Entropy profile of NCR 18650 cylindrical cell at various states of health

Abstract

Entropy measurement at various states of charge (SOC) is a potential non-destructive tool for characterizing lithium-ion batteries; however, traditional potentiometric methods are time-consuming. To address this, we developed a fast potentiometric method that estimates entropy by employing charging after pulse discharging to partly eliminate voltage relaxation. This method, combined with precise mathematical processing, reduces estimation time by approximately 4.5 times compared to conventional approaches. We validated the method by comparing it with the traditional potentiometric method and applied it to obtain entropy profiles of an NCR 18650 Panasonic battery at various cycle numbers at a 1.5C rate. Additionally, we investigated how entropy changes with variations in the battery's state of health (SOH). The study established coefficients for a mathematical model representing SOC across different SOH, revealing a strong correlation for SOC estimation.

List of nomenclature and abbreviations:

• BEV

Battery Electric Vehicle

• BMS

Battery Management System

• CV

Cyclic Voltammetry

• DFT

Density Functional Theory

• DVA

Differential Voltage Analysis

• EIS

Electrochemical Impedance Spectroscopy

• ICA

Incremental Capacity Analysis

• ICE

Initial Coulombic Efficiency

• ICP-MS

Inductively Coupled Plasma Mass Spectrometry

• LIB

Lithium-Ion Battery

• LCO

Lithium Cobalt Oxide

• NCA

Lithium Nickel Cobalt Aluminum Oxide

• OCV

Open Circuit Voltage

• SEI

Solid Electrolyte Interface

• SEM

Scanning Electron Microscopy

• SOC

State of Charge

• SOH

State of Health

• TEM

Transmission Electron Microscopy

• TGA

Thermogravimetric Analysis

• XPS

X-ray Photoelectron Spectroscopy

• XRD

X-ray Diffraction

1. Introduction

The importance of LIBs has grown significantly in recent years, driven by the increasing demand for electric vehicles, renewable energy storage, and battery-powered electronic devices. As the market expands, there is a rising need to develop new battery materials to keep pace with these demands. This has also led to an increased interest in other electrochemical systems, such as fuel cells, which have already been extensively researched [1,2].

LIBs are used in a wide variety of applications, from consumer electronics to full BEVs, and play a crucial role in emerging energy systems like wind and solar power. These applications require batteries that offer not only high power and energy density but also a stable cycle life and reliable safety throughout their lifespan. To optimize the performance of these batteries, it is essential to deeply understand the electrochemical processes occurring within the cells during charging, discharging, and state of rest. Both non-destructive and destructive measurement techniques are vital for gaining these insights.

Given the rising importance of LIBs, there is an increasing emphasis on developing effective non-destructive methods for characterizing and analyzing these batteries. Accurate methods for determining SOH and SOC are crucial for BMSs, ensuring long-term reliability and safety. Currently, non-destructive techniques such as CV, charge-discharge profiles, and impedance analysis are commonly employed to characterize batteries. However, these methods alone may not provide a complete understanding of the battery's internal processes. Researchers are actively testing new battery materials to find alternatives to existing ones, even exploring the possibility of replacing common lithium-ion batteries with other systems. For instance, Rakhymbay et al. tested the cyclability of Fe-doped Na₂Mn₃O₇ as a cathode material for sodium-ion batteries [3]. Zhang et al. proposed the novel low-cost aluminum–graphite dual-ion battery which has higher power density than lithium ion battery [4]. Wang et al. suggested a practical room-temperature rechargeable calcium-ion battery with a high discharge voltage [5]. Yang et al. investigated the Variant-Localized High-Concentration Electrolyte without Phase Separation for its application in low-temperature conditions [6]. Silicon (Si) is regarded as a promising anode material for the next generation of high-energy-density batteries. However, its commercial development is significantly hindered by its low initial coulombic efficiency (ICE) and poor rate performance. Wang et al. have made attempts to overcome these drawbacks of the Si anode material [7]. In the realm of new electrolyte materials, numerous investigations have focused on ion conduction, design, rapid optimization, and the screening of advanced solid electrolytes to eliminate the need for liquid electrolytes [8]. Some researchers have proposed composite polymer electrolytes as an alternative [9]. Liang et al. investigated the unsuccessful fabrication of Li₃Zr₂Si₂PO₁₂ via solid-state reaction using a method that combined thermodynamic calculations with density functional theory (DFT) [10]. Lithium (Li) metal is considered the most desirable anode candidate for high-energy-density batteries due to its lowest redox potential and ultrahigh theoretical specific capacity. However, the commercialization of Li metal anodes is plagued by uncontrollable dendritic growth, infinite volume changes, and an unstable solid electrolyte interface (SEI). Zhao et al. proposed a new approach for efficiently protecting Li metal anodes, offering fresh inspiration for the development of high-energy-density Li metal batteries [11].

