The influence of surface inhomogeneity on the overcharge and lithium plating of graphite electrodes

We seek to clarify phenomena involved in the overcharge of a graphite electrode in a lithium ion battery, including lithium (Li) plating. In Baker and Verbrugge (2019 J. Electrochem. Soc.), we developed a set of equations that can be used to treat Li plating and subsequent electro-dissolution, and we analyzed how the equation system behaved for a particle of graphite, a fundamental unit of the negative (porous) electrode in lithium ion cells. In this work, we employ the same governing equations, but we render them in a two-dimensional setting to examine the graphite-electrolyte interface, allowing us to clarify phenomena involved in Li plating over graphitic electrode elements in the absence of complicating factors associated with the architecture of a porous electrode. For a variety of reasons described in the Introduction of this work, the surface of graphite is nonuniform in terms of reaction rates for Li insertion and plating, and we show that when the electrode is subjected to constant-current charging, as is commonly employed, such nonuniformities lead to early Li plating over the highly reactive surfaces. These observations underscore the importance of maintaining a uniform electrode surface, especially when the cell is to be subjected to high rates of charge.

Introduction lithium ion batteries is nonuniform, with edge planes that are more reactive and basal planes that are far less reactive [14][15][16][17][18]. In addition, the use of ceramic materials over the graphite electrode, as employed in ceramicenhanced separators [19], leads to blockage of reactions over portions of the electrode. Insulating contaminants can also occlude portions of the graphite surface [3,5]. It is thus important to understand how a nonuniform interface along the graphite surface, including substantially open as well as occluded regions, influences Li plating and overcharge of a graphite electrode. The system geometry associated with figure 1(a) may be viewed as a simple and reasonable model architecture to investigate salient features of the lithiation of graphite, including Li plating on overcharge.
What follows are sections entitled Model Formulation and Discussion of Results, after which is the Summary and Conclusions.

Model formulation
We model the lithiation of graphite, corresponding to the charging of the graphite electrode in commonly employed lithium ion batteries in accordance with the schematic shown in figure 1(a). The thermodynamics of lithiated graphite are modeled using the multi-site, multi-reaction (MSMR) model as described in [20][21][22][23]. A practical aspect of the MSMR model is that, with few parameters that can be associated with the physical chemistry of the intercalation material, a quantitative fit of the open-circuit potential results, and this is key for the overall analysis. We refer the reader to [21] for a complete discussion on the application of the MSMR model to graphite.  where x is the total fraction of the filled sites in lithiated graphite and U 1 refers to insertion-electrode equilibrium potential. The index j spans from 1 to J, where J=7 in this work for lithiated graphite (see table 2). From the first line of equation (2), we can represent the activities in the insertion electrode as a x a X x and . 3

Thermodynamics and interfacial kinetics
We found in computational simulations that there are times towards the end of lithiation of the graphite that x X, j j  X x j j is small, and for some of the galleries, j w is also small, giving rise to a small number being raised to a small power, leading to numerical problems; 2 these problems can be removed by rearranging the expression for U 1 in equation (1) to yield: The function y ℓ ( ) is used to examine the influence of resistance variations over the particle surface on the plating of lithium. For uniform resistance over the surface, y 1, = ℓ ( ) the current distribution over the graphite surface is uniform, and the problem is 1D, with no variations in y. The physical significance of the function y ℓ ( ) is that it allows one to investigate surface inhomogeneities. When y ℓ ( ) varies with y, the problem is 2D. In this work, we examine the influence of a passivated surface adjacent to an active surface.
We now consider the lithium deposition reaction That is, we employ the convention that U 0.
Li 0 = For the activity of lithium a Li , we seek a function with the following attributes: only one unknown parameter is needed that has physical significance ( , ref G as will be described below), and (iv) the relation characterizing the a Li versus Γ is monotonic and (v) continuously differentiable. A simple function with these attributes is where the lone parameter is , ref G a reference surface concentration (e.g. with units of mol cm −2 ) that can be associated with the number of monolayers of Li needed to achieve bulk Li activity; i.e. after Γ exceeds Γ ref , the activity of the Li deposit asymptotes exponentially to that of bulk Li. More discussion and references underpinning the use of equation (8) can be found in [13]. Equation (8) is consistent with underpotential deposition of metals [24][25][26], including Li on carbon [15,27] and various other substrates [28]. Equations (7) and (8) where, as noted in equation (2), E e .
)/ For differential voltage spectroscopy, the following relation is helpful: The kinetics of equation (6) The total current density normal to the electrode surface j T is given by At the interface between the lithiated graphite and the Li deposit, we postulate  figure 1(c)). The corresponding rate expression is taken to be r y ka a k a . 1 6 The exchange reaction cannot take place when the Li deposit is separated from the graphite by an insulating layer, corresponding to y 0.
where r T is the total exchange rate between deposited Li and vacant sites in the graphite. In this work, we examine currents below the 1C rate, and we ignore degradation of the plated Li, consistent with findings in [7], as noted in the Introduction.

