Simulation of the Effect of Contact Area Loss in All-Solid-State Li-Ion Batteries

A 1-D model was constructed to simulate the discharge process of an all-solid-state Li-ion battery. The model includes a Li metal negative electrode, a 1500 nm-thick LiPON like solid-electrolyte, referred as Li₃PO₄, and a 320-nm thick LiCoO₂ positive electrode, as shown in Figure 1a.

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At the interface of electrolyte/negative electrode (x = 0), the electrochemical charge transfer reaction is

The net current was described as Butler-Volmer kinetics

where k_neg (SI unit: mol/(m²·s)) is the rate constant of the reaction 1, R is the molar gas constant, F is the Faraday constant (96485 C·mol⁻¹), T is the temperature, is the concentration of mobile Li, C_{Li, all} is the total concentration of Li in the electrolyte, ∝_neg is the charge transfer coefficient for reaction 1, η is the overpotential, and A₀ is the contact area that is set it is 1 cm² in this 1-D model.

The overpotential, η, is defined as

where the φ_s is the electric potential that is the same as the electrode potential, φ_ep is the solid-electrolyte potential, and E_eq is the equilibrium potential, which is set to be 0 for the negative electrode. Since the electron conductivity is much higher than the ionic conductivity in LiCoO₂, it is considered as a good electronic conductor.⁵⁰ Therefore, the φ_s is assumed to be the same everywhere in LiCoO₂.³⁰

After Li⁺ being generated at the interface of electrolyte/negative electrode, only a fraction of Li⁺ can be mobile in the solid electrolyte, and this is limited by the diffusion carrier's concentration. In this case, only a fraction (denote as δ) of the total amount of Li⁺ is mobile at equilibrium is assumed,

where C₀ is total amount of Li and is the concentration of uncompensated negative charge.

Therefore, it is assumed that a portion of mobile Li⁺ would kinetically bond with the solid electrolyte and become immobile. The reactions can be expressed as

where Li₀ is the immobile Li bonded with anions in solid-electrolyte, n⁻ is the uncompensated negative charges, k_d is the dissociation rate, and k_r is the reverse reaction rate. So the overall reaction rate can be written as

The relationship between k_d and k_r can be described as

So the net production rate of mobile Li-ion that can pass through the solid electrolyte is obtained by

The transport of Li-ion in the solid electrolyte is driven by diffusion and migration, expressed by the Nernst-Planck equation and Fick's second law,

where is the flux of Li⁺, is its diffusion coefficient, ∇φ₁ is the electrical potential gradient, and r is the net production rate of mobile Li obtained in Equation 8.

At the interface of solid-electrolyte/positive electrode the charge transfer reaction can be expressed as

with the same Butler-Volmer expression^15,30 the net current is

where and are the maximum and minimum concentration of Li in the positive electrode, respectively, and the k_pos is the rate constant of the reaction. The overpotential, η, is defined as the same as Equation 3, but E_eq is the equilibrium potential of LiCoO₂ and it depends on the concentration of Li.

After the charge-transfer reaction, Li would be produced and intercalate into the positive electrode (denote as Li_s), and its diffusion through the positive electrode is driven by the concentration gradient, as

where , , and are the diffusion coefficient, concentration, and the concentration gradient of Li_s. The reaction rate, , inside the positive electrode is 0, except at the solid-electrolyte/positive electrode interface (x = 1500 nm), where (mole·s⁻¹m⁻²), would be correlated with the current.

where I_pos is the current (A), i_pos is the current density (A/m²), ν is the stoichiometric coefficient of Li, and n_e is the number of participating electrons in the reaction, both values are one here.

With this 1D model, a discharging process under a constant current is simulated. This process means as the constant current passing through the two interfaces, x = 0 nm and x = 1500 nm, the same amount of Li-ion will be generated and released at these two interfaces, respectively. All the Li⁺ and Li_s are bounded between x = 0–1500 nm, and x = 1500–1820 nm, respectively. The electroneutrality is applied in the electrolyte that gives the concentration of Li-ion equals to the negative charge. The boundary and initial conditions in this model are listed in Table I. The concentration profile of Li⁺ and Li_s in the solid-electrolyte and the positive electrode, and the electrical potential at x = 1500 nm are solved. The boundary condition is the controlled value of current (C-rate) at the both end of the solid-electrolyte. Since for all-solid-state Li-ion batteries, the range of C-rate usually is between 0.1 C to 10 C,^2,26,51 the C-rates chosen to be studied in this research are 0.1, 1, 5, 10 and 20 C. The input open-circuit voltage (OCV) of Li_xCoO₂, where x varies in the range of 0.5 to 1, is given in Figure 3c. Initially, Li_0.5CoO₂ gives the starting voltage around 4.2 V. The simulation would end at the desired cutoff voltage. The parameters used and listed in Table II are based on the research of Danilov and Notten et al.,¹⁵ where they have fitted to the experimental data.

