Quantitative performance analysis of graphite-LiFePO4 battery working at low temperature

Chemical Engineering Science 118 (2014) 74–82

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Quantitative performance analysis of graphite-LiFePO4 battery working at low temperature Seongjun Bae, Hyeon Don Song, Inho Nam, Gil-Pyo Kim, Jong Min Lee, Jongheop Yi n World Class University Program of Chemical Convergence for Energy & Environment, School of Chemical and Biological Engineering, Institute of Chemical Processes, Seoul National University, Seoul 151-742, Republic of Korea

H I G H L I G H T S








Simulation of graphite-LiFePO4 battery is performed by pseudo 2-D numerical analysis. Method for the optimization of the electrode working at low temperature is proposed. Relationship between parameters and performances is analyzed. Performance profile varied with intrinsic and extrinsic properties is drawn. Particle radius which maximizes the performance at low temperature is calculated.

art ic l e i nf o

a b s t r a c t

Article history: Received 22 February 2014 Received in revised form 21 July 2014 Accepted 22 July 2014 Available online 30 July 2014

Although recent development of lithium ion batteries with high energy and power density enables one to actuate electrical vehicles, many challenges concerning the operational sensitivity in extreme climate conditions still exist. In this work, we investigate the effect of three independent parameters on the performance of a lithium ion battery under low temperatures via computational modeling: (1) the diffusion coefficients of lithium ion, (2) the lithiation rate constants at the surface of active material, and (3) the particle radius of active material. From the computational approach, the regions associated with maximum capacity and voltage value are identified in the parameter space. To simulate the operation of battery at low temperature, the particle radius which is independent of temperature is chosen as the control parameter, and the temperature dependence of diffusion coefficient and reaction rate constant is also calculated. By correlating the three parameters with the temperature, the optimized particle radius which can exert high capacity and voltage under low temperature was found. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Lithium ion battery Particle radius Diffusion coefficient Lithiation rate constant Low temperature Graphite

1. Introduction Lithium ion batteries (LIBs) have been used as the power supply in portable electronic devices and next-generation electric vehicles (EV) owing to their high energy and power density compared to other energy storage systems. Generally, lithium ion batteries are composed of two electrodes (anode and cathode) and electrolyte. The electrolyte in LIBs is a mixture of lithium ion salt like LiPF6 and makes short circuit by lithium ion conducting. Electrodes are involved in redox reactions with lithium ion, which determines capacity and voltage of device. If the LIBs are exposed in low temperatures, many temperaturesensitive problems of electrodes and electrolyte cause the decline of energy and power density. Due to these limitations, a variety of

n

Corresponding author. Tel.: þ 82 2 880 7438. E-mail address: [email protected] (J. Yi).

http://dx.doi.org/10.1016/j.ces.2014.07.042 0009-2509/& 2014 Elsevier Ltd. All rights reserved.

attempts have been made to analyze the kinetic mechanism and to enhance the performance of LIBs at low temperatures. It is known that the most critical problem of LIBs is the freezing of electrolyte at low temperatures. Mixtures of solvents, co-solvents, newly developed types of electrolyte salts and electrolyte additives have been used to address this issue (Zhang et al., 2009; Smart et al., 2010; Yaakov et al., 2010; Liao et al., 2013). As a result of these efforts, currently available LIBs can operate at 213 K (Zhang et al., 2009). However, other problems including high charge transfer resistance and capacity loss still remain. Previous studies indicate that charge transfer resistance is increased at temperatures below zero (Jansen et al., 2007; Abraham et al., 2008), and loss of capacity occurs at temperatures below 253 K on graphite anodes (Fan and Tan, 2006). The main reasons for the high charge transfer resistance and capacity loss at low operating temperatures are the low reaction rate of lithiation at the surface of the active material and the slow diffusivity of lithium ions inside the active material. Therefore, previous work has been focused on the enhancement of lithiation reaction rate and lithium ion diffusivity.