Transport phenomena and material balances
We employ well-known equations to describe transport by diffusion and migration in the electrolyte phase based on irreversible thermodynamics (see [23] and references therein): The material balance on the salt concentration c reflects our assumption of a constant transference number t . 0 + For equimolar counter diffusion of lithium in the host graphite, the total concentration of sites within the graphite c T is constant, and the flux of lithium-filled sites is given by We shall assume the diffusion coefficient D s is a constant, reflecting the Li diffusion coefficient value at infinite dilution (x→0). A material balance yields A material balance on lithium at the interface z=0 yields For all times, the boundary condition for the potential over the graphite surface is given by The average current density over the graphite surface is given by: which constitutes a boundary condition for current-controlled operation, for which i avg is specified and V must be chosen (calculated) so that the average current density matches the specified (input) value. For potential control, V(t) would need to be specified instead of equation (24), and equation (24) would be used to determine the average current density. The net charge of Li in the graphite host per unit length in the dimension perpendicular to y and z (e.g. C cm −1 ) can be obtained from

Q t Fc aL L a x t y z dy dz
Fc aL x t where Γ avg is the average surface concentration of deposited Li. The total charge of Li is given by For the initial conditions in this work (equilibration of the system at 1 V versus a Li reference electrode), both the initial concentrations x 0 and 0 G are so small they can be ignored, but we retain the terms for completeness. A useful check on the efficacy of the numerical solution is to verify that the second line of equation (27) is satisfied.

Remaining boundary conditions
We apply insulating boundaries at the left and right of the domain and lithium cannot pass into the current collector for the graphite: At the Li counter electrode and at the graphite/current-collector interface, the salt flux is zero and the potential is grounded at the lithium counter electrode: Last, a material balance on deposited Li at the electrode surface is given by: For numerical calculations, it is convenient to replace x with U 1 as the dependent variable. Specifically, x and dx/dU 1 in terms of U 1 are provided in equation (2) above. The material balance on x, equation (21), becomes and equation (22) can be written as The initial conditions correspond to the initial value of U 1 . At the left and right of the volume element x y dx dU U y U y y y a 0 or 0 for 0 and and, because lithium cannot pass into the current collector for the graphite electrode,

Nondimensionalization and scaling
We first introduce the following definitions: In addition, an overbar over a potential corresponds to f multiplied onto the potential, and an overbar over a current density corresponds to the current density divided by i . The thermodynamic relations and reaction-rate expressions can now be stated as ,a n d , The dimensionless fields equations for F and c correspond to For the values provided in For the system geometry (see figure 1(b)), we can express equation (33) as and at τ=0, At the electrode-electrolyte interface, z z 0, = = and we have three equations for the three dependent variables , F U , 1 and U :  (2) and (9) to determine x 0 and , 0 G respectively.
Last, we note that in the limit of large k j (see equation (17)), reaction (15) becomes facile. To avoid the evaluation of r T in the limit of large k , j we sum equations (49) and (54) to obtain a total Li material balance along the electrode surface: where i avg y , is the average current over the range 0 to y . To determine potential V , we first note that the potential is taken to be uniform in the current collector,

