Multiscale Simulations of Li Ion Conductivity in Solid Electrolyte - The

LETTER pubs.acs.org/JPCL

Multiscale Simulations of Li Ion Conductivity in Solid Electrolyte Maria L. Sushko,* Kevin M. Rosso, Ji-Guang (Jason) Zhang, and Jun Liu Pacific Northwest National Laboratory, Richland, Washington 99352, United States ABSTRACT: Optimizing solid electrolyte design for its application in Li ion and Li metal batteries requires a fundamental understanding of the mechanism of ion and electron transport in the material at the nano- to micrometer scales. We have performed simulations of Li+ and electron conductivity in lithium phosphorus oxynitride, one of the most widely used solid electrolytes, using novel hierarchical multiscale models. By comparing the results of one- and three-dimensional models, we show that for this material with complex nonlinear Li+ diffusion pathways, three-dimensional description is essential for reproducing experimentally measured conductivity. We also suggest some basic principles to design optimum electrolyte tailored for low- and high-temperature regimes. SECTION: Energy Conversion and Storage

S

caling down battery dimensions and nanostructuring of electrode materials lead to complex multiscale electrochemical phenomena. There is a significant theoretical effort in multiscale modeling at the electrode-to-battery level, and several models linking these two length scales have been proposed.1
7 On the other side of the spectrum, atomistic quantum mechanical and molecular dynamics simulations were employed to study elementary ion and electron transport in bulk energy storage materials.8
10 However, very little is known about the charge transport at the mesoscale, that is, at the length scale of individual grains in nanostructured materials or in thin films with a thickness in the nanometer range. Recently, we have proposed a multiscale model that connects atomistic and mesoscales11,12 and provides a missing link for multiscale modeling of battery performance from first principles.13 Our hierarchical hybrid multiscale simulation technique for modeling coupled ion and electron transport in nanostructured energy storage materials uses a multiphysics approach. In this approach, instead of formal consecutive upscaling, we introduce novel types of collective long-range interactions along with short-range effects of the finer-scale models. The fine scale-model uses the results of quantum mechanical simulations of elementary charge transport. The collective long-range electrostatic and excluded volume interactions are introduced on the mesoscale (10
300 nm) via classical density functional theory coupled with the Poisson
Nernst
Planck formalism for dynamic effects. Here, we extend this approach from the charge-transport model in one-dimensional (1D) channels to a three-dimensional (3D) model of ion and electron conductivity in nanoparticles or nanofilms and apply the model to gain fundamental understanding of temperature-dependent conductivity in solid electrolyte materials. In particular, we focus on understanding the electrochemical performance of lithium phosphorus oxynitride (LixPOyNz, or LiPON),14
17 one of the most widely used solid electrolytes for thin-film batteries.18 r 2011 American Chemical Society

Atomistic simulations of Li+ transport in LiPON revealed several pathways for Li+ diffusion. Li+ may either diffuse via a vacancy diffusion mechanism or through interstitial channels.19 The latter mechanism was found to be more efficient and incurs energy barriers as low as 0.2 eV, as opposed to 0.6 eV for the vacancy migration mechanism. To determine the validity of 1D and 3D models for the description of Li+ transport in LiPON, we have performed simulations of the temperature and nanoparticle size dependence on Li+ conductivity. In the 1D model, Li+ is restricted to hop between the I0 interstitial sites, along the b crystallographic direction, while in the 3D model, it can explore other paths, which include several intermediate sites. 1D Diffusion. Direct hopping between I0 sites (Figure 1) leads to relatively low conductivity on the order of 10
10 S/cm for nanochannels under 25 nm in length. The low conductivity reflects the relatively high energy barrier (0.21 eV) for direct Li+ hops between I0 sites and the large distance between these sites (0.41 nm),19 which decreases the probability of ion hopping. The conductivity increases with temperature mainly due to thermally induced increases in the probability of hopping (Figure 2). The increase in conductivity is more rapid in smaller nanochannels. Moreover, the conductivity at a given temperature for shorter nanochannels is higher than that for longer ones. This result is consistent with our previous 1D simulations of the conductivity in rutile and anatase TiO2 nanoparticles.12 We have shown previously that for nanoparticles shorter than the Debye screening length in the material, most ions (and electrons if present) are involved in constant flow, while for larger nanoparticles, a fraction of the mobile charge carriers accumulates at the boundary forming a space-charge zone and are not available for the flow. Received: July 29, 2011 Accepted: August 26, 2011 Published: August 26, 2011 2352

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Figure 1. Arrays of interstitial equilibrium (I0) and metastable (II0, II*) sites for Li+ diffusion in LiPON. Sites I0 are shown as large blue spheres, II0 as small yellow spheres, and II* as small gray spheres.

Figure 2. Li+ conductivity in LiPON nanoparticles along the b direction in a 1D model. Channel lengths are 12.2 (1), 18.3 (2), and 24.4 nm (3).

