Perspective pubs.acs.org/cm
Pushing Nanoionics to the Limits: Charge Carrier Chemistry in Extremely Small Systems Joachim Maier* Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany S Supporting Information *
ABSTRACT: Size effects in ionic systems are of extraordinary importance for ion transport as well as mass storage. This constitutes the field of nanoionics. Even though the impact of size can be manifold, the interfacial symmetry break and its consequence of carrier redistribution is to the fore. The contribution goes beyond the mesoscopic case, where any contribution of unperturbed bulk defect structure has disappeared, and extrapolates into the regime of atomistic sizes. Even though these extrapolations represent pure thought experiments, much can be learned from them. They refer to the transition of heterolayered systems to layered crystals, of composites to mixed crystals as well as of nanocrystalline state to the amorphous state. Resulting effects on thermodynamics, transport and storage are discussed. KEYWORDS: nanoionics, ion transport, storage, interfaces, thermodynamics
1. INTRODUCTION In recent years, the field of nanoionics experienced a real boost autocatalysed by exciting results and an increased number of scientists interested in it. The attraction of the field does not only lie in the availability of new degrees of freedom in materials design, such as size and phase distribution topology (Figure 1). It also relies on the improved characterization and preparation methods. Most importantly, many of the results are of great relevance for the field of energy storage that
separately experiences an enormous push for reasons that do not need to be explained.2,3 Beyond that, nanoionics also proved to be of relevance for the fields of sensors and actuators, in particular as far as chemical sensors and resistive switches are concerned.4,5 Besides obvious differences (quantum-mechanical vs classical treatment), there is a far-reaching conceptual parallelity between nanoelectronics and nanoionics. In the first case electron transport and its being influenced by interfaces and confinement is to the fore, in the second exactly the same applies but for ionic transport. It can be anticipated that nanoionics will become a field of similar technological significance, perhaps not so much for information research, this is the electronic domain, but for energy research.3,6−8 [Yet be aware of the important recent development in the fields of sensors, actuators and switches.9 ] The present paper not only aims at setting out the power of size effects for materials research by highlighting recent results but also will particularly explore the limits of down-sizing (section 4). This exploration includes (i) elaborating the case of a completely interfacially controlled mesoscopic situation, in which no part of the material remains unaffected by the interfacial influence. (ii) It will go beyond that point and address the conceptual questions that arise if the heterogeneity introduced becomes a regular constituent of the materials phase. Then the distinction made in Figure 1, namely, the
Figure 1. Besides varying simple state parameters, such as pressure and temperature (electric fields etc.), the variation of morphological complexity evolved, in addition to the well-known strategy of varying chemical complexity, to a further powerful method in materials engineering. The orthogonality of the two parameters becomes blurred in the nanoregime.1 Reprinted from ref 1 with kind permission of Verlag C. H. Beck. © XXXX American Chemical Society
Special Issue: Celebrating Twenty-Five Years of Chemistry of Materials Received: July 2, 2013 Revised: September 5, 2013
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distinction between chemical and morphological complexity, is blurred. Even though the latter transition may not be experimentally relevant in all cases, much is to be learned from a sheer thought experiment.
2. SIGNIFICANCE OF POINT DEFECT CHEMISTRY FOR THE CHEMISTRY OF MATERIALS Let us begin with elaborating the significance of defect chemistry for solid state research.10 Not only are point defects (which with a certain concentration will be present also in equilibrium) the relevant charge carriers responsible for charge and mass transport, they also represent the chemical excitations. Understanding the chemistry of materials requires understanding of not just the perfect structure but in addition, and quite frequently to an even higher degree of importance, the point defect chemistry. This is best visualized by comparison with the situation in well-known aqueous chemistry.11 The perfect structure here is given by the undissociated states of H2O while the point defects in this structure are the dissociated ions H3O+ and OH−. This is not only a metaphor but, apart from the liquid nature and the covalent character, we refer physicochemically to an identical situation. This also directly implies that the point defects are the relevant centers when it comes to reactivity. Even a pertinent ionotropic acid−base concept for ionic crystals must be based on point-defect chemistry.12 Let us summarize this by symbolically stating
Figure 2. Explicit and implicit size dependence of resistive and capacitive elements that are required to describe (electro)chemical transport or storage. The explicit size dependence refers to the direct geometrical influence (cf., Table 1), the implicit size dependence to the dependence of the effective materials parameter in the case of a heterogeneous object (cf., text). Tuning the parameters by size effects can even lead to a switching-over to an alternative mechanism.