Numerical analysis
We employed the finite difference method to solve the equation system with a Fortran program. For all calculations presented in the work, 161 mesh points were used for the z z and coordinates, and 160 mesh points for the y coordinate. The mesh spacing will be discussed in the context of figures 7 and 8. A variable timestepping routine was used for short times, involving very small time steps, which were increased with time to (and subsequently held at) 0.0005, t D = where t D refers to the dimensionless time step. We found that smaller time steps and finer meshes did not affect the results as plotted in this work. In addition, the equations were solved using COMSOL Multiphysics (finite element) software 3 , specifically, the mathematics module. The COMSOL routine also used variable time stepping and variable element sizes. We found that we obtained identical results upon comparing calculations from the two routines.

Discussion of results
The parameters and properties for the base case are provided in table 1, and table 3 provides the relevant dimensionless groups that govern the problem for the base case. The thermodynamic parameters for the MSMR model of lithiated graphite are identical to those of [21] with the exception of the LiC 12 to LiC 6 phase transition ( j=1): for this gallery, two percent of the capacity is split off to yield ideal, Nernstian behavior as the graphite completes lithiation (see j=7), consistent with the formation of an ideal solution as x 1  and x 0. H  The base case involves the charging of a large graphite particle at the 1 h rate (the 1C rate), which is known to be a challenge for lithium ion traction batteries today if a full charge is desired (i.e. charge to full lithiation of the graphite). Prior models [1][2][3][4][5][6] have assumed R R 0,  ) (see equation (20)). In addition, we see that differentiation of the ocv curve ( U Q Q versus ¶ ¶ / ) yields a sharp valley (a pronounced minimum) as (i) the graphite fills with Li and x 1,  leading to a sharp decline in the potential and U Q, ¶ ¶ / followed by (ii) Li plating, causing U V 0 ,  a constant, and causing U Q ¶ ¶ / to transition from a large, negative value to nearly zero. These observations underscore the relevance of evaluating U Q ¶ ¶ / to approximate the onset of Li plating.
). In addition, because R R 0, the potential associated with plated Li, U 2 , remains near the cell potential V, as the current density j 2 is nearly zero (see equation (13)), as is the overpotential , as F is small (due to the small value of R R 5.84 10

=´-/
). We now focus on the corresponding concentrations x, Γ, and Θ. In figure 3, x(0) is the fraction of filled sites at the graphite surface z 0, = which can be contrasted with x(1), the fraction of filled sites at the current collector z 1. = When the graphite is nearly filled with Li at the surface, near 0.3, t = only about 25% of the sites are filled with Li near the current collector. To examine the behavior of the equation system over a wide range of conditions, the simulations are carried out until 1, t = at which time only about 60% of the sites are filled with Li near the current collector, and the average state of lithiation x avg (see equation (25)) is just over 75%. At 1, t = G is about 0.6, which means that the amount of Li plated corresponds to 60% of the Li capacity of the graphite. To an observer, the total dimensionless charge passed would be Q x 0.75 0.6 1.35 avg avg = + G » + = (see equation (38)), which, if the observer were to assume that first all of the graphite fills with Li and then Li plating begins, might be interpretable as 35% overcharge, but the ratio of plated Li to intercalated Li is about 0.6/ (0.6+0.75), or 44% overcharge. It is clearly important to be able to determine the individual currents j j and 1 2 to avoid overcharge, and modeling of the electrochemical system can assist in the endeavor.  The curves for times 0.5 t > are shifted to lower potentials but are otherwise like those for 0.5.

