# A comparison of lithium diffusion between Graphite/LFP 18650 power and energy cells with low and high current pulses

William Wheeler, Damien Voyer

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William Wheeler, Damien Voyer. A comparison of lithium diffusion between Graphite/LFP 18650 power and energy cells with low and high current pulses. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0248. hal-05088271

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# A comparison of lithium diffusion between Graphite/LFP 18650 power and energy cells with low and high current pulses

William Wheeler, Damien Voyer

EIGSI Lab, EIGSI La Rochelle

26 Rue Francois de Vaux de Foletier

La Rochelle, France

Tel.: +33 / (0) - 548170339

E-Mail: william.wheeler@eigsi.fr

URL: https://www.eigsi.fr/

## Keywords

«Battery», « Battery impedance measurement», «Diagnostics», «Time-Domain Analysis», «Battery electrochemical model».

## Abstract

This study gives a new estimation model to evaluate the voltage-increase due to a step-up current in both lithium-ion power and energy cells. The step-up current analysis is similar to the GITT technic developed in the literature, but with tests at low or high C-rate and for a longer time. The power cell voltage-increase follows a square root of time dependency which is commonly observed in the literature. On the contrary, a superimposed transient overvoltage in the energy cell is observed. The transient overvoltage has the capacity to distort the characterization of diffusion as modelled by existing models. This paper proposes a phenomenological approach to model the transient overvoltage in the energy cell. This work gives a method to model the phenomenon and is a first step to clarify the origin of the phenomenon. It can also help improving the estimation of the diffusion parameters.

## Introduction

Lithium-ion battery diagnosis is a key feature to understand the behavior of energy storage systems for their entire lifespan. GITT have been used in several studies as a method to characterize battery cells overpotential or diffusion coefficient [1-3]. Studying and monitoring the evolution of diffusion throughout battery aging, in electric vehicles or stationary application, can help improve existing aging models and can help identifying diffusion limits and causes of the apparition of detrimental deposits such as lithium plating.

GITT has been tested in many studies with low C-rate, between C/10 and C/20, which is commonly used to comply with assumption of existing diffusion model [3]. Some authors use GITT with higher C-rate such as 0.4C with pulse test from five minutes to two hours long [1]. At higher C-rate, other mechanisms such as lithium diffusion in the electrolyte, interparticle mechanisms or composite resistance may become significant [3-7]. As a result, GITT using high C-rate may not be used to determine the diffusion coefficient. Nevertheless, the outcomes obtained from GITT and EIS are not identical [8]. Diffusion estimation also helps building accurate models of batteries in use, thereby enabling higher description fidelity in real-use test scenarios.

The C-rate is mostly limited by the cathode material due to the low electronic conductivity and lithium-ion transport resistance in both solid and liquid phase [9]. It has been demonstrated that high C-rate imply a non-uniform State of Lithiation (SoL) in the electrode. A higher average SoL in particles facing the separator is found wherase the SoL decrease with depth, converging at a certain depth [9,10]. The distribution of SoL in the electrode depth is due to solid state transport, whereas liquid state transport limits particles' charge and discharge [10].

This paper aims to investigate the use of GITT technique to evaluate the diffusion of energy and power lithium-ion Graphite/LFP cells. Low and high C-rate are tested in order to evaluate the changes in the evolution of the cell voltage and attempt to identify liquid and solid state transport. First, an experimental setup is described to test and compare experimental results for a power and an energy Cell. Multiple C-rate are tested between 35% and 50% of State of Charge (SoC). Second, a modified Randles circuit is proposed to take into account specific impedance phenomenon for all C-rate in the energy cell. The
modified model helps identify an additional overvoltage in the experimental results. The corresponding model in the Laplace domain provides a first insight into the phenomena.

# Experiments

## Experimental setup

Experiments are conducted on two cells, a power cell and an energy cell. Both cells are tested under the same conditions. Both cells are 18650 Graphite/LFP at room temperature, around 24°C. The first cell is a power cell, reference APR18650m1B, with a nominal capacity of 1.1 Ah, allowing a continuous charge current of 3.6C and a maximum of 27C continuous discharge. The second is an energy cell, reference NCF 18650-1500mAh-3.2V, with a nominal capacity of 1.5 Ah and allowing continuous charge current of C/2 and a maximum of 3C continuous discharge.

