TOPICAL REVIEW Diffusion and Ionic Conduction in Nanocrystalline Ceramics Paul Heitjans and Sylvio Indris Institut f�� ur Physikalische Chemie und Elektrochemie, Universit�� at Hannover, Callinstr. 3-3A, 30167 Hannover, Germany E-mail: heitjans@pci.uni-hannover.de, indris@pci.uni-hannover.de Abstract. We review case studies of diffusion in nanocrystalline ceramics, i. e. polycrystalline non-metallic materials with average grain sizes typically in the range from 5 nm to 50 nm. The experimental methods applied are on the one hand tracer diffusion or conductivity methods which are sensitive to macroscopic transport, on the other hand NMR techniques which, complementarily to the previous ones, give access to microscopic diffusion parameters like atomic hopping rates and jump barrier heights. In all cases the diffusion properties of the samples, whether single phase systems or composites, are dominated by their grain boundaries and interfacial regions, respectively. In principle, all experimental techniques allow one to discriminate between contributions to the diffusion from the crystalline grains and those from the interfacial regions. Corresponding examples are presented for SIMS and impedance measurements on oxygen conductors. NMR studies for various nanocrystalline lithium ion conductors reveal that two lithium species with different diffusivities are present. Comparison with the coarse grained counterparts shows that the slower ions are located inside the crystallites and the faster ones in the structurally disordered interfacial regions. Investigations on composite materials exhibit phenomena which can be explained by the percolation of fast diffusion pathways being formed by the interfaces between the two components. PACS numbers: 66.30.-h, 76.60.-k, 81.07.Bc, 82.45.Xy Submitted to: J. Phys.: Condens. Matter Contents 1 Introduction 2 2 Classification of nanostructured materials 3 3 Preparation and characterization of nanocrystalline ceramics 6 4 Fundamentals of diffusion and ionic conduction 10

2 Diffusion and Ionic Conduction in Nanocrystalline Ceramics 5 Outline of diffusion measurement techniques 13 5.1 Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Tracer diffusion method . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.3 Impedance spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.4 NMR line shape spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 16 5.5 NMR spin-lattice relaxation spectroscopy . . . . . . . . . . . . . . . . . . 17 6 Experimental results 18 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.2 Tracer diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.3 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.3.1 Single phase systems. . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.3.2 Composites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.4 NMR line shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.5 NMR spin-lattice relaxation rate . . . . . . . . . . . . . . . . . . . . . . . 31 7 Conclusions 34 1. Introduction The advent of nanostructured materials in recent years has inspired many new developments in solid state physics, solid state chemistry and materials science. Due to a tailored microstructure these materials can show new mechanical [1���4], electrical [5��� 9], magnetic [10���15], optical [16���20], catalytic [21, 22] and thermodynamic [23���25] properties. This is due to the increased fraction of structurally disordered interfacial regions, enhanced surface area or quantum confinement effects. Besides the investigation of the structural features of nanostructured solids per se, the challenge is to study those macroscopic properties and to interrelate them to the microscopic structure. Diffusion processes in these solids are often important for this interrelation. They are influenced by the microstructure and, in turn, determine a number of macroscopic properties. Whereas earlier reviews concentrate on diffusion studies in single crystalline or coarse grained ceramics on the one hand [26] and nanocrystalline metals on the other hand [27, 28], this review reports on experimental investigations of diffusion in nanocrystalline ceramics. The first theoretical approaches concerning diffusion in nanocrystalline materials so far deal with metallic systems [29���31]. Ceramics are traditionally defined as the products of firing nonmetallic minerals at high temperatures [32]. Some categories are silicate ceramics, nonsilicate oxide ceramics, non-oxide ceramics, glass ceramics and ceramic composite systems. They are characterized by high chemical and thermal stability but poor mechanical properties, brittleness in particular. In this review we refer to the more general definition of ceramics as non-metallic, inorganic materials [33, 34]. The macroscopic behaviour of ceramics such as plastic deformation, sintering or reactivity is often governed by atomic diffusion

