On Veracity of Macroscopic Lithium-Ion Battery Models

The mass and charge transport equations in the electrolyte phase are³⁶

subject to

on the solid-electrolyte boundary Γ^ε, respectively. In (2),

(m²sec^{− 1}) and (Ω^{− 1}m^{− 1}) are the interdiffusion coefficient and the electric conductivity in the electrolyte, respectively; k (A · m · mol^{− 1}) is the electrochemical reaction rate constant that describes the kinetics of lithium-ion transfer on Γ^ε; (V) is the open circuit potential; is the maximum concentration of lithium that can be stored in the active particle; t₊ is the transference number, is assumed constant,³⁷ and f_± is the activity coefficient; n_e is the outward unit normal vector to pointing from the electrolyte toward the active particle; F and R are the Faraday and the universal gas constants; T is temperature.

The mass and charge transport of lithium ions in the electrode (solid) phase are governed by the material balance and electroneutrality equations³⁶

subject to

respectively. In (4)-(5), (m²sec^{− 1}) is the interdiffusion coefficient in the electrode, (Ω^{− 1}m^{− 1}) is the electric conductivity in the electrode, and n_s is the outward unit normal vector to Γ^ε pointing from the active particle toward the electrolyte.

The transport processes occurring at the pore-scale include heterogenous reaction on the electrode-electrolyte interface , and diffusion and migration in the electrode and electrolyte phases, and , respectively. The characteristic time scales associated with the heterogenous reaction, ionic diffusion, and ionic migration over a macroscopic length scale L are

respectively. In (6), and , j = {e, s}, are characteristic values of the interdiffusion and electric conductivity tensors and in the electrode (j = s) and the electrolyte (j = e), respectively. We define the dimensionless Damköhler and electric Péclet numbers as

They provide information about the relative magnitude of ion transport processes in the electrolyte and the electrode phases. Let and , j = {s, e} be the dimensionless lithium-ion concentration and electrostatic potential in the active particles (j = s) and the electrolyte (j = e). Then, the mass and charge transport equations can be cast in dimensionless form as follows.

The dimensionless form of mass and charge transport in the electrolyte (1)-(2) is given by

subject to

on Γ^ε, respectively. In (1) and (2), the dimensional spatial and time scales are nondimensionalized by the macroscopic length L and the diffusion time in the electrolyte phase respectively, i.e. and ; and are the dimensionless interdiffusion coefficient and the electric conductivity in the electrolyte. Also,

where is the dimensionless open circuit potential. We emphasize that and represent the rescaled (nondimensional) electrolyte and electrode phases, with Γ^ε the interface separating them.

Similarly, the dimensional transport equations in the electrode phase , (4) and (5), can be cast in dimensionless form

subject to

respectively.

In the following we use multiple-scale expansions to derive a mean-field (continuum, macroscopic) approximation of the pore-scale equations and to identify conditions under which continuum equations are valid descriptors of pore-scale dynamics.

Lithium transport in the electrolyte phase described by (8a)–(9b) can be homogenized, i.e., approximated up to order ε², by the following effective mass and charge transport equations (see Appendix A)

and

where

provided the following conditions are met:

• (1)  
  ε ≪ 1,
• (2)  
  Da_e < 1,
• (3)  
  Pe_e < 1,
• (4)  
  Da_e/Pe_e < 1,
• (5)  
  .