To gain a deeper understanding of these challenges, both destructive and non-destructive characterization methods are employed. Destructive methods, including XRD, SEM, TEM, XPS, ICP-MS, and TGA, provide valuable insights but come with the drawback of destroying the battery, rendering it unusable for further analysis. Non-destructive methods, such as CV, ICA, DVA, battery capacity measurement, and EIS, are crucial for evaluating a battery's performance and longevity across various SOH [[12], [13], [14], [15]]. These techniques have highly developed interpretation methodologies. At present, numerous techniques are employed for in situ analysis to comprehend the process of electrode reactions and the degradation mechanisms in lithium-ion batteries [[16], [17], [18], [19]]. Nevertheless, these methods frequently require the integration of customized cell models and electrode materials possessing specific morphology, which can be inconvenient in various scenarios. However, entropy measurement techniques are not as well-developed, due to limited data on the entropy profile of batteries in different states and chemistries. Despite this limitation, entropymetry techniques could be useful for battery characterization because the entropy of a material is depend on its structural order [20]. As the entropy of a material increases, the level of chaos in its structure also increases.

Choi et al. used the thermodynamic parameter entropy—a measure of a system's disorder—to monitor changes in the structure of a battery electrode without damaging the battery. By analyzing entropy profiles, they gained insights into the atomic arrangement, such as cation mixing (where metal ions swap places in the lattice) and the presence of metal vacancies in the structure. Notably, they observed that doping the LiCoO₂ cathode with nickel increased the system's disorder, which in turn improved the stability of the framework and extended its cycle life [21] The authors were able to directly relate the measured entropy profiles of LiCoO2 to the phase change mechanism described by Reimers et al [22]. Kramer et al. utilized entropy measurements to diagnose NCA cathode material in conjunction with different anode materials, such as graphene and silicon [23] Consequently, entropy measurement shows great potential for non-destructive structural diagnosis of batteries. However, its main drawback is the lengthy measurement time required.

One reason for the lack of available data on entropy profiles is that measuring entropy can be a time-consuming process. In fact, according to several articles, entropymetry measurements can take anywhere from 12 to 36 h to complete for just one SOC of a battery [24,25]. Entropy of the battery can be determined by eq. (1):

$$\Delta S=n*F*\frac{dE}{dT}$$ (1)

where n – number of electrons, F – Faraday constant, $\frac{dE}{dT}$ – gradient of potential to temperature. The most commonly used method for computing entropy is the simple potentiometric method, which has been validated in many studies [21,23]. This method involves measuring the open circuit voltage of the battery at different temperatures to obtain the gradient of potential to temperature (dE/dT). However, this method can be time-consuming because the battery needs to reach equilibrium before the open circuit voltage can be accurately measured. Achieving equilibrium can require a significant amount of time.

Nicolas et al. developed a fast method for measuring entropy by measuring the heat generated by a battery during charge and discharge cycles. This method requires data on the heat capacity of the battery and the difference in heat generation between charging and discharging. The researchers validated their method using an LiFePO4/C battery [26]. Another group of researchers, A. Murashko et al. also used heat generation measurements to compute entropy, but they employed a different fitting method from that used by the previous researchers. They validated their model using a simple potentiometric method for computing entropy [27].

Zhenfeng et al. have presented a novel technique that improves the speed and precision of potentiometric measurements. Their method involves reducing the open circuit relaxation time by making positive adjustments during the battery's state of charge until the appropriate level is reached. To determine the battery's entropy, the authors collected relaxation data under two conditions: 1) at a constant temperature, and 2) while varying the temperature [25].

There are limited data of entropy profiles of batteries; however, there are already investigated entropy profiles of the following cells: LiNi_0.5Mn_1.5O₄/Li [24], LiCoO2/Li [21], Li1+yMn2 yO4/Li [28], NCA 18650 cylyndrical [29], LiFePO₄/C [30], LiCr_yMn_2-yO₄ [31], and Li/MnO₂ [32].

In summary, entropic potential measurements are a vital tool for enhancing the cyclability of lithium-ion batteries by offering deeper insights into the thermodynamic processes that affect their long-term stability and performance. The aim of this paper is to improve the entropy measurement method by reducing the measurement time. We believe that decreasing the measurement time while maintaining high accuracy will facilitate the collection of more entropy data. This expanded data set can enable researchers to develop precise methodologies for using entropy measurements in battery diagnostics.

This study proposes the development of a potentiometric method based on positive adjustment, which partially utilizes the relaxation data of open-circuit voltage at different temperatures. After this partial utilization, mathematical processing was applied to evaluate entropy, which will be discussed in detail further in the paper. The effectiveness of this method is confirmed by comparing it with a simpler potentiometric method, which is thoroughly explained in the results and discussion section. By employing this validated technique, the entropy profile of the battery at different cycle numbers was gathered. Based on these entropy profiles, it was determined that a linear regression model for SOC estimation is effective across various SOH.

Briefly, this article brings a new look to the existing literature in the following areas.

