pubs.acs.org/JPCL
Size Effects on Liþ/Electron Conductivity in TiO2 Nanoparticles Maria L. Sushko,* Kevin M. Rosso, and Jun Liu Pacific Northwest National Laboratory, Richland, Washington 99352
ABSTRACT TiO2 nanoparticles are the important components of nanostructured electrodes for Li-ion batteries. High Li-ion conductivity of the nanoparticles is the key design requirement for these materials. Using multiscale theoretical modeling, we study the influence of nanoparticle size on its Liþ conductivity and reveal the fundamental mechanism for the dramatic increase in conductivity with the decrease in the size of the nanoparticles. We show that the competition between Liþ and electron accumulation at the nanoparticle boundaries competes with the steady ion and electron fluxes. For nanoparticles smaller that 20 nm, the balance is shifted toward the steady charge transport, hence high conductivity, while for larger nanoparticles charge separation prevails. Size effects are also manifested in the change in the nature of charge transport from dual ionic/electronic for small nanoparticles to predominately ionic for larger ones. SECTION Electron Transport, Optical and Electronic Devices, Hard Matter
Here we focus on elucidating the size effects on Liþ ion conductivity in TiO2 nanoparticles. From the fundamental viewpoint, bulk titania is an example of a material with an ambipolar (or concerted) Liþ/electron polaron-hopping conductivity, which has been extensively studied theoretically, using quantum mechanical16-19 and molecular dynamics20 methods, and experimentally.19 However, little is known about the ion/electron conductivity in TiO2 nanoparticles. On the application side, TiO2 nanaparticles show great potential as the functional elements of nanocomposite anodes for lithium-ion batteries,21,22 and the fundamental understanding of the size effects on the conductivity is crucial for the rational design of these materials. To study the ambipolar ion and electron transport in TiO2, we developed the multiscale model schematically shown in Figure 1b, specifically for the rutile polymorph as a case study. In this model, Liþ ions hop between the equilibrium interstitial sites or stationary points in the c-channels. The barriers for Liþ hopping between the stationary points were taken as those calculated quantum mechanically.17 Charge-compensating electrons are modeled as hopping along the same pathway with barriers calculated from molecular dynamics simulations.20 The ion and electron flux was calculated using the Poisson-Nernst-Planck formalism coupled with classical density functional theory. The calculated diffusion coefficients for both ions and electrons in c-direction are equal to 10-6 cm2/s. This result agrees well with the experimental and computational data for rutile.17,23 We found a complex dependence of the ion and
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anostructured materials show great potential as highcapacity and high rate capability electrodes for rechargeable lithium ion batteries.1-5 Further progress in this important field requires fundamental understanding of the mechanism of charge transport in nanoparticles and nanocomposite materials. Experimental evidence suggests that the conductivity in nanostructured materials is dramatically different from that in corresponding largegrain polycrystalline or single-crystal materials.6-9 The main reason for this effect is that nanostructuring introduces such a high density of the interfaces that the conductivity may become interfacially controlled. Size effects can be also manifested in these systems via changing the nature of conductivity. One of the examples is CeO2-x, in which coupled O2- and electron conductivity is associated with the formation of Vo•• vacancies. In nanocrystalline ceria, predominantly electronic conductivity is observed, while it is ionic in large grain polycrystalline CeO2-x.10,11 Several models for the conductivity in nanostructured materials have been proposed. In the neutral layer model, enhanced conductivity is attributed to the decrease in the defect formation energy at the boundaries of the nanocrystals.12 The space charge model considers charge redistribution between structurally distinct core and interfacial regions as the main reason for the size effect on conductivity.13 In this model, conductivity is more likely to occur through the space charge zone near the boundary of the nanocrystals (Figure 1a). However, conductivity through the bulk region with the bulk mobility of the charge carriers is also possible. The space charge model was successfully used to interpret experimental data for nanocrystalline ceria and nanosized CaF 2/BaF 2 heterostructures.10,11,14,15
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Received Date: April 22, 2010 Accepted Date: June 8, 2010 Published on Web Date: June 11, 2010
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DOI: 10.1021/jz100520c |J. Phys. Chem. Lett. 2010, 1, 1967–1972
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the potential drop along the channel on the external voltage (Figure 3). It follows from the figure that the effective potential drop felt by charge carriers decreases with the external voltage, leading to the decrease in Liþ current. Similar IV curves are observed for the 60c (17.754 nm) channel (Figure 2a). However, the conductivity in this longer channel has mainly an ionic character. Liþ ion currents are approximately an order of magnitude larger than the corresponding electron currents. This difference is dictated by the larger potential drop across the longer channel, which requires higher concentration of electrons along the channel boundary to provide a compensating field. Therefore, fewer electrons are available to be involved in the steady flow. It is noteworthy that the size of these channels is smaller than the Debye length in the c-direction, which is equal to 20.03 nm for ion and electron concentrations of 0.01 M. Therefore, the external potential cannot be fully compensated by charge separation in the channel, and the prevailing mechanism for counterbalancing the excess electrostatic free energy in the channel is the steady ion and electron flux. For longer channels, a complex combination of these effects can be expected. Indeed, the Liþ current-voltage curves are qualitatively different for the channels longer than the Debye length (90c = 26.631 nm and 120c = 35.508 nm). For small absolute values of the external potentials, i.e., |V| : 0, r > γσ þ
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There are no short-range interactions acting between Liþ ions or electrons. In some calculations, we have also introduced the stationary points for electrons. However, we found that the results do not depend on whether this additional interaction is introduced. This result is in line with reported data that electron diffusion is coupled to the Liþ diffusion via electrostatic interactions.20 Therefore, the short-range interactions for the electrons are not present in all calculations reported here. 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 σþ/10. Convergence was considered to be achieved when the difference between the next solution of the system of equations and the previous one become smaller than 10-6.
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SUPPORTING INFORMATION AVAILABLE Figure showing the dependence of the equilibrium density of Liþ ions at the channel boundary on the bulk Liþ density for nanochannels of various length. This material is available free of charge via the Internet at http://pubs.acs.org
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AUTHOR INFORMATION (18)
Corresponding Author: *To whom correspondence should be addressed. E-mail: maria.
[email protected].
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ACKNOWLEDGMENT The development of the PNP-cDFT software is 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-AC0576RL01830.
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