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Mechanism of Li+/Electron Conductivity in Rutile and Anatase TiO2 Nanoparticles Maria L. Sushko,* Kevin M. Rosso, and Jun Liu Pacific Northwest National Laboratory, Richland, Washington 99352, United States ReceiVed: August 23, 2010; ReVised Manuscript ReceiVed: October 8, 2010
Concurrent Li ion and electron conductivity in rutile and anatase TiO2 nanoparticles was studied using multiscale simulations. We show that charge transport in titania nanoparticles is determined by the competition of charge redistribution toward the particle boundaries and constant Li+ and electron fluxes. In nanoparticles smaller than the Debye length, the constant flux prevails and the conductivity has a dual ionic and electronic character, while for larger nanoparticles conductivity becomes predominately ionic. Simulations revealed that the temperature dependence of Li ion conductivity in anatase is very weak, while in rutile the conductivity decreases with temperature in small nanoparticles and increases in large nanoparticles. Introduction The ability of different titania polymorphs to store significant amounts of lithium makes it an attractive material for anodes in Li ion batteries. Titania-based nanocomposite materials are particularly promising, as they not only show superior capacity but also have significantly improved stability compared to single-phase anode materials.1-3 Nanostructuring introduces new effects arising from the high surface area to volume ratio for each component of the composite. This leads to qualitatively different charge transport properties compared to the bulk. According to a space charge model, the presence of the grain or particle boundaries in the material tends to induce partial migration of the mobile charges to the boundary region, where they accumulate to form a so-called space charge zone (Figure 1).4-14 The conductivity in the charge carrier rich space charge zone is therefore substantially higher than in the bulk material, nominally yielding increasing conductivity with decreasing particle size in nanocrystalline materials.10,14 In our previous publication, we provided direct theoretical evidence for the space charge model for rutile nanoparticles: distributions of interacting Li+ and electron charge carriers under external voltage showed carrier accumulation at the particle boundaries.15 Here we extend these simulations by comparing the conductivity in two different titania polymorphs and in different crystallographic directions, as well as by examining the temperature dependence of the conductivity. Extensive research on Li+ and electron transport in bulk titania polymorphs strongly suggests that Li+ migrates principally as an interstitial cation, and electron transport occurs by small polaron hopping along the metal sublattice.16-21 Upon Li+ insertion, charge-compensating electrons localize on Ti4+ cations in the lattice, manifesting as observable Ti3+ states in X-ray photoelectron spectra.22,23 Strong electrostatic interaction between Li+ and the localized electron leads to the expectation of concurrent hopping of pairs of these charged species.24 Energy barriers for site-to-site migration of Li+ and electrons and the overall conductivity strongly depend on the polymorph structure and the crystallographic direction for transport. Energy barriers for the collective transport can be estimated from Li+ ion mobilities in the lattice, an experimental observable. In particular, the * To whom correspondence
[email protected].
should
be
addressed.
E-mail:
Figure 1. Space charge model of TiO2 nanoparticles and the unit cells of rutile and anatase structures. O atoms are shown as red spheres, Ti atoms are shown as gray spheres, and the interstitial sites, that can be occupied by Li+, are shown as black spheres.
