A modified pulse charging method for lithium-ion batteries by considering stress evolution, charging time and capacity utilization

Abstract

The stress evolution, total charging time and capacity utilization of pulse charging (PC) method are investigated in this paper. It is found that compared to the conventional constant current (CC) charging method, the PC method can accelerate the charging process but will inevitably cause an increase in stress and a decrease in capacity. The charging speed for PC method can be estimated by the mean current. By introducing stress control, a modified PC method called the PCCC method, which starts with a PC operation followed by a CC operation, is proposed. The PCCC method not only can accelerate charging process but also can avoid the stress raising and capacity loss occurring in the PC method. Furthermore, the optimal pulsed current density and switch time in the PCCC method is also discussed.

Keywords

fast charging method / pulse charging / stress evolution / charging time / capacity utilization

Introduction

Charging time for lithium-ion batteries is a key factor for commercialization and popularization of electric vehicles. Currently, fully recharging an electric vehicle could take long several hours [¹], which considerably reduces the motility of vehicles and the interests of customers. Simply increasing the charge current indeed accelerates the charging process, but also causes fast degradation of the batteries and consequently reduces their life-time [²–⁴]. To overcome this issue, properly designed fast charging methods are urgent to be developed.

Pulse charging (PC) method, which includes multiple high current pulse stages separated by low current relaxation stages (as shown in Fig. 1), is a typical fast charging method for lithium-ion batteries [⁵–¹¹]. Interestingly, in literature, some results show that PC brings benefit to lithium-ion batteries but others show that impact of current pulses is detrimental. For instance, Li et al. [⁵], de Jongh and Notten [⁶], and Purushothaman and Landau [⁷] suggested that PC reduces charging time, concentration gradients, leading to a better active material utilization and improvement of life-time. However, Savoye et al. [⁸] found that current pulse profiles does not show any benefit on cell performance, but is detrimental to the electrochemical performance. The long-term ageing test performed by Beh et al. [⁹] also indicated that PC would lead to lower capacity and efficiency compared with conventional CCCV charging method, i.e., constant-current (CC) followed by constant-voltage (CV) charging. Some other simulations [¹⁰] and experiments [¹¹] also suggested that PC method does not have advantages over CC method with the same mean charging current. Argument about the PC method for lithium-ion batteries has never come to a stop.

The existence of these contradictory results may be due to not fully or comprehensively understanding of some intrinsic physical mechanisms. Heuristically, Li et al. [⁵] suggested that microcracking caused by diffusion-induced stresses can be suppressed by PC and therefore electrochemical performance can be improved. This indicates that the mechanical stress plays a crucial role in the PC method. Actually, diffusion-induced stresses and subsequent mechanical failure have effects on electrochemical performance of lithium-ion batteries, including mass transfer [¹²,¹³], electron transfer [¹⁴], voltage profile [¹⁵], capacity [¹⁶] as well as life-time [³,¹⁷–¹⁹]. Stress evolution might be a key to the puzzle of the PC method. However, this has not been discussed sufficiently.

In fact, stress evolution in electrode materials highly depends on charge/discharge operations. For instance, Cheng and Verbrugge [²⁰] found out that the stress increases until a steady-state in a CC operation, whereas it increases initially and then decreases with time in a CV operation. Zhang et al. [²¹] performed a similar work for layered electrodes, and suggested a charging protocol, i.e., CC first and CV followed.

The stress evolution for fast charging method is more complicated than that under single operation, because the fast charging usually follows complex procedures. In our previous work, we developed a model to investigate the stress evolution for multistep charging method [²²]. It is found that a properly designed procedure, which consists of an initial CC stage of high current, a CC stage of low current and a CV ending stage, is helpful not only in accelerating the charge process but also in controlling the mechanical stress. As a classic type of fast charging method, PC can also be properly designed in a similar way by considering the stress evolution. Introducing stress analysis into consideration will also provide new insights of PC method.

In this paper, we will provide a model to discuss the stress evolution for PC. Discussions will be given for design reference by considering the capacity, charging time and mechanical stability of the battery. Based on calculation results, the reason for the contradictory results based on mechanical analysis will be discussed. In addition, an optimized PC method, i.e., PCCC, will be suggested.