In (20) and (21), the dimensionless effective reaction rate constant in the electrolyte phase, , is determined by the pore geometry,

and the dispersion tensors are given by:

The closure variable has zero mean, , and is defined as a solution to the local problem

Constraints 1)–4) ensure separation of scales. While constraint 1) is almost always met in practical applications since the pore size is generally much smaller that the electrode dimension, constraints 2)–4) depend on the relative importance of the diffusion, electromigration, and reaction mechanisms, i.e. they impose constraints on the transport regimes that can be appropriately modeled by the continuum scale Equations (20) and (21) within errors of order ε². These conditions are summarized in the phase diagram in Figure 1, where the line β = 0 refers to Da_e = 1 and the half-space β > 0 refers to Da_e < 1 because ε < 1; the line α = 0 refers to Pe_e = 1 and the half-space α_e < 0 refers to Pe_e < 1; the line α + β = 0 refers to Da_e/Pe_e = 1; and the half-space underneath this line refers to Da_e/Pe_e < 1. Constraint 5) is not necessary for scale separation, but facilitates the derivation of the effective parameters (23) and (24). As shown in Appendix A, this constraint allows one to interchange the surface and volume averages, and , within errors on the order of ε².

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In Appendix B, we show that the microscale reactive transport processes described by (11)–(12) can be homogenized, i.e., approximated up to order ε² in the solid phase with an effective mass and charge transport equations

and

for x ∈ Ω, provided the additional conditions

• (1)  
  Da_s < 1,
• (2)  
  Da_s/Pe_s < 1,
• (3)  
  ,

are met. In (26) and (27), the dimensionless parameter is defined by (23) and the effective diffusion and conductivity tensors are given by

The closure variable has zero mean, , and is defined as a solution of the local problem

Constraints 1)–2) ensure scales separation and depend on the relative importance of the solid phase diffusion, conduction, and reaction mechanisms of transport. Condition 3) simply facilitates the derivation of the effective tensor (28). These constraints are summarized in Figure 2.

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The constraints previously identified impose conditions on the relative magnitude of the three main processes controlling lithium-ion transport at the microscale, i.e. diffusion, electromigration and heterogenous reaction at the electrolyte-electrode interface. The constraints Da_e < 1 and Da_s < 1 require that the intercalation reaction be slower than diffusion processes both in the electrolyte and the electrode. Similarly Pe_e < 1 requires that diffusion processes in the electrolyte are faster than elecromigration. Both conditions guarantee that lithium ions are uniformly distributed, i.e. well mixed, both in the pore-space occupied by the electrolyte and within electrode pellets at the unit cell scale. Under well-mixed conditions, or when lithium-ion concentration is locally uniform, a dual-continua macroscale model can describe processes at the micro-scale within errors of order as prescribed by the homogenization procedure. On the other hand, under diffusion-limited conditions, or high resistance to mass transport, concentration gradients are formed at the sub-pore scale, and the predictivity of continuum scale models, which replace pore-scale quantities with their spatial averages, cannot be guaranteed any longer. Our findings are consistent with the widespread observation that classical macroscopic approximations loose predictive power under high C-rate^c operating conditions,³⁹ when a strong current imbalance between electrodes generates sharp concentration gradients at the sub pore level. The importance of lack of subpore scale mixing was already pointed out in Ref. 9, where subgrid concentration gradients were associated with generation of highly localized heat of mixing. The constraints Da_e/Pe_e < 1 and Da_s/Pe_s < 1 suggest that elecromigration can play a favorable role in improving the sub-pore scale mixing in presence of high mass transfer resistance, or diffusion-limited regimes. Finally, the dependence of Pe_e and Pe_s on the operating temperature, see (7), demonstrates that isothermal conditions are not sufficient to guarantee macroscale model accuracy: operating the same battery at a higher temperature may lead to the violation of Pe_e < 1 and/or Pe_s < 1, once a critical temperature is overcome. A more thorough analysis of temperature-dependent breakdown for different battery chemistry is discussed in the next section as well as in Ref. 40. In the following section, we discuss the implications of our findings on existing dual continua models.

The battery cell parameter data used in this case study, and reported in Table I, are based on a variety of electrode and electrolyte compositions at room temperature (T = 298 K).^45–51 The dimensionless parameters α, β, γ and δ in the electrolyte and electrode phases, readily calculated from (7) and (19), are reported in Table II and plotted on the corresponding phase diagram for the electrolyte and electrode, Figure 3 and 4, respectively.

