#### List of symbols for diffusion coefficient equation derivation

*y*

*I*

^{−1}]

*I*_{0}

^{−1}]

*t*

*E*

*τ*

*D*

^{2}s

^{−1}]

*c*_{i}

^{−3}]

*x*

*S*

^{2}]

*z*_{i}

*q*

^{−19}C

*L*

*V*_{M}

^{3}mol

^{−1}]

δ

*m*_{i}

*M*_{i}

^{−1}]

*N*_{A}

^{23}mol

^{−1}

*F*

^{−1}

Δ*E*_{s}

Δ*E*_{t}

### 1. Introduction

### 2. Principle of GITT

### 2.1 Diffusion coefficient equation derivation

_{3}Sb [18]. The designed cell comprises a pure compound of mobile species, an electrolyte, and a working electrode, as shown in Fig. 1(a). This method is based on the idea that the number of ions passing through the electrolyte can be calculated from the current. That is, the GITT method is derived from Fick’s law by calculating the number of mobile ions moving at the interface between the electrolyte and the electrode phase boundary, correlating with the transient and steady-state voltage measurements.

*I*

_{0}) is applied at time

*t*

_{0}, a sudden voltage step occurs from

*E*

_{1}to

*E*

_{2}owing to the current flux in the form of an IR drop (the voltage increases or decreases when the current is positive or negative, respectively). Here, we consider a situation in which the current value is positive. During the current duration time (

*τ)*, the mobile ions (lithium ions, Li

^{+}) are deintercalated from the host material, and the voltage changes to

*E*

_{3}. After the constant current pulse, the voltage drops again owing to an IR drop. During the relaxation period, the composition of the electrode becomes homogeneous by Li

^{+}diffusion in the material, and the voltage reaches another equilibrium state,

*E*

_{4}. This becomes another initial potential (

*i.e*., new

*E*

_{1}) for the next galvanostatic titration step. A schematic of the mobile ion movement during the GITT experiment is shown in Fig. 1(d).

^{+}from the redox reaction at the electrode, an electric current (

*I*) can be expressed as Equation (1) using Fick’s 1

^{st}law:

*D*is the chemical diffusion coefficient of species

*i*[cm

^{2}s

^{−1}],

*c*

*is the concentration of species*

_{i}*i*[cm

^{−3}],

*x*is the distance coordinate [cm],

*S*is the contact area of the electrolyte and electrode interface [cm

^{2}],

*z*

*is the charge number (*

_{i}*e.g*.,

*z*

*= 1), and*

_{Li}*q*is the elementary charge (1.602×10

^{−19}C). Equation (1) expresses that the external current (

*I*) is equivalent to the number of charge carriers transported in the electrode at the phase boundary with the electrolyte (where

*x*= 0 in Fig. 1(a)). Because a constant current pulse (

*I*

_{0}) is used in the GITT experiment, Equation (1) is expressed as Equation (2) for LIBs:

*c*

*⁄ ∂*

_{Li}*x*is a function of distance (

*x*) and time (

*t*), and the diffusion coefficient can be determined by solving the second-order differential equation:

^{nd}law of diffusion. Equation (4) is the initial condition at equilibrium, where the concentrations of Li

^{+}are homogeneous throughout the electrode before applying the current. Equations (5) and (6) are two boundary conditions at

*x*= 0 and

*x*=

*L*, respectively. Equation (5) is transformed from Equation (2), and Equation (6) is a condition in which ions are impermeable at a finite boundary. The solution of the differential Equation (3) under conditions (4 )-(6) is known [19] and can be expressed as follows for

*x*= 0:

*ierfc*(

*z*) = [π

^{−0.5}exp(−

*z*

^{2})] −

*z*[1 −

*erf*(

*z*)], and

*erf*(

*z*) is the error function. Assuming that the

*t*«

*L*

^{2}⁄

*D*, Equation (7) can be approximated as follows:

*V*

*is the molar volume of the material (cm*

_{M}^{3}mol

^{−1}), and

*N*

*is Avogadro’s number (6.022×10*

_{A}^{23}mol

^{−1}). Inserting Equation (9) into Equation (8) yields Equation (10). Subsequently, expanding by

*dE*yields Equation (11), and the diffusion coefficient can be expressed as Equation (12) by arranging Equation (11). This result implies that the diffusion coefficient is determined by the three variables of cell voltage, time, and stoichiometry deviation.

*F*is Faraday’s constant (96485 C mol

^{−1}), and

*dE*⁄

*dδ*and

*E*vs.

*dE*⁄

*dδ*and

*E*

*=*

_{s}*E*

_{4}–

*E*

_{1}(steady-state voltage change) and Δ

*E*

*=*

_{t}*E*

_{3}–

*E*

_{2}(total voltage change during the current pulse) (Fig. 2). Additionally, the stoichiometry change due to the current pulse in the titration step of Li

^{+}can be expressed as Equation (15).

