Moving Boundary and Non Dimensional Modeling and Gitt With Phase Transformations

Measurement of the Diffusion Coefficient of Lithium in Tin Thin Films Including Phase Transformation Effects

Eddie C. W. Fok, and John D. W. Madden

Department of Electrical & Computer Engineering, The University of British Columbia,

Vancouver, British Columbia, Canada

In this work, use of the galvanostatic intermittent titration technique to extract lithium ion diffusion coefficients in tin thin films is studied. The measured results are first analyzed under the traditional solid solution assumptions. The change of phase in the electrode is then modeled using a moving boundary model, which more accurately predicts the potential transients during the galvanostatic pulse.

Introduction Lithium alloys are promising materials for use in lithium-ion batteries due to their high lithium storage capacities. Alloys such as lithium-silicon and lithium-tin have capacities of 4200 mAh/g and 990 mAh/g, respectively. These values are significantly higher than the 372 mAh/g capacity of the carbon materials that are commonplace in today’s lithium ion batteries. In order for these and other novel electrode materials to be incorporated in commercial batteries, many fundamental properties of the materials need to be characterized. The solute diffusion coefficient is one such property, which has important scientific and practical implications. The lithium ion diffusion coefficient is most often measured using the Galvanostatic Intermittent Titration Technique (GITT) (1-3) or electrochemical impedance spectroscopy. Traditionally, GITT results are analyzed with the assumption that the inserted species forms a solid solution with the host. This means any phase change effects are neglected. In this paper, measurement of the lithium ion diffusion coefficient in tin thin films is considered. First, the governing equations for both the single phase and phase transformation models are discussed. Then, the experimental procedure is presented. Finally, the results section discusses the use of the single phase model to analyze the GITT results. Further, the phase transformation model is shown to be better at predicting the potential transients that are observed during the GITT experiments.

10.1149/05330.0131ecst ©The Electrochemical SocietyECS Transactions, 53 (30) 131-142 (2013)

131) unless CC License in place (see abstract). ecsdl.org/site/terms_use address. Redistribution subject to ECS terms of use (see 103.21.126.86Downloaded on 2014-04-16 to IP

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Theory

The charge and discharge process of a tin film electrode can be modeled as a 1D system, with thickness L. In this section, the single phase model will first be presented, followed by the phase transformation model. Single Phase Model The concentration of lithium ions, C, in the electrode is governed by the diffusion equation

dC/dt = Dd2C/dx2 [1] where D is the lithium ion diffusion coefficient, and x is the position in the thickness direction of the electrode. The boundary conditions specified at the film surface (x=0) and current collector interface (x=L) are

dC(x = 0, t)/dx = -j/nFD [2] and

dC(x = L, t)/dx = 0 [3] where j is the applied current density, n is the charge number of electroactive species (for a Li-ion electrode, n = 1), and F is the Faraday constant. The electrode is assumed to start at equilibrium with concentration C0

C(x, t = 0) = C0 [4] As shown in (1), for small time t (t << L2/D), the lithium diffusion coefficient can be approximated using

D = 4/π(VMJ/nF)2[(dE/dδ)/(dE/dt1/2)]2 [5] where E is the electrode potential (vs. Li/Li+), VM is the electrode’s molar volume, and dE/dδ is the change in electrode potential due to the stoichiometry change as a result of the current pulse. To facilitate comparison between the single phase and phase transformation models, equations [1]-[4] can be non-dimensionalized using

α = x/L [6]

τ = Dt/L2 [7]

V = (C-C0)/C0 [8]

J = jL/nFDC0 [9]

ECS Transactions, 53 (30) 131-142 (2013)




where α, τ, V, and J are respectively the dimensionless position, time, concentration, and current density. Equations [1]-[4] becomes

dV/dτ = d2V/dα2 [10]

dV(α = 0, τ)/dα = -J [11]

dV(α = 1, τ)/dα = 0 [12]

V(α, τ = 0) = 0 [13]

The surface concentration V(α = 0, τ) of the system of equations [10]-[13] can be shown to be given by n=∞

V(α = 0, τ) = 2Jτ1/2 Σ [ierfc((2n+α)/2τ1/2) + ierfc((n+1-α)/τ1/2)] [14]

n=0

where ierfc(x) is the integral of the complement of the error function. The electrode potential transient, ∆E(τ), is related to the surface concentration by

∆E(τ) ≈ (∂E/∂V)V(α=0, τ) [15]

So for small changes in concentration, the electrode potential transient is roughly proportional to the concentration at the surface of the electrode. Phase Transformation Model In the phase-change model, equations [1]-[4] are modified to include the presence of two phases, and a phase boundary between them. The moving boundary model that will be used in this work was first proposed by Wagner in (4). The phase boundary location ξ splits the electrode thickness into regions I (0 ≤ x ≤ ξ) and II (ξ ≤ x ≤ L). The diffusion equation applies individually to both phases

dC/dt = DId2C/dx2 (0 ≤ x ≤ ξ) [16]

dC/dt = DIId

2C/dx2 (ξ ≤ x ≤ L) [17]

where DI and DII are the lithium ion diffusion coefficients in phases I and II, respectively. The phase boundary movement satisfies the conservation of lithium ions at the interface, which can be written as

Cddξ/dt = DI(dC/dx|x=ξ-) – DII(dC/dx|x=ξ+) [18]

where Cddξ is the lithium ion per unit cross section generated or consumed by a displacement of dξ in the phase boundary.





