Effect of Concentration on the Energetics and Dynamics of Li Ion

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Effect of Concentration on the Energetics and Dynamics of Li Ion Transport in Anatase and Amorphous TiO2 Handan Yildirim, Jeffrey Greeley,* and Subramanian K. R. S. Sankaranarayanan* Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, United States ABSTRACT: We report on the energetics and dynamics of Li diffusion in bulk anatase and amorphous TiO2 using molecular dynamics (MD) simulations and density functional theory (DFT) calculations. Using MD simulations, for both anatase and amorphous TiO2, diffusion characteristics are first studied for an isolated Li ion, followed by simulations of Li concentrations ranging from 10% to 100% in order to explore the concentration effect on the diffusivity. The isolated Li diffusion mechanism, revealed from the MD simulations, occurs via zigzag hops between the octahedral sites in anatase. The corresponding barrier for this process obtained from DFT-NEB calculations is 480 meV. MD simulations also show that isolated Li ion diffusivity is much slower in the amorphous TiO2 than in anatase TiO2. DFT-NEB results for the diffusion in amorphous titania indicate that Li encounters deep energy wells within the amorphous network that are in the electronvolt range, confirming our MD observation of low Li diffusivity. A monotonic decrease in diffusion barriers with increasing Li concentration is observed in the case of amorphous titania whereas a non-monotonic variation is seen in anatase, with the lowest barrier observed at 50% Li concentration. At low Li concentrations (75% for amorphous titania, the Li diffusivity in amorphous is found to be much higher than in anatase. Our MD simulations suggest that the underlying reason for these differences is related to changes in diffusion mechanism. Our simulations therefore indicate a strong correlation between Li ion concentration and the observed transport characteristics, offering new insights into ion conduction mechanisms that are of importance to solid-state devices used for energy storage applications.

I. INTRODUCTION Transition metal oxides are among the group of materials that are promising as electrodes for high-energy-density batteries.1 One of the important requirements for these materials to be employed as electrodes is their ability to intercalate Li ions reversibly. Their open structures give reasonable Li capacities, and they also possess good electronic and ionic conductivity. TiO2 is one such oxide that has shown promise for battery applications,2
9 in addition to its traditional uses in solar cell applications10 and electrochromic devices.11 Anatase TiO2, in particular, has been extensively studied for battery applications, with microcrystalline TiO2 anatase used as an anode material for lithium rechargeable batteries.12 The maximum insertion ratio of Li via intercalation is reported to vary from x = 0.5 to 1 depending on the synthesis procedures and experimental conditions.13 The electrochemical experiments, though, have consistently reported x = 0.5 to be the maximum insertion ratio. Once Li ions are intercalated into TiO2 anatase, they proceed to vacant interstitial sites. It is reported that lithium insertion in anatase induces a spontaneous phase separation into lithium-poor (Li0.01TiO2) and lithium-rich (Li0.6 TiO2) domains.14 Another study reported that intercalation proceeds as a single-phase reaction up to x = 0.05
0.1, with further insertion producing a two-phase equilibrium of Li-poor and Li-rich regions. The Li-rich phase is reported to be an orthorhombic distortion of anatase with composition Li0.5TiO2.15 r 2011 American Chemical Society

Because of the significant interest in using titania for Li ion battery electrodes, diffusion of Li in titania has attracted a great deal of attention.16 Ion mobility is of prime importance for the electrochemical behavior of batteries and is directly related to the rate at which the batteries are charged and discharged. This information can be derived from the calculated diffusion coefficients. Questions of how Li ions diffuse in and out of the host lattice, the role of the host lattice morphology, diffusion mechanisms, and their contributions to Li ion diffusivity must be answered to gain further insights into the atomistic details of battery operation. Earlier studies, most of which involve static transition state searches, have attempted to answer some of these questions. However, there is also a need for dynamical studies that can offer insights into ion transport mechanisms, the temporal evolution of Li ion diffusion in the host structures, and the effect of temperature and other environmental factors on diffusion characteristics. Additionally, we note that the concentration dependence of Li ion diffusivity has not been thoroughly explored via a dynamical study, and little fundamental mechanistic understanding of this dependence exists. Differences in the preparation procedure for various titania samples, together with the way that the diffusion is probed Received: March 16, 2011 Revised: May 10, 2011 Published: July 14, 2011 15661

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The Journal of Physical Chemistry C experimentally, can lead to variations in the measured diffusion coefficients for Li ions. The Li diffusion coefficient in anatase is reported to vary significantly, within the range of 10
10
10
17 cm2/s.17
22 The macroscopic diffusion coefficient for Li in anatase obtained with chronoamperometry on chemical vapor deposited (CVD) films is reported to be 2  10
15 cm2/s for insertion and 6  10
15 cm2/s for extraction.20 Another study reported the diffusion coefficient for single-crystal anatase to be 2  10
13 cm2/s for insertion and 6  10
13 cm2/s for extraction.23 An NMR study by Wagemaker et al. reported the microscopic diffusion coefficients at room temperature to be 4.7  10
12 cm2/s in anatase and 1.3  10
11 cm2/s in lithium titanate.24 Diffusion activation energies from similar studies were reported to be 0.54 eV (insertion) and 0.78 eV (extraction)20 and 0.60 eV for diffusion.25 The diffusion barrier obtained from macroscopic intercalation data is also reported to be 0.5 eV.24 The diffusion mechanism of Li ions in anatase and in lithium titanate is reported to occur via jumps between the vacant octahedral sites.26 Atomistic simulations are essential to understand the differences in the reported diffusion coefficients and energy barriers and to elucidate the fundamental ion transport mechanisms. Computational studies for Li ion diffusivity in anatase have also been reported.25,27
31 Lunell et al.25 performed calculations by constraining Li ions to remain equidistant to four Ti atoms and reported the Li ion diffusion barrier corresponding to jump between two octahedral sites to be 0.60 eV for x = 0.5 with the unrestricted Hartree
Fock (UHF) method, while the barriers were reported to be 0.51 and 0.56 eV for x = 0.0625 and x = 0.5 with the semiempirical intermediate neglect of differential overlap (INDO) method. A DFT study30 performed with the pseudopotential approach for different Li concentrations showed the diffusion barrier to be ∼0.5 eV for x = 0.5, increasing to 1 eV at x = 0.75. Studies by Tielens et al.,28 however, reported Li ion barriers in anatase to decrease with increase in concentration for the same concentration range. Olson et al.27 also reported the barriers to range between 0.45 and 0.65 eV for x e 0.1. A recent MD study performed using the potential shell model has reported the diffusion barrier for an isolated Li ion to be 390 meV.29 This study also reported the isolated Li diffusion coefficient in anatase to be on the order of 10
10 cm2/s. It should be noted that diffusion of Li ions in amorphous TiO2 has not yet been studied. There seems to be a growing interest in utilizing the amorphous materials for Li ion battery applications,32
36 particularly TiO2.37
43 Investigating ionic diffusion in amorphous systems is not trivial, as disordered materials do not provide well-defined initial and final states for transition state searches with static calculations. However, by considering an ensemble of such states within the structure, one can gain at least qualitative insight into the potential energy surface of the amorphous network. In this study, ionic transport in anatase and amorphous TiO2 are investigated in detail using MD and DFT simulations. The diffusion characteristics of Li ions are systematically studied for both isolated Li ions and for Li concentrations ranging from 10% to 100%. Our aim is to gain atomic scale insight into the role of concentration, temperature, and morphology on Li ion diffusion characteristics. Our study will expand on the available understanding of Li ion diffusion characteristics in TiO2 anatase, and it will for the first time report the strong correlation that exists between Li ion concentration and its diffusion characteristics in amorphous TiO2. The atomistic simulations are used to make

