volmer kinetics for each material.
Experimental Setup: The study simulated a battery with a graphite anode and a blended cathode containing gradient mass fractions of NMC811 (20%, 40%, 60%, and 80%), with the remainder being LCO. Simulations were conducted under constant-current charge/discharge conditions (1C rate) at 298K.
Sensitivity Analysis: The robustness of the model was verified by introducing ±20% fluctuations in the solid-phase diffusion coefficient ( $D_s$ ) to assess the impact on the State of Charge (SOC) evolution.

Key Contributions

Novel Modeling Framework: The paper introduces a rigorous 1D modeling strategy that distinguishes the mass transfer and reaction behaviors of different materials within a single region without increasing geometric dimensionality. This overcomes the limitations of equivalent-parameter methods.
Micro-Macro Coupling: The study elucidates the chain-like governing mechanism linking "material intrinsic properties" to "microscopic mass transfer polarization" and finally to "macroscopic electrochemical response."
Quantitative Blending Analysis: It provides a systematic analysis of how varying NMC811 fractions affects intra-particle lithium concentration gradients and average electrode State of Charge (SOC), moving beyond simple linear superposition assumptions.

Key Results

Non-Monotonic Concentration Polarization: The simulation revealed a prominent non-monotonic evolution of microscopic concentration polarization as the blending ratio varied.
- Intermediate Ratios (40%–60% NMC811): The internal particle concentration difference ( $\Delta c_s$ ) anomalously rebounds to a global maximum (approx. 1500–1581 mol/m³ at 1800s). This is attributed to a "two-phase kinetic mismatch" where the severe overlap of the open-circuit potential (OCP) platforms of LCO and NMC811 triggers local current flux disruption and multi-phase kinetic mismatch. This forces the macro-electrode into a kinetically limited zone with high diffusion stress.
- Terminal Ratios (20% and 80% NMC811): These ratios mitigate the kinetic mismatch. At 20%, the system behaves similarly to the stable LCO matrix with suppressed gradients. At 80%, the high-rate NMC811 component establishes a dominant fast ion-transport framework, smoothing out the gradients and achieving a synergistic optimization.
Average Electrode SOC Evolution:
- Pure NMC811 demonstrated superior capacity utilization compared to pure LCO due to higher diffusion coefficients and a wider potential window.
- In blended systems, the macroscopic capacity (net charge increment) increased monotonically with NMC811 content. However, the 40%–60% range, while offering moderate capacity, suffered from the most severe internal diffusion kinetics mismatch.
- The study identifies a design trade-off: 20% NMC811 is optimal for long-life/safety (low stress), while 80% is optimal for high-energy density, whereas the 40%–60% range should be avoided due to kinetic traps.
Model Robustness: Sensitivity analysis confirmed that ±20% fluctuations in the diffusion coefficient resulted in negligible impact on the SOC evolution after the initial 100 seconds of discharge, demonstrating the model's high predictive robustness.

Significance and Claims

The paper claims to establish a high-throughput numerical strategy for electrode ratio optimization by refining the electrochemical continuum theory for multi-component porous structures. By successfully capturing the independent kinetic attributes and coupled mass transfer-reaction behaviors of blended cathodes, the study provides a digital-numerical toolkit for the composition screening and performance forecasting of next-generation blended electrodes. The findings offer a quantitative basis for engineering trade-offs between energy density and service stability, specifically advising against intermediate blending ratios that induce severe kinetic mismatches. The work does not propose new experimental applications but rather provides a validated simulation framework to guide the design of blended cathode systems.