Letter pubs.acs.org/JPCL
Kinetic Monte Carlo Study of Ambipolar Lithium Ion and Electron− Polaron Diffusion into Nanostructured TiO2 Jianguo Yu,† Maria L. Sushko,* Sebastien Kerisit, Kevin M. Rosso,* and Jun Liu Pacific Northwest National Laboratory, Richland, Washington 99352, United States S Supporting Information *
ABSTRACT: Nanostructured titania (TiO2) polymorphs have proved to be promising electrode materials for next-generation lithium ion batteries. However, there is still a lack of understanding of the fundamental microscopic processes that control charge transport in these materials. Here, we present microscopic simulations of the collective dynamics of lithium ion (Li+) and chargecompensating electron−polarons (e−) in rutile TiO2 nanoparticles in contact with an idealized conductive matrix and electrolyte. Kinetic Monte Carlo simulations are used, parametrized by molecular-dynamics-based predictions of activation energy barriers for Li+ and e− diffusion. Simulations reveal the central role of electrostatic coupling between Li+ and e− on their collective drift diffusion at the nanoscale. They also demonstrate that a high contact area between the conductive matrix and rutile nanoparticles leads to undesirable coupling-induced surface saturation effects during Li+ insertion, which limits the overall capacity and conductivity of the material. These results help provide guidelines for design of nanostructured electrode materials with improved electrochemical performance. SECTION: Energy Conversion and Storage; Energy and Charge Transport
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Among more sophisticated KMC methods are the so-called self-learning techniques, which use on-the-fly identification and evaluation of activation energy barriers.31 These methods can be tailored to account for, for example, significant elastic transformations in the material induced by transport of vacancies or ion insertion and would be most efficient for studying electrode stability upon a charge−discharge process. In this study, we use KMC to address the problem of coupled electron and ion transport into rutile TiO2 nanoparticles in variable contact with an idealized conductive matrix (e.g., carbon) as the electron source and with idealized electrolyte as the Li+ source. We examine Li ion concentration ranges inside of the nanoparticles below maximum capacity, where Li+induced phase transformations between bulk titania polymorphs do not occur.32 Hence, a standard KMC model is adopted, in this case, informed by previously reported molecular dynamics free-energy simulations of elementary Li+ and electron−polaron (e−) transport kinetics.33−40 Using this model, we show that the mechanism of Li+ and e− drift diffusion in the nanoparticles under constant current conditions depends on the extent of contact with the conductive matrix due to the change in the nature of Li+/e− coupling from predominantly at the surface to predominantly inside of the nanoparticle. Coupling at the surface prevents achieving high concentrations of ions and electrons in the nanoparticle and thus is an undesirable effect for electrode performance. To prevent surface coupling and, therefore, increase electrode capacity, a small contact area with the carbon matrix is required.