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systematic comparisons that are aimed at exploring the influence of Li ion concentration on ionic transport in ordered vs disordered titania. The correlation between the ionic transport, concentration, and morphology is of great interest to researchers developing solid state devices for energy storage applications,32
43 and to the best of our knowledge, such information is currently unavailable. The paper is organized as follows. We first provide a detailed discussion of the simulation methodologies. This is followed by a discussion of the simulation results, which are divided into the three subsections. The first subsection is devoted to MD and DFT results of isolated Li ion diffusion in anatase and amorphous TiO2. In the second subsection, we report on the effect of concentration on Li ion diffusion characteristics in anatase. In the third subsection, we discuss the effect of concentration on diffusion characteristics in amorphous TiO2, and we provide a comparison of Li ion diffusivity in these two morphologies. The results are then summarized in the conclusions.

II. THEORETICAL DETAILS 2.1. MD Simulations. In this study, our interest lies in exploring the concentration dependence of Li ion transport in both ordered and disordered TiO2. The diffusion of Li ions is expected to be affected by contributions from the dynamics of its environment. MD simulations performed with well-parametrized potentials are excellent techniques for accurately describing these dynamical effects on diffusion.44 The first step in our MD simulations is to construct a TiO2 anatase bulk computational cell. The basic unit cell of TiO2 anatase contains four titanium and eight oxygen atoms. In order to generate the input crystal structure for the MD simulations (consisting of 4320 atoms), we repeat the unit cell by factors of 6,6,6 along the a, b, and c axis, respectively (see the structure in Figure 1a). The MD simulations are performed using the DLPOLY-MD package.45 Our calculations make use of the shell potential model46 in which the polarizable ion consists of two particles, core and shell, that share the ion’s charge, and are connected via a spring constant. Oxygen atoms are treated as being made of a core and a shell, and core
shell is linked by a harmonic spring, k, and interacts via the core
shell interaction potential Uc-s = k  rc-s2 in which rc-s is the core
shell separation distance. Oxygen core
shell spring constants, k, are set to 44.3 eV Å
2. The shells are adiabatically in motion with the core; the mass is assigned to the core, while the charge is assigned to the shell. Assigning a small mass47 of 0.2 au to the shells ensures that there will not be any exchange of energy between vibrations of the oxygen core and shell and the vibrations of Li ions. Using a small mass for the shell, however, requires the calculations to be performed with a time step as small as 0.2 fs48 in order to keep the total energy stable. The atoms are treated as point particles that interact via both long-range Coulomb forces and short-range interactions. The short-range interactions are represented by the Buckingham potential, which is given by parametrized functions. Buckingham potential parameters are described in Table 1. The potential refers to the electron
charge cloud repulsion and van der Waals attraction forces. Recently, this potential was applied to study the isolated Li ion diffusion in TiO2 anatase and rutile.29 Note that there is no specific potential available for amorphous TiO2, but a MD study49 reported that the Buckingham potential reproduces well the properties of amorphous TiO2 that are reported by an experimental study.50 Thus, for simulating 15662

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The Journal of Physical Chemistry C diffusion in amorphous TiO2, we have employed the same potential used for the anatase.29 The details of the shell model, and the parameters used for Buckingham potentials for Li in TiO2 polymorphs, are discussed in detail in a recent publication by Kerisit et al.29 Their study has shown the reliability of the potential in obtaining several properties of TiO2 bulk polymorphs, including phase transitions.51 2.2. Initial Configurations. In order to evaluate the effect of concentration on Li ion transport, we incorporate Li into the structures at randomly chosen sites. Electrochemical insertion of Li ions is reported to create reduced Ti ions (Ti3+).52 This is due to the fact that, when Li ions enter the TiO2 lattice, electrons get to the conduction band in order to compensate the charge of Li+ ions. These electrons reduce the Ti4+ to Ti3+. Thus, in our study, incorporation of Li into the TiO2 lattice is also followed by addition of the same number of Ti3+ ions (the potential properties of which are provided in ref 29). These Li-loaded structures are then subject to further energy minimization using the conjugate gradient method to allow the ions to reach their minimum energy sites. We evaluate the effect of morphology on Li ion diffusion characteristics by performing systematic comparisons between the diffusion in anatase and in amorphous TiO2. One of our main interests in this study is to gain an understanding of how Li diffusion characteristics differ in a disordered versus a crystalline structure. To explore diffusion in the amorphous TiO2, we obtain an amorphous structure by first starting with the anatase (see Figure 1a). As a first step, we melt the crystal by heating it to 7000 K. We performed the calculations within the Berendsen NPT ensemble to allow the system volume to be adjusted according to the chosen temperature. The melt obtained at 7000 K is then subjected to a cooling process to obtain an amorphous structure. The cooling procedure is performed at a rate of 2.5  1013 K/s. In the first stage, the system temperature is reduced to 3000 K, then to 1000 K, and finally, in the last stage, the temperature is reduced to 350 K. We further relax the structure obtained at 350 K for another 300 ps (NPT). The amorphous structure obtained from these simulations is shown in Figure 1b. The radial distribution functions (rdf) for each pair in the system, the bond lengths, and the bond angle distributions are analyzed. The results show the properties expected for an amorphous structure50—in particular, there is some degree of short-range order in the rdf’s, together with wide bond length and bond angle