heterogeneities or inhomogeneities, an example may be the effective conductivity of a composite object, but also (ii) to a power of the decisive path length (L) itself (e.g. the size of the object). Accordingly, size reduction can act through varied impact of the heterogeneity/homogeneity of the object on materials constants, but also via the geometrical influence (cf. size dependence of R or C for a homogeneous object). Let us start with the latter point. The simplest examples are electrical resistors (R) which, for one-dimensional nanostructuring and for transport in that direction, scale with L (we can ignore ballistic transport), and electrical capacitors (C) which then scale with L−1. As a consequence the electrical relaxation time τ = RC is independent of L. Less trivial is the case of diffusion controlled mass storage, for example, chemical storage of Li in an intercalation compound (such as graphite). Here the relevant resistive element is determined by the motion of Li+ and e−, coupled through electroneutrality, and the capacitor is a chemical capacitor that describes how much Li is stored on variation of the chemical potential of lithium. While the resistance is proportional to L as usual, the capacitance is now proportional to L as well,15,16 leading to the well-known result that the diffusion relaxation time τ is proportional to L2. This sensitive L-dependence of τ is a major driver for nanostructuring electrode materials.6,17 For a diffusion coefficient of 10−10cm2/s, a down-sizing of a particle from 1 mm, for which complete filling by the electroactive component, for example, lithium, would take several years, to 10 nm reduces the filling time to milliseconds. To give a striking example: It was long believed that unlike anatase18 (or β-TiO219,20), rutile does not store Li for thermodynamic grounds. Rutile nanoparticles however (at a size where structural variations are still absent) store Li to a similar extent as anatase showing that it was the poor kinetics that had to be overcome.21 An intermediate case is met if mass storage is controlled by the interfacial reaction. While the chemical capacitor is the same and hence proportional to L (mass storage then occurs predominantly in the bulk), the resistive component (determined by the interface) is independent of L as long as the interfacial zone is comparatively thin leading to a relaxation time that is proportional to L, that is, being also size dependent but not as sensitive as in the example above. In such a case, down-sizing does not affect the resistive part but does proportionally reduce the amount to be stored.
chemistry of materials = chemistry of perfect structure + chemistry of defect structure
The general term “defect chemistry” then also includes higher-dimensional defects which are characteristic for the solid state, and their impact on point defect concentrations. Of special interest are boundary effects. It turns out that point defect chemistry at boundaries is not only indispensable for a full understanding of the charge carrier chemistry of a sample but that it often dominates the overall properties. Hence introduction of interfaces13 (or dislocations)14 can be used as a powerful procedure to generate new functional materials (termed heterogeneous doping or higher-dimensional doping). This is particularly exciting if the distance of neighboring interfaces becomes so tiny that the total sample becomes mesoscopic. Such aspects constitute the field “nanoionics”.