anostructuring of electrode materials has been proven to significantly enhance electrode performance, as manifested in higher conductivity, power, and stability.1−12 Introducing a high density of grain boundaries in the system changes the fundamental mechanism of ion and/or electron transport compared to that in the corresponding bulk material. For example, charge separation to the boundaries becomes an important process that can compete with otherwise highly correlated ion and electron diffusion.13−21 The impact of effects such as these strongly depends on the size of the grains and the structure of the boundaries between them, in ways that cross length scales from that of individual grains upward to grain networks possessing various possible architectures. To take better advantage of increasing material conductivity through nanostructuring, a fundamental understanding of transport in these complex systems is required. This poses a challenge for theory because one must consider nonperiodic nanoscale systems while retaining atomistic information on elementary ion and electron drift rates including their mutual interaction. Kinetic Monte Carlo (KMC) is one of the more efficient scale-bridging methods, which we adopt here for this charge-transport problem. KMC approaches for modeling ion and electron transport can be roughly divided into two categories, (1) those that rely on empirical data for diffusion coefficients and diffusion rates22−24 and (2) those in which these properties are calculated using finer-scale molecular dynamics or quantum mechanics techniques.25−29 A major implementation challenge of KMC methods is that unless a complete catalogue of all possible events is created, the simulation will never realize all possible processes, which limits the predictive value. To overcome this challenge, several transition-state search algorithms have been proposed.30 © 2012 American Chemical Society
Received: May 3, 2012 Accepted: July 20, 2012 Published: July 21, 2012 2076
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surrounding electrolyte by diffusion through the carbon layer without hindrance. The UCC setup represents electrode materials consisting of nanoparticles fully coated with thin carbon films.45 In our model, both the carbon matrix and electrolyte are implicitly represented for simplicity. Figure 2 shows KMC dynamics snapshots of charge insertion into a ∼9 nm rutile grain (20 × 20 × 30 unit cells) with carbon coating localized on one of the (001) surfaces. In the early stage of the charging process, that is, for simulation times up to 0.08 ns, both Li+ and e− are far apart, and the Li+/e− coupling is weak. Subsequent Li+ and e− insertion and diffusion inside of the particle leads to an increase in the electrostatic interaction between them and, therefore, to an increase in the degree of correlation of their diffusion. For comparison, we also carried out simulations of hypothetical cases without explicit Li+/e− coupling for insertion of Li+ only or e− only (Figure 3). In the case where only lithium ions are present, the occupation ratio curves for Li+ are the same regardless of whether the carbon coating is localized or uniform because it is assumed that the carbon coating does not affect the injection of Li+ into the grain. During the early stage of the charging process, the occupation ratios of either Li+ or e− show no significant dependence on the extent of carbon coating and on whether explicit Li+/e− coupling is taken into account. This corresponds to the early stage of the charging process whereby Li+ and e− are each dilute in the solid and the Li+/e− coupling is negligible. This is followed by a stage in which charge diffusion is fast without explicit Li+/e− coupling and slow when electrostatic coupling is introduced, regardless of whether the carbon coating is localized or uniform. In this stage, Li+ and e− are no longer isolated as more charge carriers are injected into the grain, and the charge coupling effect becomes important. Such charge coupling will slow down the diffusion of both carriers from the surface into the grain. It leads to surface saturation, as indicated by the sharp breaking of the curves (Figure 3) and the occupation map of the Li+ sites on the nanoparticle surface (last panel in Figure 2). In this regime, the charge coupling effect decreases Li+ and e− mobility. This decrease in diffusion rate is more pronounced for Li ions, which have approximately 10 times lower attempt frequency for their diffusion than electrons (Table 1). When diffusion of ions and electrons is correlated, in addition to the effective attempt frequencies becoming dominated by the slowest process, the mobilities also become dependent on the simultaneous availability of the unoccupied sites for both species. This explains the larger effect of correlation for larger occupation rates (Figure 3). Finally, the last stage of the charging process is charge saturation as no more unoccupied sites are available. We note that in our model, the interaction distance for Li+/ − e coupling is limited to the second nearest neighbors. The inclusion of long-range Coulomb interactions between charge carriers may lead to a net increase/decrease in the thermal activation energy barrier for electrical or ionic transport and hence may lead to a pronounced change in the carrier mobility.33 As a result, an undesirable buildup of charges will oppose the influx of new carriers. However, because rutile is a high dielectric constant material (ε ≈ 170), the screening effects would significantly reduce the long-range Coulomb interactions. Moreover, additional screening of the electrostatic interactions between neighboring Li+ arises from Li-induced lattice deformations,35 rendering the effective Coulomb interaction short-ranged and justifying the use of the shortrange explicit Li+/e− coupling in our model. For the remainder
To get a deeper understanding of the mechanism of charge transport at the nanoscale, we have studied the effects of the explicit Li+/e− coupling, the interfacial contact area with the carbon matrix, and the charge injection flux on the diffusion of charge carriers into rutile grains. To investigate the influence of the interfacial contact area between the rutile grain and the conductive “carbon” matrix, two simplistic models were considered (Figure 1). (a) The first
Figure 1. Schematic models of ambipolar diffusion into grains. (a) Localized/partial carbon coating (LCC): carbon is coating one of the (001) surfaces of the grain to simulate point contact electron supply. Li+ ions are uniformly supplied to all other faces from the surrounding electrolyte. (b) Uniform/full carbon coating (UCC): carbon is also uniformly coating all other faces of the grain to simulate uniform electron supply. Li+ ions are uniformly supplied to all other faces from the surrounding electrolyte by diffusion through the carbon layer.