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distributions compared to those for anatase. Note that the rdfs obtained in our simulations compare well with those reported in the earlier experimental50 and theoretical study for amorphous TiO2.49 For studying the effect of Li ion concentration on the diffusion energetics and dynamics, we load Li into the amorphous TiO2 by randomly chosen sites within the amorphous network. The way Li ions are incorporated into the structures ensures that they are evenly distributed; the loaded structures are thus fairly uniform. We have tested multiple initial random distributions of Li ions and amorphous structures, but no significant differences in diffusion coefficients were found. In Figure 1, c and d, we present the optimized structures for 50% Li loaded in the amorphous and anatase TiO2, respectively. 2.3. Simulation Analysis. Determining any diffusion parameters from dynamical simulations requires collection of sufficient statistics. This task quickly becomes very time consuming when a fine time step is used in the simulations. In our simulations, in order to ensure that sufficient statistics are collected for deriving the diffusion parameters, we perform several simulations for each concentration and extended the simulations to time scales on the order of a few nanoseconds. Throughout the study, MD simulations are performed within the NPT ensemble for temperatures ranging from 300 to 1400 K for both ordered and disordered structures. The effective barriers and prefactors are derived from high-temperature simulations, as Li ion diffusivity is limited to only a few jumps for low temperatures (below 700 K), as will be discussed later. The diffusion parameters for each concentration are then extracted from the Arrhenius plots that are constructed from the results of the 1 ns simulations for Table 1. Buckingham Potential Parameters Used in This Study (Buckingham Potential Form: Vij = Aij  exp(
rij/Fij)
Cijrij
6) ion pair (ij)

Aij (eV)

Fij (Å)

Cij (eV 3 Å6)

Li
Li

38533.955

0.100

0.00

Ti
Li

33089.570

0.127

0.00

Ti
Ti

31120.528

0.154

5.25

Ti
O

16957.710

0.194

12.59

Li
O

15465.549

0.167

0.00

O
O

11782.884

0.234

30.22

Figure 1. Structures containing 4320 atoms used in the simulations for (a) TiO2 anatase, (b) TiO2 amorphous, (c) 50% Li-loaded amorphous, and (d) 50% Li-loaded anatase. Red, gray, and green spheres represent oxygen, titanium, and lithium, respectively. 15663

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both anatase and amorphous TiO2. In order to construct the Arrhenius plots, the diffusion coefficients are obtained from the slopes of the averaged (over all Li ions) mean-square displacements (MSD) calculated at different temperatures. A plot of the MSD versus time gives a measure of vibrational motion and diffusion of each atom in the system. As time goes to infinity, the MSD behaves linearly for a random walk. The MSD as a function of time for each Li ion in the system is calculated and analyzed. For each time t, the MSDs of all Li ions are summed and averaged to determine the diffusion coefficients for a given Li ion concentration and temperature. The MSD of each Li ion is calculated using its position as a function of time:   1 ½rm ðt + t0 Þ
rm ðt0 Þ2 ð1Þ MSD ¼ Æδr 2 æ ¼ N m

∑

The average self-diffusion coefficient (D) of the Li ions for a given concentration is then obtained from the gradient of the MSD plots using Einstein’s relation, given by D¼

1 dÆδr 2 æ 6 dt

ð2Þ

The first part of our simulations is to extract diffusion mechanisms for an isolated Li ion in anatase TiO2. An earlier dynamical study for isolated Li ion diffusion in anatase showed that direct MD simulations cannot reveal Li ion diffusion in the anatase at low temperatures (300 K) due to the high correlation time.29 A similar result was also reported in an experimental study.24 Our results, obtained from 2 ns MD simulations for an isolated Li ion in TiO2 anatase at 300 K, also show that the diffusion is not accessible from direct MD simulations under the given conditions. The simulations performed at 500 K for 2 ns also predict a relatively small number of jumps that are not sufficient for a quantitative description of its diffusion. In order to reveal Li ion diffusion characteristics, we thus turn to hightemperature simulations that allow us to collect sufficient statistics to determine diffusion parameters. We determine additional quantitative estimates for isolated Li ion diffusion barriers in anatase using static transition state search simulations with DFT (see section II.4 for details); although such simulations do not provide explicit dynamical information, they nonetheless provide useful ab initio results to which diffusion activation barriers from the MD simulations may be compared. Studying diffusion characteristics in amorphous titania is much more complicated than in anatase. The nonexistant order in amorphous materials does not reveal a priori any particular diffusion path, and/or differentiate between possible paths. To best of our knowledge, there are no available studies on Li ion diffusion in amorphous TiO2. We followed a similar approach as with the anatase to explore Li ion diffusion characteristics in the amorphous TiO2. We perform high-temperature MD simulations and analyze the Li ion displacements and trajectories to explore various mechanisms as well as evaluate the concentration and temperature effects on the diffusivity. To elucidate the correlation between the Li ion concentration and its diffusion characteristics in both ordered and disordered structures, the simulations are performed for Li concentrations ranging from 10% to 100% (Li to Ti ratio) and at temperatures ranging from 850 to 1400 K. For each temperature, the simulations are averaged over at least four starting configurations, and the MSDs are calculated. The diffusion coefficients are then

extracted from the slopes of these averaged MSD’s and are used to construct the Arrhenius plots. 2.4. DFT Calculations. Total energy electronic structure calculations for 96 atoms of anatase and amorphous (initially obtained from MD) structures, and for single Li-loaded structures, are performed using DFT as implemented in the Vienna ab initio simulation package (VASP).53 For these calculations, the exchangecorrelation parametrization is chosen as the Perdew
Burke
Ernzerhof (PBE) functional.54 The energy cutoff is set to 400 eV. The k-point mesh used in the study for anatase is 5  5  2. Full relaxation of ions is performed to obtain the minimum energy of each configuration of Li in the system. The calculations are terminated when the forces acting on each atom are below 0.02 eV/Å. For calculating the Li diffusion barrier, we use the nudged elastic band (NEB) method55 and optimize the total energies of a number of intermediate images of the diffusing entity along the diffusion path between the two minima. A total of eight images are chosen between the two minima for calculating Li diffusion in anatase, while we use 12 images for the amorphous TiO2, as the potential energy surface is more complicated. For the amorphous structure, six NEB calculations are performed for different final states.