<figure><img src="image_2.png" /><figcaption>Figure 1: Pseudo-OCV in charge and discharge for the power cell and the average value considered as the OCV.</figcaption></figure>

A “wake-up” cycle of 20 cycles (charge and discharge) at 1C is carried out prior experiments. Then, both cells are characterized with a low-rate charge and discharge at C/10, followed by a discharge until a SoC of 35% is reached. Then, a current pulse test is conducted in charge and discharge between 35% and 50% of SoC. Four C-rate pulse are selected: C/10, C/2, 1C and 2C in order to test the technique at low, medium and high C-rates. Finally, an EIS is performed on each cell at a SoC of 35%, with an amplitude of C/10, from 10 mHz to 10 kHz. All current pulse are tested from a SoC of 35% to a SoC of 50% (ΔSoC = 15%) to test the cell in a low OCV variation zone as depicted in Figure 1.

# EIS measurements

EIS is performed in order to determine parameters of the equivalent circuit of the cell. The result in Figure 2 is fitted using a Randles circuit as proposed in Kuhn et al. is used, considering a CPE for the double layer impedance [11]. Impedance parameters are given in Table 1.

<figure><img src="image_3.png" /><figcaption>Figure 2: EIS spectrum and Randles equivalent circuit used for parameters identification for the energy and the power cells.</figcaption></figure>

<table><caption>Table 1: Estimated EIS values for the considered Randle circuit from Kuhn with a CPE for the double layer [11].</caption><thead><tr><td>Parameter</td><td>Energy cell</td><td>Power cell</td></tr></thead><tbody><tr><td>R<sub>s</sub></td><td>12.3 mΩ</td><td>12.2 mΩ</td></tr><tr><td>R<sub>ct</sub></td><td>12.4 mΩ</td><td>3.6 mΩ</td></tr><tr><td>Q<sub>dl</sub></td><td>5.8 S. s<sup>α</sup>dl</td><td>0.71 S. s<sup>α</sup>dl</td></tr><tr><td>α<sub>dl</sub></td><td>0.42</td><td>0.8</td></tr></tbody></table>

## Cell voltage response

The cell overvoltage response to a step-up current is given in Figure 3 for a C-rate of C/10. For all cells, a rapid change of voltage occurs on the first few seconds. Then, both cells observe a two-phase voltage evolution: a sharp voltage increase in a square root of time dependency between 10 and 30 s¹/₂, then a change of the slope appears around 35 s¹/₂. The voltage evolution is linear between 10 and 30 s¹/₂ in a square root of time plot in Figure 3 (b) for the power cell. The magnitude of the voltage increase after 35 s¹/₂ corresponds to that of the pseudo-OCV evolution in Figure 1. Our experimental result can be compared to the one of Nickol et al. consisting of two phases: a fast change of the cell voltage appears in the first few seconds called “transition region”. This is mainly due to serial resistance and charge transfer
effects. This is followed by a linear evolution dependent of the square root of time. In Nickol's work, the second is used to model the diffusion [3]. However, a third zone with a slow linear slop is identified in our experiments after 35 s¹/².

The energy cell experiences a similar behavior, but with a curved behavior around the slope change at 35 s¹/². After a stabilization period, both cells follow the same slope evolution after 5000 seconds or 70 s¹/².

Considering the power cell has a linear behavior as exposed in Figure 3 (b) for a low C-rate, a difference between the voltage of both cells is computed to identify the transient overvoltage pattern. In order to compare the slight voltage difference between both cells, each cell voltage is normalized between the first and last data point of the impulse current test as exposed in Figure 4 (a). The differential voltage in Figure 4 (b) exhibits a bump shape response, with a fast increase followed by an exponential decrease over time. This appears at low C-rate such as C/10 for the energy cell.

The subsequent transient overvoltage in the energy cell around 30 s¹/² could make it more difficult to use existing techniques in the literature to determine the diffusion coefficient. For instance, a short time current impulse such as 20

<figure><img src="image_4.png" /><figcaption>Figure 3: Cell overvoltage due to a current step-up of C/10 for the energy cell in blue and the power cell in orange versus (a) time and (b) square root of time. The 2-line linear behavior of the power cell is in dashed black line.</figcaption></figure>

s¹/², may be measured due to the steeper linear behavior between 0 s¹/² and 20 s¹/². Therefore, an error in the diffusion estimation may occur. This may explain some of the difficulties to estimate the diffusion coefficient in the literature.