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Among the twelve chemistry considered in this analysis,^45–51 ten possess electrolyte effective transport coefficients, i.e. dimensionless numbers (α, β), which do not violate the applicability conditions of macroscale models, see Figure 3. The theory developed previously ensures that the homogenized equations in the electrolyte will be able to accurately capture the dynamics at the pore-scale: this is consistent with the numerical simulations performed in Refs. 45–51, where Newman-type models have been successfully used to model transport in the electrolyte phase. On the contrary, two data points (solid symbols in Figure 3), corresponding to Li₄Ti₅O₁₂ and LiNi_1/3Mn_1/3Co_1/3O₂ chemistry,^50,51 lie outside the range of applicability. For these two chemistry, it is not guaranteed that the homogenized Equations (20) and (21) describing transport in the electrolyte will be effective in capturing pore-scale transport processes. Again, this is confirmed by the results presented in Refs. 50, 51, where a Newman-type model response could not properly capture experimental data, see Fig. 6 in Ref. 51. Such a discrepancy is understandable: Li₄Ti₅O₁₂ has a very fast intercalation reaction rate (between 6 and 9 orders of magnitude faster than the other chemistry) which leads to mass transport limitations (or reaction-dominated regimes) and lack of pore-scale mixing.

Figure 4 shows the data points corresponding to the (δ, γ) values for the battery chemistry electrodes listed in Table I and calculated in Table II. All the data points lie outside the range of applicability of the upscaled equations of lithium-ion transport in the electrode phase, therefore suggesting that full pore-scale models have to be employed to accurately capture lithium-ion transport in the active particles. This is consistent with the numerical approaches used in Refs. 45–51, where no upscaled model is used in the active particles and the transport in the solid electrode is solved at the microscale. It is worth noticing that, since bounds on α, β, γ and δ have to be concurrently satisfied, the numerical simulations matched well the experiments only when the conditions on (α, β) were not violated, as discussed previously.

Figures 3 and 4 showed the distribution of the dimensionless transport parameters α, β, γ and δ at room temperature in the phase diagrams and allowed one to a priori assess the validity of Newman-type models across different battery chemistry. These models, on the other hand, might also fail when used for the proper chemistry (i.e., the chemistry for which the corresponding α and β data points fall in the applicability regime regions at room temperature) but improper operating conditions. For this reason, the veracity of the upscaled equations of mass and charge transport in the electrolyte across battery cell operating conditions is also investigated. In particular, the study conducted in this section focuses on temperature and C-rate of operation and uses the electrode-electrolyte system described in Ref. 52. In this work, the authors compared the performance of their (continuum-scale) numerical simulations with experimental data for lithium-ion cells with Li_yMn₂O₄ and LiNi_0.8Co_0.15Al_0.05O₂ cathode materials tested at different C-rate ranging from C/25 to 10C. We conduct our analysis for the case of Li_yMn₂O₄ cathode material, but similar results can be easily extended to the LiNi_0.8Co_0.15Al_0.05O₂ case.

The case study analysis relies on the premise that temperature is one of the primary factors that influences the ability of the macroscale transport equations to capture battery dynamics at high C-rate. In fact, the battery cell temperature influences the transport parameters in the electrolyte phase. Those parameters are: k (reaction rate constant), D^e (the electrolyte diffusion coefficient), and K^e (the electrolyte conductivity coefficient). When the influence of temperature on k is much more pronounced than on D^e and K^e, as we will verify in the case of a battery operating at high C-rate, diffusion is no longer the dominant mode of transport. In this case the homogenized transport equations do not accurately capture transport at the microscale.