III. RESULTS AND DISCUSSION We report our results on the energetics and dynamics of Li ion transport in anatase and amorphous TiO2. We first provide the results for isolated Li ion diffusion. Diffusion mechanisms, Li ion displacement profiles, diffusion activation barriers, and dynamics are reported along with their temperature dependence. We then discuss the effect of concentration on ionic diffusivity, including changes observed in terms of the energetics and the dynamics as we systematically increase the Li ion load. Finally, we compare the differences in diffusion characteristics associated with an increase in Li concentration for both systems to explore the influence of morphology on the diffusivity. 3.1. Isolated Li ion Diffusion in Anatase and Amorphous TiO2. 3.1.1. Analysis of Li Ion Trajectories. We begin by exploring

the diffusion characteristics of an isolated Li ion in both anatase and amorphous TiO2. MD simulations within the NPT ensemble are performed at temperatures ranging from 300 to 1400 K. For relatively low temperatures (below 700 K), the 2 ns simulation results show that Li diffusivity is limited to few jumps. The simulations performed above 700 K show that Li ion diffuses via zigzag hops from one interstitial site to another. This mechanism was also reported in an experimental study31 and in recent dynamical simulation studies.27,29 For the diffusion of an isolated Li ion in amorphous TiO2, we follow a similar procedure as for anatase. Simulations at low temperatures (300 K) show no diffusion. The simulations (2 ns) at 700 K, where some diffusion is observed, show that Li ion diffusivity is much lower in amorphous than in anatase TiO2. Isolated Li ion trajectories (in each direction, x, y, and z) in anatase and in amorphous TiO2, obtained at 700 and at 1400 K, are plotted in Figure 2a,b and Figure 2c,d, respectively. It is evident from this figure that, for the same temperature, Li ion diffusivity in anatase is much higher than in amorphous TiO2. It is seen from Figure 2a that there is well-defined diffusion along the z-axis (c-axis) in anatase, while the Li ion only vibrates and unsuccessfully attempts to diffuse in the amorphous titania (Figure 2b). As the temperature is increased to 1400 K (see Figure 2c,d), Li ion diffusivity increases in both systems, with the anatase again 15664

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Figure 2. Isolated Li ion trajectories in (a) anatase TiO2 at 700 K for 2 ns, (b) amorphous TiO2 at 700 K for 2 ns, (c) anatase TiO2 at 1400 K for 1 ns, and (d) amorphous TiO2 at 1400 K for 1 ns.

showing significantly enhanced diffusivity over the amorphous TiO2. Compared to the trajectory of Li ions in anatase at 700 K, for which the diffusion is observed mostly along z-axis, the trajectory at 1400 K indicates enhancement in diffusion both in the x and y directions (a
b plane). Li ion diffusivity in the amorphous TiO2 at higher temperatures shows similar temperature dependence, but it has a much lower magnitude than that in anatase. 3.1.2. Mechanism of Isolated Li Diffusion in Anatase. An earlier study by Kerisit et al.29 reported that the Li interoctahedron diffusion barrier is too high to be observable by direct MD simulation at low temperatures (300 K). They showed the correlation time for Li jumps to be on the order of 1 μs, in reasonable agreement with the result (47 μs) of an experimental study.24 As it is not feasible to construct an Arrhenius plot for isolated Li ion diffusion to quantify the corresponding barrier for the observed mechanism, we perform the calculations using the NEB method within DFT. The unit cell used for these calculations consists of 97 atoms (1 Li, 32 Ti, and 64 O). The diffusion path that is initially observed in our MD simulations between the two minimum energy sites is divided into eight images for the DFT-NEB calculations. The image that corresponds to the highest energy is considered as the saddle point, and the barrier is recorded as the difference between the energies of the saddle point and the initial minimum-energy site. Figure 3a shows the neighboring interstitial sites which are the initial and final sites for the Li ion hops. The minimum-energy sites and the transition state for the diffusion are indicated with

Figure 3. Li hop between two interstitial sites in anatase: (a) two neighboring octahedral sites and (b) total energy as a function of reaction coordinate for the diffusion via a zigzag hop.

the labels of A, C, and B, respectively. The resulting total energy as a function of reaction coordinate plot is shown in Figure 3b. The calculated barrier, 480 meV, for the zigzag hop diffusion is in reasonable agreement with a quantitative estimate of an isolated Li ion diffusion barrier in anatase, obtained via one-dimensional constraint simulations with the shell potential model, of 390 meV.29 Note that the small difference observed in the calculated value of the isolated Li barrier using DFT (our result) and MD (reported by ref 29) can result from the differences in the potential energy treatment within these two approaches. Both results imply that the calculated Li interstitial barrier is high, 15665

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Figure 4. Li ion diffusion in the amorphous TiO2 at 1400 K obtained from a 1 ns simulation. Small displacements of Li ions indicate low diffusivity. The green ball corresponds to Li.

thereby explaining the limited diffusivity observed at low temperatures in MD simulations. 3.1.3. Mechanism of Li Ion Diffusion in Amorphous TiO2. In Figure 4, we present snapshots extracted from the results of the 1 ns simulation for isolated Li ion diffusion in amorphous TiO2 at 1400 K. Each figure corresponds to Li ion positions at approximately 50 ps intervals. Comparison of Li ion positions between parts a and d of Figure 4 shows that Li ions diffuse a little bit within the plane perpendicular to the long axis (c-axis). We find that, while diffusing, the Li ion stays close to oxygen. Diffusion, even at this high temperature, is limited to small displacements of the Li ions. The cause for such slow diffusivity of Li ions in the amorphous samples may reflect the presence of strong Li ion binding sites in the amorphous TiO2. As the diffusion of Li is much slower in the amorphous than that in the anatase, it is not possible to collect sufficient statistics using direct MD simulations to construct an Arrhenius plot, and hence obtain temperature-dependent diffusion parameters. We therefore turn to DFT calculations to obtain qualitative information about the potential energy surface that Li encounters in the amorphous TiO2. In order to explore Li diffusion in an amorphous cell using DFT, we generate an amorphous TiO2 unit cell consisting of the same number of atoms as in the anatase (96 atoms). The amorphous cell is first prepared using MD simulations by repeating the same steps discussed above. The structure obtained is then used as the initial configuration for our DFT calculations. Performing NEB calculations requires two minimum-energy configurations to be chosen as the initial and the finals states for Li. We explore a number of Li positions in the amorphous cell and select the position that provides the lowest energy as the initial state. The final states are generated by choosing certain Li displacements from the initial configurations, and we try to keep as diverse an environment as possible for the immediate neighborhood of Li. NEB diffusion pathways are then constructed between the initial and final states using 12 images. Analysis of several of these NEB simulations indicates that Li encounters a very rough potential energy surface. Indeed, this