Analysis

Figure 1 illustrates a typical PC method which consists of multiple pulse stages and relaxation stages. A pulse stage with high current density (i_high) accelerates the charging process. However, the high current density cannot be applied for too long, otherwise the fast degradation of the batteries would be initiated [²–⁴]. The duration of the pulse stage is denoted by t_high. The relaxation stage with lower current density i_low follows the pulse stage. After each relaxation interval of t_low, the pulse stage will be reapplied and accelerate charging again. The repeated single stage in the PC process actually can be considered as a short-time CC stage.

Particles composed of active materials are the main basic components of electrodes in commercial lithium ion batteries, as shown in Fig. 2. The concentration and stress in the electrode particle are highly related to the electrochemical performance of lithium-ion batteries and have been widely focused [¹³,¹⁶,¹⁷,²⁰,²²]. Therefore, a 3D spherical active particle with radius R in electrodes is considered here. The stage-dependent boundary condition for diffusion of Li-ions is given by

(1){D{\displaystyle \frac{\partial c}{\partial r}}|}_{r=R}=\{\begin{array}{l}{\displaystyle \frac{i_{\mathrm{high}}}{F}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{pulse}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ {\displaystyle \frac{i_{\mathrm{low}}}{F}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{relaxation}\end{array},

where c is the molar concentration of lithium, D is the diffusion coefficient, F=96485.3 (C/mol) is Faraday constant, the current density i<0 represents delithiation while i>0 represents lithiation. Due to non-uniform initial concentration profile caused by past stages, the initial condition for the single elementary stage reads as

(2)c=c_{\mathrm{0}}(r)\hspace{0.17em}\mathrm{for}\hspace{0.17em}\tau =\mathrm{0.}

The diffusion of Li-ions is assumed to be governed by Fick's law [²⁰,²²,²³]:

(3)D{\displaystyle \frac{\mathrm{1}}{r}}{\displaystyle \frac{{\partial }^{\mathrm{2}}(rc)}{\partial r^{\mathrm{2}}}}={\displaystyle \frac{\partial c}{\partial \tau }},

where t is the time variable in a single elementary stage. By solving Eqs. (1–3), we have the concentration evolution follows [²²]:

(4)\begin{array}{l}\bar{c}=I\left[\mathrm{3}\bar{\tau }+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\bar{r}}^{\mathrm{2}}-{\displaystyle \frac{\mathrm{3}}{\mathrm{10}}}-{\displaystyle \frac{\mathrm{2}}{\bar{r}}}{\displaystyle \sum _{n=\mathrm{1}}^{\infty }{e}^{-{\lambda }_n^{\mathrm{2}}\bar{\tau }}{\displaystyle \frac{\sin \left({\lambda }_n\bar{r}\right)}{{\lambda }_n{}^{\mathrm{2}}\sin ({\lambda }_n)}}}\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \frac{\mathrm{2}}{\bar{r}}}{\displaystyle \sum _{n=\mathrm{1}}^{\infty }{e}^{-{\lambda }_n^{\mathrm{2}}\bar{\tau }}{\displaystyle \frac{\sin \left({\lambda }_n\bar{r}\right)}{{\sin }^{\mathrm{2}}({\lambda }_n)}}}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}\sin \left({\lambda }_n\bar{\zeta }\right)\bar{\zeta }{\bar{c}}_{\mathrm{0}}(\bar{\zeta })d\bar{\zeta }}+Q_{\mathrm{0}},\end{array}

where \bar{c}=c/c_{\max }, {\bar{c}}_{\mathrm{0}}=c_{\mathrm{0}}/c_{\max }, \bar{r}=r/R, \bar{\zeta }=\zeta /R, \bar{\tau }=D\tau /R^{\mathrm{2}}, c_{\max } is the saturation concentration, {\lambda }_n are the solutions of {\lambda }_n\cot {\lambda }_n=\mathrm{1}, Q_{\mathrm{0}}=\mathrm{3}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}{\bar{\zeta }}^{\mathrm{2}}{\bar{c}}_{\mathrm{0}}(\bar{\zeta })d\bar{\zeta }}, and the stage-dependent dimensionless current density is

(5)I=\{\begin{array}{l}{\displaystyle \frac{i_{\mathrm{high}}R}{FD{c}_{\max }}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{pulse}\hspace{0.17em}\mathrm{stage}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ {\displaystyle \frac{i_{\mathrm{low}}R}{FD{c}_{\max }}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{relaxation\; stage}\end{array}.

The normalized amount of lithium Q' is expressed as

(6)Q^{\prime }=\mathrm{3}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}{\bar{\zeta }}^{\mathrm{2}}\bar{c}(\bar{\zeta })d\bar{\zeta }}=Q{}_{\mathrm{0}}+\mathrm{3}\bar{\tau }I.