S. Bae et al. / Chemical Engineering Science 118 (2014) 74–82

Surface modification of the active material is performed by applying a surface coating, which changes the interfacial chemistry between the electrolyte and the active material (Mancini et al., 2009). The morphology of the active material is also controlled to increase the interfacial area and to decrease the diffusion length (Bai et al., 2012; Rangappa et al., 2012). While many attempts to overcome the temperature limitation associated with LIBs have been taken through the material development, optimization of material properties is also necessary to elicit the highest performance of such materials. However, there have been bottlenecks regarding the investigation of the quantitative effect of control parameters, such as the particle size of the anode/cathode material, lithium ion diffusivity and surface reactions at the electrode on LIB performance owing to the limitations associated with experimental instrumentation and techniques including the difficulty in altering the value of a particular parameter while maintaining the others constant. For example, controlling the morphology of the active materials can affect the diffusion coefficient, particle size and surface chemistry simultaneously (Peng et al., 2013). For the precise investigation of kinetics and reaction mechanism, all of the above parameters need to be independently manipulated. Since the advent of computational approaches based on concentrated solution theory and pseudo 2-D model by Newman in the early 1990s (Doyle et al., 1993), many electrochemical-thermal models have been proposed in the last decade (Wang et al., 2013; Song and Evans, 2000; Srinivasan and Wang, 2003; Ji et al., 2013). These simulation models simultaneously solve electrochemical and thermal equations describing the effect of heat generation. In addition, some parameters such as diffusivity and conductivity are temperature-dependent in these models. Despite the advances, previous studies using such models only provide qualitative information regarding the performance of LIBs. Therefore, for the rational design of electrode materials, it is necessary to quantify the performance of a battery at low temperatures. The objectives of this study are twofold. First, we identified the ‘critical boundary,’ which divides the performance profile into a

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performance-saturated region and a performance-unsaturated region by controlling the three independent parameters; diffusion coefficients of lithium ion, lithiation rate constants at the surface of active material, and particle radius of active material. Battery performance, such as capacities or potential range, maintains a maximum value when the parameters are larger than the critical boundary. In contrast, performance starts to drop rapidly when the parameters fall into the region lower than the critical boundary. Second, we estimated the particle radius to optimize both capacity and voltage under extremely low temperature conditions. The values for the diffusion coefficients and the lithiation rate constants are intrinsic properties which are highly dependent on external conditions and chemical compositions. On the other hand, particle size, an extrinsic property of a material which can be controlled at the synthesis stage, uninfluenced by environmental changes. Therefore, the use of particle radius as an optimization variable is practicable from the engineering viewpoint. We calculated the optimized particle radius in order to obtain the maximum performance even though the diffusion coefficient and the lithiation rate constant become small at low temperature. We modulated the particle radius in order to move the performance of the battery to the saturated region beyond the critical boundary. Our proof-of-concept research is based on graphite-LiFePO4 LIBs at low temperatures. Simulation results suggest the particle radius of anode and cathode materials should be lower than 31 nm and 38 nm, respectively, to maximize the capacity and voltage at 243 K.

2. Mathematical model The numerical simulation for the discharge curve of a battery, as reported by Newman (2008), is based on a pseudo 2-D Newman model (Doyle et al., 1993). In this model, a 1-D lithium ion transport model is correlated with a spherical lithium ion diffusion model of active material inside the battery. The battery system is

Fig. 1. Schematic figure of the Newman model of discharging graphite-LiFePO4 battery.