A single and constant value for the reaction rate k was considered in Ref. 52. Yet, experimental evidence shows significant cell temperature variations in terms of C-rate.^53–58 Our analysis is conducted for three different C-rate: low (C/25), medium (1C), and high (10C). Following experimental data,^53–58 the temperature increase, starting from room tenperature, can be estimated as follows: from 298 K to 299 K, from 298 K to 306 K, and from 298 K to 333 K at a discharge C-rate of C/25, 1C and 10C, respectively. We estimate the reaction rate constant k_ref at room temperature T_ref = 298 K through

where I_app (A/m²), the applied current density, is provided in Ref. 59 for each C-rate, see Table III. The electrochemical reaction rate constant k(T) for a given electrode system can be described as a function of temperature using the Arrhenius equation, as reported in Ref. 60:

where k(T) is the reaction rate constant of a given electrode at the desired temperature T. In (31), Ea_r is the electrode reaction rate activation energy. We set Ea_r = 78.24 kJ/mol at a reference temperature of 298 K.⁶¹ Using (31), we compute the values of k for different temperature conditions, which are then used to determine the parameter values α and β.

Similarly, the diffusion and conductivity coefficients, D^e and K^e, vary as a function of both temperature and the lithium concentration in the electrolyte phase. For the estimate of D^e and K^e at the reference temperature T_ref, we use the same approach used in Ref. 52, where:

where we set mol/m³ (^d). This leads to reference values of 2.41e-10 m²/s and 0.922 S/m for D^e and K^e, respectively. Since, to the best of our knowledge, no analytical dependence on temperature is available for D^e and K^e, we use instead a curve fitting procedure from Figures 13 and 14 in Ref. 62 to determine D^e(T) and K^e(T). A summary of the estimated parameters for different C-rate and temperature ranges is given in Table IV.

Based on above calculations, we determine the temperature-dependent trajectories of the data points (α,β) computed at the temperature intervals characteristic of each C-rate. Table IV summarizes the variation of parameters α and β as a function of the operating conditions for the three C-rate of interest. The data points and their variation with temperature and C-rate are schematically represented in Figure 5.

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Figure 5 (top) shows that at a C/25 rate of discharge, there is minimal temperature increase over the duration of a discharge event. The magnitude of parameter α remains invariant, while β increases slightly due to an increase in the Da_e number. The behavior of the system (as a function of temperature) is linear with β. The data points satisfy the constraints on α and β. Hence, the upscaled equations for lithium mass transport should provide an accurate description of the pore scale behavior. This is consistent with the simulation results from a continuum-scale simulator obtained in Figure 7(a) of Ref. 52, where there is a perfect match between the model and the experimental response.

At a 1C rate of discharge, Figure 4 (middle), there is a moderate increase in temperature over the duration of the simulation cycle. The magnitude of parameter α remains invariant, while β increases at a moderate rate due to a faster increase in the Da_e number. The behavior of this system is linear in α and β. At elevated temperatures, the effect of increase in the reaction rate constant k dominates any increase of D^e and K^e. The data points satisfy the constraints on α and β over the range of operating temperature conditions. Hence the homogenized set of transport equations used in Ref. 52, Figure 7(a), still provides an accurate description of the pore scale behavior, leading to good correlation with experimental data.

At a 10C rate of discharge, there is a very significant increase in the battery temperature over the operating conditions. There is a very small increase in α as the increase in D^e marginally dominates the increase in K^e, leading to an incremental increase in the Pe_e number. The reaction rate constant is 2 to 3 orders of magnitude higher than at lower C-rate; hence, the rate of decrease in β is higher than the rate of increase in α. For the LiMn₂O₄ cathode system, the macroscale transport equations are no longer accurate in describing microscale transport processes at temperatures 313 K or higher. This is because the value of α + β is less than 0, which violates one of the three constraints on these two parameters. At high operating temperatures and C-rate, the lithium manganate cathode system operates in a transport regime where the three lithium transport processes (reaction, diffusion, and electro-migration) are of the same order. In this scenario, very fast reaction kinetics lead to diffusion-limited regimes where diffusion is no longer the dominant transport mechanism in the medium. As a result, macroscale equations describing electrolyte transport are vulnerable and can be invalidated due to lack of scale separation with respect to the pore-scale.