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surface presents significant barriers that approach an electronvolt. These analyses qualitatively corroborate our MD simulation results and suggest that low Li diffusivity in the amorphous TiO2 for isolated Li atoms likely reflects a rough potential energy surface with high diffusion barriers. Recently, a similar observation has been reported from a study exploring electrochemical lithiation of single SnO2 nanowires.33 Diffusion through crystalline and amorphous Li2O phases, which were experimentally observed to form during lithiation, was explored by DFT calculations; the Li diffusion barrier in the amorphous structure was found to be higher than that for the crystalline structure. Although it is important to gain a thorough understanding of an isolated Li ion diffusion in both ordered and disordered structures, in reality, for energy storage applications, much higher concentrations of Li ions during insertion and deinsertion from the host structure are required. As the diffusion of Li ions is of primary importance for battery operations, a realistic picture requires atomistic understanding of the Li concentration dependence of diffusion parameters and of how that dependence varies with the morphology. In order to obtain further insights into the concentration dependence of Li ion diffusivity, diffusion energetics and dynamics are explored for Li concentrations ranging from 10% to 100%. In the following sections, the concentration effect on the diffusion characteristics of Li ions in anatase and amorphous TiO2 is reported. Other factors that affect the diffusivity, such as temperature and morphology, are also discussed. 3.2. Effect of Concentration on Li ion Diffusion in Anatase TiO2. 3.2.1. Li ion Trajectories and Mean-Square Displacements (MSD). In order to study the effect of concentration on Li ion diffusion characteristics, we incorporate 10%, 25%, 35%, 50%, 75%, and 100% Li into anatase TiO2. Earlier studies on anatase TiO2 have reported a maximum Li insertion ratio of ∼50% for electrochemical intercalation at room temperature.56 For completeness and comparison with the results for the amorphous case, however, we also explore Li loadings higher than 50%, as these are reported to exist at elevated temperatures.57 These previous studies have also reported that, above 50%, the Li
Li interaction becomes repulsive. We perform 1 ns MD simulations (NPT) for each concentration for temperatures ranging from 500 to 1400 K. The diffusion coefficients are calculated using eqs 1 and 2—the slopes of the averaged MSDs of Li ions. In order to determine appropriate temperatures for constructing the Arrhenius plots, we analyzed in detail Li ion trajectories, the average displacements between time steps, and the total displacements from the initial configurations. Our analysis of the Li ion displacements for the temperatures ranging from 500 to 1400 K showed that the displacements for temperatures above 700 K are sufficient for a quantitative description of diffusion. As the maximum Li intercalation is reported to be 50%, and those concentrations above 50% are difficult to access experimentally, we will first examine Li ion diffusion characteristics for concentrations up to 50%. Later, we will also briefly describe the results for higher concentrations. As a qualitative measure of Li ion mobility, the total displacement of every Li ion from their initial configurations for 25% and 50% Li concentrations at 850 and 1400 K are plotted in Figure 5, a and d. The trajectories of a few randomly chosen Li ions in the anatase at 1400 K are also shown in Figure 5b,c,e,f to illustrate the random walk characteristic of the diffusion. As is shown in Figure 5a,d, an increase in Li concentration enhances the total displacement of Li ions from their initial configurations. The temperature effect on the diffusivity is also 15666

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Figure 5. (a) Total displacement of Li ions in 25% Li-loaded TiO2 anatase at 850 and 1400 K, (b) trajectories of Li ions in 25% Li-loaded TiO2 anatase at 1400 K, (c) trajectories of Li ions in 25% Li-loaded TiO2 anatase at 1400 K (zoomed in), (d) total displacement of Li ions in 50% Li-loaded TiO2 anatase at 850 and 1400 K, (e) trajectories of Li ions in 50% Li-loaded TiO2 anatase at 1400 K, and (f) trajectories of Li ions in 50% Li-loaded TiO2 anatase at 1400 K (zoomed in). Each trajectory corresponds to the trajectory of a randomly chosen Li ion in a given concentration.

evident from Figure 5a,d; an increase in temperature yields enhancement in total displacement, as expected. The average displacement of Li ions between MD steps is enhanced by about a factor of 3 when the temperature is increased to 1400 K. We also find the total displacement of Li ions to increase about 4
5 times for some Li ions (see the 25% Li-loaded case) as the temperature is increased to 1400 K. In Figure 5b,c,e,f, the trajectories of randomly chosen Li ions from the 25% and 50% Li-loaded anatase are shown. Each random walk shown in the figures corresponds to a single Li ion diffusion during a 1 ns simulation. For comparison purposes, Li ion trajectories are plotted on the same scale in Figure 5, c and f. As is evident from Figure 5, c and f, an increase in concentration to 50% leads to an enhancement in Li ion diffusivity. This qualitative observation is in agreement with the calculated increase in Li ion displacement between time steps and with the increase in total displacement from the initial positions when the concentration is increased (see above). Interestingly, for concentrations above 50%, the effect of concentration on the diffusivity is reversed. This result may be related to the fact that 50% Li is the maximum experimentally measured intercalation ratio56 (see also discussion below). Detailed analysis of the average displacement of Li ions at these higher concentrations shows much smaller displacements of Li ions than are observed for smaller concentrations. This is also reflected in the total displacement results. In Figure 6, a and b, we plot the averaged MSD for Li ions in both the 25% and 50% Li-loaded anatase at 850 and 1400 K. The plots show that an increase in concentration from 25% to 50% leads to an increase in the slope of MSD, together with the corresponding diffusion coefficients. This is consistent with the

result, described above, that the increase in concentration enhances the total displacement of Li ions. The plots also indicate the temperature dependence of the diffusivity, and we find that for the same concentration, increasing the temperature significantly increases the slope. In the following section, we will report on the results of the effective diffusion activation barriers, prefactors, and diffusion coefficients at 300 K. The effective diffusion barriers and prefactors are determined from Arrhenius plots constructed at each concentration for temperatures ranging from 850 to 1400 K from the results of the 1 ns simulations. 3.2.2. Arrhenius Plots, Diffusion Barriers, and Diffusivities. We have determined the effective barriers and prefactors from the calculated diffusion coefficients in anatase TiO2 using Arrhenius plots. For each Li concentration, the Arrhenius plots are constructed using the averaged diffusion coefficients obtained from 1 ns simulations for six chosen temperatures, namely, 850, 1000, 1100, 1200, 1300, and 1400 K. In Figure 7, the Arrhenius plots are shown for each Li ion concentration. The plots indicate that increasing the concentration from 10% to 50% yields a successive decrease in the slopes, while the intercepts are found to be very similar. Significantly, these results can provide qualitative insights into the details of dilute Li ion diffusion and its temperature dependence—information that is not accessible with direct MD simulations (see discussion above). Our results on Li ion diffusion for low Li concentrations indicate that the diffusion barrier is quite high. If this trend is further extrapolated to single Li ions, then an even higher diffusion barrier is predicted for isolated Li in the anatase, consistent with the low diffusivity that is observed from MD simulations below 700 K. 15667

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Figure 6. Mean-squared displacements for Li ions in anatase at concentrations of 25% and 50% Li at (a) 850 K and (b) 1400 K.

