Mathematical modeling of the electrochemical impedance spectroscopy in lithium ion battery cycling

Electrochimica Acta 127 (2014) 266–275

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Mathematical modeling of the electrochemical impedance spectroscopy in lithium ion battery cycling Yuanyuan Xie, Jianyang Li, Chris Yuan ∗ Department of Mechanical Engineering, University of Wisconsin Milwaukee, WI 53211, USA

a r t i c l e

i n f o

Article history: Received 2 December 2013 Received in revised form 7 February 2014 Accepted 8 February 2014 Available online 24 February 2014 Keywords: Lithium ion battery Impedance Cycling effect Modeling

a b s t r a c t Electrochemical impedance spectroscopy (EIS) has been widely utilized as an experimental method for understanding the internal mechanisms and aging effect of lithium ion battery. However, the impedance interpretation still has a lot of difficulties. In this study, a multi-physics based EIS simulation approach is developed to study the cycling effect on the battery impedance responses. The SEI film growth during cycling is coherently coupled with the complicated charge, mass and energy transport processes. The EIS simulation is carried out by applying a perturbation voltage on the electrode surface, and the numerical results on cycled cells are compared with the corresponding experimental data. The effect of electrical double layer, electrode open circuit potential as well as the diffusivity of binary electrolyte are simulated on battery impedance responses. The influence of different SEI growth rate, thermal conditions and charging-discharging rate during cycling are also studied. This developed method can be potentially utilized for interpretation and analysis of experimental EIS results. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Lithium ion battery is widely used in consumer electronics and energy storage systems due to its superior characteristics such as high energy density, low energy-weight ratio, no memory effect and small self-discharging rate, etc [1]. These outstanding characteristics have made lithium ion battery increasingly popular in portable electronic devices, automotive/aerospace applications and green industries [2]. Typically, the fundamental structure of a lithium ion battery consists of three parts: a negative electrode, a separator and a positive electrode, where multiple transport processes including charge, mass, energy as well as chemical/electrochemical reactions would take place during battery operating. To characterize these fundamental mechanisms, the electrochemical impedance spectroscopy (EIS) method is widely used as an effective technique for distinguishing between different processes and determining their effects [3–5]. It is obtained through an active interrogation process: an altenating voltage or current load perturbation is applied to excite the battery with frequencies ranging from very low to extremely high, then the generalized resistance, impedance, is calculated using the corresponding battery current or voltage output perturbation. In principle, the EIS

∗ Corresponding author. Tel.: +1 414 229 5639; fax: +1 414 229 6958. E-mail address: [email protected] (C. Yuan). http://dx.doi.org/10.1016/j.electacta.2014.02.035 0013-4686/© 2014 Elsevier Ltd. All rights reserved.

contains a plethora of lithium ion battery behavior information with full spectra convoluted in a single EIS response curve. If the EIS curve can be de-convoluted properly, it can facilitate the understanding of fundamental mechanism such as aging effect within a lithium ion battery beneficially. Although with wide applications, the capacity fading and impedance rise of lithium ion battery over cycle operation are still major obstacles hindering its wide industrial applications. In the past years, numerous experimental EIS works have been carried out to investigate the aging effect of lithium ion battery [6–11]. M. Balasubramanian et al. [6] and D. Ostrovskii et al. [7] have studied the effect of SEI formation at the electrode-electrolyte interface on battery cycling performance. They found that the SEI formation could specifically slow down the lithium ions transport to the electrode active sites and lead to the rising on interfacial impedance. Y.Zhang et al. [8] tested the impedance responses on an extensive cycled lithium ion battery, where they characterized the contributions from different process by individual electrode EIS spectra and the equivalent-circuit analysis. Based on the cycling experiments, Bloom et al. [9] studied the cycle life of lithium ion battery and found that the large change on state of charge could lead to great loss on battery power. Asakura et al. [10,11] also carried out the experiments on charging-discharging cycles of lithium ion battery, which showed that high charging voltage could accelerate the battery aging through the impedance augmentation. These experimental works indeed can help understand the battery

