1 Introduction

Nowadays, the development of electronic devices such as laptops and smart mobile phones leads to sharply increasing demands for high capacity batteries^[1]. For this purpose, many attempts have been made to increase the capacity of lithium ion batteries by employing silicon in electrodes as its high theoretical capacity amounts to 4 200 mA·h/g^[2], which is much higher than that of graphites. Compared with active materials of traditional batteries such as graphites, LiCoO₂ and LiFePO₄, silicon exhibits a very huge deformation upon the lithiation, e.g., up to 300% volumetric swelling^[3]. The large deformation leads to high stresses in electrodes, and subsequently results in not only electrode damages such as pulverization^[4] and crack^[5] but also stress dependent electrochemical behaviors^[6-8].

Electrochemical kinetics is usually considered as stress insensitive for common active materials like graphites, because its deformation is small, and the stress is low. However, recent works have reported that, for active materials like silicon which swells significantly upon the lithiation, stress may play an important role in electrochemical kinetics and behaviors^[9]. For example, McDowell et al.^[10] observed retardation of lithiation reaction front and attributed this phenomenon to mechanical stress. Kim et al.^[6] controlled electrochemical reaction and drove current motion via electrode bending. Piper et al.^[11] and Sethuraman et al.^[12] both found that compressive stresses made a significant contribution to the overpotential and voltage hysteresis in silicon electrodes. Lu et al.^[13] proved with a designed experiment that the voltage hysteresis of a silicon electrode under open-circuit operations depended on the electrode stress and plastic deformation. They also proposed a modified Butler-Volmer equation to account for the stress dependent electrochemical kinetics and explained the experiment well.

In previous studies, effects of mechanical stresses on the electrochemical reaction and electrode potential can be well explained under equilibrium states. For example, Bower et al.^[14] characterized the stress-potential relation by introducing a stress dependent term in the Nernst equation based on the premise that both lithium concentration and stress were uniform in electrodes. Lu et al.^[13] investigated the stress induced voltage hysteresis under the open-circuit condition and found that stresses contributed to the voltage hysteresis and led to the energy dissipation during cycles. However, during charge-discharge cycles, stresses are not uniform in electrodes and vary from time to time. The impacts of stresses on electrochemical performances must be evaluated under consecutive thermodynamic states. In addition, the evolution of electrode stresses depends significantly on charge operations, e.g., galvanostatic and potentiostatic ones, indicating that the stress dependent electrochemical behaviors are also affected by charge operations^[15].

In this paper, the impacts of stress on electrochemical behaviors under different charge operations will be investigated. The effects of galvanostatic and potentiostatic operations will be discussed based on the evolution of voltage hysteresis. The voltage hysteresis reflects the energy dissipation during charge and discharge cycles, leading to heat generation and voltage loss, and therefore it is important to battery performances. An analytical model considering the coupling among stress, lithium ion diffusion, and electrochemical reaction will be established to simulate the voltage hysteresis under galvanostatic and potentiostatic operations. Based on the studies about stress effects on the voltage hysteresis, attempts will be made to tailor the evolution of voltage hysteresis with designed charge-discharge operations.

2 Methodology

This work investigates a spherical electrode particle, as shown in Fig. 1. During the lithiation, lithium ions are intercalated into the particle across the interface between the electrolyte and particle via the electrochemical reaction, and then diffuse into the particle along the radial direction. A reverse motion takes place during the delithiation, i.e., lithium ions diffuse outwards and then deintercalate from the particle surface. The electrochemical reaction is considered as stress coupled. Therefore, three sets of equations are established, i.e., reaction, diffusion, and stress.

An electrochemical reaction between lithium ions and an active material denoted by H can be described as

(1)

where k_c and k_a are the reaction velocities of cathode and anode reactions, respectively. According to the modified Butler-Volmer equation^{[13, 16]}, the reaction velocity is affected by both the electrode potential and the hydrostatic stress at the surface layer of electrodes,

(2)

where F, i_n, R_g, T, σ_h, Ω, and α represent the Faradic constant, the net current density, the universal gas constant, the temperature, the hydrostatic stress at the surface layer of electrode, the partial molar volume, and the charge transfer coefficient, respectively. E_v denotes the applied electrode potential, and E_eq is the potential at the equilibrium state. Therefore, E_v -E_eq represents the overpotential driving the reaction away from the equilibrium state. -σ_hΩ represents the energy barrier induced by the mechanical work, reflecting that intercalation of lithium ions into electrodes needs to overcome the impeding due to surface mechanics. The exchange current density i₀ can be expressed as

(3)

where k₀, c_max, c_surf, and c_Li⁺ are the reaction rate coefficient at the equilibrium state, the maximum lithium concentration in active materials, the lithium concentration at the surface layer of particles, and the concentration in electrolytes, respectively.

