Modelling the effect of anode particle radius and anode reaction rate constant on capacity fading of Li-ion batteries

This paper investigates the effect of anode particle radius and anode reaction rate constant on the capacity fading of lithium-ion batteries. It is observed through simulation results that capacity fade will be lower when the anode particle size is smaller. Simulation results also show that when reaction rate constant is highest, the capacity loss is the lowest of lithium-ion battery. The potential drop across the SEI layer (solid electrolyte interphase) is studied as a function of the anode particle radius and anode reaction rate constant. Modelling results are compared with experimental data and found to compare well


Introduction
Side reactions can cause various adverse effects leading to capacity fading in lithium-ion batteries. The aging of Li-ion batteries usually occurs due to various parameters and electrochemical reactions, and capacity loss varies between all stages during a charge-discharge load cycle, depending on various parameters such as cell voltage, electrolyte concentration, temperature, and cell current. This work shows the model for aging and capacity loss in the anode of a Li-ion battery, where the formation of a thin film of solid-electrolyte-interface (SEI) shows an adverse capacity loss of cyclable lithium. Capacity fading in a lithium-ion battery has been studied under various load conditions.
Haran et al. [1] studied the effect of various temperatures during various cycles in the capacity fading of 18650 Li-ion cells. It is observed that with an increase in temperature of Li-ion batteries, capacity fading is increased. It is observed that at temperatures higher than 55 °C, the cell ceases to operate after 500 cycles due to ongoing SEI film formation over the anode surface. Han et al. [2] studied the cycle life of commercial Li-ion batteries with LTO anodes in electric vehicles. The author also found that at 55 °C, the capacity fading in the battery is more than lower operating temperature Lithium-ion battery. Liaw et al. [3] studied the correlation of Arrhenius behavior in power and capacity loss with cell impedance and heat generation at different temperatures and state of charge cell had a strong dependence on discharge C-rate and loss of cyclable lithium. Khaleghi Rahiman et al. [19] developed a mathematical model to study cell life with various parameters. The author studied capacity loss and SEI formation in Li-ion batteries at different temperatures at different SOCs. The author postulates that cathode side reactions are accelerated at higher SOCs and temperatures. In our model, we compare the effect of anode particle radius and anode reaction rate constant on the capacity fading of a lithium-ion battery.

Model development
A 1D model of a Li-ion battery interface is created, as shown in Figure 1. The components of a Liion battery are the negative electrode, positive electrode, and separator. Graphite electrode (LixC6) MCMB is used for negative electrode material, NCA electrode (LiNi0.8Co0.15Al0.05O2) is used for positive electrode material and LiPF6 (3:7 in EC: EMC) is used as a liquid electrolyte.