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267

Nomenclature cj , cjs cp Cdl Cfl eff

Dj F ij0 fl

The concentration of species j; The specific heat capacity; The electrical double layer capacity; The SEI film capacity; The effective diffusion coefficient; The Faraday’s constant; The exchange current density;

ij

The Electrical double layer current;

Jj kf , kb Mp Rf Sj 0 t+ Uj

The Solvent reduction current; The applied charging-discharging current density; The normal operating current; The amplitude of current perturbation; The local volumetric exchange current; Forward and backward reaction rate coefficients; The molecular weight of SEI; The film resistance; The specific interfacial area; The transference number of lithium ion; The electrode-open circuit potential;

ijs I0 I¯ ˜I

ref

Us 1,j , 2,j ¯  ˜  ω eff eff j , j p  neg pos sep

The reference open circuit potential; The potentials in solid phase and liquid phase; The normal operating voltage; The amplitude of the harmonic perturbation voltage; The frequency; The effective conductivities; The density of SEI film; The phase shift; The heat conductivity; The negative electrode; The positive electrode; The separator layer

performance, but difficulties still arise from the impedance interpretation and parameter estimation. In this aspect, modeling method can provide an effective way to study the battery behavior on the impedance responses. Compared with the widely applied equivalent circuit method [5,12–14], the multi-physics modeling based EIS simulation can take the physical laws of lithium ion battery into consideration, and can avoid the ambiguity and information loss generated by the equivalent electrical circuit. Recently, one of the most popular multi-physics modeling for lithium ion battery is the pseudo-two-dimensional (P2D) model, which is based on the principles of transport phenomena, electrochemistry and thermodynamics, and is typically represented by a bunch of coupled nonlinear partial differential equations [1]. In this area, Newman and his co-workers have presented a lot of physical based P2D models for both battery components and single batteries [15–19]. With the further advancement on battery system understanding, a number of similar works have also been developed [5,20–24]. However, only few of them have been contributed to the study of battery comprehensive impedance responses. Doyle et al. [22] developed a comprehensive model to simulate the impedance responses of lithium metal battery, where a metal anode and a porous cathode electrode were applied. G. Sikha et al. [23,24] proposed an analytical model for lithium ion battery, where the analytical expression of battery impedance was developed by considering both charge and mass transport processes in an insertion electrode. F.Mantia et al. [5] also presented a general impedance model for graphite electrode of lithium ion battery, where a good

Fig. 1. Physical structure and model geometries.

agreement has been achieved between model and experiment on the effect of electrode porosity and the assumed electrochemical processes. Although these mathematical models helped improve the understanding on battery performance, the cycling induced battery aging effect was not taken into consideration, which means the battery capacity fading and the corresponding impedance variation cannot be analyzed by these developed model. In order to link the impedance simulation to the practical battery operating procedure, the objective of this study is to develop a multi-physics based EIS simulation approach to study the cycling effect on the impedance responses of lithium ion battery. The SEI film growth during cycling is coherently coupled with battery charge transport, mass transfer, thermal energy as well as electrochemical processes. The EIS simulation is carried out by applying a perturbation voltage on the electrode surface, and the numerical results on cycled cells are compared with the experimental data. Then, comprehensive numerical studies are performed to elucidate the cycling effect on the impedance responses of lithium ion battery. 2. Mathematical modeling 2.1. Lithium ion battery modeling Fig. 1 has schematically illustrated the computational domain of the model, which consists of a negative electrode (Lin C6 ), a separator, a positive electrode (Li metal) and two current collectors on the electrode ends. The applied electrolyte solution is 1.2 M LiPF6 in a mixture of ethylene carbonate (EC) and dimethyl carbonate (DMC) with volume fraction ratio of 1:1. During the chargingdischarging cycles, lithium ions are cycling transported by inserting into or de-inserting from the solid electrode particles. According to Aurbach et al. [25] and Popov et al. [26], the insertion process of lithium ions can increase the lattice volume due to an increase in space between grapheme planes, and may lead to stretching of the surface passivating film (the well-known SEI film). Since the SEI film is usually weak in flexibility, it may break down during the battery charging-discharging cycles. The continuous small-scale reactions then occur between the exposed electrode particles and the electrolyte solution, and the surface impedance is eventually increased by new SEI film generation with cycling. Meanwhile, due to the concentration and potential effects in the solution phase, the chemical/electrochemical kinetics is actually coupled with the multi-transport processes in battery including charge, mass and thermal energy [27].