This transient overvoltage needs to be addressed and modeled in an equivalent electrical circuit. As exposed in this section, longer current impulse allows the identification of the transient overvoltage in experimental data.

## Method

The proposed method aims to separate the voltage increase due to the diffusion and the transient overvoltage for the energy cell. By separating both contributions, we can obtain data to estimate the diffusion, the transient overvoltage and their respective impact on the cell voltage. The cell OCV variation and voltage increase for each impedance are represented by an equivalent electric model. This study assumes that the Randles impedance model remains constant throughout the ΔSoC window, between 35% and 50% of SoC (i.e. the values of parameters in Table 1 are kept constant).

<figure><img src="image_5.png" /><figcaption>Figure 4: (a) Normalized cell voltage change for the power and energy cells at C/10 and (b) difference of voltage between the power and energy cell.</figcaption></figure>
## Electrical representation of the cell impedance and open circuit voltage

In this study, a typical Randle circuit in Figure 5 is used to model the cell impedance composed of: a serial resistance R_s and a parallel circuit composed of a Constant Phase Element (CPE), expressed in the Laplace domain, in equation (1). Both elements in parallel model the double layer capacitance effect and the charge transfer resistance.

$$Z_{dl}(s) = \frac{1}{Qs^{\alpha}} \qquad (1).$$

The diffusion is in series with the charge transfer resistance in order to preserve the interdependence between diffusion with charge transfer and double layer impedance. A supplementary element Z_tr is added to model the transient overvoltage evolution of the energy cell.

<figure><img src="image_6.png" /><figcaption>Figure 5: Considered Randle Circuit with a serial resistance, charge transfer, double layer, and diffusion element. Another element is added to model the transient overvoltage behavior in the energy cell.</figcaption></figure>

Finally, the open circuit voltage of the cell is modelled as a voltage source where the voltage depends on the lithiation state on the surface of the graphite particle. The overall cell voltage can be described in equation (2):

$$v_{cell}(t) = v_{OCV}(SOC_{surf}) + v_{Rs}(t) + v_{diff+ct}(t) + v_{tr}(t) \qquad (2).$$

Serial resistance, charge transfer and values for the equivalent electrical parameters have been estimated with the EIS and are given in Table 1. Serial resistance, double layer capacitance, and charge transfer resistance will be treated as constant.

The test current is a rectangular pulse of amplitude I_0 from time 0 to the end of test t_end. In the Laplace domain, the current is described by equation (3):

$$I(s) = \frac{I_0}{s} (1 - e^{-s t_{end}}) \qquad (3).$$

## Diffusion model

Diffusion within the electrodes is assimilated to Nernst diffusion [12], which ensures that the voltage increase reaches a finite value once the transient regime has passed. A Finite-Length Warburg impedance is used as detailed in equation (4):

$$Z_{diff}(s) = R_{diff} \frac{\tanh(\sqrt{\tau_{diff}s})}{\sqrt{\tau_{diff}s}} \qquad (4).$$

The impedance in equation (4) can be expanded as a series of parallel RC circuits for ease of use. The result is expressed as a sum of RC circuit in series in equation (5) [13]:

$$Z_{diff}(s) = \sum_{n=1}^{\infty} \frac{R_{diff,n}}{1 + s \tau_{diff,n}} \qquad (5).$$

The expression for R_diff_n and τ_diff_n in equations (6) and (7) allow us to model the charge transfer with only two parameters:

$$R_{diff_n} = 8 \frac{R_{diff}}{\pi^2 (2n - 1)^2} \qquad (6).$$

$$\tau_{diff_n} = 4 \frac{\tau_{diff}}{\pi^2 (2n - 1)^2} \qquad (7).$$

## Transient overvoltage model

The transient overvoltage response identified in the energy cell will be modeled using an empirical approach. It will be used in the energy cell model for all C-rates. The transient overvoltage, available in Figure 4 (b), is expressed in a time dependent equation:

$$v_{tr}(t) = k_{tr} t^n e^{-\frac{t}{\tau_{tr}}} \qquad (8).$$

The equation (8) describes the experimental data and can be interpreted as a modified CPE impedance. The supplementary impedance is added to the modified Randle Circuit in Figure 5 to take into account its impact on the full cell. The transient overvoltage in the Laplace domain is expressed in a phenomenological manner with the purpose of emphasizing its origin as a product of current and impedance. In the Laplace domain, the transient overvoltage from equation (8) is:

$$V_{tr}(s) = k_{tr} \frac{\Gamma(n + 1)}{\left(s + \frac{1}{\tau_{tr}}\right)^{n+1}} \qquad (9).$$
Equation (9) can be decomposed into a product of two elements in equation (10), which can be explained as a product of an exponential decrease of current and a nonstandard impedance.