is localized/partial carbon coating (LCC), in which carbon is coating one face of the grain perpendicular to the [001] direction (i.e., along the fastest charge carrier diffusion channel)41−43 to simulate point contact electron supply. Li+ ions are uniformly supplied to all other facets from the surrounding idealized electrolyte. This corresponds to layered materials, in which single layers of nanoparticles are sandwiched between two graphene sheets.44 (b) The second is uniform/full carbon coating (UCC), in which carbon coats all faces of the grain to simulate uniform electron supply. Li+ ions are supplied to all facets, except the one perpendicular to the [001] direction at the top of the nanoparticle (Figure 1b), from the 2077
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Figure 2. Visualization of the dynamics of charge insertion into a rutile nanorod (20 × 20 × 30 unit cells; grain size ≈ 10 nm) with localized carbon coating. Ions are colored based on charge (Li+, green; Ti3+, red; Ti4+, brown; O2−, blue). Carbon (gray color) is coating one of the [001] surfaces of the grain.
localized and uniform carbon coatings (Figure 4). It was found that in 9 nm grains, e− diffusion is fast for large carbon contact
Figure 4. Time evolution of the Li+ and e− occupation rate in the nanoparticles with localized and uniform carbon coatings. The charge injection rate is 2 × 1010 per site per second, and the grain size is about 10 nm.
areas and Li+ diffusion is fast for small contact areas. Similar results were also obtained with different grain sizes. The increase in the diffusion rate of electrons for full carbon coating is associated with the increase in the effective electron injection area from one face of the nanoparticle to the whole surface. On the other hand, in both models, Li+ ions are injected through all faces except the one perpendicular to the [001] direction. The drop in the Li+ occupation rate in the case of full carbon coating is the effect of correlation between ion and electron diffusion in the nanoparticle. With full carbon coating, electrons are entering titania from all directions, which significantly increases the overall electron injection rate and decreases the effective distance between Li+ and electrons at the surface of the nanoparticle. This leads to rapid saturation of the surface sites with e− and Li+ (to balance the surface charge) followed by slow strongly correlated diffusion of ions and electrons inside of the nanoparticle. This explains the overall slow increase in the ion and electron occupation rates, which do not reach saturation over the length of the simulations (Figure 4). In contrast, in the case of partial carbon coating, surface effects are less prominent because ions and electrons are entering the nanoparticle from different facets. Then, the correlation becomes significant only when both charged species are well inside of the nanoparticle and the occupation rate reaches 0.5 (Figure 4). Therefore, the main difference in the mechanism of ion and electron transport in titania nanoparticles with different degrees of carbon coating is in the onset of the correlated flux
Figure 3. Time evolution of the Li+ and e− occupation rate with and without explicit electrostatic coupling. The charge injection rate is 2 × 1010 per site per second, and the grain size is about 5 nm.