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In the graphite anode, the total current (itot ) transferred between the solid electrode particles and the electrolyte solution is the sum of three possible current paths [23]: the Faradaic current (iF ), the electrical double layer current (idl ) and the SEI film formation induced current (ifl ), fl

ijtot = ijF + ijdl + ij ,

 ijF = ij0 FSj

z˛aj F RT

(1)

 j

 − exp

−

z˛aj F RT



∂ ∂x

∂1,j

eff −j

∂2,j

eff

where, j

= ijtot



∂x

(5)

0) 2RT (1 − t+ ∂ + F ∂x

eff

and j



∂(ln cj )

eff −j

∂x

 =

ijtot

(6)

are the effective conductivities in solid phase

0 is the transference and liquid phase respectively (j = neg, sep); t+ number of lithium ion. To simulate the battery capacity fading during the cycling operation, the loss of active lithium ions through electrochemical reduction reactions and the SEI film generation in the graphite electrode are estimated by an irreversible inward flux of lithium ions [26]:

Rf = Rf,ini + Rf (t)

(7)

⎧ ⎨ Rf (t) = L(t)/p ⎩ ∂L(t) = − ∂t



(ijF

+ ijs )

1,j − 2,j − Uj −

(11)

(ijF + ijs ) Sj

 Rf

(12)

Qrev = (ijF + ijs )T

 Qohm = eff

∂Uj

(13)

∂T

∂1 ∂x

2

 + eff

∂2 ∂x

2 +

2eff RT F

0 ) (1 − t+

∂(ln c) ∂2 ∂x ∂x (14)

where, Qrxn is the reaction heat generation; Qrev is the reversible heat production; Qohm is the Ohmic effect induced heat production. To keep the conciseness and compactness of the paper, the formulas of related parameters are summarized in Table 1. The boundary conditions and associated operating parameters are characterized respectively in the Table 2 and Table 3. The parameter definitions are listed in the nomenclature.



∂x



∂2 T dT =  2 + Qrxn + Qrev + Qohm dt ∂x

(4)

dl ). Then, the generic Ohm’s law can be utilized to layer current (ipos describe the charge transport in the solid and liquid phases [27]:

eff

(10)

As for the thermal energy transport process, it consists of the heat transfer across the battery and the heat production process [28,29],

Qrxn =

reduction current [26], ijs = −is0 exp(z˛cj Fs ). On the cathode side, the Li-metal can only dissolve or deposit lithium during the cycling operation without aging. The total current transferred between solid cathode and liquid electrolyte is defined on the electrode F ) and electrical double surface including the exchange current (ipos

−j

is the effective diffusion coefficient

j = 10−8.43−(54/(T −229−0.005cj ))−0.00022cj εbrugg j

(3)

where, ij0 is the exchange current density[28]; Sj is the specific interfacial area; j is the overpotential; Cdl,j is the electrical double layer capacity; Cfl,j is the SEI film capacity; 1,j and 2,j are the potentials in solid phase and liquid phase respectively; 1,f is the potential in the SEI film [23], 1,f = 2,j + (ijF + ijdl )Rf ; ijs is the electrolyte



eff

Dj

cp

∂ = Cfl,j Sj (1,j − 2,j ) + ijs ∂t

∂ ∂x

eff

ficient in solid materials; Dj and given by [25],

(2)

j

∂ (1,j − 1,f ) ∂t

ijdl = Cdl,j Sj fl ij

 exp

j = neg

where, cjs is the lithium ion concentration; Djs is its diffusion coef-

ijs Mp

(8)

an p F

where, Rf is the SEI film resistance; Rf,ini is the initially formed SEI layer resistance; Rf (t) is the produced film resistance in the cycling; L(t) is the SEI film thickness; p is the film conductivity; Mp is the molecular weight of SEI film. According to the lithium ion intercalation/deintercalation and diffusion process, the mass transfer in the battery also contains the transport process in solid electrode phase and in liquid electrolyte phase, which can be formulated by the Fick’s second law [28],

