# Differential voltage analysis for battery manufacturing process control

Andrew Weng
Department of Mechanical Engineering
University of Michigan
Ann Arbor, MI 48109
<EMAIL_ADDRESS>
& Jason B. Siegel
Department of Mechanical Engineering
University of Michigan
Ann Arbor, MI 48109
<EMAIL_ADDRESS>
&Anna Stefanopoulou
Department ofMechanical Engineering
University of Michigan
Ann Arbor, MI 48109
<EMAIL_ADDRESS>
###### Abstract

Voltage-based battery metrics are ubiquitous and essential in battery
manufacturing diagnostics. They enable electrochemical “fingerprinting” of
batteries at the end of the manufacturing line and are naturally scalable,
since voltage data is already collected as part of the formation process which
is the last step in battery manufacturing. Yet, despite their prevalence,
interpretations of voltage-based metrics are often ambiguous and require
expert judgment. In this work, we present a method for collecting and
analyzing full cell near-equilibrium voltage curves for end-of-line
manufacturing process control. The method builds on existing literature on
differential voltage analysis (DVA or dV/dQ) by expanding the method formalism
through the lens of reproducibility, interpretability, and automation. Our
model revisions introduce several new derived metrics relevant to
manufacturing process control, including lithium consumed during formation and
the practical negative-to-positive ratio, which complement standard metrics
such as positive and negative electrode capacities. To facilitate method
reproducibility, we reformulate the model to account for the “inaccessible
lithium problem” which quantifies the numerical differences between modeled
versus true values for electrode capacities and stoichiometries. We finally
outline key data collection considerations, including C-rate and charging
direction for both full cell and half cell datasets, which may impact method
reproducibility. This work highlights the opportunities for leveraging
voltage-based electrochemical metrics for online battery manufacturing process
control.

_Keywords_ battery manufacturing $\cdot$ formation $\cdot$ diagnostic features
$\cdot$ differential voltage analysis $\cdot$ dV/dQ $\cdot$ process control
$\cdot$ reproducibility

Graphical Abstract

##  1 Introduction

Decreasing the cost and environmental footprint of battery “gigafactories”
worldwide relies on continuously improving the manufacturing process through
data-driven process control. To place the scale of manufacturing data in
context, consider a battery factory that supplies enough cells to produce one
electric vehicle (EV) per minute (Kane, 2022). Since the number of cells
needed for each EV is on the order of hundreds for pouch cells and thousands
for cylindrical cells, the cell production rate would need to exceed $\sim$100
cells per minute for pouch cells and $\sim$1,000 cells per minute for
cylindrical cells. Every cell produced will generate voltage and current time-
series data as part of the formation process, which is the last step of
battery manufacturing when cells are charged for the very first time to create
the solid electrolyte interphase (SEI) (Peled and Menkin, 2017; An et al.,
2017; Wood et al., 2019).

Voltage and current data from the battery formation process can be
continuously and automatically collected, stored, and analyzed to develop
smart manufacturing process specifications or tolerances, ensuring that all
cells leaving the factory have a guarantee on performance, lifetime, and
safety (Liu et al., 2021a). Using voltage-based measurements is appealing from
a manufacturing standpoint since they can be measured using existing formation
cyclers. This data may be directly collected as part of the formation protocol
or immediately after the formation protocol completes but before the cells are
taken off of the formation cyclers (Weng et al., 2021). Thus, collecting
voltage-based data bears no additional capital costs to a factory. Since the
number of formation cyclers grows with the production volume, the voltage-
based metrics will also naturally scale as factory throughput increases.

When full cell voltage data is carefully collected and analyzed, they can be
used to derive physically-interpretable, electrochemical metrics, or features,
that provide insights into the thermodynamic and kinetic properties of the
cell. A slow-rate (C/20) full cell voltage curve can, for example, be analyzed
using incremental capacity analysis (ICA) (Dubarry and Anseán, 2022) or
differential voltage (dV/dQ) analysis (Dahn et al., 2012a; Lee et al., 2020a)
to understand thermodynamic cell properties such as active material losses and
lithium inventory losses. The data can be collected immediately after the
formation cycles complete. Unlike cell dissections and electrode harvesting,
voltage-based analysis is non-destructive. The same cell that is analyzed in
manufacturing can thus be tracked throughout its remaining life, either
through accelerated cycle life testing in a lab or through fleet telemetry of
real-world usage inside of an EV. The ability to establish an electrochemical
“fingerprint” on the pristine cells immediately after manufacturing can help
improve lifetime prediction models (Weng et al., 2021) which, in turn, be can
be used to improve the manufacturing process.

However, caution is needed when setting manufacturing specifications based on
voltage-based data. Without careful data collection and interpretation,
manufacturing tolerances may be set too tightly, increasing reject rates and
lowering production throughput, or too loosely, which may increase production
throughput in the short term but lead to lifetime and safety issues after
years of usage in the field. Ultimately, manufacturing specifications must
also be set based on an understanding of long-term consequences to
performance, lifetime, or safety. The voltage-based electrochemical features
can facilitate understanding trade-offs between production throughput and
long-term reliability, but only if they are carefully designed and
interpreted.

We note that voltage-based electrochemical features do not replace the need
for more advanced end-of-line metrology methods in the factory, including
X-ray (Kong et al., 2022; Qian et al., 2021) and ultrasonic imaging (Bommier
et al., 2020; Deng et al., 2020), which may be necessary for catching non-
electrochemical related cell defects. Rather, the voltage-based features
complement root-cause analyses by providing basic electrochemical metrics
about the system such as the as-manufactured electrode capacities, cyclable
lithium inventory, and negative-to-positive ratio (NPR).

Currently, factories are not taking full advantage of voltage-based
measurements at the end of line which may be relevant to long term cell
lifetime and durability, partly owing to difficulties in data interpretation.
Resolving the complex electrochemical details using full-cell voltage data
alone is inherently challenging, since these measurements reflect a
combination of thermodynamic and kinetic factors originating from nearly every
cell component and their interactions with each other. These inherent
difficulties are compounded by a general lack of clear literature guidance on
best practices for collecting and analyzing voltage data. As a result,
commonly reported electrochemical metrics derived from voltage data, such as
modeled electrode capacities from dV/dQ analysis often differ in both data
collection and analysis methods, limiting their interpretability and
reproducibility. To overcome these challenges, recent work, such as those by
Dubarry and Anseán (2022) are essential for enabling reproducible extraction
of electrochemical features derived from voltage data.

The goal of this work is thus to guide experimental and analysis
considerations needed to enable reproducible voltage-based battery
manufacturing diagnostics. To accomplish this, we revisit a popular analysis
technique through the lens of battery manufacturing: the differential voltage
analysis, or dV/dQ, model. We will detail the experimental and analysis
considerations to facilitate data reproducibility and interpretability.

##  2 A Manufacturing Case Study

Here, we showcase an example of how the differential voltage analysis method
can be used for continuous monitoring of cell electrochemical features at the
end of the manufacturing process. We highlight the various electrochemical
metrics, or features, that can be extracted using the method and their
significance for manufacturing process control.

The study consists of two cell batches made on the same manufacturing line and
made using the same active materials
($\mathrm{Ni}_{\mathrm{0.33}}\mathrm{Mn}_{\mathrm{0.33}}\mathrm{Co}_{\mathrm{0.33}}$
positive electrode and graphite negative electrode). However, the two cell
batches were made two years apart and differed in cell design parameters,
including electrode loading targets and number of layers. Cells from the first
batch, from Mohtat et al. (2021), had nominal capacities of 5.0 Ah and were
built in 2018 ($n=21$). Cells from the second batch, from Weng et al. (2021),
had nominal capacities of 2.37 Ah and were built in 2020 ($n=40$). For these
cells, we restricted the cell population to those having full cell data
collected at room temperature (RT) ($n=20$) to match the measurement
conditions from Mohtat et al. (2021). A table comparing relevant design
parameters is given in Table S1. We will refer to these two datasets as
Mohtat2021 and Weng2021, respectively.

Figure 1 shows the general analysis workflow and lists the electrochemical
features extracted by the model. In this example, full cell voltage data
($V_{\mathrm{full}}$) was collected on every manufactured cell after the
formation cycling completed (Section 4.1.1). Each dataset was analyzed using
the differential voltage analysis model (Section 3.2). The model consisted of
two pre-computed functions, $U_{\mathrm{pos}}$ and $U_{\mathrm{neg}}$,
representing the positive and negative electrode near-equilibrium reference
potential curves, respectively (Section 4.1.2). The same functions were used
to analyze data from both Mohtat2021 and Weng2021, since the batches shared
the same positive and negative electrode chemistries.

Figure 1: Summary of how differential voltage analysis can enable coordinated
battery manufacturing process control via end-of-line testing. The provided
table describes the voltage-based electrochemical features derived from the
method, including applications in manufacturing and test design
considerations.

