Figure 2: Simulation results of concentration by line scan along the x-axis at different partitioning coefficients after 0.48 ms.

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1 Executive Summary One of the critical hurdles of the success of many clean energy technologies is energy storage. For this, new and advanced computational tools to predict battery properties are very important. Several models have been at the forefront in the discovery and design of new materials. One of the models at the forefront of materials modeling is the phase field model. It is widely used for describing phase transformations and microstructural evolution in materials sciences. One of its features is the diffuse interface, i.e. a region of space where two (or more) phases are assumed to mix. The relative amount of every phase is denoted by the phase field variable φ, which is a number between 1 (total presence of a phase) and 0 (total absence of a phase) and at the interface, its value is between 0 and 1. For electrochemical systems, the interface as the reacting zone plays a very important role. On the outer surface of the electrode, toward the liquid electrolyte, we find e.g. the electrochemical double layer (EDL), composed of adsorbed molecules and ions, and an (in)organic film based on the decomposition products of solvent molecules known as the solid electrolyte inter-phase (SEI). The diffusive nature of the EDL seems to make it most suitable for a description with phase field. By adjusting the resolution of the simulation to have the entire interface contained within the mixed-phase region completely eliminates the need for a multitude of measurements or calculations of processes on an atomistic scale. This extends the applicability of phase field simulations to describe the kinetics of this complicated interface in one go. This work aims at modeling the electrode-electrolyte interface using the phase field method and providing insight into the flux kinetics of electrochemical energy storage. The classical phase field and the finite interface dissipation phase field methods are used. In the classical phase field approach, a linear temperature profile vs. concentration diagram of the electrode and electrolyte phases is assumed. The solidus and liquidus temperatures are explicitly described in terms of the concentration of lithium and the solidus and liquidus slopes, respectively. The concentration at the interface between the two phases (electrode and electrolyte) was modeled by a weighted average of the corresponding phase fractions. The partitioning coefficient (k p was obtained from the assumed solidus and liquidus slopes. From the linear approximation of the phase diagram, the partitioning coefficient was also the ratio between the two phase concentrations, which ultimately helps to rewrite the diffusion equation in terms of this quantity. { } c ċ = (1 φ β )D α ( 1 + φ β (k p 1) ) + φ k p c βd β ( 1 + φ β (k p 1) ). (1) By careful selection of this quantity, one can solve the concentration evolution equation across the interface between the two phases with known diffusion coefficients. i

2 Figure 1: Concentration evolution of electrode particles upon Li de-intercalation. The figure on the left shows at the initial time step and the right figure shows after 16 µs. View is on the yz plane. Simulation box size is (100 x)3. Figure 1 shows the average Lithium (Li) concentration change up on de-intercalation. Figure 2 shows concentration profiles at different kp by a line scan across the x-axis. Variation of the partitioning coefficient influences the rate of lithium (de)intercalation. An exponential fit of the simulation results for the concentration change with time shows a form of first order reaction where the Lithium concentration varies exponentially with time. The rate constants (kr ) at different partitioning coefficients (kp ) were calculated from the simulation results. These results show an exponential relationship with the partitioning coefficient (figure 3). The fitting parameters are shown in equation (2). kr = e b p +c a+ k b = Ae kp +c. (2) Where A = ea and a, b and c are fitting parameters with values 8.043, and 0.062, respectively. Calculating the rate constant was an important step towards a more precise approximation of the partitioning coefficient. This was because the rate constant can be expressed with the Arrhenius equation which describes the temperature dependency of reaction rates with an activation energy. Ea kr = Ae RT. (3) This relationship helps to compare the experimental activation energy of lithium for diffusion at a certain temperature and the activation energy estimated from the simulation results. The smaller the difference between the estimated activation energy of lithium to intercalate into the electrode and the experimental activation energy that is found from literature, the better approximation of a virtual phase diagram of the electrode and electrolyte. Changing the boundary condition enables us to study diffusion of Li into and out of the electrode particles. Figure 4 shows a boundary condition where a constant source of ii

3 Figure 2: Simulation results of concentration by line scan along the x-axis at different partitioning coefficients after 0.48 ms. Figure 3: An exponential relationship between the rate constant and the partitioning coefficient. The line shows the exponential fit and the rectangular dots show the data points. iii