3. SIZE EFFECTS ON ION CONDUCTION AND MASS STORAGE: NANOIONICS 3.1. Explicit Role of Size: Direct Dependence of Resistive and Capacitive Contributions on Path Length. To begin with a pertinent general consideration of size effects on transport or storage mechanisms, it must be stated that in principle all mechanisms of relevance in that context can be expressed by effective resistors and capacitors, the first referring to transport, the second to storage phenomena. These circuit elements may be purely electrical resistors or capacitors but they may also be (electro)chemical resistors or chemical capacitors, as they are met in the context of stoichiometry changes.15 As Figure 2 displays, all such elements are proportional to (i) an effective materials parameter that averages over possible B
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that electrons and ions have fast access to these particles. In other words, the electrolyte ought to easily penetrate the electrode network but also electrons must be guided to the particles. While these points are quite simple to achieve in the case of liquid electrolytes in combination with electronically well conductive particles (such as for hierarchically porous graphite),24 the situation demands clever network solutions if this is not the case. Figure 3 shows three archetypal morphological solutions with various dimensionalities of confinement. While only the processing-intensive and quite academic thin film solution25 (a) is easily compatible with solid electrolytes, the dimensionally more complex but otherwise more practical and recently realized solutions b26 and c27 prefer liquid electrolytes for high performance. Here, the beneficial role of “soggy sand electrolytes”29 should be highlighted, where the electrolyte behaves as a quasi-solid but nonetheless the liquid part of the liquid−solid composite penetrates the electrode structure (Figure 4). To enable efficient penetration of the liquid into the realistic electrode structures and fast ion transport therein33 (Figure 3c) hierarchical morphological structures are important. Such hierarchical structures are also favorable in the other cases. 3.2. Size Effects on Materials Parameters. Now let us, for the rest of the Perspective, focus on size ef fects on materials parameters (see Figure 2). (Here L affects R and C implicitly.) In our context, the most relevant quantity is the conductivity which is composed of a concentration and a mobility term. (In the case of interfacial resistors the role of the mobility is played by effective rate constants, while the role of conductivity is taken by the exchange rate.)34 In a heterogeneous situation the superposition of the various local conductivities may be complex and may be an involved function of the volume
In the case that it is only the interface that accommodates the mass (as it is the case for the “job-sharing mechanism”,22,23 see Figure 5 below), the time constant but also R and C do not change with L in an explicit way. This is more complicated in the mesoscopic case, as then the situation depends on the neighboring phase. If finally the interfacial zones strongly overlap, we move, as far as the L-dependence is concerned, toward the normal L2 functionality. All these and also more complicated cases can be handled by electrochemical network theory.15 Table 1 shows that, unlike for R’s and C’s, the LTable 1. Dimensionality (d) Dependent Exponents (n) of the Explicit Thickness (L) Dependencea of Resistors, Capacitors, and Relaxation Times for Various Electrochemical Mechanisms (ε) mechanism ε dielectric response diffusion controlled bulk storage interfacial rate controlled bulk storage interfacial storage (semi-infinite)b
Rε 2 2 1 1
− − − −
d d d d
Cε
τε
d−2 d d d−1
0 2 1 0
a
Transport in the nanosize direction. Transport to the particles and transfer into the particles is assumed to be fast (d is dimensionality of nanostructuring). Ballistic transport is neglected. bIn the finite size regime the situation also depends on the neighboring phases.
dependencies for the τ’s are not restricted to a one-dimensional confinement, that is, they do not only hold for thin films but also for nanowires and nanodots (ignoring nonlinear effects). This is the point at which morphology and hence network design enters the game. Let us, to be specific, again consider Li storage. We have seen that, in particular if bulk-diffusion is the most time-consuming step, reducing L is highly beneficial. Even though this looks like a simple solution, one has to take care
Figure 3. Three morphological master examples of (1D, 2D, 3D) nanostructured ionically and electronically connected electrode network: (a) nanotriple plates (1D), (b) coaxial nanocables (2D), (c) mesoporous nanocomposites (3D). While panel a is rather academic,25 examples for panels b and c can be found in the literature.26,27 Blue: Liquid electrolyte. Gray: Electronic interconnect. Yellow: Storage material. (The SEM/TEM pictures are from refs 26 and 28.) To enhance the access of the carriers, hierarchical structures are helpful. Reprinted from refs 26 and 28 with kind permission of the American Chemical Society and the Royal Society of Chemistry, respectively. C
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Figure 4. LHS (lefthand side), Top: The use of soggy sand, that is, liquid−solid composite electrolytes (white particles plus blue electrolyte), enables compatibility with nanostructured electrode compartments (storage materials are red and yellow). The presence of silica may also enhance ion conductivity and Li-transference number. RHS (righthand side): Similarity between ionomers, such as Nafion and soggy sand electrolytes.30,31 Part b reprinted from ref 31 with kind permission of Springer Science and Business Media. LHS, Bottom: A picture of a “soggy sand electrolyte” based on borate esters. Reprinted from ref 32 with kind permission of the Royal Society of Chemistry.