Table 1. Simulation Parameters parameter
electron polaron −1
attempt frequency (s ) activation energy (eV)a [111] [001] [001]b with e− [001]b with Li+
lithium ion
2.4 × 10
2.9 × 1012c
0.31d 0.09d 0.081/0.095f 0.163/0.059c
0.8e 0.05e 0.287/0.027c 0.041/0.055f
13c
a
Direction-dependent. bSeparating/approaching. cReference 33. dReference 47. eReference. 35. fEstimated by adding the contribution from the Coulomb interaction.
of this Letter, we will focus on the results obtained with Li+/e− coupling implicitly included, unless stated otherwise. We investigated the influence of the interfacial contact area with the carbon matrix by modeling Li+/e− diffusion with 2078
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coupled lithium/electron−polaron diffusion into nanostructured TiO2 and applied it to the case of rutile. It was found that due to electrostatic coupling between Li+ and e−, the total conductivity is determined by the slowest process, that is, by Li+ diffusion. The coupling effect is also responsible for changing the mechanism of charge transport with the contact area between the titania nanoparticle and carbon coating. For full carbon coating, the electrostatic coupling leads to the saturation of surface sites with ions and electrons, which limits the overall capacity of the materials. Localized carbon coating, on the other hand, allows for ions and electrons to diffuse well inside of the nanoparticle before the onset of the strongly correlated flux and, therefore, is beneficial for higher capacity and conductivity of the nanostructured electrode material. These findings can be expected to be qualitatively relevant to other polymorphs of titania, such as anatase, and the model can be easily adapted to other such materials in future work. The simulation results could be also of significant importance for the design of nanostructured electrodes with other promising oxide materials for next-generation electrochemical energy storage devices with high power and high energy densities.
either at the surface (full coating) or in the bulk (partial coating). This result could be of significant importance for the preparation of nanostructured electrodes. When the ionic conductivity is relatively low compared to the electronic conductivity, a small carbon coating is sufficient. This is consistent with the results reported for LiFePO4 cathodes,46 in which the ionic conductivity is the dominant factor affecting their charge/discharge performance. To explore the influence of the charge injection rate, we have performed simulations in which the charge injection rate was varied by 8 orders of magnitude, from 2 × 104 to 2 × 1012 per site per second. Figure 5 shows the electron and lithium
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COMPUTATIONAL METHODS The simulations were performed at room temperature (300 K) with the simulation cell sizes ranging from 10 × 10 × 15 (12000 ions) to 60 × 60 × 90 unit cells (2592000 ions) to simulate single grain sizes from about 5 to 30 nm. In some simulations, periodic boundary conditions (PBCs) were imposed in the [010] direction, which corresponds to an infinite rutile nanorod. Because the energy barriers for Li+ and electron diffusion are very high in the [010] direction, test simulations showed that there was no significant difference between results obtained with and without PBCs applied along the [010] direction. There are four elementary processes in the KMC model, (1) hopping of Li+ to one of the nearest empty interstitial sites along the [001] direction or to a next-nearest empty Li site along the [111] direction, (2) hopping of electron−polarons from a Ti3+ site to a nearest-neighbor Ti4+ site along the [001] direction or to a next-nearest-neighbor Ti4+ site along the [111] direction, (3) random addition of a lithium ion per site and/or per second from the electrolyte to the surface of the rutile grain, and (4) addition of an electron−polaron to a surface Ti4+ site. Unless otherwise stated, Li+ and electrons were added proportionally so as to maintain charge neutrality in the solid. Every process j is associated with a given rate h(j), either an injection rate h(j) = Fi (i = e−/Li+ and F is the flux) or a hopping rate. The hopping rate of a given process j is described by an Arrhenius equation
Figure 5. Flux (injection rate) effect. The charge injection rate varies from 2 × 1010 (10) to 2 × 108 (8) to 2 × 106 (6) per site per second with Li+/e− coupling and contacting either LCC or UCC. The grain size is about 5 nm.