The performance prediction of continuum-scale models based on the phase diagram Figure 4 (bottom) is, again, consistent with the analysis performed in Ref. 52, Figure 7(a), where the numerical solution obtained from macroscopic models cannot capture the experimental results. Under these circumstances, a multi-scale model is necessary to incorporate the effects of transport both at the pore-scale and the macroscopic scale.

The approach implemented above is significant in terms of identifying the temperature of operation and C-rate of current charge/discharge as crucial parameters dictating the dominance of one transport mechanism over the other(s) in the battery electrode/electrolyte medium. Standard Newman-type macroscopic models under scenarios similar to the one described above are invalid and may fail to capture microscale transport processes.

Substituting (18) and (19) into the mass transport equation in the electrolyte (8) leads to

where the nonlinear term in (8) is expanded in a Mclaurin series

Similarly, the interface condition (A3) can be written as

since , where

Combining (18) and (19) with the charge transport Equation (A2) and boundary condition (A4) yields to

subject to

where A₀, A₁, B₀ and B₁ are defined in (A8). Next, we compare terms of like-order of ε.

Collecting the leading-order terms in the mass transport equation and corresponding boundary condition (A5) and (A7) respectively, we obtain

subject to

Similarly, at the leading order the charge transport equation is

subject to

Homogeneity of (A11)-(A12) and (A13)-(A14), guarantees that c^e₀ and ϕ^e₀ are independent of y, i.e.

Since ∇_yc^e₀ ≡ 0 and ∇_yϕ^e₀ ≡ 0, the mass balance equation (A5) at order simplifies to

subject to the interface condition

Integrating (A17) over with respect to y, while accounting for the boundary condition (A18), and the periodicity of the coefficients on the external boundary of the unit cell ∂Y, we obtain

where is defined by (23).

Combining (A19) with (A17) to eliminate the temporal derivative, we obtain

Similarly, the charge balance equation (A9) at is

subject to

for y ∈ Γ. Equations (A18), (A20), (A21) and (A22) form boundary value problems for both c^e₁ and ϕ^e₁. Following Ref. 64 and Ref. 27 (pp. 10, Eqs. 3.6–3.7), we look for solutions in the form

Substitution of (A23) into (A20) and (A18) leads to

subject to and

where I is the identity matrix, and and are periodic vector fields. Substitution of (A23) into (A21) and (A22) leads to

subject to

Equations (A24) and (A25) define the closure variables and . The coupling of and with c^e₀(x), ϕ^e₀(x), A₀(x) and B₀(x) through the boundary value problems (A24) and (A25) is incompatible with the closure variables' general representation postulated in (A23). This inconsistency is resolved by imposing the following constraints on the exponents α and β. If we choose β > max {0, −α} and α < 0, then the term is negligible relative to the smallest term in (A24) and the nonlinear migration term ε^{− α}λt²₊K^e/c₀^e relative to D^e. Under these constraints, (A24) and (A25) simplify to

Equation (A26) can be satisfied for all x ∈ Ω if

Similarly, (A25) yields

and

Consistency of (A27) with (A28) implies

In (A29), the conductivity tensor K^e is a function of concentration c^e and potential ϕ^e. With an order ε approximation K^e ≈ K^e(c^e₀, ϕ^e₀). Then, (A29) can be simplified to

As a result, . The treatment of the closure variable is consistent with the approach employed in Ref. 65. The closure variable defines the cell problem and describes the behavior of the effective diffusion and conductivity tensors.

Recalling the definitions of Da_e and Pe_e in (19) allows us to reformulate the conditions in terms of α and β in the form of constraints 2)–4) for the electrolyte. Having identified the conditions that guarantee homogenizability, we proceed to derive the effective mass transport equation (20).