$$V_{tr}(s) = \frac{I_{tr}}{s + \frac{1}{\tau_{tr}}} \frac{\Gamma(n + 1)}{Q_{tr} \left(s + \frac{1}{\tau_{tr}}\right)^n} \quad (10).$$

The current in equation (10) does not correspond to the test current in equation (3). This result may indicate a non-uniform distribution of current in the electrode. The non-standard impedance looks like a CPE with an offset in $s + 1/\tau_{tr}$. The impedance from equation (10), in the frequency domain, gives the following result: it tends towards a constant when the frequency tends towards 0, and tends toward 0 when the frequency tends toward the infinity. This is similar to the behavior of a Finite-length Warburg element. This result may indicate a more complex phenomenon occurring in the cell, related to diffusion. The overall diffusion may correspond to a contribution of lithium ions diffusion in particles and in the depth of the electrode.

## Open circuit voltage model

The proposed model in this study makes it possible to identify transient phenomena. To make it easier to model the phenomenon, the cell pseudo-OCV is removed from the cell data to only model the impedance.

The difference of potential of each electrode versus lithium determines the cell voltage. As a result, the cell voltage is dependent on the surface potential of Graphite and LFP particles versus lithium. The state of lithiation of the particles in the electrode is considered homogeneous for a low C-rate. The voltage change, due to the stoichiometric change in particles can be removed by subtracting the pseudo-OCV from Figure 1 to the cell voltage variation. Consequently, the cell impedance total voltage increase is represented by the following equation (11):

$$v_{cell}(t) - v_{OCV}(SoC_{surf}) = +v_{Rs}(t) + v_{diff+ct}(t) + v_{tr}(t) \quad (11).$$

For higher C-rate, inhomogeneities on particle surface has to be taken into account. As a result, the surface particle voltage is described as a function of the pseudo-OCV variation with a dependency on lithium ions concentration on the particle surface in equation (12). Parameters are: $v_{OCV}(SoC)$ is the voltage variation from Figure 1, $SoC_0$ the initial SoC in the experiment and $\Delta SoC(t)$ the SoC variation during the experiment. $\beta$ is a factor to model higher concentration of lithium ions in the particle surface. This parameter is equal to one for C/10 and is higher for higher C-rate.

$$v_{OCV}(SoC_{surf}) = v_{OCV}(SoC_0 + \beta \cdot \Delta SoC(t)) \quad (12).$$

<figure><img src="image_7.png" /><figcaption>Figure 6: Voltage evolution between 35 and 50% of SoC of the energy cell under a GITT in blue, the evolution of the OCV of the cell in red and the corrected data from equation (12) in yellow for (a) a GITT pulse of C/10, (b) a C/2 pulse, (c) a 1C pulse and (d) a 2C pulse.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7: Voltage evolution between 35 and 50% of SoC of the power cell under a GITT pulse in blue, the evolution of the OCV of the cell in red and the corrected data from equation (12) in yellow for (a) a GITT pulse of C/10, (b) a C/2 pulse, (c) a 1C pulse and (d) a 2C pulse.</figcaption></figure>

The use of long-time current pulse with a low C-rate, for a $\Delta SoC$ of 15%, imply a change in the OCV of 10mV. For the C/10 current pulse, the change of lithiation on the particle surface is considered equal to the variation of transferred capacity $\Delta SoC$ of 15%. For higher C-rate, a disequilibrium in lithiation of the particle appears. The voltage measured under high current implies higher lithiation state on the surface of graphite

particles. As a result, a higher SoC on the surface particle is considered. The result from equation (11) is given in yellow in Figure 6 (a) for the energy cell for all test case. In order to retrieve a flat curve after a long period of time, the $\beta$ value is increased for higher C-rate. This led to an increase of the $\beta\Delta SoC$ value as exposed in Figure 6 (b) and (d).