experimentally measured barrier for Li+ transport in anatase was reported as 0.2 eV for interoctahedron hopping,25 while for the rutile c-channel the barrier is as low as 0.04 eV.26 Computational molecular modeling predictions for these systems give higher values ranging from 0.39 to 0.65 eV for anatase17,24,27,28 and from 0.04 to 0.22 eV for the rutile c-direction.29-33 In this paper, we employ a multiscale drift diffusion model, which links the experimental or computational data for elementary transport barriers in the lattice with a mesoscopic-scale Poisson-Nernst-Planck treatment of the conductivity and its voltage and temperature dependence, to study the mechanism of concurrent Li+ and electron transport in titania nanoparticles. Theoretical Model Diffusion channels in the TiO2 rutile and anatase polymorphs were represented as a uniform medium with the dielectric constant equal to the dielectric constant in the corresponding lattice and crystallographic direction (Figure 1, Table 1). In all calculations the bulk concentrations of Li+ and electrons were set to 0.01 M; i.e., the system was overall electroneutral. To
10.1021/jp107982c 2010 American Chemical Society Published on Web 11/08/2010
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TABLE 1: Parameters for the PNP Model for Li+/Electron Transport in TiO2 Nanoparticles rutile
dielectric constant energy barrier (eV) distance between the stationary points (nm) channel width (nm) Debye lengtha (nm) a
a-channel
c-channel
anatase c-channel
86.0 0.80 0.229 65
170.0 0.04 0.295 86
48.0 0.20 0.2960
0.295 86 10.13
0.229 65 20.03
0.3785 5.66
For the 0.01 M concentration of the mobile charge carries.
account for Li+ migration in titania polymorphs from one interstitial lattice site to another, in the model these equilibrium lattice sites in various nanochannels are represented by a finite one-dimensional array of interaction centers, or stationary points. The stationary points interact with Li+ via a square-well potential (see below) with the depth of the potential well equal to the barrier height for Li+ migration between two adjacent stationary points. Barrier heights were set to values obtained using experimental,34 quantum mechanical,32 and molecular dynamics24 methods for the bulk polymorphs (Table 1). Experimental evidence suggests that Li+ can occupy two distinct interstitial positions in anatase octahedra along the c-direction separated by 0.161 nm.25,34 However, due to Coulomb repulsion, it is unreasonable to expect these two sites to be occupied simultaneously. Therefore, in our model we only consider Li+ migration between neighboring octahedral sites with the distance of o˜ ) 0.296 nm between them and the energy barrier of 0.2 eV, as determined from nuclear magnetic resonance experiments.25 The channel width for anatase was set to a ) 0.378 nm. Because in rutile there are two equivalent Li+ sites in the a-direction (Figure 1), the width of the c-channel was set to a/2 ) 0.230 nm (Table 1). For the same reason, the distance between the stationary points in rutile a-channel is also equal to a˜ ) a/2 ) 0.230 nm. To study the particle size dependence of conductivity in rutile, we considered several channel lengths ranging from 15x to 120x, where x is the distance between the stationary points in the channels (x ) c, a˜, o˜). Li+ ions were represented as spherical particles with charge q+ ) 1 and diameter σ+ ) 0.060 nm (ionic diameter of Li+). A similar representation was used for electrons, diffusing along with Li+, with the parameters q- ) -1 and diameter σ- ) 0.001 nm. It has been shown that during the concurrent diffusion in TiO2 the interactions between Li+ and charge-compensating electrons are predominantly electrostatic and the magnitude of this attraction is substantial at short separations.24 Therefore, there is no need for introducing additional stationary points for electrons. Indeed our simulations showed that the data for concurrent Li+ and electron diffusion in rutile c-channel are independent of the presence or absence of an array of stationary points for electrons, interacting with electrons via square well-potential with the depth equal to the barrier for small electron polaron hopping conductivity. The flux of charged particles in the stationary conditions can be calculated using the Poisson-Nernst-Planck formalism:35
[
-Ji ) Di(z) A(z)
(
dFi dµi0 dµiex dφ 1 + Fi(z) qie + + dz kT dz dz dz
)]
(1)
dJi )0 dz -
(2)