*m*

*is the mass of the host material in the electrode [g], and*

_{B}*M*

*is the molecular weight of the host material [g mol*

_{B}^{−1}]. Using the simple algebraic form of Equations (13), (14), and (15), the diffusion coefficient can be calculated at each step using the following Equation (16):

*V*

*and*

_{M}*M*

*are the material properties;*

_{B}*τ*,

*m*

*, and*

_{B}*S*are the values of the experimental condition; and Δ

*E*

*and Δ*

_{s}*E*

*are calculated from the GITT experimental results. The main assumptions for the diffusion coefficient equation were that the electrode material should be homogeneous, the molar volume change of the host material is not large enough, the current should be a low value with a short duration time, and the cell voltage should be linear for (duration time)*

_{t}^{1/2}.

_{0.5}Co

_{0.2}Mn

_{0.3}O

_{2}material, which shows the changes in current and voltage over time and the calculated diffusion coefficient. Each step comprised a 10 min current pulse at a current rate of C/20, followed by 1 h of the relaxation period with no current. During the charging process, a positive current pulse increased the cell voltage proportional to the IR drop, and the voltage slowly increased owing to the deintercalation reaction. After cutting off the current supply, the voltage decreased owing to the IR drop and continued to decrease, reaching the equilibrium state by Li

^{+}diffusion. During the discharging process with a negative current pulse, the voltage changed in the opposite direction (Fig. 3(a)). The diffusion coefficient can be calculated for various SOCs in the material by repeated galvanostatic titration steps during charging and discharging processes (Fig. 3(b)).

### 2.2 Experimental condition and practical limitations of GITT

### 3. GITT Application to LIBs

### 3.1 Diffusivity comparison of materials

*etc*., are all different for each study, this approach results in a few orders of difference for the diffusion coefficient, making it difficult to compare the absolute values. Therefore, the GITT was applied to investigate the differences or changing tendencies between the comparative groups using the same experimental conditions [12,27–30]. For example, Liu et al. [27] utilized the GITT to investigate the doping effects for high voltage operation (4.5 V vs. Li/Li

^{+}) between pristine LiCoO

_{2}(P-LCO) and La- and Al-doped LiCoO

_{2}(D-LCO), as shown in Fig. 4(a). They showed that the larger La ions in the structure effectively enlarged the

*c*-axis, increasing the diffusion pathway, and Al ions prevented the order-disorder transition, which substantially lowered the Li

^{+}diffusivity from a structural point of view. The Li

^{+}diffusivity values determined via GITT for D-LCO were twice those for P-LCO at the initial charging stage and ten times greater at the final discharging stage (Fig. 4(b)), resulting in a significantly enhanced rate capability. Wang et al. [12] showed a highly improved fast-charging performance in spinel lithium manganese oxide (LiMn

_{2}O

_{4}, LMO) by introducing a considerable number of twin boundary defects in the lattice via a defect engineering approach (Fig. 4(c, d)). They demonstrated that fast charging was enabled owing to the highly improved diffusion coefficient using the GITT (Fig. 4(e)), as well as CV analysis using the Randles-Sevcik equation mentioned above.

_{0.8}Co

_{0.1}Mn

_{0.1}O

_{2}(NCM811). They elucidated the changing tendency of the diffusion coefficient during charge and discharge as follows: The slow lithium diffusion in bulk in the initial SOC is due to the rare vacancies for diffusion. As the charging progresses, newly formed vacancies by the extraction of lithium and increased size of lithium pathways facilitate the diffusion kinetics, but the rapid shrinkage of the lithium layer at the end of the charge resulted in decreased diffusivity values. At the initial stage of discharge, the presence of concentration polarization facilitated an intercalation reaction with the highest diffusivity values. This tendency was maintained, and a drastically reduced value was observed at the end of discharge owing to the decreased size of the lithium pathway and the small number of lithium vacancies. The difference in the second cycle at the initial state of charge is due to the activation of the bulk structure of the H1 and H2 phases. The diffusivity value was the same when the lithium content was the same during the 2

^{nd}cycle.

### 3.2 Open-circuit voltage (OCV) analysis

*μ*) difference of the lithium ions between the anode and cathode is termed as the OCV [45,46].

