Particle-Scale Path-Dependent Behavior of Li_xNi_0.5Mn_1.5O₄ Electrode in Li-Ion Batteries: Part II. Mathematical Modeling

The phase transition process in the LNMO electrode is described by applying Avrami equation:

where, ε_{s, j} is volume fraction of phase j; t is the time; n is set to be 1 by assuming the homogeneous phase transition in all directions of a sphere LNMO particle. Applying the equation derivation reported by Gallagher et al.,¹⁵ the phase change reaction rate k' can be further formulated by:

where, k_ji is reaction rate constant from phase j to phase i; Δc_{s, j} is the supersaturation concentration introduced to initiate nucleation;¹⁵ H(Δc_{s, j}, ε_{s, j}) is Heaviside function to nullify the phase growth rate if it is outside the reasonable concentration and phase fraction range (c_min ⩽ c_{s, j} ⩽ c_max, 0 ⩽ ε_{s, j} ⩽ 1).

Hence, the phase change rate of Ni at each stage during discharging can be expressed as following:

where, ε_{s, Li0} + ε_{s, Li0.5} + ε_{s, Li1} = 1, meaning only two of above equations are independent, and Mn phase change process is not considered in this model by setting the cutoff voltages from 4.4V to 4.95V, which is consistent with our previous experiments. The Heaviside function, H(Δc_{s, j}, ε_{s, j}), is defined as:

• (1)  
  Li₀-Li_0.5 stage, H(Δc_{s, Li0}, ε_{s, Li0}):
• (2)  
  Li_0.5-Li₁ stage, H(Δc_{s, Li0.5}, ε_{s, Li0.5}):

The material balance of lithium inside LNMO sphere particle is formulated by neglecting effects from stress, anisotropic diffusion, volume change and all other non-idealities:

where, c_s is the local lithium concentration, , N_s is the mass flux of lithium:

where, N_{s, j} = −D_{s, j}∇c_{s, j},D_{s, j} is the diffusion coefficient in phase j.

Substituting lithium concentration in each phase, the material balance can be expressed by,

where, D_{s, eff} = ε_{s, Li0}D_{s, Li0} + ε_{s, Li0.5}D_{s, Li0.5} + ε_{s, Li1}D_{s, Li1}.

Assuming lithium would never diffuse across the interface between two coexistence phases but only the phase change process exists, above diffusion equation can be further simplified as the diffusion process of each phase:

where, ∑R_j = R_Li0 + R_Li0.5 + R_Li1 = 0, and these source terms can be assumed by using the volume average assumption: . Since the volume fraction of each phase, ε_{s, j}, is not a function of particle radius, above equations can be further transformed for each phase as:

Boundary conditions are determined according to the constant current operation in the experiments. On the particle surface,

In the center of the particle:

The mathematical model is numerically solved using finite element package COMSOL Multiphysics V5.2 and MATLAB. An one-dimensional geometry in spherical coordinates is applied to represent single LNMO particle, and a constant lithium mass flux is applied on the particle surface to simulate constant current charge and discharge. Since the thickness of our thin-layer LNMO electrode is equal to LNMO particle diameter, LNMO electrode voltage then can be estimated by the modeled lithium surface concentration using a lithium metal counter electrode. To represent phase transformation window from Li₁Ni_0.5Mn_1.5O₄ (Li₁ with Ni²⁺) to Ni_0.5Mn_1.5O₄ (Li₀ with Ni⁴⁺), same voltage cutoff window as our experiment, [4.4V, 4.95V], is applied, and we defined 0% SOC at 4.4 V and 100%SOC at 4.95V to be consistent with the experimental setup. The related model parameters are estimated from experiment, and have been listed in Table I.

Figure 1 illustrates the numerical results of LNMO particle behavior (phase transition) in the charge and discharge process, and the corresponding phase distributions along particle radius. The discharge process started from the fully charged state and is the intercalation of lithium into the LNMO particle, which experiences two phase transformation stages: Li₀ → Li_0.5and Li_0.5 → Li₁. Using the developed model, LNMO particle shows its local volume fraction of Li_0.5 phase is slightly over 50% on the particle surface (r/R = 1) but only around 40% in the center (r/R = 0), when it is discharged to 75%SOC from the fully charged state. As the discharge process moves forward to 50% and 25% SOCs, the new formed phases, Li_0.5 and Li₁, both have higher volume fractions on the surface than the center region. Similarly, in the reverse process, charge is the deintercalation of lithium out of the LNMO particle, and have Li₁ → Li_0.5 and Li_0.5 → Li₀two phase transformation processes. New formed charge phases, Li_0.5 and Li₀, both also have more volume fractions on the surface than the center region. These phenomena can be explained by the interplay between the lithium diffusion in the particle and local phase-change rate which is concentration (lithium) dependent. Besides, it is interesting to note that the phase distributions at the same SOC are quite different between charge and discharge, which could directly influence the capacity utilization in the LNMO charge and discharge process, and can lead to the observed asymmetric character between charge and discharge on capacity utilization reported in our previous paper (Part I: experimental study¹¹).