By assuming that the charge transfer coefficient α is equal to 0.5, the electrode potential can be obtained from Eq.(2),

(4)

which indicates that the electrode potential depends on both the reaction rate and the stress. Even in an open-circuit case where the reaction current i_n is zero, the variation of stress level of electrodes may also lead to the variation of electrode potential.

According to the definition, E_eq is the equilibrium potential of electrodes under the stress free open-circuit condition^[13]. However, it is nearly impossible to obtain an absolutely stress free electrode in experiments due to mechanical causes such as the restriction from substrates as well as the interactions among electrode particles. In this study, we adopt the polynomial fitting function proposed by Sethuraman et al.^[17] (see Eq.(5) and Fig. 2) for the equilibrium potential of silicon electrodes. In the experiment, very long time relaxation and the potentials in both the lithiation and delithiation had been considered to minimize the errors induced by stress effects and other side reactions. The equilibrium potential is given by

(5)

where z represents the normalized capacity.

With respect to the diffusion of lithium ions within electrode particles, as the diffusion is along the radial direction, Fick's law under spherical symmetry is used to describe the diffusion,

(6)

where D, r, c, and t are the diffusion coefficient, the radius, the concentration of lithium ions, and time, respectively. It should be noted that stress coupling effects are considered only in electrochemical reactions but are absent in the diffusion as the focus is the former. In addition, saturation effects on the diffusion^[18] which are important only under high concentration conditions are also neglected.

There are two kinds of operations for a charge or discharge, i.e., galvanostatic and potentiostatic operations. For the former where an inlet flux is constant, by introducing the intercalation flux J=i_n/F, the boundary conditions can be written as

(7a)

(7b)

where i_n < 0 means the lithiation while i_n >0 stands for the delithiation. In addition, i_n for a particle of the radius R can be further expressed via C-rate ^[16] according to

(8)

where n is the value of C-rate, and C-rate is the charge rate counted in hours. For example, 1 C means that it takes one hour to fully charge a battery. The charge rate of 2 C is twice as fast as 1 C. In another word, it takes only half an hour to fully charge the same battery.

For the potentiostatic operation where the electrode is under a controlled constant potential E_v, by considering a half cell with lithium metal as the counter electrode and ignoring the voltage drop in electrolytes, according to Eq.(2), the intercalation flux can be calculated as

(9)

and the boundary conditions can be written as

(10a)

(10b)

Regarding the diffusion induced stresses in spherical particles, Cheng and Verbrugge^[19] provided the analytical solution. The radial stress σ_r and the hoop stress σ_θ can be expressed as

(11a)

(11b)

where E and ν are Young's modulus and Poisson's ratio, respectively. c_av (r)=(3/r³)∫₀^r r²c(r)dr is the average concentration in a spherical volume of the radius r. The hydrostatic stress in a sphere is given by σ_h(r)=(σ_r(r)+σ_θ (r)+σ_φ (r))/3. By considering that the radial stress σ_r is zero in the surface layer and the hoop stress σ_θ (r)=σ_φ(r), the hydrostatic stress in the surface layer can be written as

(12)

The above equations cover reaction, diffusion, and stress. In simulations, they will be solved using a finite difference method with MATLAB. At every iterative step, the equilibrium potential E_eq and hydrostatic stress σ_h at the surface layer will be evaluated according to the lithium ion concentration c(r) and then will be substituted into the modified Butler-Volmer equation to determine the boundary condition of diffusion. The concentration distribution c(r) can thus be updated according to the diffusion equation and boundary conditions.

3 Results and discussion 3.1 Galvanostatic case

Firstly, let us study the voltage hysteresis during a charge-discharge cycle under galvanostatic operations. For the detailed procedure, the lithiation starts from a lithium free state with a constant C-rate until the electrode potential drops to a cut-off voltage of 1 mV. Immediately after that, the delithiation will commence with the same C-rate until the surface layer becomes lithium free. Silicon is chosen as an active material, and the particle radius is set as 1μm. The material properties are listed in Table 1.

The simulated voltage hysteresis is shown in Fig. 3(a). The solid lines represent the overall voltages, and the dashed lines denote the portion due to stress effects. It is seen that higher C-rates lead to higher overpotentials, the more significant hysteresis, and the lower chargeable normalized capacity at the termination of lithiation. According to Eq.(4), the overpotential is mainly induced by two parts. One is to drive lithium ions across the electric double layer, and the other one is to balance the mechanical work due to the surface stress. As indicated in the figure, the stress part, represented by the dashed lines, accounts for more portions in the whole voltage hysteresis in higher rate cases. How the stress affects the voltage hysteresis and how significant the effects are should be questioned.