⎧ ⎪ ⎪ ⎪ ⎨ Solid phase : ⎪ ⎪ ⎪ ⎩ Liquid phase :

∂cjs ∂t

= Djs ∂cj

1 ∂ r 2 ∂r

∂ εj = ∂t ∂x

 r2

 eff Dj

∂cjs



∂r

∂cj ∂x

 +

0 )(iF + is ) (1 − t+ j j

F

(9)

2.2. EIS simulation The EIS response of the battery is obtained through an active interrogation process, where the prepared coin batteries (graphite/metal) are charged and discharged with a constant rate of 0.5C between 2 V and 0.05 V. After certain cycles’ operation, the battery impedance responses are measured by VersaSTAT3(potentiostat) with AC signal amplitude of 5 mV in the frequency range from 0.1 Hz to 100 kHz. During the impedance measurement, the coin batteries are maintained at 1 V (SOC ∼ = 50%) under the open circuit. The corresponding battery current responses are then recorded. The generalized resistance, impedance, is obtained based on the perturbation voltage and the battery corresponding current. Accordingly, the EIS modeling and simulation are obtained in the same procedure, where the perturbation voltage signals are applied to the introduced transient lithium ion battery model and the corresponding current perturbations can be collected through numerical analysis. The simulated EIS response can be obtained according to model input/output signals. Suppose a complex periodic voltage perturbation is applied to excite the lithium ion battery, ¯ + ˜ exp(jωt) (t) = 

(15)

˜ is the amplitude of the ¯ is the normal operating voltage;  where,  harmonic perturbation voltage; j is the square root of −1; ω is the frequency. The corresponding response of cell current density will be composed of two parts, I(t) = I¯ + ˜I exp[j(ωt −

)]

(16)

where, I¯ is the normal operating current; ˜I is the amplitude of current perturbation induced by the voltage perturbation; is the phase shift of the harmonic current perturbation with respect to the harmonic voltage perturbation. When the periodic voltage perturbation is given, the corresponding periodic current density, including the magnitude ˜I and the phase , needs to be determined.

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269

Table 1 Parameter formulas. eff

Effective electronic conductivity :



j

= j (1 − εj − ϕj )

(1)

j = 10−4 × cj (A1 + A2 ) εbrugg j

2

eff

Effective ionic conductivity :

j

A1 = −10.5 + 0.668 × 10−3 cj + 0.494 × 10−6 cj2 + 0.074T

(2) (3)

A2 = −1.78 × 10−5 cj T − 8.86 × 10−10 cj2 T − 6.96 × 10−5 T 2 + 2.8 × 10−8 cj T 2

⎧ ⎨

(4)

Overpotentials : j = 1 − 2 − Uj −

(ijF + ijs )

Rf Sj + ijs ) Rf Sj

(5)

⎩ s = 1 − 2 − U ref − (ij

F

s

(6)

Table 2 Boundary conditions. Interface:

suf/pos

pos/sep



−eff

Charge:

∂1

= Iapp ∂x x=0 −

⎧ tot ⎨ −eff ∂1

= ipos ∂x x=pos

tot ⎩ −eff ∂2

= ipos ∂x ∂c ∂x

Mass:

−

−Deff

Energy:

− ∂T ∂x

x=0

− ∂T ∂x

r=0:

∂c −Deff ∂x

Particle:

= h(T∞ − T )



x=0

=0



x=sep

x=sep

=0





= − ∂T ∂x x=sep x=pos

(iF +is ) ∂c r = R : −Deff ∂x = j zF j

c|x=sep = c|x=neg

− ∂T ∂x

− ∂T ∂x

x=sep

= − ∂T ∂x

x=neg

x=neg



1 −eff

−Deff

∂c ∂x

= 0

x=L

∂2

=0 ∂x x=L



x=L

x=L

=0

= h(T∞ − T )

involved multi-physics process, material properties, microstructures as well as various operating conditions. 2.3. Solution algorithm



(17)