4 Figure 4: A pictorial description of a fixed top bottom boundary condition (left). The source of Li is at the top of the box at N z+1 shown by the red rectangle and Li is allowed to diffuse in the three directions in a periodic way. The figure in the left shows concentration vs time of Li filling up the electrode particles at different partitioning coefficients. Figure 5: Concentration of lithium with time at different boundary conditions showing intercalation and deintercalation processes at k p=10 (left). The figure on the right shows a possible free energy of reacting species. G i a and G d a are the activation energies of intercalation and de-intercalation, respectively. Li is set at the top of the box. The rate of lithium intercalation was faster than the rate at which it deintercalates as shown in 5 left. This is attributed to the higher activation energy required for Li to overcome for deintercalation. It is interesting to find out that this result is in agreement with experimental and atomistic results. Literature values for G ranges from 0.21 to 0.61 ev [1]. With the classical phase field model, it is possible to investigate (de-)intercalation kinetics by using an equilibrium partitioning. Now the question is, in this case, the interaction at the beginning of the (de)intercalation process is not clearly known. For this reason, a model that is able to consider non-equilibrium processes at the beginning of interaction is vital. The implementation of the new Finite Interface Dissipation (FID) phase field model was an important step to consider such non-equilibrium behaviors which are of critical importance in electrode-electrolyte interaction. With this model, a flux contribution, for example for phase α, due to diffusional potential difference between the phases which is determined by introducing a new material parameter of the interface called permeability as shown in the RHS of equation (4). φ α ċ α = (φ α D α c α ) + P φ α φ β {µ β µ α } + φ α φα (c β c α ) (4) iv

5 Permeability [m 3 J -1 s -1 ] 2.5e-12 2e e-12 1e-12 5e V 0.35 V 0.4 V 0.45 V 0.5 V Li concentration [atom%] Figure 6: Permeability of the interface with Li concentration in the electrode. Li concentration in the electrolyte was set to be 0.58 at.%. i o was estimated using K e=6.37x10 4. The different colors show calculations at different overpotentials. The permeability of the interface is an important parameter of paramount importance in the FID. A more precise description of this quantity helps in gaining a deeper understanding of the kinetics of redistribution at the interface between two interacting phases. For diffusion couples, this parameter is linked to the atomic mobility. In this work, I have demonstrated a new approach to estimate this quantity for non equilibrium electrochemistry based on the current density. A detailed description on how to obtain the permeability of the electrochemical interface from Impedance Spectroscopy (IS) measurements is shown in section Figure 6 shows the permeability of the interface with Li content at different overpotentials. For a stable simulation, there must be a suitable selection of time stepping for certain diffusion coefficients and initial concentration differences. For instance, Figure 7 shows how a suitable time stepping at particular diffusion coefficient can be chosen for a stable simulation and figure 8 shows a suitable time stepping at different initial concentrations of two phases interacting with in a periodic boundary condition. After careful parametrization of material input parameters, a single spherical electrode particle is considered and the kinetics of Li intercalation from the electrolyte into the electrode particle is investigated. Figure 9 (left) shows concentration profiles at different time steps. The influence of the size and shape of electrode particles on the kinetics of intercalation was investigated. From figure 9, one can see that ellipsoidal electrode particles show faster kinetics than spherical particles. The size of electrode particles exponentially affects the kinetics of intercalation. Figure 10 shows how the size of electrode particles affects the kinetics. The smaller the size of the particle, the faster the kinetics. Based on how one initializes the system, different concentration profiles can be found. v

6 Figure 7: A choice of suitable time stepping vs diffusion coefficients for a stable simulation. Figure 8: Suitable time stepping vs initial concentration differences between two phases for a stable simulation. Figure 9: Concentration [at.%] vs. position [grid step of 10 nm] of a spherical and an ellipsoidal particle along the z-axis; different curves show concentration profile after every 10 5 steps. vi

7 Figure 10: Concentration change at the center vs. radius of particles after 50 µs. Figure 11: Concentration vs. position of a spherical particle made of Li xc 6 along the z-axis; different curves show composition profile at different diffusion coefficients with D o= m 2 s 1. After simulation times of 5 and 10 seconds. Simulation box size: (50 x) 3 with x=10 nm. vii

8 8 7 concentration of Li [atom%] P= m3J-1s-1 P= m3J-1s position [10 nm] Figure 12: Concentration vs position of two spherical particles made of Li xfepo 4 along the z-axis; different curves show composition profile at different permeability of the interface. Simulation time: 50 microseconds. Simulation box size: (50 x) 3 with x=10 nm. Figure 13, for example, shows the concentration evolution of Li x C 6 electrode particles along the xz (left) and yz (right) axes. It is shown that a five order of magnitude increase in the permeability of the interface significantly affects the concentration change across the interface. Figure 12 shows the influence of the permeability on the concentration profile. Furthermore, the influence of diffusion was observed to be minimal across the interface. Diffusion significantly affects the concentration change more at the bulk of the electrode than at the interface. Figure 11 shows the influence of a two order of magnitude change in diffusion coefficient on the kinetics of intercalation. viii

9 (a) after 2.5 s (b) after 5 s Figure 13: Concentration evolution of Li xc 6 electrode particles along the xz (left) and yz (right) axes slices after (a) 2.5 and (b) 5 seconds, respectively. Simulation box size:(50 x) 3 with x=10 nm. ix