The differential voltage model was then used to extract electrochemical
features which include positive and negative electrode capacities
($\tilde{Q}_{p}$, $\tilde{Q}_{n}$), the capacity of lithium available for
cycling ($\tilde{Q}_{\mathrm{Li}}$), the capacity of lithium lost to the SEI
($\tilde{Q}_{\mathrm{SEI}}$), electrode lithium stoichiometries when the full
cell is discharged ($\tilde{x}_{0}$, $\tilde{y}_{0}$), and cell design
information including the practical negative-to-positive capacity ratio
($\mathrm{NPR}_{\mathrm{practical}}$) and the excess negative electrode
capacity ($\tilde{Q}_{n,\mathrm{excess}}$). The ‘tildes’ on each variable name
indicate that they are estimates which may be different from the true
parameters due to the “inaccessible lithium problem,” which will be further
explained in Section 3.3. These extracted features can then be analyzed to
determine whether machine parameter adjustments are needed in upstream
manufacturing processes such as those in electrode coating or in electrolyte
fill (E-fill).

Figure 2 show an example data visualization of the analysis outputs. An
example model fitting result is also provided in Figure S1. Panel A uses box-
and-whisker plots to compare compares several example metrics derived from the
differential voltage analysis, including $\tilde{Q}_{p},\tilde{Q}_{n}$,
$\tilde{Q}_{\mathrm{SEI}}$, and $\mathrm{NPR}_{\mathrm{practical}}$. The full-
cell C/20 discharge capacity, $Q_{\mathrm{full}}$, is also provided for
reference. Since the Mohtat2021 and Weng2021 cells used different numbers of
electrode sheets, all of the capacity values reported here have been
normalized to the respective electrode areas, enabling a head-to-head
comparison.

The following sections discusses how to interpret each group of features in
the context of manufacturing process control.

Figure 2: Leveraging differential voltage analysis to non-destructively
identify electrochemical differences between two cell batches at the end of
the manufacturing line. $Q_{\mathrm{full}}$: full cell areal capacity,
$\tilde{Q}_{p}$: positive electrode areal capacity, $\tilde{Q}_{n}$: negative
electrode areal capacity, $\tilde{Q}_{\mathrm{SEI}}$ areal capacity of lithium
lost to SEI during formation, $\mathrm{NPR}_{\mathrm{practical}}$: Practical
negative to positive ratio. Data spans two separate datasets from Weng et al.
(2021) ($n=20$) and Mohtat et al. (2021) ($n=21$). Cells from both datasets
share the same chemistry
($\mathrm{Ni}_{\mathrm{0.33}}\mathrm{Mn}_{\mathrm{0.33}}\mathrm{Co}_{\mathrm{0.33}}|$graphite),
but were made in two separate batches and vary in cell design parameters. (A)
Box-and-whisker plots comparing data from the two batches, where each marker
shows modeled outputs on each individual cell. (B) correlation plots of the
metrics, where diagonal elements show histograms comparing individual metrics
across the two different batches.

###  2.1 Monitoring Electrode Capacities and Loadings

Figure 2A shows extracted values of positive and negative electrode
capacities, $\tilde{Q}_{n}$ and $\tilde{Q}_{p}$, for every cell in this study.
In a manufacturing context, the availability of data on $\tilde{Q}_{n}$ and
$\tilde{Q}_{p}$ on every batch of cells in production, without needing cell
dissections, can help to identify deviations from electrode loading set-
points, e.g. due to electrode processing variations. These electrode-level
variations would be difficult to detect by relying on full cell capacity check
data alone, since capacity variability at the electrode level may not manifest
in full cell capacity checks (Section 2.4).

We also found that the Mohtat2021 cells had, on average, 16% higher positive
electrode areal capacity and 17% higher negative electrode areal capacity
compared to the Weng2021 cells. However, according to the cell design
parameters (Table S1), Mohtat2021 cells should have only 7% higher positive
electrode areal capacity and 9% higher negative electrode areal capacity based
on differences in the active material loading targets. The gap between the the
targeted and modeled capacities suggest that the as-built cells did not match
their target loadings, and that electrode coating process parameters may need
to be adjusted.

###  2.2 Monitoring Lithium Consumed During Formation

Figure 2A also compares the amount of lithium consumed during formation,
$\tilde{Q}_{\mathrm{SEI}}$, for all cells in this study. For the Mohtat2021
cells, the mean value was measured to be 0.35 mAh/cm2, while the Weng2021
cells measured 0.12 mAh/cm2. Mohtat2021 cells thus lost nearly three times
more lithium during formation than compared to the Weng2021 cells. This result
can be verified graphically by inspecting the voltage curve alignments from
the fitting results which shows that the positive electrode is less lithiated
at 0% SOC, reflecting the fact that extra lithium was irreversibly consumed to
form the SEI and cannot return to the positive electrode on discharge (Figure
S1). The increase in lithium consumption rate during the formation of the
Mohtat2021 cells suggests a poorer quality of electrolyte or a less
passivating negative electrode formulation. Note that both cell builds shared
the same formation protocol, so differences in lithium consumption cannot be
attributed to differences in the formation protocol. This example highlights a
scenario in which the electrolyte filling process parameters may need to be
inspected, or the quality of the electrolyte itself may need to be verified.

The increased lithium consumption in the Mohtat2021 cells also helps to
explain why, although the Mohtat2021 cells had 16% higher positive electrode
areal mass loading than the Weng2021 cells, the full cell capacities of the
Mohtat2021 cells were only 3% higher on average. Thus, although the Mohtat2021
cells were designed with higher areal capacities than the Weng2021 cells, most
of the benefits to full cell capacity were lost due the increased lithium
consumption rate during formation.

###  2.3 Measuring “Margin to Lithium Plating” Using the Practical NP Ratio

Since the Mohtat2021 cells had consumed more lithium during formation compared
to the Weng2021 cells, the Mohtat2021 negative electrodes will also be less
lithiated when the full cell is charged to 100% SOC. The Mohtat2021 cells will
therefore have a wider margin of negative electrode capacity before the
negative electrode is fully lithiated and lithium plating begins to occur, a
result that can be visually seen in Figure S1. Thus, Mohtat2021 cells should,
in theory, be less prone to lithium plating compared to the Weng2021 cells.

A metric that quantifies the capacity margin before lithium plating occurs is
the negative to positive ratio (NPR). The differential voltage analysis
provides such a metric, known as the practical NPR, or
$\mathrm{NPR}_{\mathrm{practical}}$, which we develop in Section 3.4.2. The
$\mathrm{NPR}_{\mathrm{practical}}$ for Mohtat2021 cells was measured to be
1.24, compared to 1.14 for the Weng2021 cells, confirming that the Mohtat2021
cells have a higher negative electrode capacity margin to protect against
lithium plating. In order for the Weng2021 cells to achieve the same
$\mathrm{NPR}_{\mathrm{practical}}$ as the Mohtat2021 cells, either the
positive electrode loading target would need to decrease, or the negative
electrode loading target would need to increase.

###  2.4 Electrode Manufacturing Variability

Turning to Figure 2B, we next show how the differential voltage analysis
outputs can be used to identify differences in electrode manufacturing
variability from batch to batch. We immediately see that Mohtat2021 cells
showed higher variability in the full cell capacity. The histograms of
$\tilde{Q}_{n}$ and $\tilde{Q}_{p}$ reveal that the origin of the higher full
cell capacity variability stems from the positive electrode loading, not the
negative electrode loading. In fact, the negative electrode loading for
Weng2021 cells had higher variability compared to the Mohtat2021 cells, yet
this increased variability did not manifest in the full cell capacity. This
result is expected considering the fact that the practical NP ratio of all
cells are greater than 1, and hence, variations in the negative electrode
capacities will mainly manifest in the excess negative electrode capacity
rather than the full cell capacity. Thus, to decrease the full cell capacity
variability in the Mohtat2021 cells, the manufacturer should focus on
tightening the variability in the cathode manufacturing process, not the anode
manufacturing process.

###  2.5 Parameter Correlations

Figure 2B finally presents correlation plots between every individual output
variable of the differential voltage analysis model. One application of
studying correlations is to understand which electrode-level parameters
correlate to full cell capacity and hence directly impacts measurable cell
performance. The correlation matrix reveals that, within each batch of cells,
the positive electrode capacity correlates to full cell capacity. However, the
correlation breaks when comparing across different batches of cells since the
Mohtat2021 consumed more lithium during formation (Section 2.2). This result
highlights the reality that multiple cell manufacturing factors, including
electrode loadings, lithium consumption during formation, and the negative
electrode excess capacity, all play a role in determining the full cell
capacity. Achieving a certain full cell capacity outcome therefore requires
coordinated control across multiple manufacturing processes.

In summary, this section highlighted the utility of applying the differential
voltage analysis method to extract electrochemical metrics at the end of the
cell manufacturing line. These metrics can then be directly used to adjust
process upstream manufacturing process parameters such as those in electrolyte
manufacturing or electrolyte filling. All of the data presented in this
section were directly outputted from the differential voltage method without
needing any electrode-level degradation analysis. Since the method can run on
data collected during battery formation, no additional work is necessary to
extend the analysis to new cells coming off the manufacturing line, making the
method highly scalable.