fraction. For more details on this the interested reader is referred to refs 13,34,35. A theoretical framework that explains local concentration effects in a great number of nanosized systems is provided by the concept of heterogeneous doping13,29,36 (or more clearly, “higher-dimensional doping”) which is primarily a concentration-related concept and shows how the charged ions and the electrons redistribute at interfaces at the cost of local electroneutrality given the local ionic and electronic “energy levels” (more precisely: local standard chemical potentials).13,34−37 Let us imagine an electroneutral intrinsic bulk situation with equal concentrations of positive and negative defects. Imagine we are interested in the negative carrier but the positive one is hard to form, then even if the individual formation energy of the negative defect were small, achieving a high bulk concentration would be difficult. The classical method of homogeneous doping, here requiring an effectively positively charged dopant, only succeeds if the solubility (and dissociation) is sufficient. An adequate example is LiI, for which the solubility of higher valent cations is small and a significant lithium vacancy concentration and hence a significant lithium conductivity cannot be arrived at. Here higher-dimensional doping (heterogeneous doping),13,38,39 that is, introducing higher-dimensional defects in the form of interfaces, is an excellent remedy. At the interface of Li-, Ag-, Cu-halides to Al2O3 the cation is stabilized and as a consequence huge vacancy concentrations are realized but only very close to the interface (Figure 5). Analogous effects have been reported solid−liquid interfaces.29,40 (Whether conductivity enhancements observed for glass-ceramics have a similar origin is under debate).41 Once the redistributed ion is mobile in both materials, both space charge zones are affected.42−49 Exciting novel results come from the field of protonics50,51 (see also Figure 6). It was clearly demonstrated by NMR that in CsHSO4−TiO2 composites massive proton injection occurs from CsHSO4 in TiO2.51 If the volume fraction of interfaces is substantial, the effect can be very great in particular if the distance becomes so small that the interfacial zones overlap.22,52 (Not less striking are effects that primarily lead to depletion or inversion.)22,53−55
Figure 5. Qualitative sketch of two phase anomaly of effective composite conductivity and storage capacity based on redistribution processes (contact equilibrium) at the interfaces. Strictly speaking percolation effects lead to modification in the shape for the transport case. Reprinted from ref 22 with kind permission of the American Physical Society.
It is clear that the maximum concentration effect cannot exceed the situation of complete disorder. So attempts to substantially increase conductivities of already highly disordered materials by heterogeneous doping are doomed to fail, unless there are severe effects on nonidealities (e.g., breaking up associates) or mobility effects. Ionic mobility effects are generally connected with structural changes that to a certain degree have always to occur. In terms of conductivity enhancement space charge effects should dominate in lowly disordered materials, while for highly disordered materials typically the much more modest mobility variations are expected to dominate. Here, it is very helpful to compare results obtained on CaF2/BaF2 multilayer43 with ZrO2 (heavily doped)/R2O3 multilayers.56 While the first ones showed concentration variations by orders of magnitude, the conductivity variations of the latter are comparatively small and can be ascribed to mobility changes. In view of the densely 3Dpacked structures extremely high oxygen ion conductivities postulated for ZrO2/SrTiO3 heterolayers appear questionable,57 and indeed recent experiments rather speak for electronic effects.58 Effects on strain of zirconia are studied in detail in ref 59. D
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Figure 6. Compilation of highly ionically conductive systems with emphasis on interfacially dominated systems. High activation energies typically indicate weak electrolytes. Superionic systems are characterized by low activation energies. A limit of 10 S/cm seems to be obeyed. *: The value for the SFC compound is under debate [o]. The literature basis [a−p] is given by the Supporting Information. The graphs refer to measurements in open systems with the exception of Nafion and the polysulfone S220: they latter refer to constant solvent content (16 H2O per sulfonic acid group in Nafion (H+), 9 in S220 (H+), and 20 DMSO molecules per sulfonic acid group in S220 (Li+)). The soggy-sand electrolyte refers to (LiPF6, EC/ DMC): 4v/o SiO2.