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treated as galvanostatic discharge, and lithium ion concentrations of electrodes and electrolyte are calculated as a function of discharge time. The equations were solved for five main variables: solid potential Φ1 , electrolyte potential Φ2 , lithium ion concentration of the electrolyte C e , lithium ion concentration of the active material particle C s and generated current densities ji of the cathode, anode and separator (i¼p, n, s). A schematic figure describing the Newman model of discharging graphite-LiFePO4 battery is shown in Fig. 1. The governing equations are described in Doyle et al. (1993) and Fuller et al. (1994). At the electrode, the governing equation for solid potential Φ1 is ∂2 Φ σ ef f ;i 21 ¼ ai Fji ∂x

ð1Þ

where σ ef f ;i is the effective electronic conductivity of solid and ai is the specific surface area of active material particle. Potential of electrolyte Φ2 is expressed as     2κ RT ∂ ∂Φ ∂ ∂ ln C e ¼ ai Fji κ ef f ;i 2 þ ef f ;i ð1  t þ Þ ð2Þ  F ∂x ∂x ∂x ∂x where κ ef f ;i is the effective ionic conductivity of electrolyte and t þ is the transference number of lithium ions in the electrolyte. The governing equation for lithium ion concentration of the active material particle C s is   ∂C s 1 ∂ 2 ∂C s r ¼ Ds;i 2 ð3Þ ∂t ∂r r ∂r where Ds;i is the diffusion coefficient for lithium ion of active material. The lithium ion concentration of the electrolyte ϵi is

ϵi

∂C e ∂2 C e ¼ Def f ;i 2 þ ð1  t þ Þai Fji ∂t ∂x

ð4Þ

where Def f ;i is the effective diffusion coefficient for the lithium ion of the electrolyte. The boundary conditions for (1)–(4) are given in Table 1. The Butler–Volmer equation for lithiation reaction is described as αa F

αc F

ji ¼ ki ðC s;max;i C s;surf ;i Þαa C s;surf ;i αc C e αa ðe RT ðΦ1  Φ2  U i Þ  e  RT ðΦ1  Φ2  U i Þ Þ ð5Þ where C s;max;i is the maximum lithium ion concentration when all active material is intercalated. C s;surf ;i is the lithium ion concentration of particle surface, αc is the charge transfer coefficient of cathodic reaction, αa is the charge transfer coefficient of anodic reaction and U i is the open-circuit potential of cathode (LiFePO4) and anode (graphite) (Doyle et al., 1993; Wang, 2012; Ye et al., 2012). The values of charge transfer coefficient regarding cathodic and anodic reaction is 0.5 (Forman et al., 2012). And the above equation can be reduced to:   0:5F ji ¼ 2ki ðC s;max;i  C s;surf ;i Þ0:5 C s;surf ;i 0:5 C e 0:5 sinh ðΦ1  Φ2  U i Þ RT ð6Þ

Tables 2–4 show the base parameters and open circuit potentials used in the simulation. These parameters are mainly based on the findings in Wilcox et al. (2007), Doh et al. (2011), Safari and Delacourt (2011) and Forman et al. (2012). Owing to the difference between the numerically fitted and the experimental values of the diffusion coefficient on cathode, we directly utilized the experimental diffusion coefficient of LiFePO4 and FePO4 in the simulation (Prosino et al., 2002).