• (I)Development of entropy measurement techniques: The paper introduces a novel approach to reduce the measurement time of entropy profiles in LIBs, which is crucial for enhancing the accuracy and efficiency of battery diagnostics.
• (II)Non-destructive characterization methods: It highlights the potential of entropy measurements as a powerful non-destructive tool for assessing the structural and thermodynamic properties of batteries, offering insights that complement existing techniques.
• (III)Battery diagnostics and health monitoring: While the concept of improving the accuracy of SOC estimation is introduced, the study primarily focuses on providing additional entropy profile data across various SOH. This data is then used to verify a universal SOC model applicable at different SOH.
• (IV)The article demonstrates the potential application of thermodynamic concepts, such as entropy, in advancing the understanding of battery degradation mechanisms. It also illustrates how changes in entropy profiles at various states of health can be used to analyze the root causes of high or low cyclability and performance in lithium-ion batteries entropy.

The article is organized as follows.

• -Chapter 2: Materials & Methods This chapter details the materials used in the study and the methods employed. The new developed method is discussed in discussion chapter.
• -Chapter 3: Results & Discussion
• −3.1 Time Consumption of the Simple Potentiometric Method: An analysis of the time required for entropy measurement using the simple potentiometric method.
• −3.2 Review of Attempts to Minimize Time Consumption in Entropy Measurements: A discussion of previous efforts to reduce the time required for entropy measurement and their limitations.
• −3.3 Entropy Measurement Based on the Simple Potentiometric Method: Presentation of results obtained using the traditional method and their implications for battery characterization.
• −3.4 Development of New Methodology for Entropy Measurement: Introduction and validation of the novel fast potentiometric method, including a comparison with the traditional approach.
• −3.5 Review of Entropy Profile Interpretation.
• −3.6 SOC Assessment: Discussion on the application of the entropy data obtained through the fast method for accurate SOC assessment.
• -Conclusion The article concludes with a summary of key findings, the impact of the new methodology on battery characterization, and suggestions for future research directions.

2. Materials & methods

In this study, cylindrical NCR 18650 Panasonic lithium-ion rechargeable cells with Lithium Nickel Cobalt Aluminum Oxide (NCA) chemistry were used. These cells have a nominal voltage of 3.6 V and a capacity of 3.2 Ah. According to the manufacturer's guidelines, the recommended voltage limits for charging and discharging are specified to be within the range of 2.5V–4.2V.

2.1. Instrumentation

The charge and discharge profile and open-circuit voltage of various SOC were examined by a BT2000 battery tester (Arbin Instruments Co.). The temperature was controlled by a programmable temperature-controlled incubator. The temperature of the battery was measured by UT320 Series Mini Contact Type Thermometers (Maker).

Methodology of a simple potentiometric entropy evaluation method.

The method consists of the following steps.

• 1)Precycle: 0.05C rate, voltage range 2.7–4.2 V, and determine discharge capacity of the battery.
• 2)Full charging: charge to 4.2 V at a constant current of 0.05C, set as 100 % state of charge
• 3)Waiting for equilibrium of OCV for 12 h
• 4)Measuring of OCV at various temperatures: 5–30 °C
• 5)Discharge at 0.1C to appropriate SOC
• 6)Repeat steps 3, 4 and 5 till reaching final SOC.

2.2. Software

For data processing, MATLAB R2023 was utilized. Moreover, the analysis also made use of several open-access Python 3.10 libraries, including Scipy, Matplotlib, Numpy, and Pandas.

2.3. Entropy profile measurement using a simple potentiometric method

The potentiometric method is often used to measure the entropy of a battery at various states of charge, but it is a time-consuming process. To understand the origin of this problem, it will be helpful to analyze the steps of measurement using this method. It relies on measuring the cell's potential under specific temperature conditions when it reaches thermodynamic equilibrium. The temperature of the battery must remain constant during the measurement. Then, the voltage of the battery can be determined at this specific temperature. This procedure should be repeated at several temperatures to determine the voltage values at different temperatures. Based on these values, the dependence of voltage on temperature should be plotted, and the slope of this dependence should be defined by an eq. (2):

Slope = dV/dT (2)

Then, based on the value of the slope, the entropy value at a specific state of the battery can be determined using eq. (3):"

ΔS = slope*n*F (3)

Where, n – number of electrons, F – Faraday's constant.

The entropy of the battery varies at different states of charge and health. Therefore, to obtain the entropy profile of the battery (ΔS vs. SOC), the state of charge of the battery must be specified. The state of charge of the battery is defined by eq. (4):"

SOC = (remaining capacity)/(Full capacity)*100 (4)

The full capacity can be determined by charging and discharging at 0.05C within an appropriate voltage range. Subsequently, the battery is charged until it reaches maximum voltage, establishing a fully charged state with full capacity. To achieve a specific SOC, the fully charged battery is discharged until the desired SOC is reached. After collecting entropy measurements at various SOC levels of the battery, it is possible to plot the entropy profile of the battery.