0

where the current perturbation (magnitude ˜I and the phase ) can be obtained. Basically, the processes involved in lithium ion battery are strongly nonlinear multi-physics processes. When a periodic voltage perturbation is superimposed to a nominal operating voltage as shown in equation (15), the battery behavior will oscillate around a nominal operating point. If the magnitude of the perturbation is small enough, the battery behavior can be considered as linear around the normal operating point. In this scenario, the linearized EIS can be calculated using voltage/current perturbation values, ˜ exp(jωt)  ˜I exp(j(ωt −

neg/suf

x=0

I(t) exp(jωt) dt = ˜I exp(j )

Z = Zre + Zim =

⎧ ⎨ −eff ∂1

=0 ∂x x=neg

∂ ⎩ −eff 2 = −eff ∂2

∂x x=sep ∂x x=neg

∂c ∂c −Deff = −Deff

∂x ∂x x=sep

Mathematically, a exp j(ωt) term can be multiplied on both side of above equation, which then takes integrating over a full period of

yields, 2

sep/neg

))

=

˜  ˜I





cos( ) + j sin( )

(18)

As a consequence of this calculation, impedance spectrum is composed of a series of generalized resistances under different exciting frequencies, collectively contributed by the

The introduced mathematical model is numerically solved using finite element package COMSOL MULTIPHYSICS V4.3 and MATLAB. Two submodel geometries are applied: a one-dimensional lithium ion battery model and a two-dimensional electrode solid phase model. As shown in Fig. 1, the 1D battery model consists of a positive electrode, a separator and a negative electrode, while the 2D solid phase model contains two squares to stand for the solid phase in two electrodes respectively. These two submodels are coherently coupled in such way: the concentration of lithium ions obtained in the 2D solid phase model is projected to the 1D battery model, while the mass flux from 1D battery model is extracted to the 2D solid phase model boundaries. The EIS modeling is carried out based on the numerical results of the cycled battery, and its procedure is as following: the battery cycling operation is firstly simulated at constant charging/discharging rate. After certain cycles, the EIS simulation is performed by applying a voltage perturbation of 5 mV in amplitude with frequencies ranging from 10−4 Hz to 105 Hz on the positive electrode surface. The corresponding current responses

Table 3 Operating parameters. Value Positive electrode thickness (Cathode): Separator thickness: Negative electrode thickness (Anode): Gas constant: Faraday’s constant: Li-diffusivity in anode solid phase: Reference OCP of graphite electrode at 25 ◦ C: Cationic transport number: Volume fraction of anode solid phase: Applied 1C charging-discharging current: Maximum solid phase Concentration: Anode SEI film capacity: Charging/discharging duration:

2 × 10−4 m 5 × 10−5 m 1 × 10−4 m 8.314 J mol−1 K −1 96, 485 C mol−1 3.9 × 10−14 m2 s−1 1V 0.363 0.3 1.75 A m−2 26, 390 mol m−3 0.02∗ F 3 × 103 s

Value Exchange current of SEI formation: Density of SEI film: Salt diffusivity in the electrolyte: Anode particle radius: Molecular weight (SEI): Anode electrical double layer capacity: Cathode electrical double layer capacity: Bruggeman coefficient: Anode solid phase conductivity: Initial electrolyte salt concentration: Exchange current density between Li-metal and electrolyte: 0 Exchange current density (ineg ): Adjusted/Assumed value:

8 × 10−8 A m−2 2.1 kg m−3 7.5 × 10−11 m2 s−1 1.25 × 10−5 m 0.1 kg mol−1 0.35∗ F 0.33∗ F 1.5 3.8 S m−1 1.2 × 103 mol m−3 8.5 × 102 A m−2 1.5 × 101∗ A m−2 ∗

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Numerical, After 1 cycle Experimental, After 1 cycle Numerical, After 20 cycles Experimental, After 20 cycles Numerical, After 50 cycles Experimental, After 50 cycles

100

-Z'' /ohm cm

2

150

50

T 0=25oC Rate: 0.5C

10Hz 100Hz

0

0

50

Z' /ohm cm

2

100

150

Fig. 2. The comparison between numerical impedance results and the experimental data.