Table I. The conditions and dependent variables in this model.

Boundary Conditions Initial Conditions Stop Condition Solve for Description

ipos = controled value	x = 0.5 (ratio of initial to CLi,max)	When = CLi,max (at the interface of positive electrode/electrolyte)	CLi+ (x,t)	The electrolyte concentration of Li ions (mol/m3)
			(x,t)	The concentration of Li in LiCoO2 (mol/m3)
Φs, neg = 0	Φs, pos = 4		Φs, pos (t)	The external electric potential of positive electrode (V)
Eeq,neg = 0 (Ground)	Φep = 0		Φep (x,t)	The electrolyte potential (V)

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Table II. The parameters used in this model, which have been fitted to experimental data.¹⁵

Control Variable	Description	Value
CLi,total	The total electrolyte concentration of Li ions (mol/m3)	6.01*104
CLi,max and CLi,min	Maximum and minimum concentration of Li in the solid electrode (mol/m3)	23300 and 11650
DLi+	Li-ion diffusion coefficient in solid electrolyte_Li3PO4 (m2/s)	9*10−16
	Lis diffusion coefficient in positive electrode_LiCoO2 (m2/s)	1.76*10−15
δ	Fraction of mobile Li-ion at equilibrium	0.18
kd and kr	Rate constants for dissociation and recombination (mol/(m2·s))	2.13*10−5 and 9*10−9
kpos and kneg	Rate constant for reactions (mol/(m2·s))	5.1*10−4 and 0.01
αpos and αneg	Charge transfer coefficient for the reactions	0.6 and 0.5

As illustrated in Figure 1b, the imperfect contact at the solid electrolyte and positive electrode interface is considered. Although both electrode and electrolyte surfaces are rough, one can always reduce the contact to a flat surface with an effective rough surface. To represent the imperfect contact area, a contact factor, γ, is assigned varying

where A is the actual imperfect contact area, and A₀ is the perfect contact area (cross section area). γ ranges between 0–1.

As mentioned in the introduction part, in liquid electroylte,⁵² Li-ion can diffuse through the porous electrode. Fracture and disconnection of particles will only cause disconnection electronically, which means less pathways for e⁻ to react with Li⁺. However, for solid electrolyte, the disconnection would be even worse due to the fewer pathways for both of e⁻ and Li⁺ to react. Therefore, the imperfect contact at the solid-electrolyte/electrode interface may cause part of the electrode material inaccessible, especially at a higher rate. The inhomogeneous Li distribution in LiCoO₂, observed by in-operando elemental mapping of an all-solid-state battery,¹² may be partially caused by imperfect contact as well. Therefore, in the worst case, the loss of contact between electrode and solid-electrolyte interface would cause the loss of LiCoO₂ material, in which Li-ion cannot be stored. The region of the red color in Figure 1b means the area of loss of material. In this case, the imperfect contact area, A = γA₀ is incorporated into this model at the interface of electrolyte/positive electrode. More specifically, Equations 13 is modified to represent the real contact area, as

Since constant current discharge process is simulated and the C-rate is fixed, the current density at the solid-electrolyte side of the interface is not affected by the loss of contact area, but the production rate of Li on the positive electrode side of the interface would be scaled to match the same current. Therefore, the Equation 15, which is the production rate of Li, should be adjusted by γ, as

Persson's multi-scale contact mechanics theory⁴⁹ was used to establish the relationship between the contact stress and the real contact area, A, in order to provide the key parameter γ used in the above electrochemical model. The advantage of Persson's model is that it captures the fact that the real contact area depends on the observation length scale. For example, an interface may look like contact perfectly with uniform contact pressure at a macroscopic view, but at a microscopic view, the surface roughness will lead to non-perfect contact and non-uniform contact stress. By assuming that surface roughness has self-affined property,^45–48 this model can calculate the real contact area at every length scale. Thus, it is important to define the length scales for the battery systems.