3. Results and discussion 3.1. Model validation The simulation results of the numerical models were validated by comparing with experimental data reported by Wang et al. (2011), where the experimental data for three temperature values were measured with commercial LiFePO4 26650 cylindrical battery (A123Systems). The parameters we used in simulation are based on the previous work in Prosino et al. (2002), Wilcox et al. (2007), Doh et al. (2011), Safari and Delacourt (2011) and Forman et al. (2012) (Tables 2–4). As shown in Fig. 2a, 0.5 C discharge curves at 273 K, 318 K and 333 K show a good agreement between the experimental data and the simulation results of this study. The effect of particle size was not validated because it is difficult to change the size of the active material in a commercial battery. Therefore, we compared the experimental data for the Li–LiFePO4 half-cell test instead of A123systems 26650 battery test with simulation results. Fig. S.1 shows the simulation results and experimental data (Chen and Dahn, 2002; Choi and Kumta, 2007; Lee et al., 2008). The results proves that the simulation can reflect the effect of the particle size well. Though decrease of voltage in early discharging region is not fully covered by the simulation, the calculated values of voltage and capacity during mid- and final-discharging stages match the experimental results well. It should be noted that the capacity and voltage of the battery decreased at 273 K compare to those at 318 K and 333 K, all of which are confirmed by both the experimental and numerical results. These results imply that the different values of certain parameters do not always cause the change in the performance of battery. 3.2. Relationship between parameters and performances To qualitatively investigate the relationships between independent parameters and performance, we examined the effects of diffusion coefficients of lithium ions, rate constants at the surface of the active material and particle radius of cathode/anode, respectively. During discharge, only one of the parameters is varied, while the other parameters are set to the values of commercial battery (Tables 2–4). Discharge curves with various diffusion coefficients, rate constants and the particle radius of anode are shown in Fig. 3a–c. Fig. 3a shows that the diffusion coefficient affects the battery capacity. The discharge curve is nearly identical while the diffusion coefficient is higher than 0.05 μm2 s  1. The capacity starts to

Table 1 Boundary conditions of variables. Variables

Interface of the electrode-current collector

Interface of the electrode-separator

Solid potential Φ1

∂Φ1;p ∂x ¼ I; ∂ Φ2  ef f ;i ∂x ¼ 0 e  Def f ;i ∂C ∂x ¼ 0 ∂C s ∂r jr ¼ 0 ¼ 0

Φ1  σ ef f ;i ∂∂x ¼0

Electrolyte potential Φ2 Concentration of the electrolyte C e Concentration of the active material particle C s

 σ ef f ;p

Φ1;n ¼ 0

κ

s  Ds:i ∂C ∂r r ¼ R ¼ ji i

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Table 2 Base parameters for the cathode and anode of a graphite-LiFePO4 battery (Prosino et al., 2002; Wilcox et al., 2007; Doh et al., 2011; Forman et al., 2012). Parameter

Graphite (anode)

LiFePO4 (cathode)

Radius of particles (m) Lithium Ion Diffusion coefficient (m2 s  1) Reaction rate constant (mol s  1 m  2)(mol m  3)  1.5 Conductivity (S m  1) Initial cSOC Porosity Density of the active material (kg m  3) Specific capacity (mA h g  1)

3.610  6 a 8.27510–14 a 9.00910–12 a 100 a 0.84 a 0.6188 a 2260 d 372 d

1.63710  7 6.210–19 b 1.16810–12 100 a 0.16 a 0.5206 a 3600 e 170 e

a

a

c

SOC: state of charge ( ¼ capacity/theoretical capacity). Forman et al. (2012). Prosino et al. (2002). d Doh et al. (2011). e Wilcox et al. (2007). a

b

Table 3 Base parameters for a graphite-LiFePO4 battery (Forman et al., 2012). Parameter

Value

Current density Transport number Thickness of cathode (m) Separator porosity

1 C (6.44 A m  2) 0.2495 a 6.22510  5 a 0.3041 a

Conductivity of the electrolyte (S m a b

1

)

Parameter

Value

Initial concentration (mol m  3) Thickness of separator (m) Thickness of anode (m) Diffusion coefficient of the electrolyte (m2 s  1)

Ce Ce 2 Ce 3 b þ 0:1554  1000 0:0911 þ 1:9101  1000  1:052  1000 a

1040 a 1.69710  5 a 6.22510  5a 310  10 b

a

Forman et al. (2012). Adopted from the Newman’s method.