are collected at the same time. Then, the battery impedance can be calculated by the introduced mathematical approach. 3. Results and discussion 3.1. Cycling effect The model predicted impedance responses after 1 cycle, 20 cycles and 50 cycles are compared with the experimental data, which are illustrated in Fig. 2. The charging-discharging rate is set at 0.5C, and the initial operating temperature is 25 ◦ C. The double layer capacities and the exchange current density are adjusted to fit the numerical results with the experimental data, which are listed in Table 3. As shown in Fig. 2, the results obtained from the developed model agree reasonably well with the experimental data except the low frequency branch on the 1-cycle case. This is because the cell performance after 1 cycle’s operation is not stable yet, and the low frequencies induced long testing time may also have introduced some errors. Besides, it also shows that the impedance response of lithium ion battery keeps increasing with cycling due to the aging effect. On these impedance spectra, the intercept of impedance curve on the x axis is contributed by the ohmic resistances from the battery electrodes, electrolyte solution, separator layer and the current collectors. In the mid-high frequency

Fig. 3. The effect of electrical double layer capacity on the battery impedance responses.

range, a depressed arc is generated by the resistance and capacitance of the charge transfer processes and the electrochemical reactions in the battery. As for the low frequency branch, it is well known from literature that the solid state diffusion and the battery capacity are the primary reasons to form this capacitor type impedance response. Since the battery impedance responses are cumulative signals from electrodes, electrolyte solution, separator and current collectors, it is very difficult for experimental method to identify their individual contributions. In this aspect, modeling technique provide significant advantages to explore their individual roles during the charging-discharging cycles. 3.2. Electrical double layer To study the effect of electrical double layer in battery electrodes, we choose the validated numerical result after 50 charging-discharging cycles as the base case. The simulation is firstly carried out by decreasing the electrical double layer capacity of the positive electrode from 0.33 to 0.033, while other operating conditions are kept unchanged. As shown in Fig. 3, it is interesting to see that a small arc is generated on the impedance curve at the high frequency range, while the impedance responses at the mid-low frequencies are almost the same as the base case. Then, we further change the electrical double layer capacity of the negative electrode from 0.35 to 0.035 to simulate the battery impedance responses.

Fig. 4. The effect of electrode open circuit potential on the battery impedance: a) the solid phase concentration dependency; b) the temperature dependency.

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271

Fig. 5. The effect of binary electrolyte diffusivity on the impedance during cycling.

It shows that the generated small arc at the high frequency range merges with the large mid-frequency arc, and the impedance curve is almost the same as the base case. Since the impedance responses of the last case are obtained at different frequencies from the base case, these observations indicate that same impedance response may be generated by quite different electrodes, and the overlap of the impedance responses from two electrodes is determined by the electrical double layer capacities. 3.3. Electrode open circuit potential Theoretically, the open circuit potential of the battery electrode is varying with the charging-discharging cycle operation. Under low-moderate rate of discharge, Reynier et al. [30] reported that the open-circuit potential of graphite electrode is nearly linear dependent on the temperature between 0 and 23 ◦ C. At the same condition, Kumaresan et al. [31] proposed a linear temperature dependent open circuit potential by Taylor’s first order expansion around a reference temperature, and indicated that the solid-phase concentration dependency is also a very important factor. Sikha et al. [23,24] applied the concentration dependency on electrode open circuit potential to analytically study the battery impedance, but the thermal effect was neglected. Since both temperature and concentration dependency of electrode open circuit potential are important, in our developed model, we take both effect into consideration by using the linear approximation:



ref

Uj = Uj

+ (T − Tref )