Collecting the zeroth-order term in the mass balance equation (A5) and first-order term in the corresponding boundary condition (A7), we obtain

subject to

since ∇_yc^e₀ = 0. Integrating (A32) over with respect to y and using the boundary condition (A33) leads to

where , , and

Next, we recall that

Multiplying the temporal derivative of (A36) by ε, we obtain

Multiplying (A34) by ε, adding the result to (A19), and using (A37), we obtain

Combining this result with the expansions and while recalling the definitions of Da_e and Pe_e in (19) and assuming and , where ψ = {c, ϕ}, leads to

since

where is defined by (22).

Similarly, collecting terms in the charge balance equation in the electrolyte (A9) and terms in the boundary condition (A10) while accounting for ∇_yc^e₀ = 0, we obtain

subject to

We multiply by ε both Equation (A41) and its boundary condition (A42), add them to (A21) and (A22), respectively, and integrate the resulting equation over while employing the newly obtained boundary conditions. This leads to

where . Following a similar procedure to that outlined for the mass transport equation, (A43) can be written as

where is defined by (22).

Equations (A39) and (A44) govern the dynamics of and in the electrolyte up to errors of order ε².

Substituting (18) and (19) into the mass transport equation in the electrode (B1), we obtain

subject to

where A₀, A₁, B₀ and B₁ are defined in (A8). Similarly, the charge transport equation (B2) and the boundary condition (B4) combined with (18) and (19) yield to

subject to

Collecting the leading-order terms in the mass transport equation and corresponding boundary conditions  (B5) and (B6), we obtain

subject to the interface condition

Similarly, at the leading order the charge balance equation (B7) and the boundary condition yield

subject to

The homogeneity of Equations (B9)–(B12) ensures that the above boundary value problems have both a trivial solution, i.e.

At the following order, the mass transport equation (B5) can be written as

since ∇_yc^s₀ ≡ 0, and it is subject to the boundary condition

Integrating (B14) over with respect to y, while accounting for the boundary condition (B15), and the periodicity of the coefficients on the external boundary of the unit cell ∂Y, we obtain

We combine (B16) with (B14) to eliminate the temporal derivative and obtain

Similarly, the order of the charge balance equation (B7) can be simplified to

subject to

Equations (B17) and (B18) subject to (B15) and (B19) form a boundary value problem for c^s₁ and ϕ^s₁, respectively. As outlined in Appendix A, we look for a solution in the form

Substitution of (B20) into (B17) and (B15) leads to the following cell problem for the closure variable ,

subject to and

The boundary-value problem (B21) couples the pore scale with the continuum scale, in the sense that the closure variable —a solution of the pore-scale cell problem (B21) —is influenced by the continuum scale through its dependence on the macroscopic concentration c^s₀(x). This coupling is incompatible with the general representation (B20). This inconsistency is resolved by imposing the following constraint on the exponent γ, namely γ > 0. This condition ensures that is independent of c^s₀, and the cell problem (B21) can be simplified to

Similarly, substitution of (B20) into the -charge balance equation (B18) and its boundary condition (B19) leads to the following cell problem for the closure variable ,

subject to and

where is a Y-periodic vector field. Separation between pore- and continuum-scales requires γ + δ > 0. Under this condition (B23), simplifies to

In (B22) and (B24), the diffusion and conductivity tensors are functions of concentration c^e and potential ϕ^e. With an order ε approximation D^e ≈ D^e(c^e₀, ϕ^e₀) and K^e ≈ K^e(c^e₀, ϕ^e₀). Then, , where is a solution of the closure problem

At the leading order, the mass transport equation in the electrode (B5)

subject to

Integrating  (B26) over with respect to y and using the interface condition  (B27) leads to

where . Similarly, the leading order of the charge transport equation is

subject to

Multiplying both (B29) and (B30) by ε, adding them to (B14) and (B15), respectively, and then integrating over , we obtain

where .

Following the procedure outlined in Appendix A and assuming that , Equations (B28) and (B31) lead to the macroscopic equations for mass and charge transport in the electrode (26) and (27), respectively.