occupation profiles for charge injection rates of 2 × 106, 2 × 108, and 2 × 1010 per site per second for both localized and uniform carbon coatings. Our results show that for charge rates higher or equal to 2 × 108, Li+ saturation is reached at 1 μs. For the electrons, the saturation time depends on the degree of carbon coating and is about 0.1 μs for the uniform and 1 μs for the localized carbon coating cases, respectively (Figure 5). As discussed above this difference is related to the difference in the mechanism of charge injection. Nevertheless, the rate of charging in both cases is largely determined by the slowest process, which is Li+ diffusion in the nanoparticle. This effect indicates that there is a certain limit for the maximum charging time for an electrode consisting of carbon-coated titania nanoparticles, which is mainly controlled by Li+ mobility inside of the material rather than by the injection rate. In summary, we have developed a kinetic Monte Carlo model to simulate the dynamics of lithium ion insertion and
⎛ −E ⎞ h(j) = ν0 exp⎜ a ⎟ ⎝ kBT ⎠
(1)
where Ea is the activation energy, kB is the Boltzmann constant, T the temperature, and ν0 the effective attempt frequency. The activation energy Ea for diffusion is direction-dependent and is equal to 0.31 and 0.8 eV for jumps in the [111] direction and 0.09 and 0.05 eV for jumps in the [001] direction,35,47 for electrons and Li+, respectively (Table 1). The effective attempt frequency ν0 along the two jump directions is taken to be equal to 2.4 × 1013 and 2.78 × 1012 s−1 for e− and Li+, respectively, as calculated using molecular dynamics and a two-state electrontransfer model.33 When high charge densities are simulated, the 2079
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electrons and Li+ can no longer be considered as noninteracting carriers, and their activation energies Ea become coupled.33 This is done via modification of the activation energies for Li+ and e− hopping to and from the sites next to the sites occupied by e− or Li+, respectively (Table 1). For simplicity, the model neglects surface effects and only considers the energy barriers calculated for the corresponding bulk materials. It is also assumed that the charge injection rates into the grain are constant and are set to 2 × 1010 per site per second for both e− and Li+, unless specified otherwise. Release of charge carriers from the surface back to sources is neglected. In the KMC simulations, every process is executed with a probability proportional to its relative rate. In particular, every process j has a relative probability w(j) = h(j)/R, where R is the total rate given by ⎛ −E m ⎞ R = ∑ h(j) = ∑ Ni*Fi + ∑ ν0 exp⎜ a ⎟ ⎝ kBT ⎠ i=1 m=1 2
p−1
*Tel: +1 509 371 7286. E-mail:
[email protected] (M.L.S.); Tel: +1 509 371 6357. E-mail: kevin.rosso@pnnl. gov (K.M.R.). Present Address †
Idaho National Laboratory, P.O. Box 1625, MS 3560, Idaho Falls, ID 83415 U.S.A. Author Contributions
All authors have contributed and given approval to this manuscript Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Laboratory-Directed Research and Development Program at Pacific Northwest National Laboratory (PNNL) under the Transformational Materials Science Initiative. PNNL is a multiprogram national laboratory operated for DOE by Battelle under Contract DE-AC0576RL01830.
Nh
(2)
j=0
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(3)
Step 5: Execute process p. Step 6: Update the simulation time by t = t − ln(r2)/R. Step 7: Go to step 1 and repeat 5 × 107 times. The current KMC model, while giving important insights into the ambipolar Li+ and e− diffusion into titania nanoparticles, suffers from several limitations. The model relies on the data for the activation energies and attempt frequencies for ion and electron diffusion in bulk material. These parameters are sensitive to the local environment around the equilibrium sites for Li+ and, therefore, might vary across the nanoparticle, depending on the particle shape and on the structure of the boundary between the grain and the carbon coating. These local variations in Li+ and e− mobilities may change the quantitative result and provide some additional pathways for charge transport in the nanoparticles. However, the dominant factors for the qualitative mechanism of charge transport and the effects of carbon coating and the injection rate are the 10fold difference in the attempt frequencies for Li+ and e− diffusion and the electrostatic interactions between the charged species. Therefore, the qualitative results of our KMC model will not be affected by a more rigorous representation of the elementary charge-transport events. See the Supporting Information for further discussion of the limitations of this approach.
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p
h(j) h(j) ≤ r1