According to Eq. (6), the capacity stored during the single elementary stage is proportional to the multiplication of current density and stage interval. Therefore, comparing to the CC process which has a constant current density of I_low, charging is definitely accelerated through pulse stages whose current density is higher.

The inhomogeneous concentration distribution expressed by Eq. (4) leads to diffusion-induced stresses. It is assumed that the active particle is isotropic and undergoes elastic deformation during charge/discharge. With satisfying stress-free mechanical boundary condition, the dimensionless radial and hoop stresses are respectively given by [²⁰,²²]:

(7a){\bar{\sigma }}_r={\displaystyle \frac{(\mathrm{1}-\upsilon ){\sigma }_r}{E}}={\displaystyle \frac{\mathrm{2}\Omega c_{\max }}{\mathrm{9}}}\left[Q^{\prime }-{\bar{c}}_{\mathrm{ave}}\left(\bar{r}\right)\right],

(7b){\bar{\sigma }}_{\theta }={\displaystyle \frac{(\mathrm{1}-\upsilon ){\sigma }_{\theta }}{E}}={\displaystyle \frac{\Omega c_{\max }}{\mathrm{9}}}\left[\mathrm{2}{Q}^{\prime }+{\bar{c}}_{\mathrm{ave}}\left(\bar{r}\right)-\mathrm{3}\bar{c}\left(\bar{r}\right)\right],

where \upsilon is Poisson’s ratio, E is Young’s modulus, \Omega is partial molar volume of active material, {\bar{c}}_{\mathrm{ave}}\left(\bar{r}\right)=\left(\mathrm{3}/{\bar{r}}^{\mathrm{3}}\right){\displaystyle {\int }_{\mathrm{0}}^{\bar{r}}{\bar{\zeta }}^{\mathrm{2}}\bar{c}\left(\bar{\zeta }\right)}d\bar{\zeta } represents the dimensionless average concentration in the spherical volume of radius r within the particle.

Particularly, the bi-axially stretched surface of the particle in delithiation process is commonly focused, because it could lead to surface crack which may result in growth of solid electrolyte interphase (SEI) and subsequent battery degradation [¹⁷,¹⁹,²⁴]. According to Eqs. (4) and (7b), the dimensionless stress at the surface, denoted by {\bar{\sigma }}_{\mathrm{1}}={\bar{\sigma }}_{\theta }(\mathrm{1}), is given by:

(8)\begin{array}{l}{\bar{\sigma }}_{\mathrm{1}}=\bar{i}\left[-{\displaystyle \frac{\mathrm{1}}{\mathrm{15}}}+{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{\displaystyle \sum _{n=\mathrm{1}}^{\infty }{\displaystyle \frac{\mathrm{1}}{{\lambda }_n{}^{\mathrm{2}}}}{e}^{-{\lambda }_n^{\mathrm{2}}\bar{\tau }}}\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}\Omega c_{\max }{\displaystyle \sum _{n=\mathrm{1}}^{\infty }{e}^{-{\lambda }_n^{\mathrm{2}}\bar{\tau }}{\displaystyle \frac{\mathrm{1}}{\sin ({\lambda }_n)}}}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}\sin \left({\lambda }_n\bar{\zeta }\right)\bar{\zeta }{\bar{c}}_{\mathrm{0}}(\bar{\zeta })d\bar{\zeta }},\end{array}

where

(9)\bar{i}=I\Omega c_{\max }=\{\begin{array}{l}{\displaystyle \frac{i_{\mathrm{high}}R\Omega }{FD}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{pulse}\hspace{0.17em}\mathrm{stage}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ {\displaystyle \frac{i_{\mathrm{low}}R\Omega }{FD}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\mathrm{relaxation\; stage}\end{array},

is a key dimensionless parameter called electrochemical load factor [²⁰,²²,²⁵,²⁶]. The illustration of the model is shown in Fig. 2. In pulse stage with a large electrochemical load factor, stress would increase rapidly. In order to offset the high stress generated in pulse stage, relaxation stage with a low electrochemical load factor should be applied. An improperly large electrochemical load factor or long pulse duration would result in an extreme stress level in the active material. According to this, the PC method should be properly designed.