3. Results & discussion

This study examined the entropy of the NCR18650 Panasonic battery at different cycle numbers. Fig. 1 displays the charge/discharge profile of the battery at a 0.05C rate, which is similar to that of a transition metal cathode. Fig. 2 shows the discharge capacity of the battery at each cycle number. It can be seen that 80 % of initial capacity is reached at around 1500th cycle. Entropy profiles of the battery were calculated at selected cycle numbers using the fast entropy estimation method. To verify this method, it was compared to the simple potentiometric method.

Fig. 1.

Charge discharge profile of the battery at various cycles. The mode of charge discharge process is as follow: C-rate = 1.5, voltage range: 2.7–4.2 V, at ambient temperature.

Fig. 2.

Dependence of the discharge capacity to cycle number of the battery at 1.5 C-rate.

3.1. Time consumption of simple potentiometric method

As mentioned before, one of the main issues with entropy measurements is the significant amount of time they consume. One of the primary research objectives of this article is to minimize this problem. The simple potentiometric method can be divided into several steps, as shown in Table 1. According to this table, the time consumption of each step is evaluated based on the review analysis. It can be noted that the second step of entropy measurements consumes a lot of time. This is due to the high relaxation time after the discharge or charge of the battery. Therefore, in this research work, we attempted to minimize this specific step by applying pulse discharge. Further discussion on this improvement of time consumption will be provided. X. Zhang et al. Gregory conducted a literature review on entropy measurents of lithium batteries. Based on their works it can be concluded that voltage relaxation time per SOC point ranges from 3 to 60 h [33].

Table 1.

Average time for each step in the entropy measurement process using the simple potentiometric method.

Step	Purpose of step	Time needed for measurement at specific SOC	Reference
1	Discharge till specific SOC	Depends on the SOC step, but it can be maximum 1 h for 10 % SOC change	[33]
2	Waiting for the thermodynamics equlibrium	For lithium ion batteries, it can be in the following range: 3h–60h	[33]
3	Measurting the voltage at specified temperature	In the range of 1h–2 h	[33]

3.2. Review of attempts to minimize the time consumption of entropy measurements

According to the literature review, there are several attempts to minimize to time comsuption during entropy measurements of the battery. The following main methods are identified: 1. Dynamic measurement of entropy [34], 2. Hybridized time-frequency method [35]. Inverse Heat Transfer Analysis Method [36]. Each method will be discussed further.

Dynamic measurement method determines entropy without waiting for the thermodynamic equilibrium. The cell's temperature continually adjusts during the measurement process, reducing the measurement time by eliminating the waiting period for equilibrium. In this method, an exponential temperature change is essential for measuring the entropy coefficient using this method. In contrast, the potentiometric method requires the temperature and voltage to be in at least two different static states, which are then compared. Unlike the potentiometric method, this new approach does not rely on thermodynamic equilibrium. It involves only a dynamic progression of temperature and voltage and utilizes their relationship.

Without waiting for the static state of the battery, the essence of measurement is based on subtracting the dynamic values of voltage and temperature to remove the effect of relaxation time. Therefore it is named as the “dynamic method [34].

The hybridized time-frequency method also focuses on minimizing relaxation time after the discharge process. After discharging to a specific State of Charge (SOC) and waiting for 10 min in a static state, a sinusoidal temperature reference is applied to the thermostat system. The sinusoidal temperature change of the battery leads to a sinusoidal change in voltage. The data collected for the temperature and voltage were used to evaluate entropy.

The previously described methods were based on measuring voltage, time, and temperature simultaneously. Then, based on these values, the slope shown in Equation (1) was determined. With the help of this, the entropy value can be determined. However, the 3rd method Inverse Heat Transfer Analysis is based on the measurement of the heat generation. The battery heat generation can be expressed by equation (5):

$$\text{Qgen}=I\left(OCV-Vcell\right)-IT\frac{dOCV}{dT}$$ (5)

Where, Q – heat generation during charge discharge process, I –current, OCV – open circuit voltage, Vcell – voltage of cell, T - temperature.

In this method, estimation of entropy is conducted by discharging the battery under four different current rates to inversely estimate the entropic coefficients, and least squares regression is conducted to optimize the derived entropic coefficients.

To sum up specifics of these three methods, it is illustrated in Table 2 below.

Table 2.

Developed entropy measurements methods to minimize the estimation time.

Method	The reduced measurement time compared to the potentiometric method	Battery type used to test the method	Ref.
1. Dynamic measurement of entropy	Reduced to around 1/13	a pouch-type lithium-ion cell with a NMC/C chemistry	[34]
2. Hybridized time-frequency method	Reduced to around 1/10	LMO/NMC pouch type LIB	[35]
3. Inverse Heat Transfer Analysis Method	Reduced by 93.7 %	21700 battery cell. Chemistry of battery is not specified	[36]

3.3. Entropy measurement based on simple potentiometric method

Initially, the entropy of the battery was calculated using a simple potentiometric method. The accuracy of the new proposed method will be checked based on the values obtained from the simple potentiometric method. The OCV of the battery was measured at different SOC by varying temperature, as displayed in Fig. 1S. Based on these measurements, the linear correlation of OCV to temperature was plotted in Fig. 2S. By using this linear correlation, the slope, which represents the potential gradient to temperature (dE/dT) in equation (1), was determined. Thus, the entropy of the battery at the corresponding state of charge can be calculated. Finally, the dependence of the entropy to SOC of the battery is depicted in Fig. 3.