Table 4 Open-circuit potential for a graphite-LiFePO4 battery (Safari and Delacourt, 2011). Electode Graphite (anode)

LiFePO4 (cathode)

a b c

Open circuit potential
a  0:1958 0:6379 þ 0:5416e  305:5309 xa þ 0:044 tanh xa 0:1088     xa  1:0571 xa þ 0:0117  0:1978 tanh  0:6875 tanh 0:0854 0:0529   xa  0:5692 c  0:0175 tanh 0:0875 3:4323  0:8428e  305:5309 xc  3:2474  10  6 e20:2645ð1  xc Þ 3:7995  3:2482  10  6 e20:2646ð1  xc Þ c b

3:8003

xa : state of charge for anode. xc : state of charge for cathode. Safari and Delacourt (2011).

Fig. 2. Correlation of simulation and experiment for 0.5 C discharge curve of commercial LiFePO4 battery at 273 K, 318 K and 333 K. Line indicates the data obtained from simulation and dot indicates experimental data. Experimental data is found in Wang et al. (2011).

decline rapidly when the diffusion coefficient is less than 0.05 μm2 s  1. No evidence of changes in voltage profile is found. As shown in Fig. 3b, capacity remains constant and the initial voltage decreases when reaction rate constant is lower than 360 (nmol h  1 m  2)(mol m  3)  1.5. Fig. 3c demonstrates that the radius of the active material particles affects both the capacity and voltage. Battery performance remains the same until the value of the parameters reach 10 μm, and then begins to decline rapidly when the value passes 10 μm. Fig. 3d–f shows the discharge curves with various parameters at the cathode. The plots show the same tendency except the order of the critical boundaries. The values for the boundary between maximum performance and low performance for the cathode are 0.5 nm2 s  1, as shown in Fig. 3d, 18 (nmol h  1 m  2)(mol m  3)  1.5 in Fig. 3e, 100 nm in Fig. 3f, respectively. The independence between voltage and the diffusion coefficient still holds even if the value of reaction rate constant is changed as shown in Fig. S.2 and S.3. Similarly, the reaction rate constant does not change the capacity with other values of the diffusion coefficient. We denote the boundary between the

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Fig. 3. Discharge curves for a graphite-LiFePO4 battery for various parameters. Controlled parameters are (a) the diffusion coefficient of the anode (graphite), (b) the rate constant of the anode, (c) particle radius of the anode, (d) diffusion coefficient of the cathode (LiFePO4), (e) rate constant of the cathode and (f) particle radius of the cathode. The arrow indicates the increase of performance.

performance-saturated region and the performance-unsaturated region as the ‘critical boundary’. The critical boundary can be mathematically defined as C:B:capacity ¼ C max  0:99

ð7Þ

C:B:voltage ¼ V max  0:99

ð8Þ

where C:B: is critical boundary, C max is maximum capacity of the battery and V max is maximum voltage of the battery. The effects of the parameters on the performance are derived from the characteristics of capacity and voltage. Capacity, which is affected by diffusion coefficient and particle radius, indicates the amount of transferred charge carriers from electrode material to external circuit. Therefore, the resistance against the lithium ion movement in the particle which is strongly related to the diffusion coefficient is an important issue determining the value of capacity. The resistance of sphere-shaped active material particles which is calculated using the volume insertion mechanism is proportional to the particle size and inversely proportional to the conductivity (Gaberscek and Jamnik, 2006; Gaberscek et al., 2007). While the capacity performance is influenced by the parameters related to the movement of lithium ion inside the particles, voltage is determined by reactions at the surface. An increase in the reaction rate, indicative of an increase in exchange current density, enhances the power density (Kato et al., 2003). In the case of a galvanostatic discharge, an enhancement in power causes an increase in voltage. Eq. (6) which describes the local exchange current density, should be multiplied by the specific surface area to obtain the generated current. Thus, voltage is affected by the reaction rate constant and particle radius which determines the specific surface area. The cathode undergoes different phase transformations which includes two delithiation mechanism; a solid-solution reaction and a two-phase reaction. In Fig. 3d, the solid-solution mechanism is observed in last 10% SOC for high diffusion rate. In contrast, the two-phase reaction is dominant in

case for low diffusion rate. As shown in Fig. 3f, we can observe the solid-solution mechanism for nano-particulated (5–100 nm) LiFePO4 and two-phase mechanism for micro-particulated (0.3 μm) LiFePO4. The results are in good agreement with the previous experiments (Tan et al., 2009; Bai et al., 2011; Yoo and Kang, 2013).