Model equations
The model equations analyze the current equilibrium in the electrolyte and electrodes, the mass balance for the lithium and electrolyte in Li-ion batteries. The Li-ion battery physics at interface analyses five dependent variables: a) s -the electric potential, b) e -the electrolyte potential, c) ∆SEI -the potential losses due to solid-electrolyte interface (SEI), d) cLi -the concentration of lithium in the electrode particles e) ce -the electrolyte salt concentration.
The domain equations in the electrolyte are the conservation of current and the mass balance for the salt according to the following [20]: where e denotes the electrolyte conductivity, f is the activity coefficient for the salt, t+ is the transport number for Li + , isum is the sum of all electrochemical current sources, and Qe denotes an arbitrary electrolyte current source. In the mass balance for the salt, e denotes the electrolyte volume fraction, De is the electrolyte salt diffusivity, and Re the total Li + source term in the electrolyte. In the electrode, the current density, is is defined as where s is electrical conductivity. The domain equation for the electrode is the conservation of current expressed as is = -isum + Qs (4) where Qs is an arbitrary current source term. The electrochemical reactions in the physics interface are assumed to be insertion reactions occurring at the surface of small solid spherical particles of radius rp in the electrodes. The insertion reaction is described as: During charging, at anode xLi + + xe -+ graphite → LixC6 at cathode LixNi0.8Co0.15Al0.05O2 → xLi + + xe -+ Ni0.8Co0.15Al0.05O2 (6) During discharging, at Anode LixC6 →xLi + + xe -+ graphite (7) at cathode xLi + + xe -+ Ni0.8Co0.15Al0.05O2 → LixNi0.8Co0.15Al0.05O2 (8) An important parameter for lithium insertion electrodes is the state-of-charge variable for the solid particles, denoted SOC. This is defined as The equilibrium potentials E0 of lithium insertion electrode reactions are typically functions of SOC. The electrode reaction occurs on the particle surface and lithium diffuses to and from the surface in the particles. The mass balance of Li in the particles is described as where cLi is the concentration of Li in the electrode. This equation is solved locally by this physics interface in a 1D pseudo dimension, with the solid phase concentrations at the nodal points for the element discretization of the particle as the independent variable. The gradient is calculated in Cartesian, cylindrical, or spherical coordinates, depending on if the particles are assumed to be best described as flakes, rods or spheres, respectively.
The boundary conditions are as follows: where RLi denotes the molar flux of lithium at the particle surface caused by the electrochemical insertion reactions. In the porous electrodes, isum denotes the sum of all charge transfers current density contributions according to: where, Av denotes the specific surface area at any node of the lithium-ion battery interface. Active specific surface area (m 2 /m 3 ) defines the area of an electrode-electrolyte interface that is catalytically active for porous electrode reactions. Equation 13 describing the total current source in the domain is a function of active specific surface area and local current in the electrode. The source term in the mass balance is calculated from: where Rl.src is an additional reaction source that contributes to the total species source. At the surface of the solid particles, the following equation is applied: where n is the number of electrons and Sshape (normally equal to 1) is a scaling factor accounting for differences between the surface area (Av) used to calculate the volumetric current density and the surface area of the particles in the solid lithium diffusion model. Sshape is 1 for Cartesian, 2 for cylindrical, 3 for spherical coordinates and Li is the stoichiometric coefficient.
A resistive film (also called solid-electrolyte interface, SEI) might form on the solid particles resulting in additional potential losses in the electrodes. To model a film resistance, an extra solution variable for the potential variation over the film is introduced in the physics interface. The governing equation is then according to where RSEI denotes generalized film resistance, which can be expressed by: where, 0 is initial film thickness, ∆ is film thickness change and SEI is film conductivity. The activation overpotentials,  for all electrode reactions in the electrode then receives an extra potential contribution, which yields where, Eeq is the equilibrium potential of a cell. The battery cell capacity, Qcell,0 is equal to the sum of the charge of cyclable species in the positive and negative electrodes and additional porous electrode material if present in the model [20].
where a and c are the anode and cathode transfer coefficient and ka and kc are reaction rate constant for anode and cathode.

Results and discussion
Capacity fading of Li ion battery is studied with the effect of various parameters. A summary of the list of parameters used for simulation is shown in Table 1. The battery cycling consists of 3 various stages of charging and discharging, as shown in Figure 2: • Discharging at constant current discharge rate at 1 C until the cell potential reaches the minimum voltage of 2.5 V.