In conventional CCCV protocol, the CV stage commonly takes considerable time in the whole charging process but only contributes very limited capacity. For example, in (LiNiCoMnO₂ + LiCoO₂)/graphite which makes a common battery material system, the CV stage in CCCV protocol takes more than 50% of total charging time but only contributes less than 20% capacity utilization [¹¹]. However, time is extremely crucial for fast charging methods, and therefore the CV stage has been abandoned in some recent studies of the PC method [²⁷,²⁸]. Hence, CV stage is also not considered in this work. The charging procedure in this work ends when surface concentration of the active particle reaches its limitation. Because of no CV stage, decrease of capacity utilization would be a problem for fast charging methods, and this will be evaluated in the following discussion.

Investigation into pulse charging method

For demonstration, we will focus on the positive electrode in which the active particle is delithiated and the surface is bi-axially stretched when the battery is charging. The maximum volume change \Omega c_{\max } is chosen as 0.07 [¹²,²⁹]. The initial concentration \bar{c}=\mathrm{1} is assumed for each charging process and the dimensionless surface concentration {\bar{c}}_s=\bar{c}(\mathrm{1}) reaches zero at the end of each charging process. In this case, the capacity for the positive electrode can be expressed as Q=1‒Q'. In order to quantitatively evaluate the charge acceleration of PC method, a standard CC with I_ref=‒2 is chosen as the reference procedure in this section. The current density of relaxation stages in all PC procedures is set as the reference value, i.e., I_low=I_ref.

The evolution of current density in PC procedures for different pulsed current density I_high is demonstrated in Fig. 3(a), in which dimensionless stage durations {\bar{\tau }}_{\mathrm{high}}=D{\tau }_{\mathrm{high}}/R^{\mathrm{2}} and {\bar{\tau }}_{\mathrm{low}}=D{\tau }_{\mathrm{low}}/R^{\mathrm{2}} are both chosen as 0.005, and dimensionless charging time is denoted by \bar{t}=Dt/R^{\mathrm{2}}. All PC procedures start with the pulse stage.

In Fig. 3(b) which demonstrates the evolution of the capacity, all solid lines with solid symbols represent PC methods and are on top of the reference (black dash line), indicating that PC method indeed accelerates the charging process. Increasing the pulsed current can further improve the acceleration. However, the accelerated charging process would lead to a decrease of the capacity utilization, as shown in Fig. 3(b). For instance, the PC procedure with I_high=‒5 can shorten approximately 60% of the charging time but causes 30% capacity decrease compared with the reference. This is quite similar to the CC procedure which simply increases the current. Therefore, three additional CC procedures are introduced in Fig. 3. The current densities of these CC procedures are correspondingly given by

(10)I={\displaystyle \frac{I_{\mathrm{high}}{\bar{\tau }}_{\mathrm{high}}+I_{\mathrm{low}}{\bar{\tau }}_{\mathrm{low}}}{{\bar{\tau }}_{\mathrm{high}}+{\bar{\tau }}_{\mathrm{low}}}},

indicating that these CC procedures have the same mean charging current with that for the corresponding PC procedures. As shown in Fig. 3(b), the evolution of capacity for PC and corresponding CC is almost identical. For example, the red solid line (with solid triangle) for PC with I_high=‒3 and the red dash line for CC with I=‒2.5 nearly overlap to each other. It agrees with the results in Refs [¹⁰,¹¹]. that PC method has no obvious advantage over CC method with the same mean charging current on the charging speed. This also indicates that the charging speed for PC method can be roughly estimated by the mean current.

In addition, capacity utilization of PC is even 3%‒5% lower than that of corresponding CC process. This is caused by the evolution of {\bar{c}}_s which is shown in Fig. 3(c). The dimensionless surface concentration {\bar{c}}_s decreases in pulse stages rapidly, and recovers in relaxation stages slowly. Hence, the surface concentration in PC procedure could be lower than the one in the CC procedure as shown in Fig. 3(c), and easier to reach its limited value, which leads to earlier finish of the charging process and lower capacity utilization.

According to Eq. (7b), the stress at surface {\bar{\sigma }}_{\mathrm{1}} highly depends on the surface concentration {\bar{c}}_s. Therefore, {\bar{\sigma }}_{\mathrm{1}} rapidly increases when {\bar{c}}_s decreases in pulse stages, and it decreases in relaxation stages while {\bar{c}}_s is recovering, as shown in Fig. 3(d). Interestingly, it is found that the high stress caused by pulse stage cannot be sufficiently relaxed in relaxation stage, which leads to constantly higher stress in the whole charging process and much higher maximum stress compared with the reference (black dash line). Increasing the pulsed current density I_high aggravates this phenomenon.