##  3 Differential Voltage Analysis Method

###  3.1 Model Selection

Near-equilibrium (i.e. slow-rate) full cell voltage curves have been widely
used to understand dominant cell degradation modes such as loss of active
material (LAM) and loss of lithium inventory (LLI) (Bloom et al., 2005;
Dubarry et al., 2012; Birkl et al., 2017; Olson et al., 2023). The appeal of
these techniques is easy to appreciate: full cell data is easier and faster to
collect than materials-level characterization data which often requires cell
tear-down and electrode harvesting. Such types of analyses have been
introduced under different names, including differential voltage analysis (DVA
or dV/dQ) (Bloom et al., 2005; Smith et al., 2011), incremental capacity
analysis (ICA) (Dubarry et al., 2012; Weng et al., 2013; Dubarry and Anseán,
2022), open-circuit voltage models (Birkl et al., 2017; García Elvira et al.,
2021; Schmitt et al., 2022), and voltage-fitting analysis (Lee et al., 2020a).
While all of these methods rely on full cell voltage curve data, their data
analysis approaches are different. In differential voltage (or dV/dQ) methods,
the voltage is differentiated with respect to capacity and plotted against
capacity. The resulting data reveals peaks and troughs that can be attributed
to either the positive or negative electrode. In incremental capacity (or ICA)
methods, the capacity is differentiated with respect to voltage and plotted
against voltage. The features observed from ICA are no longer linearly
separable, but through careful data interpretation, distinct degradation modes
can still be inferred (Dubarry and Anseán, 2022). Other variations of
differential analysis have also been recently introduced which leverage
signals such as expansion (Mohtat et al., 2022) and pressure (Huang et al.,
2022), but since these methods require measurements beyond full voltage, they
are omitted from present consideration.

Since the exact method for extracting the electrode-level parameters vary from
author to author, comparing results across different papers is complicated.
Notably, some authors use full cell data alone, assigning features of interest
and graphically inferring degradation metrics such as lithium inventory and
electrode capacity losses (Dubarry and Beck, 2021), while others leverage
half-cell “reference curves” (Harlow et al., 2019) to build a model of the
full cell voltage as a function of the electrode-level parameters (Birkl et
al., 2017; García Elvira et al., 2021).

In a battery manufacturing context, a quantitative and reproducible voltage-
based analysis method is needed. The method must be quantitative in order to
resolve minute differences in manufacturing process parameters which could
have large, long-term consequences to battery performance, lifetime and
safety. The method must also be reproducible in order to scale.

Out of all of the available voltage curve analysis methods, the “voltage
fitting method”, documented by Lee et al. (2020a), provides the best balance
between mathematical rigor and interpretability. We therefore choose this
method as the basis for extracting electrochemical features for manufacturing
diagnostics. Note that, in this work, we use the words “differential voltage
analysis” and “voltage fitting analysis” interchangeably since they refer to
the same fundamental model construction.

###  3.2 Model Formulation

The differential voltage model seeks to decompose the full cell near-
equilibrium (i.e. open circuit) voltage curve into its constituent half-cell
positive and negative potential curves vs Li/Li+. The half-cell curves, also
sometimes called “reference curves” (Harlow et al., 2019), then provide
information about thermodynamic states of the cell, including positive and
negative electrode capacities, positive and negative electrode lithium
stoichiometries as a function of the full cell state of charge (SOC), and the
capacity of cyclable lithium inventory.

Figure 3: Concept illustration of the differential voltage analysis method and
the inaccessible lithium problem. (A) The full cell near-equilibrium (‘open
circuit’) voltage curve $V_{\mathrm{full}}$ (black) plotted alongside the
corresponding positive electrode (blue) and negative electrode (red) near-
equilibrium potential curves $U_{\mathrm{pos}}$ and $U_{\mathrm{neg}}$.
Parameters with tildes indicate estimated model parameters due to the
“inaccessible lithium problem” (see Section 3.3). Gray lines and variables
indicate true model parameters that cannot be ascertained using the native
model. (B) The corresponding differential voltage (dV/dQ) curves, illustrating
how full cell dV/dQ peaks can be attributed to features from the positive and
negative electrodes.

We now proceed with a quantitative treatment of the model, using Figure 3 as a
visual guide. The fundamental state equation of the model is an assertion of
voltage conservation between the half-cell near-equilibrium potentials and the
full cell near-equilibrium potential given by

$V_{\mathrm{full}}(q)=U_{\mathrm{pos}}(y)-U_{\mathrm{neg}}(x),$ (1)

where $V_{\mathrm{full}}$ is the full cell near-equilibrium (i.e. open-
circuit) voltage and $U_{\mathrm{pos}}$ and $U_{\mathrm{neg}}$ are the
positive and negative electrode near-equilibrium potential curves versus
Li/Li+, respectively. $q$ is a vector of capacities in the full cell domain
and is related to the cell SOC, $z\in[0,1]$, by the relation

$z=\frac{q}{Q_{\mathrm{full}}}$ (2)

where $Q_{\mathrm{full}}$ is the measured capacity of the full cell.
$x\in[0,1]$ and $y\in[0,1]$ are vectors of lithium stoichiometries in the
negative and positive electrodes, respectively. Note that $x$ and $y$ are
sometimes also reported as $\theta_{n}$ and $\theta_{p}$ by other authors. The
lithium stoichiometries are governed by the half-cell reactions

$\displaystyle
x\mathrm{Li^{+}}+x\mathrm{e^{-}}+\mathrm{C}_{6}\rightarrow\mathrm{Li}_{x}\mathrm{C}_{6}$
$\displaystyle\qquad\qquad\mathrm{(Negative,charging)}$ (3)
$\displaystyle\mathrm{LiM}\rightarrow\mathrm{Li}_{y}\mathrm{M}+(1-y)\mathrm{Li^{+}}+(1-y)\mathrm{e^{-}}$
$\displaystyle\qquad\qquad\mathrm{(Positive,charging)},$ (4)

where the first reaction represents the intercalation of lithium into
graphite, and the second reaction describes the release of lithium from any
intercalation positive electrode such as (NMC), nickel cobalt aluminum (NCA),
iron phosphate, etc.

Equation 1 assumes that both electrodes are at near-equilibrium, meaning that
the current densities used to acquire the curves are sufficiently small so
that over-potential contributions can be neglected. Note that while some
authors attempt to use $V_{\mathrm{full}}$ to extract kinetic information such
as resistance increase (Dubarry et al., 2012), we focus our analysis on
extracting only thermodynamic quantities to limit the number of modeled output
parameters and improve solution uniqueness. To ensure negligible kinetic
effects, $V_{\mathrm{full}}(q)$ must be collected at a sufficiently slow rate
(see Section 4.1.1). The reference half-cell curves $U_{\mathrm{pos}}(y)$ and
$U_{\mathrm{neg}}(x)$ may be obtained experimentally, e.g. via coin cell
testing (see Section 4.1.2).

The terms in Equation 1 can be mapped to a common, full capacity basis $q$.
Charge conservation requires that this mapping be affine (linear plus an
offset). We can therefore define the following basis transformation equations:

$\displaystyle x(q)$ $\displaystyle=x_{0}+\frac{q}{Q_{n}}$ (5) $\displaystyle
y(q)$ $\displaystyle=y_{0}-\frac{q}{Q_{p}},$ (6)

where $Q_{n}$ and $Q_{p}$ are the total negative and positive electrode
capacities, respectively, and $(x_{0},y_{0})$ are the negative and positive
electrode stoichiometries in the fully discharged state. These equations are
derived under the boundary condition that $q=0$ corresponds to the fully
discharged state, with $(x,y)=(x_{0},y_{0})$. Equations (5, 6) can be inverted
to obtain

$\displaystyle q(x)=Q_{n}(x-x_{0})$ (7) $\displaystyle q(y)=Q_{p}(y_{0}-y),$
(8)

which can be used to map lithium stoichiometries to the shared capacity basis.
Note that, under this construction, $q$ is allowed to take on negative values.
For example, when $x=0$, $q=-Q_{n}x_{0}<0$, according to Equation 7. Negative
$q$ values can be interpreted as “virtual capacities,” that is, capacity
present in the electrodes but is inaccessible from the perspective of the full
cell due to the full cell minimum voltage constraint.

We can also define a complementary set of variables describing the lithium
stoichiometries in the fully-charged state, $(x_{100},y_{100})$, which can be
written as

$\displaystyle x_{100}$ $\displaystyle=x_{0}+\frac{Q_{\mathrm{full}}}{Q_{n}}$
(9) $\displaystyle y_{100}$
$\displaystyle=y_{0}-\frac{Q_{\mathrm{full}}}{Q_{p}}.$ (10)

$Q_{\mathrm{full}}$ is a known quantity from the full cell data and is not
considered to be a modeled parameter. Hence, $(x_{100},y_{100})$ are not
independent model parameters.

In summary, knowledge of four parameters

$\theta=\\{Q_{n},Q_{p},x_{0},y_{0}\\}$ (11)

provides a complete description of the model from Equation (1). These
parameters bear physical meaning, since ($Q_{n},Q_{p}$) are related to the
total electrode capacities and $(x_{0},y_{0})$ predict lithium stoichiometry-
dependent electrode potentials. These model parameters can then be used to
derive additional electrochemical features relevant to battery manufacturing
diagnostics, as will be described in Section 3.4.