This effect reduces the conductivity of larger nanoparticles. For the 0.001 M Li+ in LiPON, the Debye length varies from 14.0 nm at 213 K to 19.6 nm at 298 K. We also observe an increase in Li+ concentration at the boundary with an increase in LiPON channel length from 12.2 to 24.4 nm. However, due to the relatively low Li+ hopping rate in 1D LiPON nanochannels, the concentration of ions in the space-charge zone remains small for all channel lengths. Therefore, the fundamental mechanism of conductivity does not change within the bounds of the channel lengths studied, and the decrease in the conductivity at a given temperature is just the consequence of a smaller effective concentration of mobile ions available for constant flux in longer channels. 3D Diffusion. In a 3D model, apart from sites I0, intermediate sites II0 and II* were also introduced (Figure 1). According to quantum mechanical DFT simulations, the most energetically favorable path for Li+ diffusion is a zigzag path via the II0 and II* sites.19 Our mesoscopic simulation showed that diffusion along the b direction is strongly preferred over the diffusion along a and c directions, in good agreement with atomistic data.19 Moreover, Li+ current is much higher at the border of the nanoparticle and gradually decreases toward the center, as evidenced by the Li+ flux in the b direction mapped in the (a,c) plane (Figure 3).

LETTER

Figure 3. Li+ flux in a LiPON nanoparticle with dimensions 2a  15b  5c or 2.106 nm 9.18 nm 2.456 nm at 298 K. The flux distribution with respect to the (a  c) plane is shown.

Figure 4. Li+ conductivity in LiPON nanoparticles along the b direction in a 3D model. Particle sizes in the b direction are 3.06 (1), 6.12 (2), 9.18 (3), 12.24 (4), 61.2 (5), 122.4 (6), and 244.8 nm (7).

The conductivity in this model is 4 orders of magnitude higher than that in the 1D model and similar to the conductivity of 3.0  10
7 S/cm observed experimentally20 (Figure 4). This indicates that the constraints to a 1D channel are not valid for this material with essentially nonlinear character of Li+ diffusion. To gain a deeper understanding of the mechanism of charge transport in LiPON, we analyzed the temperature evolution of various components of Li+ free energy (Figure 5). The temperature dependences of the electrostatic correlation, hard sphere repulsion, and Coulomb repulsion between Li ions are monotonic for all nanoparticle sizes studied. Electrostatic correlation becomes more attractive (less positive) with decreasing temperature, leading to a more coherent Li+ flow through the material. At the same time, both the excluded volume and Coulomb repulsion between flowing Li ions decreases with decreasing temperature (Figure 5), allowing for a higher net concentration of Li+ to pass through a unit area. Interestingly, the short-range interactions, which reflect the effective barriers for the elementary ion transport, have highly nonlinear temperature dependence for the 100b nanoparticles with a minimum at around 278 K. This nonlinearity reflects the decrease in energy cost for elementary ion transport due to the increase in electrostatic correlation interactions at lower temperatures, while at higher temperatures, thermal energy becomes comparable with 2353

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LETTER

Figure 5. Temperature dependence of the contributions to the Li+ free energy in LiPON nanoparticles, obtained in a 3D model: electrostatic correlation (a), short-range (b). The particle sizes in the b direction are 6.12 (empty circles) and 61.2 nm (filled circles).

the barrier heights for the elementary ion transport, which decreases the effective barrier and lowers the short-range energy. The latter effect may dominate over the thermal distortion of the correlated ion flow in some materials at high temperatures, leading to an increase in the conductivity. For example, in our previous study of Li+ and e
transport in TiO2 nanoparticle, the conductivity in the temperature range of 298
1500 K was found to decrease with temperature in small rutile nanoparticles, while it increased in larger nanoparticles.12 In LiPON, the transition to increasing temperature dependence of the conductivity occurs for nanoparticle sizes above 100b. For these nanoparticle sizes, the conductivity is almost independent of temperature below 273 K and strongly increases with higher temperatures, indicating, as mentioned above, thermally induced decreases in the effective barrier height for elementary charge transport and short-range free energy. In contrast, for smaller nanoparticles, the effect of temperature on short-range free energy is very small. This component of the free energy monotonically decreases with temperature increases from 223 to 298 K by approximately 2  10
8 eV, reflecting a small increase in the residence time of Li ions at their equilibrium sites at lower temperatures. However, this effect is negligible compared to the much larger changes in electrostatic correlation and Coulomb free energies, which determine the changes in the overall Li+ conductivity. In summary, our results show that because of the essentially nonlinear conductivity pathway for this material, a 3D model should be used as opposed to a 1D model, which confines the conductivity path to direct hopping between the equilibrium interstitial states along the b direction. We have shown that the conductivity in LiPON strongly depends on temperature and on the size of nanoparticles/film thickness. The increase in conductivity at lower temperatures was found to be due to the increase in electrostatic correlation interactions between Li+, which promotes the directed flow of ions through the material. The size of the nanoparticles/film thickness determines the effective concentration of Li ions available for the constant flow in this material. In particular, a LiPON film with a thickness smaller than 10 nm would provide the highest conductivity. These findings can guide the design of solid electrolyte films for low- and high-temperature battery applications.