3.3. Quantifying of the critical boundary at various parameters To quantify the performance drop at the critical boundary, we obtained the capacity and the voltage profile with various independent parameter values. Because the battery performance is determined by the inferior electrode between anode and cathode, the parameters of one electrode are fixed at their nominal values while the profile of the parameters of the other electrode can be determined (Tables 2 and 3). Critical boundary is set at the point where the decrease of performance is greater than 1% of the maximum performance. Fig. 4a shows the log-scale capacity profile as a function of various parameters of the anode, and the capacity reaches the maximum point when the diffusion coefficient is larger than the critical boundary. As the profile passes the critical boundary, the capacity is significantly decreased. Particle radius also influences the position of the critical boundary. Reduced particle radius causes a decrease in the performance drop region. The short diffusion length indicates greater diffusion rate which is proportional to the diffusion coefficient. Thus, the capacity loss caused by a decrease in the diffusion coefficient can be counterbalanced by the reduction of particle radius. The capacity profile of the cathode, shown in Fig. 4b is similar to the plot in Fig. 4a. At the critical boundary, the slope of capacity profile of cathode is steeper than that of anode. The effect of the particle radius of cathode is higher than that of anode owing to the 10,000 times larger diffusion coefficient of anode than that of cathode.

S. Bae et al. / Chemical Engineering Science 118 (2014) 74–82

Fig. 4. Log-scale capacity profile as a function of particle radius and diffusion coefficient of (a) the anode (graphite) and (b) the cathode (LiFePO4). Other Parameters which are not described in this graph are the base values used in the simulation.

The voltage profile in Fig. 5 is similar to the plot in Fig. 4. In Fig. 5, rate constant and particle radius are selected as parameters. The voltage with sufficiently large rate constant remains at the maximum value, and performance begins to decline after passing through the critical boundary with reduced diffusion coefficient. An increase in particle radius results in a faster drop in performance. The surface to volume ratio is inversely proportional to the particle radius. Therefore, an increase in particle radius causes a decrease in particle surface area, which has a direct effect on the exchange current and lithiation reaction rate. The rate constant should become higher to offset this effect. Capacity and voltage profiles are qualitatively similar but quantitatively different. The performance profile has a different shape because capacity is affected by the movement of lithium ion inside the particle and voltage is affected by the lithiation at the particle surface. Fig. 6 shows the contours for fixed capacity of the anode and cathode as a function of the diffusion coefficient and particle radius. Interestingly, the value of the diffusion coefficient is proportional to the square of the particle radius. Fig. 7a shows contours for the voltage of the anode as a function of rate constant and particle radius. The shape of the voltage contour is different from that of the capacity. Given the same voltage, the rate constant is proportional to the particle radius. A graph of the cathode in Fig. 7b is similar to Fig. 7a.

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Fig. 5. Log-scale voltage profile as a function of particle radius and rate constant of (a) the anode (graphite) and (b) cathode (LiFePO4). Other Parameters which are not described in this graph are the base values used in the simulation.

The performance profile and the relationship between parameters can be used as a blueprint for the design of a battery with optimized performance. The contour line allows for performance to be quantified by providing a correlation between the parameters and battery performance. It is possible to achieve the decent architecture of a battery with maximized performance by modulating the parameters associated with the critical boundary. 3.4. Determining the critical particle radius at 243 K We classify the three independent parameters into two groups, intrinsic and extrinsic properties. The diffusion coefficient and reaction rate constant are intrinsic properties which are characteristics of the material. The intrinsic properties are sensitive to the environment such as temperature and can only be controlled indirectly. On the other hand, particle radius, an extrinsic property can be directly controlled during synthesis and not affected by environmental factors during operation. Thus, capacity and voltage can be relatively easily changed by modulating the particle radius compare to modulating the diffusion coefficient or the reaction rate constant. The critical boundary provides the optimal condition required to achieve the maximum performance, as performance does not increase further past the critical boundary. Therefore, the optimal particle radius can be calculated from the critical boundary under extremely low temperature.