Note that the choice of model parameters are not unique. For example
$\\{Q_{n},Q_{p},x_{100},y_{100}\\}$ is an equally valid set of parameters. For
the remainder of this work, we will continue to use the
$\\{Q_{n},Q_{p},x_{0},y_{0}\\}$ basis with the recognition that the model can
always be reformulated using another set of four basis states using Equations
(9,10).

The goal of the model is then to identify a unique combination of model
parameters $\theta$ that provides the best fit against the measured voltage
data. The optimization problem can be solved by implementing an error function

$\mathrm{E(q:\theta)}=\underbrace{U_{\mathrm{pos}}\left(y_{0}-\frac{q(y)}{Q_{p}}\right)-U_{\mathrm{neg}}\left(x_{0}+\frac{q(x)}{Q_{n}}\right)}_{\mathrm{V_{\mathrm{full}}(q)}}-V_{\mathrm{meas}}(q),$
(12)

where $V_{\mathrm{meas}}(q)$ is the experimentally-measured full cell voltage
curve and the domain of $q$ is now restricted to the full-cell domain spanning
$(0,Q_{\mathrm{full}})$. The model returns an optimal parameter-set which
minimizes an error function of the form

$\theta_{\mathrm{opt}}=\operatorname*{argmin}_{\theta}\sum_{q}\left(\lambda|\mathrm{E}(q:\theta)|^{2}+(1-\lambda)\left|\frac{d\mathrm{E}(q:\theta)}{dq}\right|^{2}\right).$
(13)

The first term inside the summation is the voltage error and the second term
represents the differential voltage error. $\lambda\in[0,1]$ is a weight
parameter, i.e. when $\lambda=1$, the contribution from the differential
voltage signal is ignored. The optimization problem can be solved using
standard non-linear least-squares solvers which are available in most
scientific programming languages such as Python or Matlab. Note that, in
Figure 3, plotted full cell voltage data corresponds to the modeled data, and
the measured voltage data is omitted for simplicity. A real-world comparison
of measured versus modeled full cell voltages is available in Figure S1.

Variations the optimization method exist in the literature. For example, Lee
et al. (2020a) set $\lambda=1$ and implemented additional constraints based on
full cell voltage limits, i.e.
$V_{\mathrm{max}}=U_{\mathrm{pos}}(y_{100})-U_{\mathrm{neg}}(x_{100})$ and
$V_{\mathrm{min}}=U_{\mathrm{pos}}(y_{0})-U_{\mathrm{neg}}(x_{0})$. With this
approach, the model-predicted full cell capacity is guaranteed to match the
measured value. However, solution convergence becomes more challenging in the
presence of these additional constraints which could make the solution space
ill-conditioned. Dahn et al. (2012b) included the differential voltage (dV/dQ)
curves in the error function to assign preferential weight the phase
transition features (Figure 3B) and to negate the effects of impedance
mismatch between the full cell and coin cell form factors. However, these
dV/dQ methods reported previously are generally not fully automated, often
requiring manually adjusting model parameters through graphical user
interfaces (Dahn et al., 2012b) and evaluating goodness-of-fit by visually
inspecting peak alignments. Overall, a “one-size-fits-all“ optimization scheme
is not known to exist yet. The specific details of the optimization may
therefore need to be tuned to perform optimally for a given system.

###  3.3 Model Extension: Accounting For the Inaccessible Lithium Problem

So far, we have laid the groundwork for the differential voltage analysis
method. In this section, we outline a fundamental challenge with interpreting
and reproducing model outputs known as the “inaccessible lithium problem.”
This issue was alluded to by Truchot et al. (2014) and recently expanded upon
by Lu et al. (2021). In this section, we first describe how this issue affects
interpretation of the model outputs in a manufacturing context. We then
propose corrections to the differential voltage analysis model to clarify the
interpretation of the model outputs.

The “inaccessible lithium problem” is summarized as follows. The model state
equation assumes that $U_{\mathrm{pos}}$ and $U_{\mathrm{neg}}$ are defined
for the entire range of lithium stoichiometries spanning $y\in[0,1]$ and
$x\in[0,1]$, respectively. Yet, half-cell potential curves are generally
experimentally unattainable over the entire range of stoichiometries. For
example, for layered oxide positive electrode systems such as NMC, full
delithiation would render the host material thermodynamically unstable,
impacting the reversibility of lithium intercalation reaction. Full lithiation
of the layered oxide particles is also experimentally challenging due to
massive kinetic limitations towards the fully lithiated state
(Phattharasupakun et al., 2021). These kinetic limitations explain why it is
so difficult to “fully discharge” a positive electrode half cell; even when
discharging at small C-rates, the potential tends to rebound above 3V vs
Li/Li+ after resting.

Near-equilibrium potential data on layered-oxide positive electrodes can only
be feasibly obtained within a restricted stoichiometry window, which is
determined by the voltage limits of coin cell data collection. Defining
$y_{\mathrm{min}}$ and $y_{\mathrm{max}}$ as the minimum and maximum
observable lithium stoichiometry in the positive electrode, it must be the
case that $y_{\mathrm{min}}>0$ and $y_{\mathrm{max}}<1$. An analogous
situation applies to the graphite negative electrode, with
$x_{\mathrm{min}}>0$ and $x_{\mathrm{max}}<1$, though the resulting errors
from assuming $x_{\mathrm{min}}\approx 0$ and $x_{\mathrm{max}}\approx 1$ are
generally understood to be more benign, since lithium transport in graphite is
generally more facile.

To proceed with the analysis, practitioners must either implicitly or
explicitly assume that the tested range of potentials (e.g. 3.0V to 4.3V for
layered oxide positive electrodes) correspond to the full range of lithium
stoichiometries, i.e.

$(x_{\mathrm{min}},x_{\mathrm{max}},y_{\mathrm{min}},y_{\mathrm{max}})\approx(0,1,0,1),$
(14)

and proceed with the analysis method as described in Section 3.2. However,
this simplification will lead to errors in the model outputs. To demonstrate
the magnitude of this error, we compare the positive and negative electrode
capacities calculated using differential voltage analysis from Section 2 to
the true electrode capacities, which can be calculated by combining the
theoretical capacities with the cell design parameters, according to:

$Q=m\cdot f\cdot n_{\mathrm{faces}}\cdot A\cdot Q_{\mathrm{theor}},$ (15)

where $Q$ is the electrode capacity in Ah, $m$ is the electrode areal loading
target in g/cm2, $f$ is the mass fraction of active material in the electrode,
$n_{\mathrm{faces}}$ is the number of active electrode faces, $A$ is the
electrode area per face, excluding overhanging regions, in cm2, and
$Q_{\mathrm{theor}}$ is the theoretical capacity of the material in Ah/g.

The resulting calculations, summarized in Table 1, show that the estimated
positive electrode capacity is only 54% to 55% of the true positive electrode
capacity, while the estimated negative electrode capacity is 87% to 89% of the
true negative electrode capacity. This result was demonstrated on both the
Mohtat et al. (2021) and the Weng et al. (2021) datasets, suggesting some
level of generality. The gross underestimation of positive electrode capacity
can be primarily attributed to the fact that the layered oxide
$\mathrm{Ni}_{\mathrm{0.33}}\mathrm{Mn}_{\mathrm{0.33}}\mathrm{Co}_{\mathrm{0.33}}$
positive electrode material used in these works retain significant lithium
inventory even above the coin cell upper cut-off potential of 4.3V vs Li/Li+,
and consequently, $y_{\mathrm{min}}\gg 0$.

| Positive Electrode | Negative Electrode
---|---|---
Source | Mohtat2021 | Weng2021 | Mohtat2021 | Weng2021
Number of active faces | 28 | 14 | 28 | 14
Area per face (cm2) | 79.20 | 79.20 | 79.56 | 79.56
Loading (mg cm-2) | 18.50 | 17.23 | 8.55 | 7.85
Theoretical capacity (mAh g-1) | 279.5 | 279.5 | 372 | 372
Active material fraction | 0.94 | 0.94 | 0.95 | 0.97
Total capacity (Ah) | 10.78 | 5.02 | 6.73 | 3.16
(A) Capacity, theoretical (mAh cm-2) ($Q$) | 4.86 | 4.53 | 3.02 | 2.83
(B) Capacity, from fitting (mAh cm-2) ($\tilde{Q}$) | 2.66 | 2.46 | 2.70 | 2.46
Percent of theoretical capacity observed ($\tilde{Q}/Q$) | 55% | 54% | 89% | 87%
Table 1: Comparison of (A) electrode theoretical areal capacities versus (B)
practical areal capacities derived from the differential voltage analysis. The
capacities predicted by the differential voltage analysis are lower than the
theoretical values due to the ‘inaccessible lithium problem’ (Lu et al.,
2021).