<figure><img src="image_9.png" /><figcaption>Figure 8: Curve fitting results for the energy cell for each GITT pulse: C/10 in blue, C/2 in red, 1C in yellow and 2C in purple versus (a) time and (b) square root of time. The dashed line represents the model result for the overall impedance of the cell voltage. The overall voltage increase is composed of the diffusion (dotted line) and the transient overvoltage dashed-dotted line).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 9: Estimation results for the diffusion and transient parameters for each GITT pulse for (a) the diffusion resistance coefficient, (b) the diffusion time constant, (c) the transient amplitude coefficient and (d) the transient time constant. The linear dependency $k_{tr}$ versus the C-rate in (b) is represented with a dashed line.</figcaption></figure>

The same procedure is performed for the power cell in Figure 7. However, the power cell exhibits an exponential voltage evolution without a transient overvoltage, unlike the energy cell. The voltage transient is influenced by the current impulse amplitude and cell type. The pseudo-OCV compensation allows to explain the slow voltage increase after 35 s¹/². The procedure allows fitting the model to the corrected data in order to obtain the diffusion and transient overvoltage parameters.

## Results and discussion

The experimental data is fitted with the proposed model for the energy cell using the equivalent electric model from Figure 5. Diffusion, charge transfer and double layer capacitance are simulated using Fourier transform as no analytical solution can be derived from the model. Serial resistance, double layer capacitance and charge transfer resistance are estimated based on EIS experimental results. The current used in the experiment is described by equation (3). The transient overvoltage is modelled with the temporal equation (8) proposed previously. The model is only applied during the GITT positive current pulse. The Figure 8 (a) expose the fitting result for each GITT pulse versus time. The result is in good agreement with the experimental data. The Figure 8 (b) exposes the same results versus square root of time. The diffusion overvoltage has

a linear behavior between 5 and 20 s¹/² as expected in existing models in the literature. The diffusion coefficient can be estimated based on this estimation. Furthermore, the transient phenomenon can be analyzed independently. The transient and diffusion parameters estimation is given in Figure 9. The time constant and resistance estimation for the diffusion is given in Figure 9 (a) and (b). Both results are highly dependent on C-rate, which was expected as the high non linearity of the diffusion. However, a parabolic behavior seems to appear, especially for the diffusion resistance estimation. Moreover, the time constant of the diffusion seems very low for all C-rate over C/10. This result need further investigation to understand the very low time constant in high C-rate experiments. The transient overvoltage coefficient $k_{tr}$ in Figure 9 (c) and the transient time constant in Figure 9 (d) are also dependent on the C-rate. The $k_{tr}$ factor for the transient estimation has a linear dependency with C-rate. This is in good agreement with our assumption as $k_{tr}$ represent the product of the current $I_{tr}$ and the impedance factor $Q_{tr}$ in equation (10). As a result, $Q_{tr}$ is considered as independent of the current.

In addition, the product of the time constant $\tau_{tr}$ in Figure 9 (d) with the corresponding tested C-rate give a result of 100 for all GITT pulses (i.e. a reciprocal function multiplied by a factor 100 appears between C-rate and $\tau_{tr}$). So, the transient amplitude response is linearly dependent on the
tested current and the time constant is only SoC dependent. Finally, the results reveal that the two parameters τtr and τdiff follow the same trend, despite the fact that no link is considered between the two elements in the study.

## Conclusion

This paper presents the voltage response of an energy cell to a long-time current impulse, which experiences a transient overvoltage superimposed on the diffusion. In contrast, the power cell is only affected by the diffusion. The paper reveals a method to identify transient overvoltage and proposes a phenomenological approach to model it. The method helps identifying the voltage response and isolate the contribution from both the diffusion and the transient phenomenon on the cell overvoltage. The results indicate that the C-rate affects all parameters, but the Qtr value is found to be independent. Furthermore, the time constants of the diffusion and the transient overvoltage follow the same evolution with respect to the C-rate.

The transient response needs further investigation to fully understand its behavior and its origin in energy cells. The proposed model of the transient overvoltage as a product of a decreasing exponential current and a non-standard impedance will be studied in more details. Finally, the correlation between the diffusion and the transient phenomenon will be investigated to understand their potential link. Hence, a new diffusion model will be developed to identify their links.

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