Therefore, in some cases that the active material has low damage toleration, PC would be detrimental to the mechanical stability of the electrode and leads to poor electrochemical performance. This explains the experimental results in Refs. [⁸,⁹,¹¹] in which the positive electrode materials are LiFePO₄, LiMn₂O₄, or LiNiCoMnO₂. However, for some materials which are not very sensitive to mechanical behaviors in charging process, the effect of PC on stress may not be important. For instance, when the positive electrode material is LiCoO₂ [⁵,⁶] whose maximum volume change \Omega c_{\max } is negative [¹²,²⁹], it would lead to compressive stress at the surface and no onset of surface fracture will occur when the PC method is applied. In other words, the mechanical stability of LiCoO₂ is barely affected by the PC procedure. Obviously, we provide a theoretical explanation of the contradictory results about PC method in the existing literature.

Besides the pulsed current, stage durations {\bar{\tau }}_{\mathrm{high}} and {\bar{\tau }}_{\mathrm{low}} are also key factors in PC method. Here, {\bar{\tau }}_{\mathrm{high}}={\bar{\tau }}_{\mathrm{low}}={\bar{\tau }}_0 is assumed for demonstration purpose and by this assumption, the mean current density would reduce to I_mean=(I_high+I_low)/2. The effect of stage duration {\bar{\tau }}_0 is shown in Fig. 4, in which I_high and I_low are chosen as ‒4 and ‒2, respectively, and therefore the mean current density is ‒3.

As shown in Fig. 4(b), the effect of {\bar{\tau }}_0 on capacity is insignificant. All solid lines with solid symbols almost overlap with black dot line which stands for the CC process with I=‒3. Reducing the stage duration or increasing the pulse frequency can promote the capacity utilization, which is of benefit to PC method. However, the capacity utilization is still much lower than the reference value.

The evolution of the surface concentration is considerably affected by stage duration as shown in Fig. 4(c) and so is the evolution of stress as shown in Fig. 4(d). With decreasing {\bar{\tau }}_0, the stress evolution becomes even more approaching to the black dot line of CC with I=‒3, and the maximum stress slightly decreases but is still much larger than the reference (black dash line). This explains the experimental data of battery cycle life where slightly faster capacity fade is observed for larger stage duration [¹¹].

The above discussions suggest that the classic PC method accelerates the charging process efficiently, but decreases the capacity and increases the stress significantly and inevitably. The large pulsed current density I_high causes fast increasing of stress and serious waste of capacity utilization, but the charging acceleration of PC method with too small I_high is insignificant and meaningless. Reducing the stage duration time {\bar{\tau }}_0 can improve the PC method, but the effect by doing so is quite limited. Thus, a new scheme is required to modify the PC procedure for a balance of charging acceleration, stress reduction and capacity promotion.

Modification of pulse charging method

Note that the stress increases monotonically in CC procedures and reaches its maximum value at the end of the charging process. We denote the maximum value of stress in the reference CC procedure as {\bar{\sigma }}_{\mathrm{ref}}. As shown in Fig. 3, the stress in the PC process is not always higher than {\bar{\sigma }}_{\mathrm{ref}}. It can be seen the stress is lower than {\bar{\sigma }}_{\mathrm{ref}} during a few early periods of the PC procedure but becomes higher than {\bar{\sigma }}_{\mathrm{ref}} in the rest of the charging process. In the analysis of multistep CC (MCC) method in Ref. [²²], a CC stage with low current is applied after a high current stage for suppressing the increasing of stress. Inspired by this, we propose here a PCCC method which starts with a standard PC operation followed by a CC operation, as shown in Fig. 5. The charging can be accelerated in the first PC operation, and the increasing of stress can be suppressed in the second CC operation. The switch from PC to CC occurs when {\bar{\sigma }}_{\mathrm{1}}={\bar{\sigma }}_{\mathrm{ref}}, which indicates that the proposed PCCC method introduced stress control into the whole charging process. The time that PC switches to CC is denoted by {\bar{t}}_{\mathrm{switch}}.