We thus conclude that ignoring the inaccessible lithium problem (Equation 14)
will likely lead to inaccurate reporting of true electrode capacities and
stoichiometries, especially for layered oxide positive electrode materials
such as NMC. This issue could affect the ability to directly use modeled
electrode capacities to convert into electrode loadings using Equation 15.
Unfortunately, the inaccessible lithium problem is a fundamental limitation of
the coin cell data collection process used to initialize $V_{\mathrm{pos}}$
and $V_{\mathrm{neg}}$ in the model.

Knowing that the inaccessible lithium problem can lead to large errors in the
model outputs ($x,y,Q_{n},Q_{p}$), we reformulate the model by defining
‘tilde’ variables ($\tilde{x}$, $\tilde{y}$, $\tilde{Q}_{n}$,
$\tilde{Q}_{p}$), to represent estimated parameters. The state equation can
then be re-written as:

$V_{\mathrm{full}}(q)=U_{\mathrm{pos}}(\tilde{y})-U_{\mathrm{neg}}(\tilde{x})$
(16)

where

$\displaystyle\tilde{x}(q)$
$\displaystyle=\tilde{x}_{0}+\frac{q}{\tilde{Q}_{n}}$ (17)
$\displaystyle\tilde{y}(q)$
$\displaystyle=\tilde{y}_{0}-\frac{q}{\tilde{Q}_{p}}.$ (18)

This modified system is identical to the original system, but recognizes the
fact that the modeled lithium stoichiometries and electrode capacities may, in
general, be different from the true values which are not observable due to the
limitations in coin cell data collection for $U_{\mathrm{pos}}$ and
$U_{\mathrm{neg}}$.

The estimated parameters can then be related to the true parameters by the
following relations:

$\displaystyle\tilde{Q}_{p}$
$\displaystyle=Q_{p}(y_{\mathrm{max}}-y_{\mathrm{min}})$ (19)
$\displaystyle\tilde{Q}_{n}$
$\displaystyle=Q_{n}(x_{\mathrm{max}}-x_{\mathrm{min}})$ (20)
$\displaystyle\tilde{y}_{0}-\tilde{y}_{100}$
$\displaystyle=\frac{y_{0}-y_{100}}{y_{\mathrm{max}}-y_{\mathrm{min}}}$ (21)
$\displaystyle\tilde{x}_{0}-\tilde{x}_{100}$
$\displaystyle=\frac{x_{0}-x_{100}}{x_{\mathrm{max}}-x_{\mathrm{min}}}.$ (22)

This system of equations has more unknowns than equations, highlighting the
difficulty in recovering the true parameters from the model alone. Ideally,
the true lithium stoichiometries may be experimentally measured, e.g. via
inductively coupled plasma optical emission spectroscopy (ICP-OES)
(Kasnatscheew et al., 2016). True capacities may also be calculated using
information about the electrode design (Equation 15). Without more
measurements, further assumptions are needed to fully resolve the estimated
model parameters, e.g. $\tilde{x}_{100}\approx x_{100}$ and
$\tilde{y}_{0}\approx y_{0}$.

Despite the data interpretation challenges introduced by the inaccessible
lithium problem, we highlight several applications of the model outputs which
remain unaffected by this problem. First, calculations of differences in
electrode capacities are not affected. This is because the error terms in
Equations 19 and 20 cancel when taking ratios of capacities. This fact
explains why literature reports of losses of active material (LAM) and our
analysis of electrode capacity loading differences from Section 2.1 are still
valid even though the inaccessible lithium problem may render absolute values
of $Q_{n}$ and $Q_{p}$ inaccurate. Second, the inaccessible lithium problem
also does not impact the process of searching for an optimal set of model
parameters (Equation 12) since the domain for optimization is strictly only
within the full cell capacity window, i.e. $q\in(0,Q_{\mathrm{full}})$.
Missing data beyond the limits of the observable full cell voltage data is
therefore inconsequential to the optimization.

Overall, the inaccessible lithium problem effectively introduces an optional
post-processing step in which the model output parameters may be converted
into true parameters using Equations 19, 20, 21 and 22. This final step would
require additional input not provided for by the differential voltage model.
Yet, even without knowing the true model parameters, the estimated model
outputs remain useful for estimating differences in parameters such as
electrode capacities. The main contribution of the model extension, including
the introduction of the tilde variables, is thus to clarify that the model
outputs must be interpreted as estimates of the true values due to the
limitations in constructing $U_{\mathrm{pos}}$ and $U_{\mathrm{neg}}$ from
data.

###  3.4 Extended Model Outputs for Manufacturing Diagnostics

This section defines and discusses several additional electrochemical features
relevant for battery manufacturing process control: the lithium consumed
during formation ($\tilde{Q}_{\mathrm{SEI}}$) the practical negative-to-
positive ratio ($\mathrm{NPR}_{\mathrm{practical}}$) and the total cyclable
lithium inventory ($Q_{\mathrm{Li}}$). These features can be calculated
directly from the optimized parameters of the modified model
$\tilde{\theta}_{\mathrm{opt}}=\\{\tilde{Q}_{n},\tilde{Q}_{p},\tilde{x}_{0},\tilde{y}_{0}\\}$.
In light of the previous discussion on the “inaccessible lithium problem”, we
make a distinction between estimated model parameters, denoted with tildes,
and true model parameters. To aid understanding, expressions in this section
will be given for both true values and estimated values, where appropriate.

####  3.4.1 Lithium Consumed During Formation

The capacity of lithium consumed during formation, $\tilde{Q}_{\mathrm{SEI}}$,
is analogous to the first cycle Coulombic efficiency metric (Mao et al.,
2018), defined as ratio of discharge and charge capacity during formation.
However, unlike Coulombic efficiency, which requires slow-rate charge-
discharge cycles to calculate, $\tilde{Q}_{\mathrm{SEI}}$ provides a
consistent measure of lithium consumption even when the formation protocol
consists of complex charge-discharge cycles involving multiple C-rates and
partial SOC windows, as is the case with modern formation protocols (An et
al., 2017; Wood et al., 2019). Conveniently, since $\tilde{Q}_{\mathrm{SEI}}$
can be obtained through the analysis of voltage data after formation
completes, this metric enables the recovery of information about lithium
consumed during formation even if formation data is missing.

We define the estimated capacity of lost lithium inventory due to formation,
$\tilde{Q}_{\mathrm{SEI}}$, as

$\displaystyle\tilde{Q}_{\mathrm{SEI}}$
$\displaystyle=q(x_{\mathrm{min}})-q(y_{\mathrm{max}})$ (23)
$\displaystyle=q(\tilde{x}=1)-q(\tilde{y}=0)$ (24)

This formula can be understood by considering that, before formation, the
positive electrode stoichiometry is at its highest value, $y_{\mathrm{max}}$,
since all of the cyclable lithium has not yet left the positive electrode. For
the same reason, the negative electrode stoichiometry is at its lowest value,
$x_{\mathrm{min}}$. After formation completes, the lithium lost to the SEI
will not return to the positive electrode during discharge, causing the
highest possible stoichiometry value in the positive electrode to decrease by
$q(x_{\mathrm{min}})-q(y_{\mathrm{max}})$. Equation 23 can alternatively be
interpreted as the capacity of unoccupied lithium sites in the positive
electrode when the negative electrode is fully delithiated.

We note that the calculation assumes that the positive electrode voltage curve
and capacity is unchanged during formation. Previous studies have shown that
this assumption may not be true (Kang et al., 2008), which would introduce
another error contribution to this calculation which warrants further studies.
From a manufacturing standpoint, if the positive electrode system is known to
be the same from batch-to-batch, then $\tilde{Q}_{\mathrm{SEI}}$ may still be
used to detect differences in the SEI formation process.

####  3.4.2 Re-Interpreting the Negative to Positive Ratio

The negative to positive ratio (NPR) is a common term used in battery design
and generally refers to the ratio of negative to positive electrode
capacities. The optimization of NPR is sometimes referred to as ‘capacity
balancing’ (Reuter et al., 2019), not to be confused with the process of
balancing series-connected cells. Conventional wisdom suggests that the NPR be
must greater than one to prevent over-lithiation of the negative electrode
during charging. Increasing NPR also implies less utilization of the negative
electrode, which could be advantageous in silicon-containing systems where
volume expansion is a major contributor to cycle life degradation (Luo et al.,
2022). However, increasing the NPR requires putting in more negative electrode
active material into the cell without increasing usable energy content, which
decreases the cell energy density. Adding more negative electrode loading
could also increase the surface area for SEI reactions which could be
detrimental to calendar life. The NPR is therefore a critical cell design
metric that should be optimized and tracked during manufacturing.

In this section, we first review the most commonly-held definition of the NPR
and identify some conceptual gaps preventing a clear interpretation of this
metric. We then propose a revised, more practical definition of the NPR which
enables a more physically-grounded assessment of the lithium plating risk.