Fig. 3.

Entropy profile of the battery at various state of charge (SOC).

It is noticed from Fig. 3 that entropy changes significantly for different SOC. This behavior is nonlinear and its shape is similar to large anisotropic unit volume changes happening in the LiNixCoyAlzO2 (NCA) type battery. Upon de-lithiation of the NCA cathode, the lattice parameter of the c-axis is increased up and until 4 V, where it suddenly dropped upon further de-lithiation, while the lattice parameter of the a-axis was steadily decreasing. Fig. 5 depicts the same behavior since cathode de-lithiaiton indicates an increase in the state of charge. Since entropy is related to a degree of randomness, microscopic changes happening inside a cell can be observed by noninvasive technique such as entropymetry. This nonlinear behavior is reported for the first time to the best of our knowledge. It can also be noticed from Fig. 3 that the value of entropy in 80–100 % SOC does not change significantly. It can be inferred that this value conforms to the characteristics of a first-order two-phase transformation, which is often accompanied by a large volume change, resulting in large internal stress, particle fracture, and deterioration of cycle performance. This behavior also occurs at various SOH of the battery.

Fig. 5.

Verification of the fast entropy estimation method by comparison with the simple entropy estimation method. R² = 0.9994.

3.4. Development of new methodology for entropy measurement: the fast potentiometric method

Kashiwagi et al. studied the entropy profiles of Mn₂₋yO₄ spinel structures with various types of metal doping. Given the lengthy measurement time of 10 h per step, they used a quadratic function to estimate the self-discharge of the half-cells, which allowed them to correct the results accordingly [37]. Thomas et al. expanded on this method and managed to shorten the measurement time for entropy profiles somewhat. They achieved this by identifying the drift background using a similar charge/discharge procedure, but without incorporating temperature variations [38]. However, this method assumes ideal thermodynamic behavior, which could pose issues if the state of charge is affected by temperature variations between charge steps.

In our study, we took an even simpler approach. Following Hudak et al., we suggest that all the information needed to eliminate drift from the temperature-program response curves is already embedded within those curves, as long as the temperature program is carefully designed [39]. In contrast to Pei et al., who recently introduced a sophisticated extrapolation method to predict equilibrated OCV using a second-order RC circuit model [40]. We propose that having an exact value of the equilibrium OCV is not crucial; instead, accurately measuring the voltage/temperature slopes is sufficient. These measurements can then be used with extrapolation techniques to account for changes due to voltage relaxation.

Previously, we discussed that the estimation time is very high due to long relaxation time after discharge of the battery. It is because the open circuit voltage (OCV) tends to decrease after the termination of the charging process, while OCV tends to increase after the termination of the discharging process. This behavior is primarily due to polarizations caused by the low diffusion of ions in the cathode or anode material. The low diffusion of ions creates nonuniform ion concentrations, leading to a shift in potential.

To partly minimize the issue of high relaxation time, we propose a charging after pulse discharging method. This method involves applying a low-rate, moderate-duration charging process immediately after discharging. Theoretically, this process accelerates activation depolarization by removing electron accumulation from the electrode surface, thereby reducing the overall relaxation time and making the entropy measurement process more efficient [25]. After partially eliminating the relaxation of OCV, the OCV is then measured at various temperatures, as shown in Fig. 3S.

In Fig. 3S, the OCV of the battery measured at different SOC by varying temperatures is shown. Here, it is noted that the partial polarization is not removed fully, even if positive adjustment while achieving the appropriate SOC is reached. Therefore, the additional mathematical processing of values is necessary. Once the application of positive current is completed, the battery voltage is rest and measured at the initial temperature for 1 h.

Next, the temperature is lowered to a value of T₁. At this temperature, the battery voltage is measured for 2 h. Subsequently, the temperature of the battery is raised to a value of T₂. At this temperature, the battery voltage is measured for 1 h. Processing of Results:

Based on this experiment, two key voltage values are obtained.

• 1.Voltage at Initial Temperature (T1): The value of the voltage measured at the first temperature.
• 2.Voltage at Final Temperature (T2): The value of the voltage measured at the final temperature, labeled as E₁.

To mathematically process the results.

• •The voltage data at the initial temperature T₁ is linearly extrapolated to the time when the OCV is measured at the final temperature T₂.
• •The extrapolated voltage value is referred to as E₂, while the actual measured voltage at T₂ is E₁.

The obtained values are then used in eq. (6) to calculate the entropy. To provide a clear understanding of the mathematical process, this is illustrated in Fig. 4.

$$\Delta S=n*F*\frac{E_2-E_1}{T_2-T_1}$$ (6)

Where.