∂Uj ∂T

+



cj ∂cj

c

∂Uj

ini

impedance curve, significant change can be noted. Similarly, the effect of temperature dependency is also studied by reducing it to a half, while the other operating parameters are kept unchanged. As shown in Fig. 4b, both the mid-high frequency arc and the lowfrequency branch are significant changed in comparison with the base case. Because the temperature variation is coherently coupled with the charge, mass and chemical processes in the battery, reducing temperature dependency of open circuit potential can directly lead to the underestimation of SEI film formation rate and the battery fading process, and will be eventually reflected on the battery impedance spectra. 3.4. The diffusivity of binary electrolyte Fig. 5 has illustrated the effect of the diffusivity of binary electrolyte on the battery impedance responses during cycling. Based on the validated numerical cases, we applied two different effective diffusion coefficients, 0.1Deff and 10Deff , in the modeling. The applied operating conditions are the same as the base cases. It shows that after 1 cycle operation the differences on the battery impedance among three cases are concentrated at the mid-low frequency range: a smaller diffusion coefficient can lead to a larger midfrequency arc, and a lower increasing trend of the capacitor type impedance responses at the low frequencies. Because the

(19) r=R ref

where, Tref is the reference temperature (25 ◦ C); Uj circuit potential under the reference temperature;

∂Uj ∂T

is the open

is the deriva-

tive of the open circuit potential to the temperature [31];

∂Uj ∂cj



is cini

the derivative of the open circuit potential with respect to the solid phase concentration at the initial concentration [23]. The effect of these two linear dependencies on the battery impedance responses during cycling are studied in Fig. 4. The validated numerical impedance responses are utilized as the base case, while the first simulation is carried out by reducing the linear concentration dependency,

∂Uj , ∂cj

to a half. Since decreasing the con-

centration dependency can reduce the influence of active species transport process, it can be seen from Fig. 4a that the depressed impedance arc in mid-high frequency range is slightly increased after cycling operation, but on the low frequency branch of the

Fig. 6. The effect of SEI film growth rate on the impedance with cycling.

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Y. Xie et al. / Electrochimica Acta 127 (2014) 266–275

if the diffusivity of binary electrolyte becomes worse, the battery aging effect will be intensified with cycling.

3.5. SEI growth rate

Fig. 7. The Bold plots of different SEI growth rates in the cycling operation.

binary electrolyte transport process determines the surface concentration of lithium ions on the solid electrode particles, the diffusivity of binary electrolyte can generate great influence on the solid state diffusion process. After 20-cycles operation, it shows that the mid-high frequency arc of 0.1Deff case is increased much larger than the other two cases, while the impedance pattern at low frequencies are not changed much. These observations indicate that

The SEI film serves as a crucial passivating layer by isolating the graphite electrode from the electrolyte solution, of which the growth rate is determined by the exchange current of side reduction reactions [20,26]. Fig. 6 has illustrated the effect of the SEI growth rate on the battery impedance responses, where three exchange currents conditions are compared. The validated numerical impedance responses after 1 cycle and 50 cycles of charging-discharging are used as the base cases. The impedance responses are firstly simulated by reducing the exchange current to its 50%, then by increasing it to 2 times. It can be seen that higher exchange current of side reactions can lead to a larger midfrequency arc on the impedance spectrum, and this effect can be intensified with cycling operation. This is due to the passive effect of SEI growth on the battery multiple transport processes. Besides, it can also be noticed that the differences on the low frequency branch of three cases are reduced with cycling. This phenomenon indicates that cycling operating can lead to a more stable transport process of active species into solid phase in the graphite anode. In order to find more details from the effect of SEI film growth on the battery impedance responses, the magnitude and phase shift of the simulated impedance responses (Bold plots) are presented in Fig. 7. It is clear that the differences among three cases primarily appear at the mid-low frequency range. As the charging-discharging cycle goes on, it is also clear that both the impedance magnitude and the phase shift angle are increased.

Fig. 8. The effect of operation temperatures on the battery impedance responses during cycling.

Y. Xie et al. / Electrochimica Acta 127 (2014) 266–275

Fig. 9. The effect of cooling conditions on the battery impedance responses during cycling.