Issues With Conventional Definitions of NPR. To begin, consider a common
definition of the NPR based on areal loadings and theoretical electrode
capacities:

$\mathrm{NPR}_{\mathrm{theor}}=\frac{m_{n}Q_{n,\mathrm{theor}}}{m_{p}Q_{p,\mathrm{theor}}}.$
(25)

In this equation, $m_{n}$ and $m_{p}$ are the electrode active material mass
loadings in grams, and $Q_{n,\mathrm{theor}}$ and $Q_{p,\mathrm{theor}}$ are
the theoretical capacities in units of mAh/g. Applying this formula to the
example
$\mathrm{Ni}_{\mathrm{0.33}}\mathrm{Mn}_{\mathrm{0.33}}\mathrm{Co}_{\mathrm{0.33}}|$graphite
systems from Mohtat et al. (2021) and Weng et al. (2021) yields calculated NPR
values of 0.62 for Mohtat2021 and 0.63 for Weng2021 (Table 1). This result
would suggest that these cells designs have undersized negative electrode
capacities and are at risk for lithium plating. However, this assessment
assumes that all of the theoretical capacity of the
$\mathrm{Ni}_{\mathrm{0.33}}\mathrm{Mn}_{\mathrm{0.33}}\mathrm{Co}_{\mathrm{0.33}}$
positive electrode (279.5mAh/g) can be utilized, which is not true due to the
inaccessible lithium problem (Section 3.3). This definition of NPR, while
theoretically valid, is therefore not practically useful in a manufacturing
context.

Repeating the calculation but using the estimated values $\tilde{Q}_{n}$ and
$\tilde{Q}_{p}$ to define

$\mathrm{NPR}_{\mathrm{conventional}}=\frac{\tilde{Q}_{n}}{\tilde{Q}_{p}}$
(26)

leads to yet another definition of NPR found in literature. This definition
partly excludes the inaccessible lithium in the positive electrode since
$\tilde{Q}_{p}$ and $\tilde{Q}_{n}$ are bounded by observable potentials in
the coin cell data used to construct $U_{\mathrm{pos}}$ and
$U_{\mathrm{neg}}$, respectively. However, this definition remains ambiguous
because the calculated value of NPR will change based on the choice of
potential ranges chosen for the half-cell reference potential curves which can
vary across different datasets. NPR definitions based on
($\tilde{Q}_{n},\tilde{Q}_{p}$) therefore do not guarantee reproducibility
across different authors and datasets.

Definition of Practical NPR. In this section, we develop a practically
relevant and consistent definition of the NPR, which we call “Practical NPR.”
We first return to the original motivation of comparing relative loadings of
positive and negative electrode capacities from a cell design perspective,
which is to determine whether the negative electrode will become over-
lithiated when the cell is fully charged to 100% SOC. A sensible boundary
condition is to set the NPR equal to unity if the negative electrode is
completely lithiated when the full cell is at 100% SOC (i.e. $x_{100}=1$). An
NPR greater than 1 would then indicate that there exists some excess negative
electrode capacity at 100% SOC, which can serve as a margin to protect again
lithium plating during charge.

Under this construction, we realize that a practical NPR definition must then
be dependent on two factors: the voltage at which 100% SOC is defined, and how
much lithium was consumed during formation. The voltage at which 100% SOC is
defined becomes relevant since higher voltages requires more utilization of
both the positive and the negative electrodes, thereby decreasing the excess
negative electrode capacity. How much lithium was consumed during formation
becomes relevant because more lithium consumed during formation decreases the
negative electrode lithium stoichiometry in the fully charged state,
$x_{100}$, which effectively increases the negative electrode excess capacity.
In fact, since SEI formation continues over the life of the cell, $x_{100}$
will continue to decrease, which will effectively increase the negative
electrode excess capacity over life. Loss of negative active material, on the
other hand, could have the opposite effect by shrinking the negative electrode
curve (Dubarry et al., 2012).

Overall, a practical NPR definition cannot be a static metric, but one which
changes after formation and over the life of a cell based on the competition
between different degradation modes such as lithium inventory loss and active
material loss.

Given the above realizations, we now define the Practical NPR as

$\mathrm{NPR}_{\mathrm{practical}}=1+\frac{\tilde{Q}_{n,\mathrm{excess}}}{Q_{\mathrm{full}}},$
(27)

where

$\displaystyle\tilde{Q}_{n,\mathrm{excess}}$
$\displaystyle=q(x_{\mathrm{max}})-q(x_{100})$ (28)
$\displaystyle=q(\tilde{x}=1)-q(\tilde{x}_{100})$ (29)

represents the measured excess capacity in the negative electrode when the
cell is fully charged. The Practical NPR correctly accounts for the
sensitivity to the full cell upper cutoff voltage window, where increasing
voltage windows leads to decreasing NPRs (i.e. less protection against lithium
plating). The definition is also sensitive to changes in the electrode
stoichiometries ($x_{100}$) over life, and is therefore able to account for
the increase in the Practical NPR over life as lithium loss shifts $x_{100}$
to lower values over life.

Applying the Practical NPR definition to the dataset from Weng2021 and
Mohtat2021 yields values of 1.14 and 1.24. This calculation shows that, after
formation, the Mohtat2021 cells have more negative electrode excess capacity
than the Weng2021 cells. The difference between the calculated practical NPRs
can be attributed to more lithium consumed during formation for the Mohtat2021
cells (Section 2.2), which increased the negative excess capacity (Section
2.3). By contrast, none of the other definitions of NPR could distinguish this
fact, since those definitions do not consider the lithium lost during
formation.

A summary comparison of different NPR calculation methodologies is provided by
Table 2.

Definition | NPRtheor | NPRconventional | NPRprac
---|---|---|---
Equation | $\frac{m_{n}Q_{n,\mathrm{theor}}}{m_{p}Q_{p,\mathrm{theor}}}$ | $\frac{\tilde{Q}_{n}}{\tilde{Q}_{p}}$ | $1+\frac{\tilde{Q}_{n,\mathrm{excess}}}{Q_{\mathrm{full}}}$
Ref. | Eq. 25 | Eq. 26 | Eq. 27
Excludes inaccessible lithium? | N | Y | Y
Considers effect of lithium lost to SEI? | N | N | Y
Mohtat et al. (2021) | 0.62 | 1.02 $\pm$ 0.009 | 1.14 $\pm$ 0.009
Weng et al. (2021) | 0.63 | 1.00 $\pm$ 0.014 | 1.24 $\pm$ 0.013
Table 2: Comparison of various definitions of negative to positive ratio
(NPR). Numerical values show mean $\pm$ standard deviation. Values were
calculated based on differential voltage analysis results performed on the
datasets from Mohtat et al. (2021) and Weng et al. (2021).

####  3.4.3 Total Cyclable Lithium Inventory

The total cyclable lithium inventory in the system is imperative to track over
life, since lithium inventory depletion is a primary reason for capacity loss
in standard lithium-ion batteries (Birkl et al., 2017; Dubarry and Anseán,
2022). Lee et al. (2020a) proposed that the total cyclable lithium inventory
can be accounted for from the equation

$\tilde{Q}_{\mathrm{Li}}=\tilde{x}_{0}\tilde{Q}_{n}+\tilde{y}_{0}\tilde{Q}_{p}.$
(30)

In this equation, the first term captures the lithium trapped in the negative
electrode due to the full cell minimum voltage constraint. Meanwhile, the
second term includes both the cyclable lithium within the full cell operating
voltage window and the inventory of lithium above the full cell maximum
voltage. To gain a deeper intuition into Equation (30), we can rewrite the
same equation on the basis of the shared capacity vector $q$ as

$\displaystyle\tilde{Q}_{\mathrm{Li}}$
$\displaystyle=\underbrace{[q(y_{\mathrm{min}})-q(y_{100})]}_{V>V_{\mathrm{max}}}+\underbrace{[q(y_{100})-q(y_{0})]}_{\mathrm{Cyclable}}+\underbrace{[q(y_{0})-q(x_{\mathrm{min}})]}_{V<V_{\mathrm{min}}}$
(31) $\displaystyle=q(y_{\mathrm{min}})-q(x_{\mathrm{min}}).$ (32)

Equation 31 shows that the total lithium inventory consists of three
components: the lithium available in the positive electrode above the full
cell upper cut-off voltage (first term), the lithium available within the full
cell voltage window (second term), and the lithium available in the negative
electrode below the full cell lower cut-off voltage (third term). Note that if
$x_{\mathrm{min}}\approx 0$ is assumed, then the third term drops out and
$Q_{\mathrm{Li}}$ becomes simply a statement of the total lithium inventory
remaining in the positive electrode after discounting for lithium lost to the
SEI. By combining Equations (23) and (32), it can be verified that

$\tilde{Q}_{p}=\tilde{Q}_{\mathrm{Li}}+\tilde{Q}_{\mathrm{SEI}},$ (33)

which clarifies the fact that all of the lithium inventory in the system
originates from the positive electrode.