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Fig. 6. Contour line for capacity profile as a function of diffusion coefficient and particle radius for (a) the anode and (b) the cathode. 2.25, 2.2 and 2.15 mAh are selected as the basis.

The diffusion coefficient and rate constant follow the Arrhenius equation and their value changes with temperature as (Bernardi et al., 1985). Ds;i ðTÞ ¼ D s;298:15; i e  ki ðTÞ ¼ k 298:15; i e 

Ea; D; i 1 1 R T  298:15

ð

Ea; k; i 1 1 R T  298:15

ð

Þ

Þ

ð9Þ

Fig. 7. Contour line for the voltage profile as a function of rate constant and particle radius for (a) the anode and (b) the cathode. 3.25, 3.2 and 3.15 V are selected as basis.

Table 5 Activation energy for intrinsic properties (Yu et al., 1999; Takahashi et al., 2002; Allen et al., 2007; Yamada et al., 2009).

ð10Þ

where D 298:15; i is the diffusion coefficient of cathode and anode electrode (i¼p,n) at 298.15 K, Ea; D; i is the activation energy for diffusion of cathode and anode electrode (i¼ p,n), k 298:15; i is the lithiation rate constant of cathode and anode electrode (i¼p,n) at 298.15 K and Ea; k; i is the activation energy for lithiation reaction of cathode and anode electrode (i¼ p,n). Table 5 includes the activation energy for diffusion and lithiation reaction on each electrode. The diffusion coefficient and the rate constant at an extremely low temperature, until 243 K, was calculated using the experimental value of diffusion coefficient and rate constant at standard temperature. In this case, we can calculate the particle radius which recovers the performance drop because the three parameters are correlated with battery performance represented by capacity and voltage. Fig. 8a shows the capacity as a function of particle radius of the anode at various temperature values. The value of the diffusion coefficient at a given temperature is calculated using the Arrhenius equation. The critical boundary for capacity (1% lower than the highest capacity) is fixed at 2.246 mAh for the anode and 2.239 mAh for the cathode. As shown in Fig. 8b, the difference between the plot for the anode and cathode can be attributed to the different values of diffusion coefficient and the activation

Cathode Anode

Activation energy for diffusion

Activation energy for reaction

39 kJ mol  1 a 5.1 kJ mol  1 c

29 kJ mol  1 58 kJ mol  1

b d

a

Takahashi et al. (2002). Allen et al. (2007). c Yu et al. (1999). d Yamada et al. (2009). b

energy. Low value of diffusion coefficient leads to the plot with a steep slope and low activation energy makes the difference between the slope at 243 K and the slope at 298 K small according to the Arrhenius equation. At standard temperature, 298 K, the radius of the active material on the graphite anode should be less than 7.2 μm and the radius of the active material on the LiFePO4 cathode should be less than 230 nm to maximize the capacity. At 243 K, the radius of the active material on the anode should be less than 5.7 μm and the radius of the active material on the cathode should be less than 38 nm to maximize the capacity. Fig. 9a is the voltage profile as a function of particle radius for the anode at various temperature values. The critical boundary of the voltage is fixed at 3.26 V which is 1% lower than a voltage of commercial LiFePO4 battery, 3.3 V. The data shown in Fig. 9b is