Space charge zones may have the advantage that the counter carrier to the accumulated one is depleted and hence counter carrier transport is depressed. In Nafion the counterions to the protons are covalently bonded to the organic backbone,62 in the “soggy sand electrolytes”29 the counterions (e.g., to Li+) are attached by adsorption forces to the second phase particles (Figure 4). Notwithstanding this favorable situation, and as mentioned, the examples studied so far mostly rely on improved carrier concentrations. It is, however, worth designing interfacial situations in which mobilities, in particular owing to structural variations, are optimized. There are indications of decreased migration thresholds for ionic transport even in interfacial core layers of quite conductive systems.40,63 Generally speaking it is not to be expected that fully interfacially controlled systems will break the conductivity threshold mentioned for bulk systems, rather the introduction of interfaces might enable high overall conductivities in systems, where bulk conductivities are poor. At this point it is worth going back to Figure 5 and to consider the storage analogue to the transport anomaly in twophase system.22,23 Such systems may store compounds (A+B−) even though both constituent phases might not do so (even if neutral storage in the interfacial zone is excluded). The point is that one phase might store A+, while the other phase might store the counterion (B−). As in the transport case, the possibility of an exclusive increase of one species in a given phase can be exploited regardless of the solubility of the
Of substantial interest is the assessment of the maximum conductivity effect. Not only in terms of carrier concentration but, with certain restrictions, also in terms of mobility the molten state may offer an estimate for the maximum conductivity to be achieved. Even though there are good arguments for the possibility of having even higher mobilities in certain solids, a conductivity of 10 S/cm is presumably the upper limit, which seems to be supported by the materials world synthesized so far (see Figure 6).60 Figure 6 particularly highlights interfacially dominated systems. These systems provide very good ion conductors at room temperature. In particular, the highly disordered stacking fault system realized in AgI nanoplates61 shows the highest room-temperature ion conductivity for a binary material achieved so far. Such conductivities are only surpassed by compositionally and structurally complex materials such as sodium-β-alumina, Rbsilver iodide or Rb14Cu16I7Cl13. But the compilation shows that, given well-suited structures that allow for high mobilities, much room is left for creating superionic systems based on interfacedriven disorder. In exceptional cases, liquid state mobilities may be even surpassed in the solid state. Note that in the molten state owing to short-range order ions are always surrounded by counterions that move in the opposite direction. If ions are dissolved in a liquid solvent, then they are within a solvent shell that needs to be dragged along. This is partly different in solids where typically a single ion is conductive. Nonetheless, necessary reorganization processes affect the migration barrier. E
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Figure 7. LHS: While in a macroscopic ensemble a certain energy gap suffices to “localize” the system in the ground state situation (left level scheme), for atomistic species (right level scheme) the gain in configurational entropy is great enough to lead to a distribution over ground state and excited state at nonzero temperatures. The nanoscopic case (center level scheme) takes an intermediate position. RHS: These configurational effects lead to smearing out of transitions that are of first order in the macroscopic case. Reprinted from ref 8 with kind permission of Wiley-VCH.