• •n is the number of electrons involved in the reaction,
• •F is the Faraday constant,
• •E₂ is the extrapolated voltage at temperature T₂,
• •E₁ is the voltage at temperature T₁,
• •T₂−T₁ is the temperature difference between the two states.

Fig. 4.

Mathematical processing used to obtain the entropy value using the fast method. The region labeled with the digit “1″ indicates where the voltage is stabilized after partial mitigation of relaxation through positive adjustment.

The fast potentiometric method consists of the following steps.

• 1)Precycle: 0.05C rate, voltage range 2.7–4.2 V, and determine discharge capacity of the battery.
• 2)Full charging: charge to 4.2 V at a constant current of 0.05C, set as 100 % state of charge.
• 3)Discharge at 1 C to decrease SOC by 11 %
• 4)Charge at 0.2C to increase SOC for 1 %, so overall SOC is decreased by 10 %. Steps 4 and 5 are to minimize the partial polarization of electrodes which leads to decreasing of relaxation time
• 5)Measuring of OCV at various temperatures: 15–30 °C
• 6)Repeat steps 2, 3, 4 and 5 till reaching final SOC.

The values of entropy obtained through the fast potentiometric method are confirmed by comparison with the results obtained through the simple potentiometric method, as depicted in Fig. 5. The coefficient of determination (R²) was found to be 0.9994. According to Table 3, it can be noted the error of estimation for the each SOC. We can conclude that the method shows high accuracy in the following region of SOC: 20–80 %.

Table 3.

Developed entropy measurements methods to minimize the estimation time.

SOC	$\Delta S$ measured by simple potentiometric method	$\Delta S$ measured by proposed fast measurement	Error (%)
90	9,60	12,54	23
80	10,50	9,87	6
70	5,10	5,61	9 %
60	6,76	6,61	2 %
50	19,42	19,23	1 %
40	14,33	13,80	4 %
30	6,76	5,53	22 %
20	−5,60	−10,84	48 %

Table 4 illustrates the overall estimation time for a specific SOC using various measurement methods. Based on this, we can conclude that the new proposed method is 4.5 times faster than simple potentiometric method.

Table 4.

Estimation time difference between two methods.

Step	Simple potentiometric method	Proposed fast measurement
Reaching of specific SOC	Around 1 h	Around 10 min
Voltage measurement at various temperatures	Around 18 h	Around 4 h
Sum	Around 19 h	Around 4 h 10 min

Based on the high accuracy demonstrated by the fast potentiometric method, as depicted in Fig. 4 and Tables 3 and it was employed for further analysis of entropy at various SOH levels. Once the fast potentiometric method was confirmed, entropy profiles of the battery were obtained for various cycle numbers, as shown in Fig. 6. The figure highlights a rapid change in entropy within the 40–60 % SOC region, which is believed, corresponding to a significant phase transition that was previously identified through differential capacity analysis of the charge-discharge profile at the 3.3–3.7 V region (Fig. 4S) [8]. As the battery degrades, the amplitude of the peaks in the entropy profile decrease, which is consistent with findings reported by Hye J. in the case of a LiCoO₂ battery. This behavior is attributed to the high ordering of the structure as battery degradation increases, leading to a decrease in entropy, due to an increasing ordering of the microstructure of the material [41].

Fig. 6.

Entropy profile of NCR18650 PANASONIC battery at various cycles of battery at 1.5C rate.

Overall, the shape of the entropy profile of the battery depends on the entropy profile of the cathode and anode materials. According to Fig. 5, it can be noted that the overall shape of the battery was changed slightly, as instead the magnitude of the entropy is slightly shifted in the following cycling numbers: 0–177 cycles. This means there is little change in the microstructure of the cathode and anode parts. The shifting of the entropy value can be explained by lithium loss by the following possible reasons: electrolyte decomposition, oxidation of electrolyte, lithium plating, formation of Li grains, and/or solvent co-intercalation. These processes are related to a reaction happening between the electrolyte – cathode and the electrolyte – anode interphases. Due to these reactions, the microstructure of the cathode and anode does not change, but the content of active lithium is decreased, which leads to a shift in entropy profile. At the same time, from the 0^th – 177th cycles, the abrupt change in capacitance is seen, which is explained by the lithium loss processes discussed above. According to Fig. 2, the discharge capacity of the battery changed slightly between the 177–277th cycles. This is possibly due to the stoppage of the chemical process between the electrolyte and anode interphases. However, the entropy profile and discharge capacity again start changing during the 277–1478 cycles. In this region, it is mainly due to a microstructure change of the cathode and anode materials since the entropy profile shape is not just shifting, but its shape is also changing.