As before, the boundary conditions are

dC(x = 0, t)/dx = -j/nFD [18]

dC(x = L, t)/dx = 0. [19]

and the initial conditions have become

C(α, τ = 0) = C0 (0 ≤ x ≤ ξ) [20]

C(α, τ = 0) = Cd (ξ ≤ x ≤ L) [21] As in the single phase model, equations [16]-[21] can be made dimensionless using equations [6]-[9], and introducing

ε = ξ/L [22]

λ = D2/D1 [23]

cd = Cd/C0 [24]

where ε is the dimensionless phase boundary position, λ is the ratio of lithium ion diffusion coefficient of phase 2 to that of phase 1, and cd is the dimensionless form of Cd. The dimensionless equations become

dV/dτ = d2V/dα2 (0 ≤ α ≤ ε) [25]

dV/dτ = λd2V/dα2 (ε ≤ α ≤ 1) [26]

dV(α = 0, τ)/dα = -J [27]

dV(α = 1, τ)/dα = 0 [28]

cddε/dτ = (dV/dα|α =ε-) – λ (dV/dα|α=ε+) [29]

V(α, τ = 0) = 0 (0 ≤ α ≤ ε) [30]

V(α, τ = 0) = cd (ε ≤ α ≤ 1) [31]

Unlike the set of equations [10]-[13], there is no closed-form solution for equations [25]-[31]. In this work, the finite-element solver COMSOL Multiphysics is used to numerically calculate the solution to this set of equations.

Experimental Copper foil (lithium-ion battery grade, MTIXTL) was degreased by sonication for 15 minutes in acetone. The oxide was removed by placing the foil in 10% (v/v) H2SO4 for 5 minutes. The foil was sealed onto a PVC plate with electroplating tape (3M) so that tin is





coated on one side. The tin deposition bath consisted of 137 g/L potassium pyrophosphate, 36 g/L tin pyrophosphate, and 0.3 g/L gelatin. The electrodeposition was performed at a constant current density of 2.5 mA/cm2 for 5 minutes at room temperature with stirring. This deposition procedure was taken from (5). The tin thickness was about 1 µm. The tin electrode was a disc of 1 cm diameter cut from the tin-coated copper foil. A custom designed setup was built to hold the electrodes inside a stainless steel coin cell case (MTIXTL). The reference and counter electrodes were lithium films (Sigma-Aldrich). Glass microfiber filters were used as separators between the working and reference electrodes, as well as between the reference and counter electrodes. The electrolyte solution was 1 M LiClO4 in propylene carbonate (BASF). Approximately 200 µL of electrolyte solution was used to wet the separators. Before the GITT test, the electrode was cycled between the potential limits of 0.8 V and 0.05 V vs. Li/Li+ for 5 cycles at descending current densities of 100 µA/cm2, 80 µA/cm2, 60 µA/cm2, 40 µA/cm2, and 20 µA/cm2. The GITT procedure consisted of pulses of 40 µA/cm2 for 6 minutes, followed by a rest period of 40 minutes. The anodic direction test was performed until the electrode potential reached below 0.01 V. The cathodic direction test was performed until the electrode potential went above 1.2 V. Electrochemical tests were performed using an Autolab PGSTAT101 (Metrohm AG), controlled using the NOVA software. Cell assembly and testing were performed in an argon-filled glovebox. Scanning electron microscopy and Auger microscopy was performed using a Thermo VG Microlab scanning Auger microscope.

Results A scanning electron microscope image of the tin surface is shown in figure 1. The film appears to be relatively flat and without significant features. Scanning Auger microscopy before and after a series of Ar+ sputtering proved that the deposited tin was chemically pure.





Figure 1. Scanning electron microscope image of the tin surface. Potential of the tin film electrode during the 60 µA/cm2 discharge and charge is shown in figure 2. The curve shows plateaus during both discharge and charge. These indicate the occurrence of phase transformation reactions in the electrode. By comparing the potentials of these plateaus to those in (6), specific reactions can be assigned to each plateau. During discharge, the first three plateaus are associated, respectively, to the reactions

Sn + Li+ + e- → Li2Sn5 [32]

Li 2Sn5 + Li+ + e- → LiSn [33]

LiSn + Li+ + e- → Li7Sn3 [34] Below ca. 0.4 V, the extended slope is due to the closely spaced transformations between the phases Li7Sn3, Li5Sn2, Li13Sn5, Li7Sn2, and Li22Sn5.





