Visualizing non-equilibrium lithiation of spinel oxide via in situ transmission electron microscopy

Spinel transition metal oxides are important electrode materials for lithium-ion batteries, whose lithiation undergoes a two-step reaction, whereby intercalation and conversion occur in a sequential manner. These two reactions are known to have distinct reaction dynamics, but it is unclear how their kinetics affects the overall electrochemical response. Here we explore the lithiation of nanosized magnetite by employing a strain-sensitive, bright-field scanning transmission electron microscopy approach. This method allows direct, real-time, high-resolution visualization of how lithiation proceeds along specific reaction pathways. We find that the initial intercalation process follows a two-phase reaction sequence, whereas further lithiation leads to the coexistence of three distinct phases within single nanoparticles, which has not been previously reported to the best of our knowledge. We use phase-field theory to model and describe these non-equilibrium reaction pathways, and to directly correlate the observed phase evolution with the battery's discharge performance.

. Synchrotron XRD patterns of (a) pristine phase (x = 0) and (b) lithiated phases (x = 2). By fitting the experimental spectra using Rietveld refinement, the pristine phase is determined to be spinel structure with lattice parameter a = 8.390 Å, and the lithiated phase to be LiFe 3 O 4 with lattice parameter a = 8.476 Å. The entire nanocrystal displays the rocksalt structure (SAED reflection spots) along with the formation of a number of nanosize Fe particles (SAED diffraction rings). Fe nanoparticles can also be seen as darker contrast in TEM and HRTEM images. Scale bars, (a) 50nm; (b) 5nm; (d) 10 nm; (e) 5 nm. Figure 6. (a) Schematics showing two different modes of STEM imaging by collecting either high-angle scattered electrons using an annular dark-field (ADF) detector, or low-angle scattered and directly transmitted electorns using a bright-field (BF) detector. The ADF-STEM image (b) contains highly scattered electron signals and gives a better sensitivity of Z (atomic number) contrast. The BF-STEM image (c) contains mostly coherent electron signals and is more sensitive to the diffraction contrast caused by local strains. Direct comparison between simultaneously acquired (b) ADF-and (c) BF-STEM images of the same sample region suggests that a better contrast of the lithiation front (indicated by red arrows) can be revealed by BF-STEM imaging. Scale bars, 20 nm.