####  3.4.4 Note on Degradation Diagnostics: From Absolute Capacities to
Capacity Losses

The differential voltage analysis method may be repeated on full cell voltage
datasets collected over the course of a cycle life or calendar aging test for
the aged cell system. Taking
$\\{\tilde{Q}_{n}^{\prime},\tilde{Q}_{p}^{\prime},\tilde{Q}_{\mathrm{Li}}^{\prime}\\}$
to be model outputs from the aged cell, the following familiar quantities can
be defined (Sulzer et al., 2021):

$\displaystyle\mathrm{LAM}_{\mathrm{NE}}$
$\displaystyle=1-\frac{\tilde{Q}_{n}^{\prime}}{\tilde{Q}_{n}}$ (34)
$\displaystyle\mathrm{LAM}_{\mathrm{PE}}$
$\displaystyle=1-\frac{\tilde{Q}_{p}^{\prime}}{\tilde{Q}_{p}}$ (35)
$\displaystyle\mathrm{LLI}$
$\displaystyle=1-\frac{\tilde{Q}_{\mathrm{Li}}^{\prime}}{\tilde{Q}_{\mathrm{Li}}},$
(36)

where $\mathrm{LAM}_{\mathrm{PE}}$ is the loss of active material in the
positive electrode, $\mathrm{LAM}_{\mathrm{NE}}$ is the loss of active
material in the negative electrode, and $\mathrm{LLI}$ is the loss of lithium
inventory. Conveniently, these equations are equally valid for both true model
parameters (e.g. $Q$) and observed model parameters (e.g. $\tilde{Q}$), and
therefore, the ‘inaccessible lithium problem’ introduced in Section 3.3 does
not affect the numerical outcome of the study of capacity losses over the
lifetime of a cell.

Finally, we note that the definition of lithium inventory loss given by
Equation 36 does not provide any information on where the lithium is
physically lost. It is generally understood that a primary pathway for lithium
loss is through electrolyte reactions at the negative electrode to form the
SEI. However, lithium can also be irreversibly trapped in the negative and
positive electrodes due to electrical isolation of particles, which could
occur either due to physical fracturing of active material particles (Yang et
al., 2012; Zhou et al., 2019; Liu et al., 2021b) or islanding of entire
particles due to loss of adhesion to the binder material (Müller et al.,
2018). A more general expression for the loss of lithium inventory can be
expressed as the sum of lithium trapped in each component:

$\mathrm{LLI}=\mathrm{LLI}_{\mathrm{pos}}+\mathrm{LLI}_{\mathrm{neg}}+\mathrm{LLI}_{\mathrm{SEI}}+\mathrm{LLI}_{\mathrm{plating}}.$
(37)

In this equation, the first two terms represent lithium trapped inside
lithiated but electrically-isolated positive and negative electrode particles,
the third term represents the lithium trapped in the negative electrode SEI
(Sulzer et al., 2021), and the last term represents dead lithium lost to
lithium plating (Yang et al., 2017). It is important to recognize that
differential voltage analysis analysis cannot be used to decompose
$Q_{\mathrm{Li}}$ or $\mathrm{LLI}$ into their constituent parts. Differential
voltage analysis thus only reveals the quantity of lithium lost without
providing information about where the lithium was lost. This limitation is
fundamental to the technique, since different breakdowns of $Q_{\mathrm{Li}}$
yield identical voltage features (Birkl et al., 2017). More advanced,
materials-level characterization, will be needed to fully understand the
sources of lithium loss.

##  4 Discussion

###  4.1 Data Collection Considerations

Collecting high-quality lab data is a starting point for ensuring that the
differential voltage analysis results are reproducible. We therefore dedicate
this section to discussing basic experimental considerations to enable
reproducible differential voltage analysis for a given cell chemistry, using
Figures 4 and 5 as guides.

####  4.1.1 Obtaining Full-Cell Near-Equilibrium Potential Curves

Differential voltage analysis requires the cell voltage curves to be collected
at near-equilibrium conditions so that kinetic effects may be ignored. Without
an assumption of near-equilibrium, Equation 1 would require additional terms
to capture overpotential effects, complicating the analysis. A natural
question that follows is “at what C-rate does the voltage data needed to be
collected at to constitute near-equilibrium conditions?” This question is
especially salient in battery manufacturing contexts, where the speed of
diagnostic tests need to be as fast as possible.

In work by previous authors, C-rates ranging between C/20 and C/30 are most
commonly used to obtain voltage curves at near-equilibrium conditions. For
example, Dubarry and Baure (2020) used C/25 curves for both differential
voltage analysis and incremental capacity analysis, Lu et al. (2021) used C/30
curves, and Keil et al. (2016) used a fixed current value which translated
into C-rates ranging between $\sim$C/10 and $\sim$C/30, depending on the cell
type.

Using our own datasets, we studied the effect of C-rate on the shape of the
voltage and differential voltage curves in Figure 4. Panel A shows an example
charge dataset collected on an
$\mathrm{Ni}_{\mathrm{0.6}}\mathrm{Mn}_{\mathrm{0.2}}\mathrm{Co}_{\mathrm{0.2}}|$graphite
system, with C-rates ranging between 1C and C/100. The result shows that the
C/20 charge rate achieves a capacity of 2.50Ah, which is within 1% of the
capacity of 2.52Ah measured at C/100. The corresponding differential voltage
curves, shown in Panel C, shows that the C/20 rate provides peak features that
are closely aligned with the near-equilibrium limit at C/100. (Note that, in
this panel, each line is offset by an arbitrary constant $\alpha$ for
clarity.) Therefore, in our dataset, C/20 appeared to be a reasonable C-rate
to capture a near-equilibrium condition.

While C-rates ranging between C/20 and C/30 appear to provide viable input
datasets for past work, this result may not always be true for all cell types
in manufacturing. Notably, cells having higher energy densities may experience
more kinetic limitations, and cells having larger form factors could
experience more electrode-level lithium concentration heterogeneities. In both
cases, even slower C-rates may be required to resolve the features in the
differential voltage plots. Work by Mohtat et al. (2020) further suggested
that the performance of voltage fitting methods deteriorates at higher C-rates
due to peak smearing. Overall, more work is needed to quantify the effect of
C-rate on the analysis outputs as well as the generalizability of C-rate
setpoints across different cell chemistries and cell designs.

The directionality of charge must also be considered. Figure 4B,D show that
the voltage and dV/dQ features on discharge are, similar, but not identical,
to those on charge. At C/100, the low-SOC peaks, which correspond to graphite
features, appear to have sharper peaks on discharge. An additional peak also
appears in the discharge data, which can be attributed to the graphite 2L
staging reaction (see Figure 5C). Practically speaking, the presence of the
charge-discharge asymmetry suggests that the differential analysis outputs may
be different depending on whether the charge or discharge curves are used. The
extent of these differences should be further studied, which we leave as
future work.

Figure 4: Full cell data collection example for differential voltage analysis.
(A, C) Full-cell constant-current charge curves collected at C-rates ranging
between 1C and C/100 and their corresponding differential voltage curves. (B,
D) The same curves, but collected on discharge. $Q_{0}$ is the nominal cell
capacity (2.5 Ah). An arbitrary offset $\alpha$ has been added to each dV/dQ
curve for clarity.

####  4.1.2 Obtaining Half-Cell Near-Equilibrium Potential Curves

Electrode near-equilibrium potential curves can either be obtained
experimentally from scratch using coin cell testing (Hu et al., 2021; Murray
et al., 2019) or downloaded from open-access databases such as LiionDB (Wang
et al., 2022). In either case, due caution is necessary for several reasons.

First, the majority of datasets report half-cell potentials versus lithium
stoichiometries, but since there is no common voltage range setting, different
half-cell datasets will lead to different modeled outputs for electrode
capacities and stoichiometries (see Section 3.3).

Second, the experimental procedures used to obtain the curves may differ, with
some authors using continuous currents, while other authors obtaining ‘quasi-
static’, or ‘pseudo-OCV’ curves, by combining data collected after some
voltage period at each SOC, similar to the galvanostatic intermittent
titration (GITT) technique (Chen et al., 2020). While the latter method could
theoretically provide a more accurate near-equilibrium potential curve, fewer
data points are typically collected to ensure the test finishes within a
reasonable time. As a result, some characteristic features related to phase
transitions in the negative electrode (Figure 4C) may be lost (Hess, 2013).

A third reason for exercising caution when using half-cell data obtained from
literature is that the current direction is sometimes not reported, yet
current direction materially impacts the features seen in the voltage curves.
Figure 5 shows that the current direction influences the voltage (and
differential voltage) features for both a graphite negative electrode (Panels
A,C) and an
$\mathrm{Ni}_{\mathrm{0.33}}\mathrm{Mn}_{\mathrm{0.33}}\mathrm{Co}_{\mathrm{0.33}}$
positive electrode (Panels B,D). At the negative electrode, a hysteresis gap
between the lithiation and the delithiation curves exists, which persists
across consecutive charge-discharge cycles. The characteristic differential
voltage feature occurring at $\sim$2.3 Ah, corresponding to the stage 2
transition, is sharper and higher in magnitude in the lithiation direction.
The staging reactions 2 through 4 are also distinctly different. In
particular, the staging reaction 2L appears sharper on lithiation and the
stage 4L reaction appears sharper on delithiation (Hess, 2013).

