# UHPC 101 Part 5 – Coulombic Efficiency: the Breakdown

## Introduction

The UHPC 101 series has thus far been all about Coulombic Efficiency (CE); how it relates to cell cycle life, how the CE of cells can be normalized to account for different cycle times due to either different cycling rates or different states of charge (CIE/hr) and the two degradation metrics that make up CE: capacity fade and charge end-point capacity slippage.

In this post, the intention is to demonstrate, both graphically and mathematically, how capacity fade and charge end-point capacity slippage add up to give the CE of a cell. The first part of this post derives the CE expression explicitly in terms of capacity fade and charge end-point capacity slippage, while the second part uses a fictional case study to demonstrate how a decomposed CE measurement can be interpreted.

The goal is to show the wealth of information that high fidelity CE measurements hold, well beyond predicting which cell will have the longest lifetime.

## Part 1

The graph above shows fictional data that has been modified to exaggerate both capacity fade and charge end-point capacity slippage. For a given cycle $n$$n$, the charge end-point is labelled ${q}_{c}^{n}$$q_c^n$, and the discharge end-point is labelled ${q}_{d}^{n}$$q_d^n$. The discharge and charge end-point capacity slippages are the differences between the end-points of cycle $n$$n$ and cycle $n-1$$n-1$: $\mathrm{\Delta }{q}_{c}^{n}={q}_{c}^{n}-{q}_{c}^{n-1}$$\Delta q_c^n = q_c^n - q_c^{n-1}$ and $\mathrm{\Delta }{q}_{d}^{n}={q}_{d}^{n}-{q}_{d}^{n-1}$$\Delta q_d^n = q_d^n - q_d^{n-1}$, respectively. The capacity fade, ${Q}_{f}$$Q_f$, for cycle $n$$n$ is the difference between the discharge capacity of cycle $n-1$$n-1$ and cycle $n$$n$: ${Q}_{f}={Q}_{d}^{n-1}-{Q}_{d}^{n}$$Q_f = Q_d^{n-1}- Q_d^n$.

Interestingly, ${Q}_{f}$$Q_f$ can also be written as the difference between the discharge and charge end-point capacity slippages: ${Q}_{f}=\mathrm{\Delta }-\mathrm{\Delta }$$Q_f = \Delta q_d^n-\Delta q_c^n$. To see this, imagine sliding the discharge curve of cycle $n$$n$ to the left by an amount $\mathrm{\Delta }{q}_{c}^{n}$$\Delta q_c^n$ such that the charge end-points of cycle $n$$n$ and cycle $n-1$$n-1$ lie on top of each other. The difference in discharge end-points then gives the capacity fade, ${Q}_{f}$$Q_f$, and is the same as the difference between the discharge and charge end-point capacity slippages. The CE for cycle $n$$n$, which is the discharge to charge capacity ratio, can now be written in terms of the capacity fade, ${Q}_{f}^{n}$$Q_f^n$, and the charge end-point capacity slippage, $\mathrm{\Delta }{q}_{c}^{n}$$\Delta q_c^n$, by noting that the cycle $n$$n$ discharge capacity is the cycle $n$$n$ charge capacity minus the discharge capacity slippage, $Q_d^n = Q_c^n - \Delta q_d^n$:



And, as described above, the discharge end-point capacity slippage can be expressed in terms of the capacity fade and charge end-point capacity slippage:

$C{E}^{n}=1-\frac{\mathrm{\Delta }{q}_{d}^{n}}{{Q}_{c}^{n}}=1-\frac{{Q}_{f}^{n}+\mathrm{\Delta }{q}_{c}^{n}}{{Q}_{c}^{n}}$

This is ultimately a very useful expression because the values ${Q}_{f}^{n}$$Q_f^n$ and $\mathrm{\Delta }{q}_{c}^{n}$$\Delta q_c^n$ can easily be extracted from cycling data for each cycle number and immediately gives insight into how the cell is degrading, whether electrolyte reduction or oxidation, for example. Before considering an example, it is worth showing how the corresponding expression for CIE/hr is obtained. The CIE is defined as: $CIE=1–CE$$CIE = 1 – CE$, and the CIE/hr is obtained by dividing by the time, in hours, per cycle. Thus,

$CIE/hr=\left(1-CE\right)/hr=\left(\frac{{Q}_{f}^{n}+\mathrm{\Delta }{q}_{c}^{n}}{{Q}_{c}^{n}}\right)/hr=\frac{{Q}_{f}^{n}/hr+\mathrm{\Delta }{q}_{c}^{n}/hr}{{Q}_{c}^{n}}$

This expression shows that the CIE/hr can easily be obtained from the capacity fade and charge end-point capacity slippage simply by normalizing each of them by the time per cycle. This allows comparison between cells that were cycled at different rates and/or to different states of charge causing the cycle time to be different.

## Part 2

The following graph shows fictional data to demonstrate the importance of breaking up a CE measurement. Consider a hypothetical situation where two cells have identical CE, as shown in the top panel. If no further analysis was done, one would conclude that these cells have identical electrochemical performance.

However, by differentiating capacity fade and charge end-point capacity slippage, it is revealed that these two cells are achieving the same CE in different ways; the cell in black has more capacity fade (center panel), the one in red has more charge end-point capacity slippage (bottom panel).

This begs the question: is one of these two cells better than the other even though they have identical CE*? The answer to that question is generally not straightforward, however, examining other cycle metrics can give clues. For example, if the cell with larger charge end-point capacity slippage (red) also had a comparably increased impedance, it is possible that salt from the electrolyte was being consumed during electrolyte oxidation, which would likely lead to worse performance over time compared to the cell with larger capacity fade (black).

Breaking down the CE this way and cross-referencing with other cycle metrics (e.g. impedance growth – to be the subject of a future post), provides additional information that can be used to make informed decisions not only about which cell is better compared to the other, but also about how cells degrade and how chemistries can be improved.

*Even though this is a fictional scenario – cells would typically have at least very small differences in CE – this example nonetheless represents a real challenge that arises when testing cells carefully, especially when comparing cells that have very similar electrochemical performance.

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