The contact mechanics is at multi-length scale, as shown in Figure 2a, where the largest contact area A₀ and the corresponding length is L = A₀^1/2, assuming isotropic in x-y direction. In the reciprocal space, the smallest wavelength is q_L= 2π/L. Therefore, any wavelength, q, can be defined via a magnification, ξ, as q = ξq_L. For the longest length scale and the smallest wave length, ξ is 1. The smallest length scale for contacting would be observed in the atomic scale, which sets the upper limit of wave length q.

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The surfaces are composed of asperities for both the film-type or bulk-type battery. The surface roughness power spectrum can be calculated by the Fourier transform of the height-height correlation function:⁵³

where C(q) is the power spectrum of wavelength q; h(x) is the height of the surface above a flat reference plane that is chosen for h(x) = 0. The angular bracket < · · · > is the ensemble averaging operator. Solid surfaces are approximately self-affine fractals, which means if observing the surface in different magnification, it looks the same because the surface pattern repeats. That being said, when the length scale changes in the self-affine region, C follows a power law of q, as C∝q^m× 10^I, as shown in Figure 2b, where m is the slope and I is the y-intercept. The lower limit of q or ξ defines the scale when the self-affine property is not held anymore. In this case, the self-affine property is considered to be non-distinguishable. For example if the diffusion-length of Li at 1 C rate is used to be the observed length scale, Li distribution can be considered as uniform at this scale, then the contact is not self-affined anymore. This length scale is estimated to be 3.6 μm, based on . So the lower limit of the q is defined by the magnification ξ of around 2700 for A₀ = 1 cm² in this model. The contact area for this length scale was computed.

C(q) would be obtained from the surface information, such as root-mean-square (RMS) roughness and peak-valley value. As other research investigated the surfaces by atomic force microscopy (AFM), for the film-type cathode, such as LiMn₂O₄, LiCoO₂, LiNi_1/3Co_1/3Mn_1/3O₂, the RMS roughness is around 10–20 nm.^54–58 On the other hand, for the bulk-type cathode, the RMS roughness is around 100–200 nm, such as LiFePO₄, CoO + Co₃O₄, and LiNi_0.8Co_0.2O₂.^59–61 Currently, C(q) is not available for all-solid-state battery electrode and solid-electrolyte surface yet. Therefore, we rationalize C(q) for the observed RMS based on the surface analysis by Flys et al.⁴⁷ In their study, they have used AFM to investigate the surface profile and obtained C(q) for samples prepared with different RMS roughness, ranging from 2 to 120 nm. The result of C(q) at RMS roughness of 15 and 120 nm from the research of Flys et al. were chosen to represent the film-type and bulk-type cathode, respectively. Figure 2b shows the power spectrum-wavelength logarithm relationship for these two different RMS roughness. Also, it indicates that the rougher surface, which is the RMS roughness of 120 nm, has a larger slope, m, and smaller y-intercept, I.

Based on the Persson's contact mechanics theory,⁴⁹ the stress distribution at the magnification, ξ, which refers to any arbitrarily chosen length scale, can be defined as

where P(σ, ξ) is the probability of stress distribution, σ(x, ξ) is the interface stress distribution at any position x in the spatial coordinate, and δ is the delta function. By this definition, the real contact area A, which is projected on the interface-plane, can be directly calculated from this stress distribution as the following

The integration of stress should be equal to the applied load. Therefore, the stress distribution can be obtained by solving,⁶²

where E and ν are the effective Young's modulus and Poisson's ratio averaged from the two materials. C(q) is the power spectrum obtained from Equation 19.

where subscript 1 and 2 refer to the two solid materials. In this research, the solid electrolyte material Li₃PO₄ and LGPS; and the cathode materials LiCoO₂ and TiS₂ are chosen to be calculated. Their elastic properties are listed in Table III.

The different properties of the contact, such as elastic contact, adhesion or elastoplastic contact can be addressed by changing the initial and boundary conditions of Equation 22. Since both Li₃PO₄ and LiCoO₂ are ceramic materials, the contact is considered to be elastic without adhesion in this study. For the initial condition, which means the lowest magnification ξ = 1, the contact area would be the cross-section area and looks like a flat and full contact. So the stress distribution P(σ,1) = δ(σ–σ₀), σ₀ represents the applied pressure. There are also two other boundary conditions along the σ-axis are necessary. For elastic contact, P(σ, ξ) would be 0 when σ → ∞, when σ < 0 the P(σ, ξ) should be 0 as well since there is no adhesion. With the initial and boundary conditions, the stress distribution of Equation 22 can be solved, and the real contact area in Equation 21 can be further obtained by

In this simulation, the battery is discharged at a constant current, and the cell voltage is the potential difference between the positive electrode and negative electrode. Since the negative electrode is grounded, the potential, φ_{s, neg}, is maintained as 0. The cell voltage is determined by the potential at positive electrode, φ_{s, pos}, which is affected by the equilibrium potential of positive electrode, φ_eq, the over potential and the solid-electrolyte potential. The local Li concentration at the solid-electrolyte/cathode interface decides the equilibrium potential, φ_eq. Therefore, it is necessary to first check the concentration of Li on the cathode surface when the contact area is not perfect.