t =
The corresponding traces in x are shown in the lower plot. There is more variation in x with z than there is in U 1 with z ; the plateau-like patterns in figure 4 are due to the influence of the different galleries (phases) in graphite, as described in the context of figure 2.
Sensitivity to the geometric parameters a and L G is shown in figure 5. Other than changes to a and L , G the parameters are identical to those employed for the plot of figure 3. The values for a and L G were chosen by first reducing L G from 10 μm (figures 3) to 5 μm (figure 5) and then increasing a so as to keep the graphite capacity the same; by equation (10), we see that the graphite capacity at equilibrium is given by Fc aL x U . T G ( ) Because the C-rate for the current density is inversely proportional to a, the 1C rate is halved to 0.408 mA cm −2 . The resistance R d is halved, and the results look like those of figure 3, but with reduced impedance. As seen previously, the minimum in dV/dτ near 2 t = corresponds to the time when U 2 approaches 0 V. The graphite is substantially   (42)). As noted in table 3, for the 2D investigations in this work y ℓ ( )is 0.9 for y 0 0.5,  < the more reactive surface, and 0.1 for y 0. 5 1, < < indicative of a kinetically-hindered surface. Plots analogous to those in figure 3 for the 1D case are provided in figure 6 for the 2D case. Other than introducing variations in y , ℓ ( ) all parameters are the same for the 1D and 2D cases, including the 1C charge rate, and the potentials U 1 , U 2 , and V (as well as dV/dτ) are associated with the left ordinate, while the concentrations , G Θ, and x are associated with the right ordinate. The ordinate axes are the same for the figures, but the abscissa ranges from 0 1   t in figure 3 and 0 0.5   t in figure 6. For the 2D For the 2D results of figure 6, relative to their 1D counterparts in figure 3, the cell voltages V and the ocv's at y 0, = U 0, 0 1 ( )and U 0 , 2 ( ) are shifted to lower values due to the increased impedance. Similarly, the concentrations at y 0, = x 0, 0 ( ) and 0 , G( ) are shifted to higher values, associated with more Li intercalation and plating, and, at y 1, = x 1, 0 ( ) and 1 G( ) are shifted to lower values due to the impeding surface for

Summary and conclusions
We show how the model equations developed in [13], can be applied to examine a graphitic electrode element and overcharge that leads to lithium plating. Two new features of this work and [13] involve the development of an expression for the activity of Li on graphite, equation (8), and the exchange of Li between plated and inserted Li, The vertical arrow to the right of the plot reflects increasing times for the curves. reaction (15) and equation (16). The equation system involves multiple electrochemical and nonelectrochemical reactions, and significant diffusion resistance within the graphite when high charge rates are employed, as investigated in this work. Dependent variable transformations are described (equations (33), (50), and (54)) that simplify the numerical analysis. Results obtained from two different software routines were compared (and found to agree), as described in the brief 'Numerical analysis' subsection.
We find that if the amount of deposited Li needed to obtain the activity of bulk Li is significantly less than the capacity of the graphitic host, i.e. if L c , where the reference surface concentration of plated lithium ref G (mol cm −2 ) reflects the surface concentration above which the activity of deposited Li rises quickly to that of bulk Li, and L c G T is the total amount of Li that can be stored in the graphitic host per unit geometric area, then the calculated results of the model equations, as plotted in this work, are close to those of 0.
ref G = To be specific, equation (8) provides the activity of plated Li, which tends to that of bulk Li for .  [1][2][3][4][5][6], have assumed that the plated Li does not react directly with graphite vacancies at the interface (see reaction (15)), even though the plated Li resides directly on the graphitic surface ( figure 1(c)), i.e. atoms of deposited Li are within angstroms of vacant sites within the graphite. For this reason, we compare the conventional approach of no Li exchange (k 0 j = for all j in equation (17)) to that of nearly facile exchange of Li (k j  ¥ for all j in equation (17)). We find that during constant current charging the time traces of the calculated electrode potential, as well as those for the opencircuit potentials for the Li insertion and plating reactions are very sensitive to the ratio of the resistance to Li diffusion in graphite to that of the Li exchange reaction, R R . / For a variety of reasons, the surface of graphite is nonuniform in terms of reaction rates for Li insertion and plating, as overviewed in the Introduction, and we show that when the electrode is subjected to constant-current charging, as is commonly employed, such nonuniformities lead to early Li plating over the highly reactive surfaces (see figures 6 through 9). These results underscore the importance of maintaining a uniform electrode surface, especially when the cell is to be subjected to high rates of charge.