potential of the particle but also of a component within the particles is varied by capillarity, not only phase transition temperatures but also critical solubilities, that is, critical compositions, are varied.70 Naturally such variations are sensitively perceived if the transition from a nanoparticle to a molten state (1/r ̅ = 0) is concerned, while for a transition from a solid phase α to a solid phase β typically variations in r ̅ and γ ̅ are small. Here, then often effects of stress and strain are dominant.72 Carter, Chiang, and co-workers73,74 presented a thermodynamic approach that can explain the significant narrowing of the miscibility gap in the LiFePO4−FePO4 system that is of paramount interest for Li-based batteries. At the twophase contact LiFePO4 will, according to the different molar volumes, be under compressive strain and FePO4 under tensile strain: Compressive strain in triphylite LiFePO4 results in decreased solubility and tensile strain in heterosite FePO4 increases Li solubility leading to a strongly decreased miscibility gap. (Because of the geometry these coherency effects are strongly size dependent.74,75) Wagemaker76 et al. explained the disappearance of the gap by the smearing out of interfacial zones as demanded by the Cahn−Hillard gradient effect. In the LiFePO4/FePO4 system considerable attention is currently paid to the mechanism of phase transition.77 While the considerations in ref 75 consider motion of planar interfaces through the electrode particles, there is support for core−shell and inverse core−shell situations.78 Delmas et al.,79 Dreyer et al.,80 and Ceder et al.81 give arguments for the frequent observation that in nanoscale arrangements particles are either in one or the other phase and interfaces within particles are frequently not observed. Recent literature even points toward the possibility of observing staging phenomena.82 Ref 83 stresses the role of amorphicity in that context. For crystals with high surface energies the amorphous state may be thermodynamically favorable if the crystal size is tiny. 84 Obviously, this technologically highly relevant system offers insight into the richness of effects occurring on the nanoscale. What is commonly observed in nanosystems, is the smearing out of phase transformations. From the viewpoint of capillarity, this may be explained by size distributions and size variations.23 A relevant example is the nonflatness of discharge curves in macroscopically nonvariant multiphase systems. But also in small systems with zero curvature sharp interface formation can be problematic.76 An even more fundamental aspect involves the configurational entropy. Not only do the statistics of charge carriers need correction but also configurational entropy effects for the
counter species in that phase. The storage phenomenon is tackled again at the end of the paper. So far we considered concentration variations of charge carriers (c) owing to the necessary redistribution at the boundaries (effect of the boundary conditions on the RT ln c term of the chemical potential), yet we ignored effects on the formation (f ree) energy or more precisely on the standard chemical potentials (ground-state of the chemical potential, termed “energy levels” for the sake of brevity below). The significant variation of energy levels of electrons on down-sizing is well-known as quantum-size effect. Particularly striking are color changes of small crystals as a function of size which stems from band gap variation. (Another striking effect is the extremely low work function of alkali metal suboxides which becomes intelligible if these structures are viewed as extremely confined metal clusters embedded in an insulating oxide matrix.64) In wide-band semiconductors band gap reduction already starts to happen in the normal nanosize regime. Yet for small polarons this effect is restricted to tiny sizes. But what about confinement effects on the ionic energy? Let us consider a vacancy as a typical ionic charge carrier. As the elimination of an ion affects also the neighborhood via relaxation, the corresponding energy level will perceive the interfaces and are expected to be a function of size indeed. Recent calculations on SrTiO3 showed that for T = 0 K this purely energetic effect is reserved to atomistically small size.65 It is, however, expected to be of larger range if vibrational contributions to energy and entropy are taken into consideration. A further point in these considerations is that at small spacings positive and negative defects may not be sufficiently separated as to complete dissociation becoming possible. This phenomenon, well-known for electronic carriers in semiconductors, is certainly also important for ionic defects in ionic conductors.66 In the case of curved particles another size effect of significance, namely, the capillary effect, comes into play in the nano-range. In contrast to liquids, the mechanical situation is, owing to rigidity and anisotropy, more complicated (appearance of surface stress).67 For the sake of simplicity we will largely assume liquid-like behavior nonetheless. The capillary pressure effect is then given by 2γ/r ̅ ̅ (γ ̅ = mean surface tension, r ̅ = mean radius).8,68−70 This will increase the free energy level of the defect by (2γ/r ̅ )v ̅ d, vd being the partial molar volume of the defect under consideration. Assuming γ ̅ to be within 0.1 and 1J/m2 the effect will be typically much less than 1 eV. Nonetheless such effects can be measured by calorimetry or by electrochemical cells.71 As the chemical F