Figure 3a and 3b shows the variation of the local concentration of Li, , with time at the Li₃PO₄/LiCoO₂ interface (x = 1500 nm) under different contact area and two separate discharge rates, 1 C and 10 C, respectively.

Comparing Figure 3a and 3b, at the same discharging time, the concentration of Li is higher at 10 C than at 1 C, since higher C-rate means larger current, which causes more Li produced on the LiCoO₂ surface. If the diffusion rate is not as fast as the production rate, Li is accumulated at the LiCoO₂ surface. In both rates, at the same discharging time, the local concentration of Li in LiCoO₂ (Li_s) at the cathode surface increases when the contact area is reduced. It is because the constant current imposed the same amount of Li produced at LiCoO₂ surface, so the local concentration of Li is higher when the contact area is reduced. The local Li_s concentration would then be continuously increased to the maximum Li concentration, which is 23000 mol/m³ for LiCoO₂.

When the local concentration of Li on LiCoO₂ surface increases with reduced contact area, it results in the reduction of OCV and the potential of LiCoO₂. Therefore, the battery voltage will reduce due to the loss of contact area, this is further analyzed in the cell discharge voltage vs. capacity plot.

Figure 4 shows the results of discharging curves, comparing the effects of the different discharging rate ranging from 0.1 to 20 C and the contact area, which is expressed by contact factor, γ, varying from 1 to 0.75. The cutoff range of 3.7–4.2 V for the battery voltage is used because most of the batteries operate between these values. Q_out is the discharging capacity per unit area, which is calculated by

where t is the discharging time (s). The dashed line stands for the critical capacity, which is defined as 80% of the maximum capacity since typically the battery for transport application would be seen as "end of life" when the capacity is less than 80%.

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Figure 4a shows the discharge curves with the perfect contact (γ = 1) at different C-rates. It shows the discharge curves for 0.1 C and 1 C are very similar. With increasing discharging rate, both the cell voltage and capacity are reduced. This result is similar to other models with perfect contact area.^27,30 With the current model and parameter setting, the discharge capacity can still be maintained above 80% of the maximum capacity even at 20 C with perfect contact area.

Figure 4b and 4c show the discharge curves with imperfect contact at C = 1 and 10, respectively. At the same discharging rate, both the discharge voltage and capacity drop with the reduced contact factor, γ. The drop of voltage can be considered as the increase of interface resistance and ohmic loss due to loss of interface contact area. The decrease of capacity is rate dependent. For example, if discharging at 1 C, the battery would lose 20% of the capacity if the contact area is less than 80% of the cross-section area (γ = 0.8). If discharging at 10 C, the battery would be at the "end of life" if the contact area is less than 85%. This is because of the accumulation of Li on cathode surface due to reduction of contact area and the high rate. Since the diffusion rate of Li_s is not as fast as the reaction rate, the high surface concertation of Li_s leads to less utilized electrode particles at the cutoff voltage, thus lost contact area and higher C-rate give less discharge capacity.

The definition of discharging capacity depends on the cutoff voltages, which is also known as the depth of discharge of the battery. Figure 5 compares the capacity loss due to contact area loss at two cutoff voltages, namely 3.8 V and 4.0 V. Here, the 100% of remaining capacity stands for the capacity that is obtained at 0.1 C discharge. Both higher rate and loss of contact area reduce the capacity. Interestingly, the loss of capacity due to contact area loss shows almost a linear relationship. At the cutoff voltages of 3.8 V, the slopes of all the lines are almost 1; and at the cutoff of 4.0 V, the absolute value of the slopes are all smaller than 1. For example, the slope of 0.1 C is −0.34 and the slope of 20 C is −0.26. Thus, at the cutoff of 4.0 V, the magnitude of slope is larger at lower C-rate.