1 d dφ ε(z) A(z) )e A(z) dz dz
(
)
∑ qiFi(z)
(3)
i
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), respectively, A(z) is the cross-section of the channel, φ is the electrostatic potential, µ0 is the ideal and µex is the excess chemical potential of Li+ and electrons, 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 calculation of the electrostatic potential. We use classical density functional theory for evaluation of the chemical potentials of the charged species. In this model the total free energy is divided into two parts: the ideal (Fid), which includes the contributions from the configurational entropy of the noninteracting species and bonding enthalpy if any, and the excess free energy (Fex), 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 (C), electrostatic correlations (el), hard sphere repulsion (hs), and short-range interactions (sh) with the stationary points:
Fex ) FCex + Felex + Fhsex + Fshex
(4)
These free energies are calculated as follows (see ref 36 for more details):
Fid ) kT
r FR(b) r - 1]db r ∑ ∫ FR(b)[ln
(5)
R)+,-
FCex )
e2 2ε i,j)B,+,-
∑ A
Felex ) Felex[{FRbulk}] - kT
f) qiqjFi(b) r Fj(r′ f db r dr′ f |b r - r′ |
∫ dbr ∑
∆CR(1)el ×
A
∑
R)+,-
kT f (FR(b) r - FR ) db r dr′ ∆Cij(2)el × 2 i,j)+,f f) - F bulk) r - Fibulk)(Fj(r′ (| b r - r′ |)(Fi(b) j bulk
(6)
(7)
Fhsex can be expressed as an integral of the functional of b)) using the fundamental measure weighted densities (nω(r theory:37
Fhsex ) kT
r b r ∫ Φhs[nω(b)]d
(8)
The short-range interactions between Li+ and the stationary points (denoted by “s”) are given by
Fshex )
1 2
A dbr dr′f ∑
i,j)+,s
f)φ (| b f Fi(b) r Fj(r′ ij r - r′ |)
(9)
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Figure 2. Diffusion coefficient for Li ions in 30c rutile c-channel as a function of the distance from the channel boundary. The external potential is equal to -5.32 mV (a) and -53.2 mV (b).
where the potential φij is the square-well potential with the depth ξij
{
∞, r < 0 φij(r) ) ξij, σ+ < r 0, r > γσ+
(10)
As mentioned above, there are no short-range interactions acting between Li+ ions and electrons, since these interactions are dominated by Coulomb attraction and electrostatic correlation.15,24 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. We used a uniform grid of points separated by the distance of σ+/10. Convergence was considered to be achieved when the difference between the next solution of the system of equations and the previous one becomes smaller than 10-6. Unless noted otherwise, the temperature used in the calculations was 298 K. We note that the use of a one-dimensional model for charge transport in TiO2 nanoparticles imposes certain limitations on the possible conductivity pathways. However, the model is appropriate for rutile, for which experimental evidence points to negligible Li+ diffusion in the ab plane.26 For anatase the limitation is more severe, as in principle in the real material Li+ can hop to any of the four equivalent octahedral sites and consequently its diffusion pathway may be nonlinear. Nevertheless, according to molecular dynamics data a symmetric linear pathway is energetically more favorable.34 Results and Discussion Li+/Electron Diffusion in Rutile and Anatase Nanoparticles. Using our model, we performed a comparison between coupled Li+ and electron diffusion in rutile nanoparticles along the c- and a-directions and in anatase nanoparticles along the c-direction. The main differences between these three cases are the different energy barriers for carrier transport (0.04 eV for the rutile c-direction,31,32 0.8 eV for the rutile a-direction,32 and 0.2 eV for anatase),25 the different dielectric constants (170, 86, and 48, respectively), and the different distances between the stationary points (Table 1). These differences are readily reflected in the calculated Li+ diffusion coefficients. The highest diffusion coefficient of 10-6 cm2/s is obtained for the rutile c-direction, while the diffusion coefficients for the rutile