Figure 3(b) shows the lithium concentrations at the termination of both the lithiation and delithiation with respect to three C-rate operations. The whole evolutions are not presented here for the sake of clear illustration. Consistent with the reported literature^[24], operations of higher rates lead to higher concentration gradients in simulations. Interestingly, it is also observed that the higher rate lithiation terminates at relatively lower concentration levels, which indicates higher overpotentials adverse to lithium intercalation. To account for the stress effects, Fig. 3(c) shows the evolution of the hydrostatic stress in the surface layer of particles during a cycle, as well as the overpotential induced by stresses. At the beginning stage of lithiation, the compressive stress arises quickly because the expansion of the surface layer is restricted by the inner region of lower concentration. After that, the compressive stress becomes steady as the concentration gradient is nearly constant. During the delithiation, the stress switches quickly to the tensile one due to the shrinkage of surface layer, and finally becomes steady again. According to Eq.(4), compressive stresses hinder the lithiation while facilitate the delithiation, while tensile stresses facilitate the lithiation but hinder the delithiation. Therefore, the obtained stress hysteresis would affect the voltage hysteresis. According to the right y-axis of Fig. 3(c), the max overpotential induced by stress, represented by σ_hΩ/F, increases from about 0.05 V to 0.1 V and 0.2 V with respect to the increase in the C-rate from 0.25 C to 0.5 C and 1 C, accounting for 31%, 51% and 66% of the overall overpotential at the end of lithiation, respectively. Therefore, it can be concluded that the stress induced hysteresis accounts for more shares in the overall voltage hysteresis in higher rate cases. Our conclusion agrees well with the report by Sethuraman et al.^[17] that 40% energy lost between charge and discharge is accounted for by mechanical dissipation.

Figure 3(d) further shows the voltage hysteresis under the 1 C rate and different stress states. Three cases are investigated, i.e., a free standing active particle with stress effects ignored, a free standing particle with stress effects considered, and a particle subject to the 1 GPa pressure which represents the compression in electrodes. It is seen that the higher compressive stress in the lithiation leads to the higher overpotential and the lower chargeable capacity at the termination of lithiation. During the delithiation, the stressed cases show the higher voltage than the case with stress ignored as the tensile stress in the delithiation impedes deintercalation of lithium ions. The voltage in the delithiation stage of 1 GPa pressure case (magenta line) is a bit lower than that of stressed particle without compression (blue line). This is because the compression from the adjacent domain decreases the tensile stress in the surface layer of the active particle. Generally speaking, the stress effects lead to the higher voltage hysteresis and the lower chargeable capacity, which means that quite a part of energy has been dissipated to balance the stress and elastic work. In addition, previous studies^[15-16] revealed that the lithiation induced stress depended on the electrode size. Electrodes with the larger size would induce the higher stress and the more significant stress induced voltage hysteresis.

3.2 Potentiostatic case

The potentiostatic operation is under the controlled constant potential. However, in reported studies, simulations of the potentiostatic operation usually adopted a simplified boundary condition that the lithium ion concentration in the surface layer of active materials was constant and the concentration was usually set as saturation^[19]. This condition is not so applicable in real operations as it is hardly maintainable due to the giant current density and overpotential^[9].

In this study, we will simulate the potentiostatic operation with more practical boundary conditions provided by Eq.(10) based on the modified Butler-Volmer equation^[13]. According to the detailed procedure shown in Fig. 4, the lithiation will start from a lithium free state with a constant applied potential of 0.24 V which is equal to the open-circuit potential corresponding to a normalized capacity of 0.68, and terminate once the intercalation current density is less than 3× 10^-5mA/m². After that, the applied potential will be immediately switched to 0.51 V, corresponding to the open-circuit potential for a normalized capacity of 0.07, to start the delithiation. Finally, the operation stops when the deintercalation current is less than 3× 10^-5mA/m². The particle radius is still set as R=1 μm.

Figure 5(a) shows the evolutions of surface hydrostatic stress and distributions of lithium ion concentration during a cycle. Figure 5(b) illustrates the evolution of the corresponding overpotentials. At the beginning of lithiation, the interfacial intercalation reaction overruns the lithium diffusion within the particle due to the high overpotential between the equilibrium potential and the applied potential. The concentrations of lithium ions near the particle surface arise quickly, leading to a steep concentration gradient as well as the quickly increasing compressive stress in the surface layer (see the point A in Fig. 5(a)). After that the interfacial reaction gradually slows down with the decreasing overpotential whereas the lithium diffusion within particles plays a more important role. Therefore, the distribution of lithium ions within the particle becomes more and more uniform till the end of lithiation where both intercalation current and hydrostatic stresses nearly vanish. During the delithiation, the hydrostatic stress takes on a similar evolution in the tensile side, i.e., increases quickly at the beginning due to the ultrafast interfacial reaction flux and decreases gradually afterwards.