Moreover, according to the OCV profile at various SOH (Fig. 7), the two main phase transition regions at 40–60 % and 70–90 % SOC can be seen. These phase transitions can also be seen in the entropy profile depicted in Fig. 6. These transitions can be explained by two different chemical reactions happening in the cathode: oxidation of Ni, and oxidation Co. So, there is a two phase solid solution reaction. Overall, it can be noted that as SOH of the battery decreases, the valley seen at 100th cycle between 40 and 60 % SOC is disappearing. This is due to cation mixing of Ni and Co which further leads to the single-phase solid solution reaction. The shape of the ideal single phase and two phase solid solution reaction looks similar to our observation (Fig. 5S). Therefore, the entropy gradually decreases with no peaks or valleys at the lowest SOH (the 1477th cycle) (see Fig. 6).

Fig. 7.

OCV profile of NCR18650 PANASONIC battery at various cycles of the battery.

It can also be observed that the fluctuations in the entropy profile curves within the SOC regions of 30–60 % and 60–90 % decrease as the SOH of the material decreases. L. Cells et al. measured the entropy profiles of NCA cathode chemistry in 18650 cylindrical cells at various SOH, and a similar behavior was noted: the fluctuations in the entropy profile also decreased as the SOH declined [42]. To conduct more comprehensive studies, it would be beneficial to include in-situ X-ray diffraction studies. This approach could be the focus of future research by our team. Although we have not conducted X-ray diffraction studies, we have performed a literature review on interpreting entropy profiles across various SOH in Section 3.5.

3.5. Review of entropy profile interpretation

To better interpret entropy profiles of the battery, it is crucial to have a thorough understanding of the degradation mechanisms. The degradation process can manifest differently across various components of the battery, such as the cathode, anode, and electrolyte. This degradation can occur under different operational modes, including charging and discharging cycles or through calendar aging. Additionally, these processes can be influenced by various conditions, such as temperature, C-rate, and depth of discharge. To gain deeper insights into the relationship between entropy profiles and the specific conditions or components of battery degradation, it is essential to consider how these variables interact and impact the overall health and performance of the battery.

The shape of the entropy profile of a battery directly depends on the entropy profiles of both the cathode and anode materials. It should be noted that during the discharge of a LIB, reduction occurs at the cathode while oxidation takes place at the anode. Therefore, the total change in entropy for a cell during discharge is defined by Ref. [42]:

$$\Delta S={\Delta S}_c+{\Delta S}_a$$ (7)

where ${\Delta S}_c$ - is the entropy change of the cathode material during the reduction reaction, ${\Delta S}_a$ is the entropy change of the anode material during the oxidation reaction.

However, the electrolyte in the battery can indirectly influence the entropy profile through the degradation of the battery. The electrolyte can affect the microstructure of the cathode and anode, which, in turn, impacts the entropy profile as defined in Eq. (7).

K. Maher et al. measured the entropy profile of an LCO/graphite LIB cell at various cycle numbers. They proposed an interesting idea that changes in entropy values at specific SOCs can correspond to the respective parts of the battery: anode or cathode. For instance, they observed that the entropy value at 5 % SOC did not change significantly after 1000 cycles. They hypothesized that the change in entropy at 5 % SOC is linked to the microstructural changes in the anode. This hypothesis was validated by XRD analysis of the anode, comparing its initial state with its state after 1000 cycles. XRD analysis showed minimal changes, indicating that the anode's structure remained relatively stable [43]. Another study by this researcher investigated the entropy profile of an LCO/graphite cell by applying high-voltage charging between 4.2 V and 4.9 V cut-off voltages. They found that the entropy and enthalpy profiles varied dramatically with the applied cut-off voltages. Additionally, they reached the same conclusion as in their previous work: the changes in thermodynamic properties were tentatively related to crystal structure deterioration at the anode (0 % SOC) and the cathode (80 % SOC) [44].

In addition, it is important to note that the degradation mechanism can vary depending on conditions such as temperature, C-rate, and depth of discharge. M. Voit et al. investigated the change in the entropy profile of LCO/graphite under various C-rates. They found that the entropy profile remained unchanged at appropriate C-rates, even in the presence of some capacity loss. For instance, LiCoO₂ electrodes cycled 20 times at C/2 in half-cells experienced significant loss in specific capacity but showed no measurable changes in the entropy profile. Based on this result, they proposed that the degradation in performance was due to increasing kinetic barriers to lithiation and delithiation, rather than structural or thermodynamic changes in LixCoO₂. This suggests that entropymetry has potential for revealing the causes of degradation [45]. We have reviewed some interpretations of entropy profiles concerning the components of LIBs and various degradation conditions. However, there is still no comprehensive guide for interpreting entropy profiles. We believe this is due to the lack of an extensive database of entropy profiles for each part of the battery across various SOH and degradation conditions. As discussed in the Introduction, this gap is largely due to the lengthy process involved in conventional entropy measurement methods. We believe that our developed method could provide a significant advancement in the field of entropymetry interpretation.