a-direction and the anatase c-direction are significantly smaller at 2.7 × 10-10 and 1.7 × 10-11 cm2/s, respectively. These results are consistent with the experimental and computational data for these systems. In particular, the experimentally measured diffusion coefficient along the c-direction in rutile is 10-6 cm2/s,26 while the calculated values using quantum mechanical32 and molecular dynamics24 methods are 10-6 and 9 × 10-5 cm2/s, respectively. Similar to our results, the diffusion coefficient in the a-direction obtained using plane-wave DFT methods is significantly lower and equal to 10-14 cm2/s.32 There is high variability in the experimental data for anatase, and the measured diffusion coefficients are reported as 10-13 cm2/s for single crystals and 10-10-10-17 cm2/s in anatase films.25,38-45 Our calculations showed that the average diffusion coefficient (D) for Li+ is almost independent of the externally applied electrostatic potential in the range from 0 to -100 mV. However, variations of the diffusion coefficient along the channel are observed in the low potential regime (Figure 2a). These variations correlate with the distance between Li+ ions and the stationary points, i.e., the diffusion coefficient increases when the ion is approaching the stationary point and it decreases when the ion is leaving the stationary point. This effect results in a 3 × 10-9 cm2/s variation in the diffusion coefficient. In the regime of high external potential, these spatial variations of the diffusion coefficient disappear (Figure 2b). Another important feature of the diffusion coefficient under the higher potential condition is its increase close to the boundaries of the nanochannel. The latter reflects the accumulation of charge carriers at the boundaries in conjunction with the constant Li+ and electron flow through the nanochannel. Current-Voltage Characteristics for Rutile and Anatase Nanoparticles. We have shown previously that the currentvoltage (IV) characteristics for the c-channel in rutile nanoparticles have a complex shape determined by the interplay of charge accumulation at the channel boundaries and constant ion and electron fluxes.15 In this regard, the IV characteristics for the anatase and rutile a-channels are qualitatively similar to those of the rutile c-channel. The shape of the curves depends on the relative volume of the bulk and the space charge zones. When the length of the channel is smaller than the Debye length, the space charge zone fills the whole nanoparticle, while distinct space charge and bulk zones are present in larger nanoparticles. These variations in the charge distribution inside the nanoparticles lead to changes in the mechanism for conductivity. In smaller nanoparticles at low external potentials most mobile
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Figure 3. IV characteristics for Li+ and electron (e-) transport in anatase c-channel (a) and rutile a-channel (b) for various channel lengths. Note that the IV curve for e- in anatase 15o˜ channel practically coincides with the corresponding Li+ IV curve and not shown for clarity.
charges (Li+ and electrons) are involved in a steady flux. However, charge accumulation at the boundary becomes an increasingly important mechanism for compensation of the external potential at higher voltages. The efficiency of the compensation is directly linked to the barriers for elementary Li+ and electron migration, which increases from rutile-c, to anatase, and then to rutile-a. As a consequence, conditions are such that the external potential is overcompensated by very fast migration in the rutile c-channel,15 fully compensated in the anatase c-channel, and slightly undercompensated in the rutile a-channel. This leads to the decrease,15 independence, and increase in the Li+ current with increasing voltage, respectively (Figure 3). In large nanoparticles, charge redistribution to the
boundaries becomes the major mechanism for compensation of the external potential. Most of the charge carriers, in particular more mobile electrons, are accumulated at the channel boundaries and are not available for the constant flow. Therefore, in this regime both Li+ and electron currents are relatively small and Li+ conductivity prevails. A complex combination of charge redistribution and constant flux for the intermediate size nanoparticles results in IV curves with a maximum at a certain values of external potential (Figure 3). Since in anatase and the rutile a-channel the Debye screening length is significantly smaller than that for the rutile c-channel due to smaller dielectric constants (Table 1), the transition between the predominantly space charge and constant flux regimes takes place at smaller
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Figure 4. Temperature dependence for Li+ and electron conductivity in rutile (a) and anatase (b) c-channels. The external potential is -53.2 mV.