S. Bae et al. / Chemical Engineering Science 118 (2014) 74–82

Fig. 8. Capacity profile for various temperatures as a function of (a) particle radius of anode (graphite) and (b) particle radius of cathode (LiFePO4). Diffusion coefficient of each simulation is calculated using the Arrhenius equation. Other Parameters which are not described in this graph are base values during simulation. The critical boundary is set as 2.246 mAh for anode and 2.239 mAh for cathode to obtain the particle radius which permits the maximum capacity on the battery at a given temperature. Inset figure is a magnification of the critical boundary.

similar to that in Fig. 9a, except for both the slope of the curve and particle radius at the critical boundary. In order to obtain 3.26 V as an operating voltage of battery, the radius of the active material on the anode should be less than 6.1 μm and the radius of the active material on the cathode should be less than 570 nm. At 243 K, to achieve 3.26 V, the radius of the active material on the anode should be less than 31 nm and the radius of the active material on the cathode should be less than 40 nm. The tendency of performances can be changed by current densities. In this regard, we examined the value of particle radius for critical boundary (critical particle radius) at various C-rates in Fig. 10. As a current of 0.5 C is smaller than that of 1 C, the diffusion and reaction of lithium ion in the active material would be slower. Therefore, the critical particle radius of 0.5 C is larger than that of 1 C. The critical particle size required to maintain 99% of the saturated capacity of the anode at 298 K and 243 K are 10.1 μm and 7.2 μm, respectively, and the values for the cathode at 298 K and 243 K are 3.2 μm and 54 nm. In the case of voltage, the critical particle radius for the anode and cathode at 298 K and 243 K are 16 μm, 80 nm, 3.6 μm and 180 nm, respectively. For 2 C, on the other hand, an opposite effect to the critical particle radius is observed. The critical particle radius of the anode for the capacity decreases to 5.5 μm at 298 K and to 1.8 μm at 243 K. For the cathode, the value is 160 nm at 298 K and 18 nm at 243 K. The critical particle radius for a voltage of 2 C are 2.9 μm for the anode

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Fig. 9. Voltage profile of various temperatures as a function of (a) particle radius of the anode (graphite) and (b) particle radius of the cathode (LiFePO4). Rate constant of each simulation is calculated using the Arrhenius equation. Other Parameters which are not described in this graph are base values used during the simulation. The critical boundary is set as 3.26 V to obtain the particle radius which performs the maximum voltage on the battery at certain temperature. Inset figure is the magnification of the critical boundary.

at 298 K, 14 nm for the anode at 243 K, 72 nm for the cathode at 298 K and 5.1 nm for the cathode at 243 K. Consequently, the smaller radius of anode and cathode particles is more advantageous for the increase of capacity and voltage.

4. Conclusion A computational method to quantify the effect of critical parameters on battery performance at low temperatures was proposed. The particle radius was treated as the main factor in controlling the performance because it is controllable, independent of external conditions. We studied the performance profile as a function of the particle radius and intrinsic properties such as diffusion coefficient or reaction rate constant, in order to examine the contribution of three independent parameters on battery performance. Performance profile shows a critical boundary where the performance drops by greater than 1%. This boundary was used as the criterion for determining the optimal particle radius. The optimal particle radius that can circumvent the performance drop under low temperature could also be calculated. In the case of 1C-rate, the particle radius of the anode should be less than 5.7 μm to maximize the capacity and less than 31 nm to maximize the voltage. Particle radius should be less than 38 nm for the largest capacity and less than 40 nm for the highest voltage for the cathode.

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Fig. 10. Critical particle radius for the anode and cathode at 298 K and 243 K for (a) capacity and (b) voltage as a function of C-rate. Critical particle radius is the particle radius for which the capacity is 99% of that for a saturated performance or when the voltage is 99% of 3.3 V.

Acknowledgements This research was supported by the Global Frontier R&D Program on Center for Multiscale Energy System funded by the National Research Foundation under the Ministry of Science, ICT & Future, Korea (NRF-2011-0031571) and this research also was supported by the Supercomputing Center/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2013-C1-024).

Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.ces.2014.07.042.

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