Table 2. Effective Diffusion Barriers and Prefactors for Li Ion Diffusion in TiO2 Anatasea concn

Eeff (meV)

D0 (cm2/s)

10

496 ( 15

2.18  10
3

25

455 ( 13

1.79  10
3

35

418 ( 9

1.24  10
3

50

406 ( 13

1.15  10
3

75

510 ( 49

2.07  10
3

100

819 ( 49

17.0  10
3

a

The deviations from the calculated barriers are determined from the linear regression.

Figure 7. Arrhenius plots for Li ion diffusion in TiO2 anatase for Li concentrations of 10%, 25%, 35%, 50%, 75%, and 100%.

For each concentration, the effective diffusion barriers (Eeff) are calculated from the slopes of the Arrhenius plots. These barriers correspond to an average of the activation barriers for all the mechanisms observed in the given simulation time and temperature ranges. The prefactors (D0) that carry information on the vibrational contributions to diffusion are extracted from the intercept of the Arrhenius plots. Our results provide important information for the concentration dependence of both energetics and dynamics that is currently not available in the literature. The calculated barriers and prefactors for Li ion diffusion in anatase are summarized in Table 2. We find that an increase in Li ion concentration from 10% to 50% leads to gradual decreases in the barriers, while prefactors are found to vary within a factor of 2 as the concentration changes. The prefactors are found to be similar to the usual value that is commonly reported in the literature.58 The overall decrease in the barrier when the concentration is increased from 10% to 50% is about 90 meV. Note that such a decrease in the barrier can lead to a 2 orders of magnitude increase in the diffusivity evaluated at 300 K. Decreases in Li ion diffusion barriers with increasing lithium concentration in titania have been previously reported. It has been shown, using kinetic Monte Carlo (KMC) and DFT calculations, that Li diffusion in spinel (Li1+xTi2O4) is also concentration dependent, and the barriers decrease with increasing concentration, and the cluster expansion within kinetic Monte

Carlo simulations for Li diffusion in LixTiS2 predicts diffusion coefficients varying by several orders of magnitude with Li concentration.59 A dynamical simulation performed with a potential shell model reported that the Li ion migration path in anatase runs between octahedral sites with corresponding barriers of 0.45 eV (x = 0.1), 0.57 eV (x = 0.06), and 0.65 eV (x = 0.03) eV.27 An experimental study by Wagemaker also reported a decrease in activation barriers with the increase in Li concentration.60 This study shows that the barrier for lithiated anatase (x = 0.03) is higher than that for lithium titanite (x = 0.5). This trend is reported to originate from the stronger electronic screening between the occupied interstitial sites with high concentrations of electrons.60 An earlier DFT calculation30 reported Li barriers in anatase to be 0.5 and 1 eV for x = 0.5 and x = 0.75, respectively. The experimental study by Wagemaker et al. reported the macroscopic diffusion barrier to be ∼0.5 eV.24 The theoretical study25 using UHF and INDO methods reported the barrier for Li hopping between the interstitial sites to be 0.6 eV (x = 0.5) using HF, and 0.51 eV (x = 0.0625) and 0.56 eV (x = 0.5) using INDO. Calculations using a plane-wave pseudopotential approach reported a decrease in Li diffusion barriers from 1.31 to 0.67 eV as the concentration increased from x = 0.063 to x = 0.5.28 Our results on the Li concentration dependence of the barriers are in good agreement with those studies reporting a decrease in barriers upon an increase in concentration. We also examined the change in the prefactors upon increase in concentration, and the results are summarized in Table 2. The change in the prefactors is not more than a factor of 2 with the increase in concentration. Note that the contribution from the barriers to the diffusivity overcomes that of the prefactors, since the barriers enter into the exponential in the rate equation (see eq 3). Thus, any small change in the barriers is expected to 15668

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Figure 8. Amorphous TiO2 structures used in the calculations: (a) amorphous at 350 and 0% Li, (b) 25% Li, (c) 50% Li, and (d) 75% Li. Inset figures are the zoomed versions of each Li concentration case. Red, green, and gray balls represent oxygen, lithium, and titanium, respectively.

dominate changes in the diffusivity, which is given by   ΔE DðTÞ ¼ D0 ðTÞ exp
kB T

ð3Þ

In eq 3, D0 and D are the prefactor and the diffusion coefficient, ΔE is the diffusion activation barrier, and kB is the Boltzmann constant. From our calculations, we can infer that, as the change in the prefactors is negligible upon increase in concentration, the enhanced Li ion diffusivity results from the gradual decrease in the barriers. Using these calculated diffusion barriers and prefactors, we can extract diffusion rates at 300 K. For instance, the diffusion coefficient for Li ions with 10% Li loading is found to be about 5.25  10
12 cm2/s. The increase in concentration to 50% then leads to the diffusion coefficient of 1.02  10
10 cm2/s, indicating 2 orders of magnitude increase in Li ion diffusivity. Note that the increase in the barriers obtained above 50% Li causes a sharp decrease in the diffusivity (10
17 cm2/s for 100% Li loading). The diffusion coefficients for Li in TiO2 anatase have been reported by several experimental studies. For anatase films, the macroscopic diffusion coefficients are reported to vary from 10
10 to 10
17 cm2/s.17
22 The diffusion coefficient of Li is reported to be about 10
13 cm2/s by Kavan et al. in single-crystal anatase.23 Li diffusion at the microscopic scale is determined by Wagemaker et al.,24,61 and the diffusion coefficient is reported to be on the order of 10
12 cm2/s. A recent dynamical study reported the diffusion coefficients for Li diffusion in anatase bulk at 300 K to be 3.4  10
10 cm2/s.29 Our calculated diffusion coefficients are in agreement with these reported values. Let us comment on the changes observed in diffusion characteristics for high Li concentrations. Our simulations for the 75% and 100% Li concentrations showed relatively large and opposite deviations in the Arrhenius plots from those at lower concentrations. Although surprising, this result may be related to the fact that the maximum Li intercalation ratio in anatase is 50%.56 This observation both indicates that these high Li loads may not be physically relevant and suggests that, if lithium loadings higher than 50% could somehow be achieved in anatase, significantly different physical and chemical interactions might exist compared to the lower Li loadings. In particular, these trends may result from the fact that the energetically preferred sites for Li ions in the anatase are the octahedral sites, and the mechanism for diffusion (for all the temperature ranges of