effective particle sizes. The change in the qualitative shape of the IV curves occurs for nanoparticle sizes between 15o˜ and 30o˜ (4.44 and 8.88 nm) and between 30a˜ and 60a˜ (6.89 and 13.78 nm), respectively, rather than between 60c and 90c (17.75 and 26.62 nm) in the rutile c-channel.15 Due to the high-energy barriers for Li+ transport in anatase and in the rutile a-direction, the Li+ currents are of the order of 10-3 and 10-5 pA, respectively, which is 1-3 orders of magnitude smaller than that for the rutile c-channels. Temperature Dependence of Ion Conductivity. To examine the effect of temperature, we calculated the temperature dependence of the conductivity in rutile and anatase c-channels. These calculations revealed qualitatively different trends for small and large nanoparticles. Both Li+ and electron conductivities in the 30c rutile c-channel decrease with temperature (Figure 4a). The electron conductivity drops approximately 10 times upon increasing the temperature to 600 K. The decrease in Li+ conductivity is slower and the slope of the conductivity curve changes at 600 K. At this temperature, the thermal energy becomes larger than the barrier for Li+/electron hopping conductivity, which facilitates ion and electron thermal motion in the nanoparticle, enhancing entropic disorder, and results in partial disruption of the space charge region, as seen from the decrease in Li+ density at the boundary (Figure 5a). This increases the fraction of charge carriers available for constant flow and, therefore, reduces the rate of the decrease in conductivity. The increase in thermal disorder at elevated temperatures is also reflected in the temperature dependence of various components of the free energy for Li+ and electrons
(Figure 5b). The increase in the intensity of thermal motion leads to partial disruption of electrostatic correlations in the channel, to the increase in Coulomb repulsion, and to slight increase in repulsive hard sphere interactions. The free energy of hard sphere repulsion is 3 orders of magnitude smaller than the other contributions, though. Hence it has only a minor effect on the energy balance in the system. A similar trend is observed for the electron conductivity in the c-channel of anatase with the length of 15o˜: the electron conductivity monotonically decreases with temperature. However, Li+ conductivity exhibits two plateaus in the temperature ranges from 298 to 1000 K and 1100 to 1600 K (Figure 4b). Because the energy barrier for transport in anatase is higher than the thermal energy in the temperature interval considered here, the influence of thermal effects on the carrier flux is weaker than in rutile. For longer channels the conductivity through the bulk region plays the major role. Therefore, the temperature effect is mainly manifested in lowering the effective energy barrier for transport. This effect is significant in rutile, resulting in the monotonic increase in Li+ and electron conductivities (Figure 4a). However, for anatase both Li+ and electron conductivities are approximately independent of temperature due to larger energy barriers for elementary charge migration in the lattice (Figure 4b). The temperature dependence of the conductivity (σ) also allows calculation of the effective mesoscopic activation energy for ion and electron transport in TiO2 nanoparticles.8,11 The effective activation energy for charge transport in the nanopar-
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Figure 5. Density distribution of Li+ along rutile c-channel as a function of temperature (a) and the temperature dependence of various components of the free energy (b). Curves 1 and 2 in part b correspond to Felex for Li+ and electrons, respectively, curves 3 and 4 to FCex for Li+ and electrons, and curve 5 to Fhsex. The channel length is 30c. The external potential is -53.2 mV.