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nanoparticle itself may become important.8,37,85,86 It is instructive to look at Figure 7 (LHS). The first level scheme refers to a macroscopic particle being in the low (free) energy state. (Note that these free energy levels do not include configurational effects being mirrored by the occupation of the levels; they hence represent more precisely standard chemical potentials.) The strongly bound entity is assumed to be so large that the probability to find the system in a state of higher energy be completely negligible, such as for a macroscopic solid below the melting point. On the contrary, the third level scheme of Figure 7 shows an ensemble of isolated single atoms; here a perceptible fraction of ensemble constituents occupies the higher level. More relevant examples may be ions in solids, a fraction of which is defective even though this state has a higher energy or the relevance of mass action laws when it comes to chemical interactions. For nanosized ensemble constituents, therefore, the statistical variation from one particle to the other may not be completely negligible and the energy may fluctuate from one nanoparticle to the other and mass action laws have to be considered (center scheme). As Figure 7 RHS indicates, this effect that also leads to smoothening of phase transitions and in particular to the fact that distinction of first order and second order transitions become meaningless,85 is however restricted to extremely small particles. (Let us view those nanoparticles that are at high energy levels as defective clusters. Then both the configurational as well as the nonconfigurational part of the chemical potential of this defective cluster strongly vary if the cluster size increases. See, for example, ref 87) This variation of particle properties is particularly interesting as far as ionic point defects therein are concerned. One example is the fluctuating electron density in a distribution of sensor particles.4 Another example concerns ion transport. If the tiny particle, even though exhibiting a substantial number of structural unit cells, is not large enough as to include more than a single “defect chemical elementary cell” with obvious consequences for the carrier concentration distribution. The transport anisotropy in LiFePO4 offers an example of relevance for ionic mobility. Perfect LiFePO4 exhibits fast ion conduction along channels predicting a higher ionic conductivity along c-axis than b-axis. While results with nanosized particles support this view, experiments on large single crystals show similar values in both directions. This isotropization can be ascribed to antisite defects. For large crystals, the probability to find a defect in the channel that is sufficient to block the path, approaches unity. The chance for undisturbed transport increases obviously with decreasing grain size.88 It was reported in ref 89 that a coating of LiFePO4 nanocrystals with an ion conducting Li-phosphate glass is helpful in allowing for a fast filling. In addition to the better utilization of the fast diffusion pathways, interfacial defects may be beneficial for ion transfer. Remarkable in this context is the prediction obtained by modeling that a significant interfacial Li+ redistribution occurs at such a contact, which provides a further example of the thermodynamically required space chare redistribution.42
Figure 8. Transition of semi-infinite composite systems to finite systems and finally to artificial complex mixed phases.
Figure 9. Single crystal is decomposed to individual clusters (LHS) or nanocrystals (RHS). The subdivision can be elegantly described within Hill’s thermodynamic approach.86
still separate individual phases (first and second column in Figure 8). But let us then, in thought experiments, even cross the thermodynamic border of phase transformation and consider the very limit that these “interfaces” eventually become inherent layers of the newly formed complex single phase and that the former interfacial atoms become regular therein (3rd column). Such thought experiments are not decoupled from reality, since the above-discussed conceptual points, as well as preparational features reveal that such transitions are rather blurred on the nanoscale. Should some of them only be realizable in a thought experiment, they nonetheless would bring about considerable conceptual insight as we will see. As regards size reduction, we essentially distinguish four size categories: (I) Extended systems where interfacial effects are negligible (bulk systems, infinite boundary conditions). (II) Small systems in which interfaces perceptibly contribute to the overall properties but are independent of each other (semiinfinite boundary conditions, first column in Figure 8). (III) Tiny mesoscopic systems in which interfacial zones overlap in terms of defect structure (finite boundary conditions, second column in Figure 8). (For the capillary effect such a distinction between II and III is not feasible.) (IV) Nearly atomistic systems with ground structures significantly different from I throughout (3rd column in Figure 8). The first step in the transition from III to IV may be characterized by the interfacial core structure becoming dominant throughout and may then be termed “structural overlap”,8 thereby the term “phase” loses its