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In order to qualitatively explain the asymmetric behavior of LNMO charge and discharge capacity, Figure 2 illustrates the variation of average volume fractions of each phase (Li₀, Li_0.5 and Li₁) in a full charge and discharge cycle. It is clear to see that there is no overlap in phase fractions between charge and discharge process of the LNMO electrode. In other words, the present three phases' volume fractions in the LNMO are different in the charge and discharge process although the total amount of lithium inside are same. This demonstrates the path dependent behavior of phase transformation inside LNMO electrode. Meanwhile, it indicates that the path-dependent behavior observed on thin LNMO electrode is originated from both the transport limitation (lithium diffusion) and the slow phase transformation process.

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Because of the lithium diffusion in solid phase, fast discharge (or charge) rate on LNMO particle can lead to lithium accumulation (or reduction) on its surface. Since local phase transformation rates are highly dependent on the lithium concentration according to chemical kinetics, this inevitably causes local phase transformation rates are higher on the surface than the core region of LNMO particle. Hence, phase coexistence and distributions in the LNMO particle will be affected by the competing nature between diffusion and phase transformation. To explore this competing effect, in Figure 3, we simulated Li₀ → Li_0.5 phase change process (ε_{s, Li0} + ε_{s, Li0.5} = 1) in a constant rate discharge of a fully charged LNMO particle. As the lithium flux goes into the particle, Li_0.5 phase will be firstly generated on the LNMO particle surface while the local volume fraction of Li₀ decreases. Three different Li₀ → Li_0.5 transformation rates in forms of ratio to the lithium diffusion coefficient (k/D, 2k/D and 10k/D) are applied in the model respectively. Along the normalized particle radius, a clear phase boundary between Li₀ phase and Li_0.5 phase could show up at r/R = 0.6 when 10k/D is applied. Besides, the coexistence region of Li₀ and Li_0.5 in the particle (0 < ε_{s, Li0} < 1, 0 < ε_{s, Li0.5} < 1) increases instead of a clear phase boundary when k/D and 2k/D are applied to the model. These observations demonstrate that the competition between phase change limitation and transport limitation determines the phase distribution and coexistence as well as the existence of phase boundary. Since LNMO electrode undergoes two phase change processes (Li₁↔Li_0.5 and Li_0.5↔Li₀), two distinct phase boundaries (Li₁/Li_0.5 and Li_0.5/Li₀) can only exist when the lithium diffusion is much slower than the phase change rates of Li₁/Li_0.5 and Li_0.5/Li₀ simultaneously, otherwise, phase distribution would show coexistence character throughout the LNMO particle in the charge and discharge operation.

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Our model has demonstrated that one consequence of transport limitation and phase change limitation interaction is the path dependent behavior of phase transformation and distribution inside LNMO particle, which results in the difference of lithium concentration on the particle surface in charge and discharge, and is the origin of the electrode path-dependent voltage behavior. In our previous experimental study,¹¹ we found thin-layer LNMO electrode did not only have path-dependent voltage behavior, but also show SOC dependent character on this path-dependence. To explore this behavior, in this section, we simulated two operation procedures: fast discharge after slow discharge/charge, and fast charge after slow discharge/charge. Both operation procedures are started from fully charged (discharged) state to fully discharged (charged) state, and the switch between fast C-rate and slow C-rate is set at a given SOC, where three SOCs, 25%, 50% and 75%, were simulated respectively. The cutoff voltage window, [4.4V, 4.95V], is applied the same as that in the experiment, which represents SOC variation from 0%SOC (Li₁ phase) to 100%SOC (Li₀ phase).

As shown in Figure 4, the discharge path-dependence exists at all three SOC conditions, moreover, numerical results show that 75%SOC has the largest path-dependence on capacity utilization in comparison with the other two. This can be understood by that both the diffusion process and the phase change are rate limiting, and a larger SOC takes longer time to get fully discharged and hence has more time for local phase change reactions taking place to capture lithium. The results of charge path-dependence are illustrated in Figure 5. Similarly, modeling results show that charge path-dependence exists at all three SOCs. However, different from discharge, the charge path-dependence on capacity utilization at 25%SOC is the largest. This can also be explained by the limited phase change rates as well as the diffusion process: the charging process from 25%SOC to fully charged state (100%SOC) takes a longer time than from 50%SOC and 75%SOC, and thus more phase change reactions can take place, and more charge capacity can be obtained. Here, it has to be emphasized that our model is trying to explain the particle scale path-dependence as well as its SOC dependent behavior of LNMO electrode, while thick LNMO electrodes involve complicated porous electrode effect and are beyond the scope of this work.

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Our above numerical studies have demonstrated that slow phase transformation process in LNMO particle can lead to the SOC dependent character of the path-dependent voltage behavior. If this were true, long relaxation time would have important impact on the voltage behavior since it can provide the phase transformation time needed. In Figure 6, the effect of relaxation was simulated on the LNMO discharge behavior: a thirty-minutes rest was applied when the operation is switched from slow charge/discharge at 1C to fast discharge at 5C. Comparing with 'none rest' condition, the path-dependence behavior decreases as the difference of capacity utilization becomes smaller between 'discharge after charge' and 'charge after discharge'. This is because long relaxation time provided more time to phase transformation reactions, and the lithium distributions on the surface and inside LNMO particle are thus different before and after the relaxation, although the total lithium amount is the same. This also can well explain the vanish of path-dependence in both charge and discharge in our experiments by applying sufficient long relaxation period.

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