Figure 2. Potential of the electrode during discharge and charge at 60 µA/cm2. The discharge GITT results are shown in figure 3. Current pulses lead to the downward potential peaks in the plot. The subsequent potential reversal is due to the relaxation in the experimental procedure. At the ends of these relaxation periods, the rates of potential changes are approximately 10 µV/s. This illustrate that the 40 minutes relaxation period allows the electrode to sufficiently relax before the subsequent discharge pulse. If the relaxation period is increased significantly, the leakage current becomes an issue. The electrode potential at the end of each relaxation period gives an approximate open circuit voltage of the electrode at that state. Three plateaus in the open circuit voltage can be identified at ca. 0.74 V, 0.66 V, and 0.50 V. These three plateaus correspond to the reactions in equations [32]-[34].

Figure 3. Results from the GITT experiment in the discharge direction.





Figure 4 shows the GITT results for the charging direction. As in the discharge case, examination of the potential at the end of each relaxation periods shows three plateaus that are associated with phase change reactions.

Figure 4. Results from the GITT experiment in the charge direction. Using the single phase approximation, the lithium ion diffusion coefficients in the electrode can be calculated using equation [5]. The results for the discharging and charging directions are shown in figures 5 and 6, respectively. The range of diffusion coefficients found here is consistent with those reported in (3). Large dips in the diffusion coefficients can be seen at potentials close to the phase change potentials. This is due to the small change in potential after a current pulse when the electrode potential is in the vicinity of a plateau. This can be explained by the large quantity of charge that is needed to complete the transformation of the electrode from one phase to another. Since the single phase approximation does not take this phase transformation into account, the lithium diffusion coefficients are inaccurate close to the plateaus in the open circuit potential.





Figure 5. Lithium ion diffusion coefficient in the tin film, extracted from the GITT experiment in the discharge direction.

Figure 6. Lithium ion diffusion coefficient in the tin film, extracted from the GITT experiment in the charge direction.

To examine more closely the effects of phase change on the GITT results, the charging GITT data for the 0.54V plateau was investigated further. Figure 7 shows the potential transients during four consecutive galvanostatic pulses. The figure shows that each pulse results in a significant overpotential, before the potential begins to flatten. Also, each subsequent pulse produces a higher overpotential than the last.





Figure 7. Potential transients during four consecutive charging pulses of the GITT experiment (1st: cross, 2nd: star, 3rd: circle, 4th: dot) near the 0.54 V open circuit potential plateau. The single phase and phase transformation models were used to try to reproduce these features in the GITT data. The value of the dimensionless parameter J that was used for the single phase model computation is shown in table I. The single phase model results are shown in figure 8, which were computed using equation [14]. The phase transformation model system of equations [25]-[31] was solved using COMSOL Multiphysics. The arbitrary Lagrange Eulerian method was used to calculate and monitor the position of the phase boundary. The parameters used in the calculations are shown in table I. ξ0 is the initial dimensionless phase boundary position. A larger ξ0 means the phase boundary is located further away from the electrode surface.

TABLE I. Values of Model Parameters Used in the Calculations

Model Parameter Single Phase Model Phase Transformation Model J 0.1 0.1 ξ0 - 0.2, 0.4, 0.6, 0.8 cd - 1 λ - 1

The computed results for the phase change model are shown in figure 9. For all four initial phase boundary positions, there is a rapid increase in potential, followed by flattening of the potential. Also, subsequent galvanostatic pulses results in increasing overpotentials. These features were observed in the experimental results in figure 7. This can be explained by the effects of the phase boundary position on the early potential transient. The further the boundary is away from the electrode surface, the later the effect of the phase boundary and phase changes are reflected in the surface lithium ion concentration. Therefore, subsequent pulses of a moving boundary reaches a higher potential before flattening.





On the other hand, from the single phase model, a small overpotential is seen at the beginning, followed by a sloping potential transient at later times. Therefore, the phase change model predicts the GITT results more accurately than the single phase model.

Figure 8. Calculated concentration transients during the charging phase of the GITT experiment using the single phase model.

Figure 9. Calculated concentration transient during the charging phase of the GITT experiment using the phase transformation model. The four curves correspond to different starting phase boundary locations ξ0: 0.2 (solid), 0.4 (dot), 0.6 (dash), and 0.8 (dash-dot).





Conclusions

In this work, tin thin films were characterized using the galvanostatic intermittent titration technique. The lithium ion diffusion coefficient was first extracted from the resulting data under single phase assumptions. Large drops in diffusion coefficients were observed at potentials close to the phase transformation potentials of tin. This can be explained by the phase change that occurs at these potentials. A moving boundary model was numerically computed to explain the shape of the potential transients during the galvanostatic pulses. This model can be used to more accurately model the galvanostatic intermittent titration experiments.

Acknowledgments

The authors gratefully acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada.

References

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Cao, Solid State Ionics, 181, 1611 (2010). 4. W. Jost, Diffusion in Solids, Liquids, Gases, Academic, New York (1960). 5. J. Yang, M. Winter, and J.O. Besenhard, Solid State Ionics, 90, 281 (1996). 6. M. Winter and J. O. Besenhard, Electochimica Acta, 45, 31 (1999).





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Moving Boundary and Non Dimensional Modeling and Gitt With Phase Transformations