3.6. SOC assessment

Manane et al. have proposed a novel method of state of charge assessment based on eq. (8). This mathematical model was verified using Li/MnO₂ batteries [16].

$$\text{SOC}=\alpha +\beta *\Delta S+\gamma *\Delta H$$ (8)

, where $\alpha ,\beta \gamma$ – coefficients, $\Delta S$ – entropy, $\Delta H$ – enthalpy. Entropy and enthalpy can be calculated by eq. (1) mentioned before, eq. (9), and eq. (10):

$$\Delta H=\Delta G+T\Delta S$$ (9)

$$\Delta G=-n*F*E$$ (10)

eq. (3) was fitted to the entropy and enthalpy data of the cylindrical NCR 18650 battery at different SOH. Based on the R² value of the SOC assessment model, it can be observed that model functions effectively across various SOH of the battery. In this model, the coefficients depend on both SOH and nature of the battery.

For clear visibility of the SOC estimation error at various SOH levels, a bar chart was plotted in Fig. 8. Based on this chart, we observe that the SOC estimation error ranges from 0 % to 8 %. Notably, errors in the range of 6 %–8 % are predominantly indicated at 30 % SOC.

Fig. 8.

The bart chart plot of error of the estimation SOC at various SOH based on the mathematical model.

To estimate the State of Charge (SOC) using the provided mathematical model, the following input values are required.

• -Open Circuit Voltage (OCV): The OCV of the battery must be measured at various temperatures after applying the positive adjustment method.
• -Entropy and Enthalpy Values: Based on the OCV data, the entropy and enthalpy values can be calculated.
• -These calculated thermodynamic parameters are then employed in Equation (7) to assess SOC.

Additionally, it is important to note that the equation requires specific coefficients, which are dependent on the battery's chemistry and SOH, as demonstrated in the work by Yazami's team. These coefficients must be determined to accurately apply the model in real-world applications [32,46]. A list of coefficients at various SOH is provided in Table 5. However, future work should focus on identifying the relationship between the model's coefficients and the battery's chemistry and SOH to further improve the model's accuracy and applicability. When identifying the relationship between chemistry and coefficients, it's important to consider the impact of entropy changes in each component—anode, cathode, and electrolyte. Changes in the structure of each material can influence the entropy profile. However, according to the article, the most significant changes in the entropy profile are primarily associated with the cathode material [47].

Table 5.

SOC model coefficients at various state of health of the battery.

SOH [%]	α [%]	$\beta$ [mol * K/J]	γ [K/kJ]	R2
100	−371,298	0,4789	−1,1788	0,9943
96,33	−388,579	0,6120	−1,2240	0,9965
93,20	−403,533	0,6394	−1,2634	0,9968
93,15	−392,211	0,6636	−1,2300	0,9975
90,42	−397,364	0,6559	−1,2434	0,9977
88,03	−404,842	0,7588	−1,2619	0,9986
86,65	−407,797	0,6999	−1,2693	0,9990
84,84	−396,911	0,8011	−1,2353	0,9997
83,84	−416,094	0,6322	−1,2925	0,9986
79,35	−419,475	0,8370	−1,2935	0,9990

In this study, a total of 8 cells were tested to evaluate the results presented in Fig. 3. Specifically, 4 cells were used to clarify the results of the developed method by comparison with the simple potentiometric method. Additionally, 2 cells were tested to evaluate the results shown in Fig. 1, Fig. 2, Fig. 5, Fig. 6. All cells were subjected to identical testing conditions to ensure consistency in the results.

4. Conclusion

This paper introduces a novel method for rapidly estimating entropy by applying positive charging after discharging the battery until the appropriate state of charge (SOC) is reached, followed by mathematical processing of the results. The method was verified by comparing it to the traditional potentiometric approach. It was then used to obtain the entropy profile of a cylindrical NCR18650 battery at different states of health.

The results revealed a rapid change in ΔS in the 40–60 % SOC region, indicating a significant phase transition. This phase transition was further confirmed through differential capacity analysis. Additionally, two primary phase transition regions were identified at 40–60 % and 70–90 % SOC through entropy and OCV measurements.

The newly proposed fast method for entropy determination provides relatively accurate results while significantly reducing the time required. In the conducted experiments, under the specified conditions and with the particular cell used, the method was approximately 4.5 times faster than the potentiometric method for entropy measurement. This improvement in methodology can significantly advance non-destructive battery characterization using entropymetry techniques. Utilizing this new, faster method of entropy estimation, entropy profiles for various States of Health (SOH) were established. It was observed that at each SOH, a linear mathematical model for SOC estimation can be employed using the coefficients determined, as illustrated in Table 5. These findings could be instrumental in the development of new thermodynamics-based battery management systems.

Funding information

This research was funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09261149)

CRediT authorship contribution statement

Yerkin Serik: Writing – original draft, Software, Formal analysis, Data curation. Desmond Adair: Writing – review & editing, Funding acquisition, Formal analysis. Zhumabay Bakenov: Writing – review & editing, Supervision. Berik Uzakbaiuly: Writing – review & editing, Supervision, Project administration, Methodology, Funding acquisition.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Desmond Adair reports financial support was provided by Ministry of Education and Science of the Republic of Kazakhstan. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.