It is seen that the evolution of hydrostatic stress forms a hysteresis plot. The compressive stress during the lithiation impedes intercalation while the tensile stress during the delithiation impedes deintercalation, indicating that quite a part of work provided by the electric power has been dissipated to balance the stress effects. We therefore plot the evolutions of all parts of overpotential in Fig. 5(b), including the part due to stress effects σ_hΩ/F, the part corresponding to the reaction current , and the overall overpotential E_v -E_eq which is the sum of the two parts. It is observed that the overall overpotential decays all the time in either lithiation or delithiation. The ultra high overpotentials at the beginning of lithiation and delithiation induce a very large current and a high energy dissipation due to effects such as resistance and stress. This is one of the reasons why battery charging starts always with the galvanostatic operation rather than the potentiostatic one. Among the total voltage hysteresis, the stress effects contribute to a significant share. According to calculation, the stress induced voltage hysteresis accounts for an area portion of 56.6% in the total hysteresis. Therefore, it would be beneficial if the stress could be tailored during charge and discharge to minimize the energy dissipation due to stress effects.

3.3 Reducing voltage hysteresis via combined galvanostatic and potentiostatic operation

According to the above studies, the stress induced overpotential keeps increasing in the galvanostatic operation, while the opposite trend is found in the potentiostatic operation, i.e., the stress induced overpotential is very high at the beginning stage and decreases afterwards. Therefore, it would be an effective way to reduce the stress induced overpotential and the consequent voltage hysteresis by employing a combined galvanostatic and potentiostatic operation.

Figure 6 shows the evolutions of hydrostatic stress and corresponding overpotential in the combined operation. Results of galvanostatic and potentiostatic operations are also provided for comparison. In order to be consistent with the previous section, the electrode potential in the potentiostatic operation is still set as 0.24 V and 0.51 V in the lithiation and delithiation phase, respectively. The pure galvanostatic operation adopts a C-rate of 0.5 C and cuts off at the normalized capacity of 0.68, the termination point of purely potentiostatic operation.

It is seen in Fig. 6 that there are two transition points. The galvanostatic operation presents lower stresses than the potentiostatic one before the transition point, but higher after the transition. Therefore, a combined charge-discharge operation, i.e., starting with the galvanostatic operation and switching to the potentiostatic one at the transition point, could effectively reduce the stress induced overpotential and the corresponding voltage hysteresis. According to the calculated area of voltage hysteresis loops, the proposed combined procedure reduces 23.8% and 25.7% energy dissipation due to stress effects compared with those of pure galvanostatic and potentiostatic operations, respectively. In addition, the energy dissipation could be further minimized by adopting a lower C-rate at the galvanostatic phase. For example, if we adopt 0.25 C in the galvanostatic phase, the energy dissipation is further lowered by 55.7% compared with the pure potentiostatic operation.

4 Conclusions

This paper investigates the dependence of voltage hysteresis on charge-discharge operations and the stress. A reaction, diffusion, and stress coupled model is established to simulate the voltage hysteresis under charge-discharge operations.

For the galvanostatic operation, the lithiation phase and the delithiation phase induce the compressive and tensile stresses, respectively, in the surface layer of active particles. The compressive stress impedes lithium intercalation while the tensile stress hinders deintercalation, inducing the certain voltage hysteresis in the overall hysteresis. It is suggested that a part of energy provided by the electric power has been dissipated to balance the stress effects on electrochemical intercalation/deintercalation reactions. In addition, the voltage hysteresis is found more significant in higher rate cases. The higher C-rate leads to earlier terminating of operation at the cut-off voltage due to the higher overpotential induced by the larger current density and higher stresses. Among the overall voltage hysteresis, the stress induced part is more significant in high rate cases. According to our calculation for a silicon particle of 1μm radius, the percentage of stress induced overpotential in the total overpotential increases from 33% in the 0.25 C case to 67% in the 1 C case. Therefore, low rate operations are beneficial for battery performances from the aspect of minimizing energy dissipation, especially the energy dissipated due to mechanics reasons.

For the potentiostatic operation, the overpotential is very high at the beginning of either lithiation or delithiation due to the high voltage. As the operation goes on, the overpotentials decrease gradually with the decaying current. The stress induced overpotential, evolving consistently with the hydrostatic stress in the surface layer of particles, reaches a peak value very soon after the operation commences and contributes a significant share to the overall overpotential. Therefore, according to the features of galvanostatic and potentiostatic operations, a combined charge-discharge strategy, i.e., first the galvanostatic one and then the potentiostatic one, is found beneficial not only to avoid too high stress during operations to prevent electrodes from failure but also to reduce the voltage hysteresis and the energy dissipation due to stress effects.