the following system of equations "


Ji ¼ Di ðzÞAðzÞ

dFi 1 dϕ dμ0 dμex þ Fi ðzÞ qi e þ i þ i dz dz dz kT dz

!#

dJi ¼0 dz


ð1Þ

ð2Þ

  1 d dϕ ε ðzÞAðzÞ ¼e AðzÞ dz dz

∑i qi Fi ðzÞ

ð3Þ

In these equations, Ji are the fluxes for Li+ and electrons, Di(z) and Fi(z) are their diffusion coefficients and densities along the channel (z axis) in a 1D case, respectively, A(z) is the cross section of the channel, ϕ is the electrostatic potential, μ0 and μex are the ideal and excess chemical potential of Li+ and electrons, respectively, kT is the thermal energy, and e is the electron charge. In this system of equations, the first describes the flux, the second is the stationary condition, and the third is Poisson’s equation for the calculation of the electrostatic potential. We use the classical density functional theory for evaluation of the chemical potentials of charged species. In this model, the total free energy is divided into two parts, the ideal part (Fid), which includes the contributions from the configurational entropy of the noninteracting species and bonding enthalpy, if any, and the excess free energy, which has contributions from all interactions in the system. In the case of charged species in the channel, these include the free energies of Coulomb interactions, electrostatic correlations, hard sphere repulsion, and short-range interactions with the stationary points ex þ F ex þ F ex F ex ¼ FCex þ Fel hs sh

ð4Þ

These free energies are calculated as follows (see ref 21 for more details) F id ¼ kT

’ THEORETICAL MODEL Within the Poisson
Nernst
Planck formalism, the flux of charged particles in stationary conditions can be calculated using

FCex ¼ 2354

∑ α ¼þ,


e2 2ε

Z dB r F αð B r Þ½ln F α ð B r Þ
1

∑

i, j ¼ B, þ ,


ZZ

qi qj F i ð B r ÞF j ð B r 0Þ dB rdB r0 jB r
B r 0j

ð5Þ

ð6Þ

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Z ð1Þel ex ¼ F ex ½fFbulk g
kT d r Fel ΔC α ðF α ð B rÞ α B el α ¼ þ,
ZZ kT ð2Þel dB r dB r0
Fbulk ΔCij ðj B r
B r 0 jÞðFi ð B rÞ α Þ
2 i, j ¼ þ ,


∑

Division of Materials Sciences and Engineering under Award KC020105-FWP12152. PNNL is a multiprogram national laboratory operated for DOE by Battelle under Contract DEAC05-76RL01830.

∑

ð7Þ

r 0 Þ
Fbulk Þ
Fibulk ÞðFj ð B j

Fex hs

can be expressed as an integral of the functional of weighted densities (nω(r B)) using the Fundamental Measure Theory22 Z ex ¼ kT Φ ½n ð r Þ d r Fhs ð8Þ B hs ω B The short-range interactions between Li+ and the stationary points (denoted as “s”) are given by ZZ 1 ex dB r dB r0 Fi ð B r Þpj ð B r 0 Þϕij ðj B r
B r 0 jÞ Fsh ¼ 2 i, j ¼ þ , s

∑

ð9Þ where the potential ϕij is the square-well potential with the depth ξij 8 > r γσ þ There are no short-range interactions acting between Li+ ions and electrons. In some calculations, we have also introduced the stationary points for electrons. However, we found that the results do not depend on whether this additional interaction is introduced. This finding is consistent with reported data that electron diffusion is coupled to the Li+ diffusion via electrostatic interactions.23 Therefore, the short-range interactions for the electrons are not present in all calculations reported here. The experimental value of 16.6 for the dielectric constant of LiPON was used.24 The chemical potentials were evaluated analytically as the functional derivatives of the free energy over the densities of the mobile species. The system of eqs 1
3 was solved numerically using Newton’s method for a 1D system and using successive overrelaxation (SOR)25 for the 3D model. We used a uniform grid of points separated by the distance of σ+/10. The convergence criterion was a 10
6 decrease in the difference between the next solution of the system of equations and the previous one. We used generous particle dimensions in all three crystallographic directions (2a  Nb  5c, where N = 5
400), rendering the results equally applicable to thin films and nanoparticles.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The development of the PNP-cDFT software was supported by the Laboratory-Directed Research and Development Program at Pacific Northwest National Laboratory (PNNL) under the Transformational Materials Science Initiative. The study of charge transport in LiPON nanoparticles was supported by the U.S. Department of Energy, Office of Basic Energy Sciences,

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