*V*(

*x*) is the equilibrium potential (

*i.e*., OCV), where

*x*is the amount of lithium, and

*e*is the magnitude of the electronic charge. The chemical potential of lithium varies not only for the site in the crystal structure of the active material but also for the types and ratios of transition metals in alkali-ion transition metal oxides [7,45]. Therefore, the GITT method can be a powerful tool for examining the thermodynamic characteristics of materials. For example, Kim et al. [47] employed the GITT experiment combined with theoretical calculations to understand the fundamental characteristics of transition metals and their interactions in a given structure of Ni-rich layered cathode materials. As shown in Fig. 5(a), they investigated the correlation between the Ni and Mn ratios and the electrochemical behavior of the layered Ni-rich cathode for LiNi

_{0.5}Co

_{0.2}Mn

_{0.3}O

_{2}(NCM523) and LiNi

_{0.7}Co

_{0.2}Mn

_{0.1}O

_{2}(NCM721). Through first-principle calculations, they demonstrated that the equilibrium potentials based on the redox reactions of Ni

^{2+}/Ni

^{3+}are highly dependent on the Mn ratio (NCM523 and NCM721: ~3.7 and 3.5 V) because of a donor electron transferred from Mn to Ni owing to crystal field splitting. Because of the fixed chemical state of Mn

^{4+}in all structures during delithiation, the Ni

^{2+}/Ni

^{3+}redox reaction initially contributes to the delithiation process, and the Ni

^{3+}/Ni

^{4+}redox reaction is subsequently activated in the structures, resulting in NCM523 exhibiting a higher delithiation potential than NCM721. They confirmed the theoretical calculation results of the GITT measurements.

_{3}O

_{4}. They showed that a long relaxation period of over 20 days is needed to reach an equilibrium state, which indicates the existence of long-lasting chemical gradients and poor transport kinetics in conversion electrodes. Additionally, Assat et al. [33] used the GITT to examine the kinetic properties of Li

_{2}Ru

_{0.75}Sn

_{0.25}O

_{3}between the two staircase steps during the ‘staircase-like’ first charge. As shown in Fig. 5(b) insets, the time to reach equilibrium potential needed a few hours for the first 3.6 V step. On the contrary, voltage relaxation lasted for 44 h at the 4.15 V plateau, indicating an obvious kinetic difference during the first charging process. Similarly, Wang et al. [11] utilized the GITT to investigate the asymmetric behavior between charge and discharge when the O3-NaLi

_{1/3}Mn

_{2/3}O

_{2}material experienced a high potential. The discharging curve dropped gradually upon reaching the high voltage redox process over 3.3 V with the simultaneous rapid growth of the overpotential. Two different equilibrium times were observed at weakly and highly charged states, indicating that the hysteresis is triggered toward the last 50% of the charge, hence, defining two domains. This result was further consistent with the nuclear magnetic resonance analysis, in which the existence of a very broad peak is compatible with slow ion dynamics.

### 3.3 Overpotential and internal resistance analysis

*η*) and internal resistance. Considering the electrochemical reactions occurring at the electrode, an overall electrode reaction encompasses a sequence of reaction steps, such as adsorption, mass transfer, and charge transfer processes from a kinetic point of view [52]. Each step is characterized by a certain overpotential, so that the overall total overpotential can be considered as the sum of each contribution. In the GITT measurement, the total overpotential can be calculated by the difference between the measured cell voltage during the current pulse and the voltage at the end of the relaxation period, as shown in Fig. 6 and Equation (18).

_{0.5+}

*Co*

_{x}_{0.2}Mn

_{0.3-}

*O*

_{x}_{2}(

*x*= 0, 0.1, and 0.2) and calculated the overpotential values during the initial delithiation process in various SOCs (Fig. 7). A lower cation disorder of nickel ions and a wider inter-slab for lithium-ion diffusion with increasing nickel content were correlated with lower overpotential values during deintercalation. Cui et al. [53] used GITT measurements to investigate the equilibrium voltage and overpotential variation of the Li-O

_{2}batteries with

*N*-methyl-

*N*-propylpiperidinium bis(trifluoromethanesulfonyl)imide as the electrolyte and vertically aligned carbon nanotubes as the cathode. The equilibrium voltage at 60°C was 2.85 V and showed a zero-voltage hysteresis between discharging and charging; however, the overpotential analysis revealed asymmetrical polarization behavior during discharge and charge. This means that the voltage gap during the discharge and charge originates only from the kinetics and not from the asymmetrical thermodynamic reaction path.

*I*

*) using Equation (19):*

_{applied}_{2}/carbon onion hybrid materials (nano-TiO

_{2}-C) compared to anatase nano-TiO

_{2}resulted from the conductive carbon network improving the charge transfer kinetics. Internal resistance analysis can also be used to investigate the system after the gas evolution reaction at a high voltage (4.6 V) in a commercial 18650 cylindrical LIB cell [58]. The internal resistances remained similar regardless of the increase in the staying potential (4.6 V) and time; they interpreted that gas evolution has a limited influence on the degradation of the electrodes themselves. The large increase in internal resistance at the end of discharge is due to the higher ohmic resistance and concentration polarization [35]. The internal resistance analysis was also used to compare the different cell systems for LELB and ASLB, as shown in Fig. 8(b) [31].