Interestingly, as shown in Fig. 5, the total charging time of the reference CC with I_ref=−2 is 11% longer than the presented PCCC. The differences of capacity utilization and maximum stress between them are negligible (less than 1%). On the other hand, compared to the PC procedure with the same operation parameters, the PCCC process leads to 24% increase in the capacity and 23% decrease in the maximum stress. This indicates that PCCC can accelerate considerably the charging process without losing much capacity or raising high stress. Therefore, the proposed PCCC method is a promising fast charging scheme to accelerate charging process without raising the risk of mechanical failure and capacity loss.

The evolutions of PCCC in Fig. 5 are quite similar to those of MCC proposed in Ref. [²²]. The properly designed PCCC and MCC will have similar charging speed, capacity utilization and maximum stress. Interestingly, a charging protocol similar to our proposed PCCC scheme was suggested by Ref. [³⁰] in which the self-heating in charging process is considered. It indicates the proposed PCCC scheme may be able to overcome the low temperature limitation of the battery and boost the battery performance. In addition, the electrochemical heat evolution during the fast charging is also crucial for lithium ion batteries and remains for further studies [³¹,³²]. The heat generation could also severely damage the electrode. Therefore, other factors such as thermal properties might also play roles and need to be further investigated in fast charging methods including PCCC and MCC.

Charging acceleration of PCCC method is evaluated in Fig. 6, in which {\bar{t}}_{\mathrm{end}} stands for the dimensionless total charging time and {\bar{t}}_{\mathrm{ref}} stands for the constant total charging time for the corresponding reference CC process. It demonstrates three different CC processes in which I_ref are −2, −3, and −4, respectively. As shown in Fig. 6, PCCC for the larger reference current leads to a smaller {\bar{t}}_{\mathrm{end}}/{\bar{t}}_{\mathrm{ref}} which indicates a better acceleration. Therefore, for conventional CC procedures with very high current densities, the modification based on PCCC would have considerable advantages. This also indicates that PCCC would be a promising method for the high power batteries whose charging current is commonly large [¹,¹¹].

It is also found that normalized total charging time in Fig. 6 decreases rapidly at first, and then slowly increases after a critical value of I_high/I_ref, indicating that there exists an optimal pulsed current density. According to the given results, the optimal I_high is about two times of I_ref, no matter how I_ref changes. Therefore, the selection of a proper value for I_high is quite easy for PCCC method. For the optimal PCCC method with I_high=2I_ref, the dimensionless switch time {\bar{t}}_{\mathrm{switch}} can be computational calculated via Eqs. (8) and (9) and is shown in Fig. 7. So far, the operation instruction of the optimal PCCC procedure is provided.

It should be noted that the fatigue problem caused by current pulses remains open here. The mechanical fatigue induced by charge/discharge cycling is one of the main reasons to cause battery degradation [¹⁷,³³,³⁴]. But the effect of the complicated stress spectrum induced by current pulses on fatigue has not been sufficiently studied. Qualitatively, the extra pulses definitely lead to more severe fatigue, which explains the experimental result in Ref. [⁹] which stated that PC method is detrimental to the long-term performance of LiFePO₄. However, as pointed by Ref. [⁵], long-term performance of LiCoO₂ is barely affected by PC procedure. This indicates that whether PC protocol is bad to the electrochemical performance depends on the material system of electrodes, which needs further theoretical as well as experimental studies.

Conclusions

In this paper, PC strategy, which includes multiple high current pulse stages separated by low current relaxation stages, has been evaluated based on stress evolution, total charging time and capacity utilization. Analytic expressions of concentration profile and diffusion-induced stress within a spherical active electrode particle under current pulses have been provided. Comparing to the conventional CC method, PC efficiently accelerates the charging process, in which the charging speed can be roughly estimated by the mean current. However, PC method also causes increasing of stress and decreasing of capacity utilization inevitably. Increasing the pulsed current aggravates these phenomena, while decreasing the stage duration (or increasing the pulse frequency) can only slightly provide improvements on stress and capacity. Therefore, from the point of view of mechanical analysis, PC method is not an ideal fast charging method for lithium ion batteries.

In order to improve the drawbacks occurred in PC method, a PCCC method which starts with a PC operation followed by a CC operation has been proposed in this paper. The switch from PC to CC occurs when {\bar{\sigma }}_{\mathrm{1}}={\bar{\sigma }}_{\mathrm{ref}}, which indicates that stress control is introduced to govern the charging process. It has been found that the proposed PCCC method is a promising fast charging method to accelerate charging process without decreasing the battery capacity utilization and raising the risk of mechanical failure. In addition, the optimal pulsed current density, which is two times of the relaxation current density, has been suggested.

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