interest in our study) is the zigzag hop between the octahedral sites. As the Li concentration increases above 50%, the number of available low-energy preferred sites decreases. Thus, Li diffusion between the octahedral sites is inhibited as the possibility for such jumps is decreased by their availability. It is also reported that above 50%, Li
Li repulsion increases and may be an additional reason for the increase in the barrier.57 3.3. Concentration Effect on Li Diffusion in Amorphous TiO2. 3.3.1. Li ion Trajectories and Mean-Square Displacements (MSD). We have also explored the effect of concentration on Li ion diffusivity in amorphous TiO2. A recent experimental study37 on Li intercalation in amorphous TiO2 nanotubes reported a high intercalation ratio (up to 100% Li) and very promising reversibilities for both intercalation and deintercalation, suggesting that fundamental studies of Li ion diffusion in amorphous titania at high concentrations may be relevant to practical lithium ion battery technology. It is worth noting that our results on isolated Li ion diffusion in the amorphous TiO2 showed that Li is less diffusive in amorphous than in anatase titania. Our MD results showed, for amorphous samples, that even at high temperatures (∼700
1000 K), diffusion is limited to a couple of jumps. It is thus important to explore whether the Li diffusion in the amorphous TiO2 is also enhanced upon increase in concentration, as is observed for anatase. Similar to the diffusion of Li ions in anatase, we perform 1 ns MD simulations (NPT) for temperatures ranging from 500 to 1400 K. From these results, Arrhenius plots are constructed for each concentration to extract the barriers and the prefactors. To carefully select the temperatures at which the statistics are to be collected, we analyzed in detail the trajectories, the average displacements of each Li ion, and the total displacement of ions from their initial configurations. In Figure 8a
d, we present the structures used in these simulations. Starting with Figure 8b and proceeding to Figure 8d, it is seen that Li ions are uniformly distributed throughout the amorphous host, ensuring that they are exposed to each part of the amorphous network. Our analysis of Li ion displacement for the temperatures studied shows that, above 700 K, the displacement is sufficient for calculating diffusion parameters. Thus, we focus on the temperature ranges between 850 and 1400 K. In Figure 9, a, d, g, and j, the total displacement of each Li ion from the initial configurations in the 25%, 50%, 75%, and 100% Li loadings is shown for temperatures of 850 and 1400 K. The figures show a similar trend—the diffusivity is enhanced with increasing 15669

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Figure 9. (a) Total displacement of Li ions in 25% Li-loaded TiO2 amorphous at 850 and 1400 K, (b) trajectories of Li ions in 25% Li-loaded TiO2 amorphous at 1400 K, (c) trajectories of Li ions in 25% Li-loaded TiO2 amorphous at 1400 K (zoomed in), (d) total displacement of Li ions in 50% Li loaded in TiO2 amorphous at 850 and 1400 K, (e) trajectories of Li ions in 50% Li-loaded TiO2 amorphous at 1400 K, (f) trajectories of Li ions in 50% Liloaded TiO2 amorphous at 1400 K (zoomed in), (g) total displacement of Li ions in 75% Li-loaded TiO2 amorphous at 850 and 1400 K, (h) trajectories of Li ions in 75% Li-loaded TiO2 amorphous at 1400 K, (i) trajectories of Li ions in 75% Li-loaded TiO2 amorphous at 1400 K (zoomed in), (j) total displacement of Li ions in 100% Li-loaded TiO2 amorphous at 850 and 1400 K, (k) trajectories of Li ions in 100% Li-loaded TiO2 amorphous at 1400 K, and (l) trajectories of Li ions in 100% Li-loaded TiO2 amorphous at 1400 K (zoomed in).

concentration—for the Li ion diffusivity dependence on concentration, as is observed in the anatase. Figure 9 indicates that

the total displacement of ions increases as the concentration increases and reaches its highest value for 100% Li. 15670

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Figure 10. Results of the MSD of Li ions in amorphous titania for 50%, 75%, and 100% Li at (a) 850 K and (b) 1400 K.

A detailed analysis of the average displacement between time steps is performed to explore its dependence on concentration and temperature. The isolated Li ion diffusion in amorphous titania at high temperatures shows only small total displacements of about 4
5 Å during 2 ns simulations. An increase in Li concentration gradually enhances the displacement between the steps, which reaches about 5 Å for 100% loading, indicating that the concentration enhances Li ion diffusivity. To the best of our knowledge, this is the first report of the concentration dependence of Li ion diffusivity in amorphous TiO2. Figure 9b,c,e,f,h,i, k,l shows Li ion trajectories at 1400 K extracted from 1 ns simulations; these figures provide clear evidence that Li ion trajectories show more diffusivity as the concentration increases. The averaged MSD’s for Li ions at concentrations of 50%, 75%, and 100% are plotted in Figure 10a,b at 850 and 1400 K (1 ns simulations). The slopes of the MSD’s increase with the increase in Li concentration for a given temperature, and the slopes become higher when the temperature is increased to 1400 K, also indicating an increase in the diffusivity. This indicates that the diffusivity is enhanced as the concentration is increased. 3.3.2. Arrhenius Plots, Diffusion Barriers, and Diffusivities. The Arrhenius plots constructed from 1 ns simulations for temperatures ranging from 850 to 1400 K are plotted in Figure 11 for Li concentrations of 25%, 50%, 75%, 85%, and 100%. Figure 11 shows that an increase in Li concentration leads to a systematic decrease in the slopes. The resulting barriers and prefactors are summarized in Table 3. The table shows that the barriers decrease with increasing Li ion concentration, similar to what is observed for anatase. The decrease in the barriers is found to vary from one concentration to another. Within the studied concentrations, the highest decrease (∼200 meV) in the barrier is observed for the transition from 85% to 100% Li. The transition from 75% to 85% leads to an 83 meV decrease, while the transition from 25% to 50% causes an approximately 120 meV decrease. The overall decrease in the diffusion barriers from 25% to full Li loading (100%) corresponds to ∼400 meV. Such a large decrease in the barrier will be highly consequential for the diffusion rates, provided that the prefactors do not change significantly as the concentration increases. We will discuss the change in the prefactors and the diffusion rates for Li in amorphous titania and compare below the results to those obtained for anatase. It is worth noting that, for 100% Li in amorphous TiO2, we find a significant decrease in the slope of the Arrhenius plot (see Figure 11) that corresponds to a sharp drop in the diffusion barrier. This suggests that there could be structural changes to the titania which might provide more energetically favorable

Figure 11. Arrhenius plots for Li ion diffusion in amorphous TiO2 for concentrations of 25%, 50%, 75%, 85%, and 100%.