Effect of particle radius on capacity fading in lithium-ion batteries
Research on anode particle radius on capacity fading in lithium-ion batteries has been done previously. Rai [21] postulated that batteries with smaller anode particle sizes generate better capacity. The authors postulated that smaller particles(graphite) allow quicker lithium-ion intercalation and deintercalation due to the short distances for lithium-ion transport within the particles. There is no agreed-upon consensus for optimal particle size in lithium-ion batteries though particles less than 150 nm are mainly used. Wu [22] investigated the effect of silicon particle size in the micrometer range when used as a lithium-ion battery anode. The authors have found out in their study that particle size of 3μm shows better outcomes with respect to the 20 μm particle size with an initial capacity of 800 mAh/g and retention of 600 mAh/g after 50 cycles. Buqa [23] investigated three different graphite particle sizes (6, 15 and 44 µm) and showed that smaller particles could achieve better capacity retention. Several authors like Drezen [24] and Fey [25] have postulated that smaller particles improve capacity retention. Mei has [26] postulated that energy and power density increase with smaller particle sizes due to lower overpotential. Mei [26] has also postulated that smaller particle size increases the surface area for reaction. Our focus was to study the effect of the anode particle radius on the capacity fading in lithium-ion batteries taking into consideration lithium losses during cycling. Figure 3 shows the capacity fading of a lithium-ion battery with cycling for various anode particle radii. The model assumes zero lithium loss during the process of cycling. Four different anode particle radii (0.5, 1, 2 and 2.5 m) were considered for analysis. It is seen that the least capacity fading (high relative capacity) is seen for an anode particle radius of 0.5 m. Relative capacity is defined as the capacity of the battery at any point of time divided by the initial capacity of the battery. It can be seen that as the anode particle radius increases from 0.5 to 2.5 µm, the relative capacity decreases over 2000 cycles.  Figure 4 shows the capacity fading in a lithium-ion battery cycling for four different particle radii (0.5, 1, 2 and 2.5 m) with 10 percent lithium loss during cycling. During charge-discharge cycling, there is more lithium loss during initial cycles. A comparison of Figures 3 and 4 shows that the capacity loss is seen to be less without cyclable lithium loss compared to 10 % initial lithium loss as the number of cycles increases. This is clearly shown in Figure 5. It is seen from Figure 5 that there is less capacity loss of around 3 % when we go from zero percent lithium loss to 10 percent lithium loss during cycling.  The anode reaction rate constant indicates the intercalation/deintercalation reaction rate constant. Figure 6 shows that when the intercalation/deintercalation reaction rate constant is the http://dx.doi.org/10.5599/jese.1147 9 highest, the capacity loss is the lowest. With increasing intercalation/deintercalation reaction rate, the rate of lithium transport increases, effectively increasing the capacity of the battery. While Figure 6 shows the capacity loss when there is no initial cyclable lithium loss during cycling, Figure 7 shows the capacity loss when there is 10 % initial lithium loss during cycling. Figure 8 shows the comparison of the capacity losses when there are 0 and 10 % lithium losses during cycling. The figure shows that when the initial lithium loss during cycling increases from zero percent to 10 percent, there is a 4 % differential in the capacity loss due to side reactions.   Figure 9 shows the concentration of lithium ions at the anode/SEI interphase as a function of anode particle radius (4 different particle radii are shown in the figure). It is seen that the highest concentration of lithium ions at the anode/SEI interphase occurs at the smallest particle radius. Smaller anode particles allow lithium ions to intercalate and deintercalated quickly due to the short diffusion path for lithium ion transport within the particles. This leads to a higher concentration of lithium ions at the anode/SEI interphase. Figure 10 shows the concentration of lithium ions at the anode/SEI interphase as a function of the reaction rate constant for lithium intercalation. With the increasing rate constant of deintercalation, the concentration of lithium ions at the anode/SEI interphase is seen to increase.  During the charging cycle, lithium from the cathode moves to the anode and hence the concentration of lithium ions at the anode/SEI interphase increases. The lithium ions move from the anode to the cathode during the discharging cycle. Hence, the concentration of lithium ions at the anode/SEI interphase is seen to go from maximum to zero. As the battery cycles, lithium ions are lost in the intercalation deintercalation process. Hence, the concentration of lithium ions at the anode/SEI interphase is lower in the 2000 th cycle than in the 1 st cycle.

Effect of anode radius on lithium-ion concentration at the anode/SEI interphase
The potential drop across the SEI layer as a function of anode particle radius Figure 12 shows the effect of anode particle radius on the potential drop across the SEI layer. The figure analyses the effect of four different particles sizes on the potential drop across the SEI layer. The least potential drop across the SEI layer occurs when the anode particle size is the smallest. As explained earlier, smaller anode particle sizes lead to higher intercalation deintercalation rates leading to higher current densities. Given a constant power output, this indicates a lower potential drop across the cell and hence a lower potential drop across the SEI layer.  The graph shows that the potential drop across the SEI layer increases with decreasing rate constant. The anode reaction rate constant indicates the rate of intercalation deintercalation of lithium ions in the anode particles. When the anode reaction rate constant is lower, the intercalation/deintercalation of lithium ions in the anode is reduced, giving rise to a lower current density. Given a constant power output, this indicates an increased potential drop across the cell and, hence, a potential drop across the SEI layer. This is shown in Figure 13. Figure 14 shows the comparison of modelling predictions with experimental data [3,11]. Modeling predictions are found to compare well with experimental data. The model comparisons are made for 1 C discharge at 45 o C operating conditions for the lithium-ion battery cell. The parameters used for data fitting are shown in Table 1. discharge rate and 45 °C [3,11] Conclusion A 1-dimensional mathematical model is developed to study the effect of anode particle radius and anode reaction rate constant on capacity fading of a Li-ion battery. Simulation results predict that for the smallest anode particle radius of 0.5 m, capacity fading is less in comparison to 2.5 m. Smaller anode particle radii lead to faster lithium intercalation/deintercalation rates leading to higher current densities and lesser capacity fade. Smaller anode particle radii also lead to increasing anode surface area for reaction. The anode reaction rate constants are also found to play a major role in the capacity fading of lithium-ion batteries. It is found that the higher the anode reaction rate constant, the lesser is the capacity fade in the battery. Model results are compared with experimental data and found to compare well.

Nomenclature s
The electric potential at electrode