At the positive electrode, a large kinetic limitation is observed at the end
of the lithiation curve, causing the lithiation capacity to be measurably
lower than the delithiation curve across multiple cycles. The observation of
poor kinetics during lithiation of the positive electrode has been
experimentally confirmed by Kasnatscheew et al. (2016) and thoroughly
discussed in Phattharasupakun et al. (2021). Overall, charge-discharge
asymmetry exists at the material level for both the positive and the negative
electrodes. For high-fidelity differential voltage analysis, the current
direction used to obtain the electrode potential curves should be aligned
against what is used for the full cell. Doing so will improve the model’s
ability to fully describe the full cell voltage curve using the electrode
potential curves.

Figure 5: Half-cell data collection example for differential voltage analysis.
(A,C) Graphite$|$lithium half-cell lithiation and delithiation curves and
their corresponding different voltage curves. (B,D)
$\mathrm{Ni}_{\mathrm{0.33}}\mathrm{Mn}_{\mathrm{0.33}}\mathrm{Co}_{\mathrm{0.33}}|$lithium
half-cell lithiation and delithiation curves and their corresponding
differential voltage curves. All data is measured at C/50. To indicate
reproducibility, data from two consecutive charge-discharge curves have been
provided (except for the graphite delithiation curve). The direction
corresponding to full cell charging is indicated by a red arrow.

####  4.1.3 Data Logging Frequency and Data Smoothing

The data logging rate and filtering method will affect the smoothness of the
collected voltage curves and its derivatives. It is generally recommended to
over-sample than under-sample to minimize data loss. Noise in the over-sampled
data can be overcome by post-processing the data using filtering methods. A
common approach is to use a Savitzky-Golay filter (Savitzky and Golay, 1964)
which is included as part of most scientific programming languages. Note,
however, that the process of selecting filtering parameters, such as window
size and polynomial order, may introduce distortions in the data, decreasing
the reproducibility of modeled results (Feng et al., 2020; Lu et al., 2021;
Schmid et al., 2022). To improve model reproducibility, more studies are
needed to understand the degree to which data filtering strategies impact
model outputs, which we leave as future work.

###  4.2 Comments on Factory Deployment

Here, we summarize how a battery manufacturer might deploy the differential
voltage analysis method in the battery factory for online process control and
quality control applications. We also highlight the remaining knowledge gaps
that may prevent deployment today which warrant further research.

Enabling the differential voltage analysis in manufacturing requires both
half-cell and full-cell data collection. The half-cell data only needs to be
collected once. The data, once vetted, can be stored as part of the model
input parameters and be re-used to analyze all full cell data belonging to the
same family of positive and negative electrode material sets, e.g. across
multiple cell batches and production lines.

The largest barrier to method deployment is then in the acquisition of the
full cell near-equilibrium data. As discussed previously, current methods for
differential voltage analysis generally require around 20 hours in order to
collect the full cell voltage trace. If the measurement method is to be
deployed on every cell coming out of formation, then the production throughput
will decrease. Considering that the industry trend is to decrease formation
times (Wood et al., 2019), any additional process that will make the total
time on test longer may be untenable.

To improve the adoption of the method, we propose several pathways. First,
rather than collecting the full cell voltage data after formation, the voltage
data during the formation process itself may be processed for differential
voltage analysis. To our knowledge, this method has yet been reported in
academic literature, and is the subject of future work.

Second, a study should be conducted to understand the influence of higher
C-rates on the modeled parameters. Historically, some of the challenges with
differential voltage analysis arose in the analysis of aged cells, where the
effective current density increases as the cell capacity decreases. For
example, at 50% capacity retention, a “C/20” charge step would only take
approximately 10 hours to complete since the C-rate is defined with respect to
the nominal (pristine) cell capacity. In battery manufacturing, we are only
concerned with studying the electrochemical state of the pristine cells
without consideration for aging. Hence, the C-rates traditionally thought to
be necessary to enable differential voltage analysis may be too conservative
in a manufacturing context. A future study of the influence of C-rate on the
modeled electrochemical parameters will provide some necessary guidance in
this direction.

Finally, not every cell may need to be fingerprinted. For example, retaining
and measuring several dozen cells (out of hundreds of thousands built per day)
could provide enough statistical information to track day-to-day changes in
electrochemical process parameters. These measurements can still be made on
formation equipment, and would only require a small, dedicated number of
formation cyclers, to be withheld for this purpose. This partitioning method
can also encourage the adoption of more advanced characterization methods such
as computed tomography (CT) (Wu et al., 2018; Bond et al., 2022; Gauthier et
al., 2022), ultrasound imaging (Bommier et al., 2020; Deng et al., 2020), and
strain measurements (Mohtat et al., 2022). Measuring all of these parameters
on the same cell could enable a more complete perspective on the cell
electrochemical and mechanical state, enabling deeper insights.

###  4.3 Improving Method Reproducibility

Besides logistical challenges of deployment, another major and perhaps larger
obstacle preventing adoption of the differential voltage method is the
difficulties with implementation and interpretation of model outputs. These
difficulties stem from a lack of literature guidance on a standard, rigorous
method to implement the fitting model and an absence of understanding of how
sensitive the model outputs are to small differences in input datasets,
including both full cell and half cell data. This work sought to clarify the
model interpretation by enabling a quantified understanding of model errors
due to the inaccessible lithium problem (Section 3.3). We also discussed
elementary data collection considerations needed to enable a working model
(Section 4.1).

Several avenues may be taken to further improve reproducibility of the
electrochemical features derived from the differential voltage analysis
method. First, model input sensitivities need to be better understood. This
work introduced several input sensitivity considerations including C-rate and
current direction (charge versus discharge). However, a discussion on
sensitivity to temperature was not covered by this work. Moreover, a
quantitative assessment of the input-output sensitivities have not yet been
completed. A logical next step is thus to quantify model output differences
due to these differences in model inputs.

Another important consideration involves understanding the optimizer’s ability
to find the optimal solution $\theta_{\mathrm{opt}}$. Reproducibility requires
that there exists a global minimum value of the error function defined by
Equation 12 and that the optimizer can converge to this value. However, in
general, neither condition is guaranteed since the objective function is, in
general, non-convex. The solution uniqueness for the positive electrode
parameters $\tilde{Q}_{p}$ and $y_{0}$ may be especially challenging for NMC
systems which are relatively featureless (Figure 3). Some past efforts have
been made to understand how model output parameter uncertainty increased as
the data collection voltage window decreased (Lee et al., 2020b). These
studies should be expanded. A better understanding here will enable
practitioners to quantify how much of the variability in model-predicted
outputs, such as those shown in Figure 2, are due to true manufacturing
variability versus model uncertainty.

Finally, it must be recognized that all of the electrochemical features
extracted using the differential voltage model are estimates of the true
values (Section 3.3). Experimental verification of model-predicted outputs,
such as positive electrode capacity, negative electrode capacity, lithium lost
to SEI, cyclable lithium inventory, and negative to positive ratios, can be
challenging to obtain, but may be necessary to obtain confidence in the model-
predicted outcomes.

##  5 Conclusion

This work clarified how the differential voltage analysis method can be
automated to improve online battery manufacturing process control. Using an
example manufacturing dataset, we demonstrated how modeled outputs, such as
positive and negative electrode capacities, lithium lost to form the SEI, and
negative to positive ratios, can be used to gain insight into the sources and
consequences of manufacturing variability. In our example, we found that the
source of variability in full cell capacity for one batch of cells is due to
variations in positive electrode loadings. We also detected a three-fold
difference in the amount of lithium consumed consumed to the SEI between two
different cell batches, a conclusion we were able to make without observing
any data from the formation cycles or using any destructive analytical
methods.

To facilitate adoption of the differential voltage analysis method in
manufacturing, we detailed the mathematical formalism required to implement
the method as well as the input data requirements. We extended the base model
formulation to explicitly account for model parameter errors due to the
inaccessible lithium problem. Using the reformulated model, we defined an
expanded set of electrochemical features which included the lithium consumed
during formation, the practical NP ratio, the excess negative electrode
capacity, and the total cyclable lithium. When combined, these features
provide a “fingerprint” of the electrochemical state of the pristine cell
which can then be used to adjust parameters in various manufacturing steps
including electrode manufacturing and the formation process.

Besides clarifying the model implementation details, this work also recognized
the importance of obtaining high-quality experimental data, which includes
both full cell and half-cell data, to enable reproducible results. We
identified several basic data collection considerations, including the C-rate,
current direction (charge versus discharge), and data smoothing method, as
being potential factors affecting model output reproducibility.

With an improved awareness of the nuances involved with model construction,
model output interpretation, and data collection, the battery community can be
better positioned to deploy the differential voltage analysis method in
factories, driving continuous improvement in the battery manufacturing
process.

## Data Availability

This study presented datasets from multiple works. The cell test dataset from
Mohtat et al. (2021) is available at https://doi.org/10.7302/7tw1-kc35. The
cell test dataset from Weng et al. (2021) are available for download at
https://doi.org/10.7302/pa3f-4w30. All source code used to analyze the
datasets and generate the plots are available at
https://github.com/wengandrew/dvdq. The differential voltage analysis code was
written in MATLAB. The data post-processing tools and data visualizations were
developed in Python.

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