Table 3. Effective Diffusion Barriers and Prefactors for Li Ion Diffusion in Amorphous TiO2a concn

Eeff (meV)

D0 (cm2/s)

25

697 ( 33

3.01  10
3

50

579 ( 33

2.12  10
3

75

569 ( 23

4.67  10
3

85 100

486 ( 45 305 ( 43

2.35  10
3 0.27  10
3

a

The deviations from the calculated barriers are determined from the linear regression.

diffusion pathways. Our structural analyses for this concentration, obtained from 1 ns simulations, show an increase of partial long-range order within the structure at higher temperatures. We note that for the ranges of temperatures of interest in this study, the pristine (no Li) amorphous TiO2 stays disordered. Further analysis is underway to determine the possible reasons for these observations. We also summarize the change in the prefactors upon increase in Li concentration in Table 3. We find that the prefactors vary slowly with concentration, similar to that observed for anatase. As with the diffusion in anatase, we can conclude that Li ion diffusion rates are enhanced mainly due to decreases in the barriers. Note that 392 meV reductions in the diffusion barriers result in 7 orders of magnitude increase in Li ion diffusion rates at 15671

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The Journal of Physical Chemistry C room temperature. These results once again show the critical role played by concentration in enhancing Li ion diffusivity in amorphous titania. Thus, they also suggest that for such systems, in which the diffusivity is very limited, concentration can be used as a manipulative tool to enhance diffusion rates and related phenomenon such as growth. From the calculated barriers and prefactors, we extract the diffusion coefficients at 300 K. The diffusion coefficient for 25% Li-loaded amorphous titania is found to be 2.35  10
17 cm2/s, which is about 5 orders of magnitude lower than that of the corresponding concentration for the anatase. Diffusion coefficients for 75% and 100% Li loadings are 6.09  10
13 and 1.36  10
9 cm2/s, respectively. These increases in diffusivities originate from decreases in the barriers. Comparison of Li ion diffusivities in anatase vs amorphous configurations with similar Li compositions suggests that Li ions are more diffusive in anatase than in amorphous titania when the concentrations are below the thermodynamic intercalation limits. This result is consistent with our results for isolated Li ion diffusion in amorphous TiO2 and with experimental results for Li diffusion in SnO2.33 Our analysis of the Li diffusion barriers in amorphous titania shows that the dependence of diffusivity on concentration is monotonic from 25% up to 100%, while for anatase, the dependence is reversed for concentrations above 50%. For the amorphous structures, there is no diffusion mechanism between well-defined vacant sites, as in the case of anatase; so the diffusivity is not expected to be affected by reductions in the number of available sites at higher Li loadings. Moreover, the amorphous structure is not as rigid as the crystal structure, and it may allow the creation of new sites at higher Li loadings. We find that, for the same Li concentration, the barriers in anatase are lower (e.g., ∼240 meV for 25% and 173 meV for 50% Li loading) than those in the amorphous titania, with the exception of the 100% Li-loaded amorphous TiO2 case. For this particular case, the Li ion diffusion barrier drops sharply and becomes smaller than that for the anatase. Compared to the diffusion coefficients at 300 K for amorphous titania, we find the diffusion coefficients in the anatase to be higher (up to 50% loading). However, for 100% Li-loaded amorphous titania, diffusivity becomes an order of magnitude faster compared to the 50% Li-loaded anatase. This may perhaps be initiated by a new diffusion mechanism resulting from a structural transformation, which is energetically favorable. Further analysis is underway to understand this structural transformation occurring at 100% Li loading in amorphous titania and its correlation with the Li diffusion mechanism and observed diffusion barriers.

IV. CONCLUSION We explore the correlation between Li ion concentration and Li transport characteristics in anatase and amorphous TiO2 using a combination of MD and DFT calculations. Atomistic simulations performed at high temperatures for isolated Li ion diffusion indicate that the diffusivity is limited to a couple of jumps in amorphous TiO2. Li ions in anatase, however, are more diffusive at high temperatures than in amorphous titania. Due to the high diffusion barrier (480 meV), obtained from DFT calculations, of an isolated Li ion in the anatase, diffusion at 300 K is not accessible to direct MD simulations within the nanosecond time scale. The diffusion mechanism in anatase is found to be via zigzag hops from one interstitial to another interstitial site. DFT calculations for Li diffusion in amorphous TiO2 suggest that Li

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ions encounter deep wells that are in the electronvolt range. They are consistent with the low diffusivity observed in our MD simulations for diffusion in amorphous TiO2. Analysis of Li ion diffusion in anatase for concentrations ranging from 10% to 50% indicates that Li ions become more diffusive with increasing concentration. We find that, by increasing the concentration to 50%, the total displacement of Li ions can be doubled. Li ion trajectories that are extracted for each concentration show that they span larger ranges in space for higher concentrations, also indicating an increase in diffusivity. The barriers extracted from Arrhenius plots show that an increase in concentration leads to a decrease in effective barriers. For instance, the decrease in the barrier corresponding to the transition from 10% to 50% Li in anatase is about 90 meV. The decrease in the barriers thus results in 2 orders of magnitude increase in Li ion diffusion rates at 300 K. Our results also show that the prefactors vary by at most a factor of 2 with increasing concentration, suggesting that the enhanced diffusivity mainly originates from the decrease in barriers. Our results for Li ion diffusion in amorphous TiO2 for Li concentrations ranging from 25% to 100% show also that the barriers decrease with increasing concentration. We find that the decrease in the barrier corresponding to the concentration transition from 25% to 100% is about 400 meV. Similar to the results obtained for the diffusion in anatase, the prefactors do not change significantly with the increase in concentration. By increasing the concentration to full (100%) Li loading, at room temperature diffusion rates can be enhanced by up to 7 orders of magnitude because of the decrease in the barriers. In general, Li ion diffusivity is slower in the amorphous than that in the anatase except for the 100% Li-loaded case, where a structural modification may be present. Our calculations indicate that concentration can be considered as a manipulative tool for controlling ionic diffusivity, a result that is of importance to many contemporary problems associated with energy storage applications.

’ AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected], [email protected].

’ ACKNOWLEDGMENT Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. The authors also acknowledge computer time at the Laboratory Computing Resource Center (LCRC) at Argonne National Laboratory, and the EMSL, a national scientific user facility sponsored by the Department of Energy’s Office of Biological and Environmental Research located at Pacific Northwest National Laboratory. The authors acknowledge the discussions with the individuals at the Nano-Bio Interfaces Group led by T. Rajh at CNM and C. Johnson. ’ REFERENCES (1) O’Hare, D. In Organic Materials; Bruce, D. W., O’Hare, D., Eds.; John Wiley & Sons, Ltd.: Chichester, UK, 1996; pp 171
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