Supplementary Tables
Supplementary

Phase-Field Modeling
Our in situ experiments and DFT calculations have confirmed that the Fe 3 O 4 nanocrystals undergo two reaction stages during the lithiation process, i.e., from original Li-free Fe 3 O 4 to Liintercalated LiFe 3 O 4 phase, and then to a composite of Li 2 O+Fe. Here, we explain the evolution process using the electrochemistry theory based on non-equilibrium thermodynamics developed by Bazant and coworkers. 1,2 The standard phenomenological model of electrode kinetics is the Butler-Volmer equation 1 where t c   is the change rate of the local filling fraction c of Li-ion;  , the electron-transfer symmetry factor, is approximately constant for many reaction; 0 I is the exchange current;  is activation overpotential; ne is the net charge transferred from the solution to the electrode; B k is Boltzmann's constant and T is temperature. The exchange current will be written as 1 where k is the insertion/extraction rate; c s is the reaction site density; a + and a e is respectively the ionic activity in the electrolyte and the electron activity; λ 0 is the reorganization energy. The letters with tilde in Eq. (2) represent these variables are scaled to dimensionless form. For example, . In Eq. (2), ∆ε is the activation strain; c G   is the functional derivative of Gibbs free energy functional. Based on the Cahn-Hilliard phase field model, 3 where is the homogeneous free energy function; is the Cahn-Hilliard gradient energy coefficient; and    L ( L is the length scale of sample). The activation overpotential in Eq. (1) can be written as is the thermodynamic driving force. It is the consistent definition of diffusional chemical potential. For an inhomogeneous system, the simplest approximation is c . For Li-ion battery electrodes, the overpotential is is the local voltage drop across the interface; and   : is strain energy due to Liion insertion. The stress T k c B s    and the strain  are resulted from lattice distortion with Liion insertion. Obviously, the strain depends on Li-ion concentration. For simplicity, we usually assume the strain linearly depends on Li-ion concentration. Therefore, the strain energy due to Li-ion insertion is quadratic function of Li-ion concentration. Appling the above theory to Li-ion battery electrodes,  a and e a are constants and ∆ε = 0 in the electrolyte, 1 and substituting Eq. (2) and Eq. (6) into Eq. (1), we have where   where M is the Li mobility tensor, which is in general a function of Li-ion concentration. It is assumed to be a constant in our simulations.
In order to describe the chemical kinetics using the non-equilibrium thermodynamics, we need a homogeneous free energy with three local minima. The three minima correspond to to maintain the constant current. In the model, we assume that the concentration of lithium ions at the surface of sample keeps a constant, and then decreases rapidly with the gradual in-depth internal materials in initial time 0  t . A half-Gaussian distribution (the first line in Figure 5d) is used to describe the initial concentration distribution. The range of strain energy is about 10 -4 -10 -2 eV with strain 0.01 -0.1. The pair interaction energy is about 2k B T c ≈ 0.1 eV at critical temperature T c = 600 K of phase separation. 1 It implies that the strain energy with small strain less influence on phase separation. However, large strain will suppress phase separation and lead to non-crystalline formation. 5 Many parameters can affect the simulations in the phase-field model. Singh et al. 6 and Bai et al. 2 have discussed the effects of applied voltage, gradient energy coefficient and applied current. The parameters used in our simulations have referred to these studies. The barrier of free energy is on the same order of magnitude as the free energy function used by Bai et al. 2 The specific parameters are listed in Supplementary Table 2. In order to initiate the phase transformation, 1% perturbation of Li-ion concentration is induced in each cycle.
The results of phase-field simulation are shown in Figure 5c, d. Figure 5c shows the calculated discharge voltage profile, which records the voltage responses with average lithium concentration at a constant current. Figure 5d shows the Li composition profile as a function of reaction time. The time stamps of t 1 -t 14 use the uniform interval. Two reaction steps are found during the entire lithiation: first, the rocksalt LiFe 3 O 4 phase forms and undergoes a rapid growth (t 1 -t 4 ); meanwhile, the conversion reaction also occurs but at a much slow speed (t 5 -t 7 ); afterward, the conversion phase quickly expands (t 8 -t 14 ) and have the sample fully lithiated.
In our modeling, a new form of free energy function, as expressed in Eq. (10), is introduced for describing the lithiation process in Fe 3 O 4 . Therefore, we need to figure out the effects of the parameters b 1 and b 2 in Eq. (10) on the electrochemical phase transformation kinetics. We performed simulations by varying b 1 and b 2 individually, or changing both of them proportionally. Supplementary Figure 9-11 shows a series of time-sequenced Li composition profiles with the change of b 1 and/or b 2 . Comparing these figures, we found that the smaller b 1 and larger b 2 favors the growth of LiFe 3 O 4 phase; and similarly, smaller b 2 and larger b 1 promotes the speed of the conversion reaction. It is straightforward to understand since b 1 and b 2 determine the energy barrier between Fe 3 O 4 and LiFe 3 O 4 , and between LiFe 3 O 4 and Li 2 O+Fe, respectively. We also noted that b 1 and b 2 will influence the interfacial structure even with the Cahn-Hilliard gradient energy coefficient unchanged. For example, smaller b 1 and b 2 values will lead to larger interfacial widths. It is consistent with the effect that we found in previous calculations. 7 In addition to b 1 and b 2 , other variables, such as the Cahn-Hilliard gradient energy coefficient, Li + mobility in electrode, and applied current, may also affect the evolution of lithiation process. Nevertheless, despite the minor effect on reaction speeds and interfacial structures, the intrinsic nature the two-step lithiation has been successfully reproduced.
It is worth noting that Eq. (10) is an empirical formula. Some experimental phenomena, such as Li composition profiles with respect to time, evolution of phase transition, can be qualitatively described using the phase-field theory. In order to describe the experimental results of a specific material more quantitatively and accurately, the parameters of b 1 and b 2 , the Cahn-Hilliard gradient energy coefficient, and Li + diffusion coefficient, need to be determined by relevant experimental measurements.