3.6. Thermal effect In principle, nearly all transport processes in the lithium ion battery are temperature dependent. Therefore, the operation temperature of lithium ion battery has significant influence on the battery global performance. By using the validated cases at 25 ◦ C as a base, we carried out the numerical study of different operating temperatures in the battery cycling operation. After different

273

charging-discharging cycles, the simulated impedance spectra are illustrated in Fig. 8, where two operating temperature of 0 ◦ C and 55 ◦ C are chosen to make the comparison with the base cases. It can be seen that after 1 charging-discharging cycle, the impedance spectra at three different temperatures are almost the same at mid-high frequencies, but their impedance responses at low frequencies are quite different. With the cycles moving on, this difference on low frequency branch of impedance spectrum is becoming smaller, but the size of the depressed arc on the midhigh frequency range is increasing, especially under the operating temperature of 55 ◦ C. Since the solid state diffusion process is represented by the low frequency branch of impedance spectrum, these observations indicate that different operating temperatures can lead to quite different transport properties of active species into solid electrode phases. Besides, from the increasing on the mid-high frequency arc, it can also be concluded that higher operating temperature can greatly intensify the battery aging effect with cycling. The surface cooling condition of lithium ion battery is another important factor that can influence the battery performance. Fig. 9 illustrates the battery impedance responses under three surface convective heat conditions: h = 0 W/m2 K (insulation), h = 20 W/m2 K and h = 50 W/m2 K. The simulations are carried out after 1 cycle and 50 cycles respectively. It is clear that the impedance curves of three cooling conditions are almost the same after 1 cycle operating. But after 50 cycles of charging and discharging, it shows the impedance arc is much larger under insulation condition than the other two. This indicates that better cooling condition of lithium ion battery can positively reduce the cycling induced impedance increasing, and can effectively stabilize the battery performance.

Fig. 10. The effect of battery charging-discharging rate on the impedance responses with cycling.

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3.7. Charging-discharging rate

References

Different charging-discharging rate has significant influence on the battery multi-transport processes and the internal electrochemical processes. Based on the validated numerical cases, the cycling effect on the battery impedance responses under three charging-discharging rates conditions (0.5C, 0.2C and 0.1C) are compared in Fig. 10, where the other operating conditions are the same as the base cases. As shown in Fig. 10, the cycling effect induced size-increasing on the impedance mid-high frequency arc is much more obvious at 0.5C case, where the mid-frequency arc is almost unchanged after 50 cycles’ operation under 0.1C condition. This is because large charging-discharging rate can intensify the side reduction reactions between electrode particles and electrolyte solution. Since the generation process of SEI film is enhanced, the larger charging-discharging rate eventually results in the increase on battery impedance. Besides, Fig. 10 also shows that although the low charging-discharging rate can greatly reduce the influence of cycling on the battery impedance in mid-high frequency range, its effects on the low frequency branch of impedance spectrum are still significant, which implies that even under very low changing-discharging rate, the solid state diffusion process still varies with cycling.

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4. Conclusions In this study, a multi-physics based EIS simulation approach is developed for lithium ion battery, which is dedicated to study the cycling effect on the battery impedance responses. The cycling induced SEI film growth is coherently coupled with battery charge transport, mass transfer, thermal energy as well as electrochemical processes. The EIS simulation is carried out by applying a perturbation voltage on the electrode surface, of which the results on cycled cells are compared with the experimental data. During the battery cycling operation, modeling study indicates that the electrical double layer capacities in two electrodes determine the overlap of the impedance responses from two electrodes, and the dependencies of electrode open circuit potential on temperature and solid phase concentration have great influence on middle frequency arc and low frequency branch of the impedance curve. Under different diffusivities of binary electrolyte, the comparison study shows that the diffusivity of binary electrolyte can generate great influence on the solid state diffusion process, and worse diffusivity can intensify the battery aging effect with cycling. Through applying different SEI growth rates, the numerical results show that cycling operation can reduce the variation of impedance on the low frequency branch, and stabilize the transport process of active species into solid phase. The thermal effect on the battery impedance responses suggests that high operating temperature can greatly intensify the battery aging effect with cycling, but good cooling condition of lithium ion battery can positively reduce the cycling induced impedance rising. The effect of battery charging-discharging rate is finally studied. It is found that large charging-discharging rate can intensify the battery impedance rise in the mid-high frequency range with cycling. It is also found that although the low chargingdischarging rate can greatly reduce the influence of cycling on the battery impedance in mid-high frequency range, its effects on the low frequency branch of impedance spectrum are still significant.

Acknowledgements The authors gratefully acknowledge the financial support from UWM Research Growth Initiative (RGI).

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