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It can be seen from Figure 4 that the cutoff voltage of 3.8 V means the cathode surface is fully lithiated, the capacity does not change with voltage much anymore. Therefore, the remaining capacity at the cutoff voltage of 3.8 V is limited by the material available on the cathode surface. Since we have assumed loss of material due to the loss of contact area, the capacity loss drops with loss of contact area with a slope of 1. For the cutoff voltage of 4.0 V, the discharging capacity is still very sensitive to the voltage, which is related to the local Li_s concertation. The local Li_s concentration depends on the Li generation rate and diffusion rate, thus the capacity is not dominated by the amount of materials only. Therefore, the slopes for capacity drop due to loss of contact area are smaller than 1, and the slope for lower C-rate (like 0.1 and 1 C) is larger than the higher C-rate when the cutoff voltage is 4.0 V. At higher rate, less diffusion time, the role of loss contact area will be more dominating, such the slope decreases with increasing C rate.

The linear relationship may be due to the simplified 1D model and the assumption of worst scenario (losing accessible cathode material due to loss of contact), more sophisticated 2D and 3D models that can describe the local concentration change and the amount of accessible cathode material more accurately are being developed for further study. Nevertheless, the 1-D continuum model has directly correlated the loss of contact are with loss of capacity. So the next question if the loss of contact area can be recovered by the applied pressure, and how to calculate the necessary applied pressure. In the following part, calculations for the applied pressure will be discussed.

As introduced in the last section of Computational methods, for a given C(q), the contact area ratio, γ, can be calculated under different applied pressure, σ. Two all-solid-state batteries were considered, a film-type contact at Li₃PO₄/LiCoO₂ interface, and a bulk-type contact at LGPS/TiS₂, respectively. The previous battery discharge model is based on parameters fitted for the first case. It can be expected the predicted capacity loss due to contact area loss relationship is general for other all-solid-state batteries. In order to make a comparison with the experiments of Li et al.,²⁶ the second case was modeled. As introduced in the last section of Computational methods and plotted in Figure 2b, the initial C(q) has been determined based on measured RMS roughness for the thin film type and bulk-type and relationship between C(q) and RMS discussed by Flys et al.⁴⁷ After cycling, experimental studies^54,56,58,63 have reported the RMS roughness would increase. Also, based on the study of surface morphology,^45,46,64 the absolute value of m, of the plot in Figure 2b would increase, and the intercept, I, would be decreased when the RMS roughness is increased. (m, I) values were changed to mimic the effect interface roughening due to battery cycling.

Figure 6 shows the surface contact ratio as a function of the applied pressure. As Figure 6 shows, at low applied pressure, the increase of contact area with the applied pressure is nearly linear, which agrees with other contact models, such as the theories of Hertz,⁶⁵ Greenwood-Williamson,⁴³ and Bush.⁴⁴ When the contact area ratio is greater than 60% (A/A₀ = 0.6), it needs much more pressure to increase the contact area further. The two black lines stand for the contact area before cycling, where m = −2.77, I = −11.35 for film-type battery in Figure 6a and m = −3.08, I = −8.61 for the bulk-type batter in Figure 6b. Comparing the film-type and bulk-type all-solid-state batteries, at the same applied pressure the film-type has more contact area than the bulk-type. Both Figure 6a and 6b show that with increasing surface roughness (increasing m and decreasing I) due to cycling, the contact ratio decreased. Furthermore, increasing the applied pressure can increase the contact ratio.

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To demonstrate this model leads to reasonable contact area estimations, the results shown in Figure 6b can be compared with the experiments by Li et al.,²⁶ in which they investigated an In-Li/LGPS/TiS₂ all-solid-state Li-ion battery. They applied two different pressures, 19 and 230 MPa, for the fabrication process and during the operation of the battery. From the calculation, at 19 MPa, the contact area is only about 40%, which results in the poor capacity retention as shown in their experimental result. If increasing the pressure to 230 MPa, the contact area is estimated to be around 95%, and so they observed effectively enhanced capacity retention.

With this model, it is now possible to calculate the proper applied pressure for al-solid-state batteries. For example, if the initial condition for the film-type battery is under 20 MPa, as starting from point A in Figure 6a, the contact area is around 70%. After cycling, it may decrease to point B, since the surface becomes much rougher, it also means around 20% of the contact area is lost. Then to get the original contact area and the capacity back, it needs to follow the pink line, for instance, increasing the contract pressure to 60 MPa will recover the contact area to 70% again.

Therefore, based on Figure 5, the lost contact area can be estimated by the capacity drop. Then, based on Figure 6, the applied pressure needed to recover the contact area and capacity can be obtained. By connecting these two calculations, one could calculate how much contact area lost during cycling, and how much pressure should be applied to get the contact area back, to prevent the degradation of the all-solid-state Li-ion battery.