3 Diffusion and Ionic Conduction in Nanocrystalline Ceramics in these solids [35, 36]. Furthermore, diffusion in ceramics by itself is important since diffusion in ionic crystals is related to ion transport and thus to electrical conductivity. This leads to solid electrolytes, which may find applications in battery systems, fuel cells or sensors. In nanocrystalline semiconductors, additionally to the increased volume fraction of interfacial regions influencing ionic conduction, quantum effects can also play an important role. Due to quantum confinement of the electronic charge carriers in the small grains an increase of the band gap can occur. This results in a blue shift of the absorption edge in, e. g., nanocrystalline TiO2 [21], CdS2 [16] and CdSe2 [17] for crystallites smaller than about 10 nm. Because of the numerous possible applications many efforts are being made to synthesize nanocrystalline materials. However, such materials also play a role in some phenomena of daily life, for example the coloration of glass windows in old churches was achieved by inclusion of nanocrystalline metal particles in the glass matrix and the occurrence of ball lightnings is ascribed to inhibited oxidation of nanocrystalline Si/SiO particles [37]. Up to now many studies have been made concerning electronic conductivity in nanocrystalline semiconductors, e. g. [7���9]. Ionic conductivity and diffusion in nanocrystalline ceramics has been studied less intensively, in contrast to the impact for, e. g., catalytic or sensor properties of these solids. For ionic systems the formation of ionic space charge layers is expected to increase the ionic conductivity parallel to the interfaces [38, 39]. This effect will be more pronounced in nanocrystalline materials [40, 41]. The preparation of nanocrystalline materials itself can also be diffusion-controlled, e. g. when they are prepared by transformation of an amorphous phase [42]. This review is organized as follows. In section 2 we give a classification of the different types of nanostructured materials of which nanocrystalline solids represent a subgroup. The preparation and characterization of nanocrystalline ceramics dealt with here is summarized in section 3. Section 4 briefly recalls some fundamentals of diffusion and ionic conduction. An overview of the experimental techniques applied so far to nanocrystalline ceramics is given in section 5. Illustrative experimental results, obtained both on single phase and composite materials, are presented in section 6. Conclusions are drawn in section 7. 2. Classification of nanostructured materials Nanostructured or nanoscaled solids are materials with structural length scales shorter than 100 nm in at least one dimension [23,43���47]. Figure 1 gives an overview presenting a classification of nanostructured materials. A special category is assigned to nanocrystalline materials (upper row of figure 1). These are polycrystalline materials with an average grain size typically in the range from 5 nm to 50 nm. The crystalline grains are represented in figure 1 by regular lattices with

4 Diffusion and Ionic Conduction in Nanocrystalline Ceramics like crystallites unlike crystallites isolated crystallites dispersed in a matrix nano- crystals layers or rods Figure 1. Classification of nanostructured materials. The top row shows nanocrystalline materials with length scales in the nm regime in all three dimensions. The bottom row presents low-dimensional systems where the nanocrystalline regions are restricted to one or two dimensions resulting in layered and rod-shaped structures, respectively. The middle column shows composite materials consisting of two different types of grains and the right column is a generalization with isolated nanocrystals embedded in a host matrix. This matrix can be crystalline or amorphous. an atom at each lattice site. Between the randomly oriented crystallites there are grain boundaries or interfacial regions. Reducing the crystallite size to some nm and assuming that the average interface thickness ranges from 0.5 to 1 nm, the volume fraction of interfacial regions can be as high as 50 % [44,48]. So nanocrystalline materials are solids consisting of crystallites with length scales in the nm regime in all three dimensions. When these short length scales show up only in one or two dimensions the material consists of crystalline layers and rods, respectively. This is depicted in the lower row of figure 1. Diffusion in such materials will be anisotropic and one is able to discern diffusion along and across the interfacial regions by macroscopic diffusion measurements. The middle column of figure 1 shows a generalization where the crystalline regions are built of two different crystallites yielding composite materials. Furthermore the nanocrystals can be isolated and embedded in a matrix material being crystalline or amorphous (right column of figure 1). A key point is whether the grain boundaries or interfacial regions in nanocrystalline

5 Diffusion and Ionic Conduction in Nanocrystalline Ceramics (a) (b) (c) Figure 2. Sketches of (a) crystalline (with vacancy), (b) nanocrystalline, (c) amorphous solids. Figure 3. Sketch of a nanocrystalline material with crystalline grains, interfacial regions and a large volume fraction of pores [64]. materials are crystallographically well defined by the regular structure of the adjacent crystallites or are more glass-like ���, i. e. amorphous. The situation is indicated in a simplified way in figure 2 which shows hard sphere representations of single crystalline, nanocrystalline and amorphous solids. As is well known, single crystals are characterized by long-range translational symmetry and transport can only occur via point defects like vacancies or interstitials (figure 2 (a)). The concentration of intrinsic defects in thermal equilibrium increases with temperature which means for ionic crystals that the concentration of charge carriers is temperature dependent [26, 36]. This is not true for extrinsic defects which are generated by aliovalent impurities. Amorphous solids (figure 2 (c)) do not show long-range but short-range order, saying that interatomic distances and coordination numbers are still similar for all atoms. Accordingly, one may conceive nanocrystalline solids (���cum nano-grano salis���) as consisting of ��� The early notion of an even gas-like structure of the inferfaces is apparently no longer favoured in the literature [49, 50].