ticles is determined by the activation energy for the charge carrier migration in the bulk (E∞), the difference of the space charge potential at the interface and the bulk (∆φ0 ) φ0 - φ∞), and its temperature dependence:
(
E ) E∞ + e ∆φ0 +
)
∂(ln σ) 1 ∂(∆φ0) ) kB T ∂(1/T) ∂(1/T)
(11)
The right-hand side of eq 11 is the definition of the activation energy, while the terms e∆φ0 and e(1/T)[∂(∆φ0)/∂(1/T)] reflect charge carrier mobility and Boltzmann concentration profile in electric field, respectively. The second term in eq 11 increases monotonically by absolute value in the low-temperature range and saturates at high temperatures (see Supporting Information). This trend is observed for both small and large rutile and anatase nanoparticles. The maximum reduction of the bulk activation energy is 0.009 and 0.04 eV in rutile 30c and 90c, respectively. Therefore, in small nanoparticles the temperature-induced reduction of the activation energy is less than 25% for all temperatures studied, which explains a very weak temperature dependence for Li+ conductivity. In contrast, an almost complete compensation of the bulk energy barrier is achieved in larger rutile nanoparticles in the temperature range higher than 1000 K and hence a rapid increase in Li+ conductivity (Figure 4). In anatase, the temperature-induced reduction of the Li+ activation energy is similar for small nanoparticles, 0.012 eV, and larger for 30o˜ nanoparticles, 0.10 eV. Nevertheless, due to a higher bulk energy barrier for Li+ conductivity (E∞ ) 0.2 eV), the activation energy remains significant even for the highest temperatures studied. Hence, a slower increase in Li+ conductivity in 30c anatase nanoparticles is found compared to that in 90c rutile. It is noteworthy that the effective activation energies calculated directly from the temperature dependence of the conductivity (eq 11) agree well with the results obtained using ∆φ0, which confirms the consistency of our approach. Conclusions Using the multiscale computational approach we have shown that the conductivity in TiO2 nanoparticles strongly depends on the titania polymorph, the crystallographic direction for the conductivity, and the nanoparticle size. The size effect is mainly manifested in the change in the qualitative character of the conductivity from dual ionic and electronic to predominantly
ionic. We also found qualitative differences in the temperature dependence of small and large nanoparticle conductivities. The ionic conductivity in small rutile nanoparticles decreases with temperature, while it increases for large nanoparticles. 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 TiO2 nanoparticles is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award KC020105-FWP12152. PNNL is a multiprogram national laboratory operated for DOE by Battelle under Contract DE-AC05-76RL01830. Supporting Information Available: Figure S1, showing the temperature dependence of the second term in eq 11. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Liu, J.; Cao, G. Z.; Yang, Z. G.; Wang, D. H.; Dubois, D.; Zhou, X. D.; Graff, G. L.; Pederson, L. R.; Zhang, J. G. Chemsuschem 2008, 1, 676. (2) Wang, D. H.; Choi, D. W.; Li, J.; Yang, Z. G.; Nie, Z. M.; Kou, R.; Hu, D. H.; Wang, C. M.; Saraf, L. V.; Zhang, J. G.; Aksay, I. A.; Liu, J. ACS Nano 2009, 3, 907. (3) Wang, D. H.; Kou, R.; Choi, D.; Yang, Z. G.; Nie, Z. M.; Li, J.; Saraf, L. V.; Hu, D. H.; Zhang, J. G.; Graff, G. L.; Liu, J.; Pope, M. A.; Aksay, I. A. ACS Nano 2010, 4, 1587. (4) Maier, J. Prog. Solid State Chem. 1995, 23, 171. (5) Kreuer, K. D.; Fuchs, A.; Maier, J. Solid State Ionics 1995, 77, 157. (6) Sata, N.; Eberman, K.; Eberl, K.; Maier, J. Nature 2000, 408, 946. (7) Tschope, A. Solid State Ionics 2001, 139, 267. (8) Kim, S.; Maier, J. J. Electrochem. Soc. 2002, 149, J73. (9) Maier, J. Solid State Ionics 2003, 157, 327. (10) Maier, J. Nat. Mater. 2005, 4, 805. (11) Guo, X. X.; Maier, J. AdV. Funct. Mater. 2009, 19, 96. (12) Guo, X. X.; Maier, J. AdV. Mater. 2009, 21, 2619. (13) Kern, K.; Maier, J. AdV. Mater. 2009, 21, 2569. (14) Maier, J. Phys. Chem. Chem. Phys. 2009, 11, 3011. (15) Sushko, M. L.; Rosso, K. M.; Liu, J. J. Phys. Chem. Lett. 2010, 1, 1967. (16) Yagi, E.; Hasiguti, R. R.; Aono, M. Phys. ReV. B 1996, 54, 7945. (17) Lunell, S.; Stashans, A.; Ojamae, L.; Lindstrom, H.; Hagfeldt, A. J. Am. Chem. Soc. 1997, 119, 7374. (18) Nowotny, J.; Radecka, M.; Rekas, M. J. Phys. Chem. Solids 1997, 58, 927.
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