4. NANOIONICS: LIMITS AND EXTRAPOLATIONS 4.1. General Considerations. After these considerations, let us come to the heart of the presentation in the context of which we consider the hand-waiving but conceptually crucial examples displayed in Figures 8 and 9. We first increase the degree of interfacial density such that the interfaces become the dominant constituents of a multiphase system in which they G
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justification.90,91 Eventually it makes no sense to speak of interfaces in the course of former interfacial atoms becoming regular constituents of these systems. Even though one is used to make a clear-cut difference to the previous cases in terms of thermodynamics,92 there is much to be learned from a Gedanken experiment, in which we link state IV to state III. Also in practical terms it is meaningful to consider the transition in a quasi-continuous fashion: In the context of layerby-layer growth, for example, with the help of molecular beam epitaxy, one does not care about the above thermodynamic reservations. In fact this is not in contradiction with a proper thermodynamic treatment.92 The fact that many artificial structures can be built up layer by layer is closely related to the fact that the concept of a (autonomous) phase becomes blurred in regime IV. Or put differently, distinction between components and phases is lost in this size regime.85 In a first experiment (Figure 9), we decompose a bulk crystal into smaller separate pieces and investigate the thermodynamic consequences of such a down-sizing until finally clusters appear with their own chemistry. In a second experiment (Figure 8, top row), we will start out from a polycrystalline arrangement in which differently oriented grains are separated by grain boundaries. Extreme down-sizing leads to a virtually amorphous phase composed of the atoms of the former grain boundary core. In both experiments we are essentially interested in the Gibbs energy as it is reflected by the cell voltage of electrochemical cells in electrochemical equilibrium. In the other experiments (Figure 8 center and bottom rows), it is the behavior of the ionic charge carriers in terms of transport and storage that is to the fore. In these experiments, we start out with composites (center row) or heterolayers (bottom row) of two different ion conducting phases and investigate the transition to multilayered or mixed single phases. Let us consider them step by step. 4.2. From Macroscopic Crystal to an Ensemble of Clusters. Creating surfaces in a single crystal by down-sizing the crystal (Figure 9) or creating grain boundaries by misorientating subparts (Figure 8 top row) is a process in the context of which the energy increases. This energetic damage is not nullified but reduced by local relaxation which also includes point defect redistribution. The new interfaces do not only provide novel structural migration pathways, they can, and this is often the predominant effect, also give rise to adjacent zones of much higher or lower carrier concentration. At stages II or III we can model the energetic situation by standard chemical potentials (μ°) in the interfacial core and in the interior that are close to the values in the macroscopic situation (I, II). Owing to curvature however the values in the interior are modified by capillary contributions. As already mentioned, for a liquid-like thermodynamic situation the capillary contribution to μ° is characterized by a term 2(γ/r ̅ )v, ̅ where γ ̅ is the mean surface tension, r ̅ the mean radius and v the (partial) volume of the species considered. The thermodynamic transition to tiny crystallites or clusters is then done by formally allowing γ ̅ to become size dependent. (Note that for small systems, the Gibbs potential is no longer extensive and that fluctuations break equivalence of the various statistical ensembles.) There is an alternative approach developed by Hill86 that offers some advantage in the statistical treatment by dispensing with surface tension parameters and introducing a “subdivision
potential” or down-sizing potential instead. This approach makes the distinction between surface and bulk that becomes artificial at these dimensions, irrelevant. It considers an ensemble of mesoscopic subsystems and the subdivision of the total system into the subsystems appears as new degree of freedom. It takes explicit account of clusters as chemical entities to which then in loose terms, in addition of the atomic chemical potentials, a related chemical potential may be allocated. It is noteworthy to state that the treatment is different from the Gibbs treatment of a two-component system, as the two mole numbers (atomic and subsystem) are not independent of each other. (Adding a cluster implies adding constituent atoms). Rather the introduction of the subdivision potential allows for a thermodynamic access to small systems, in the same way as Gibb’s introduction of the chemical potential allowed one to treat open systems. This feature is particularly relevant in the consideration of the transition from nanosystems to heterogeneities within single phases. The renewed interest in Hill’s approach was essentially the result of the work of Chamberlin93 who showed that single phase domain structures such as occurring in ferromagnetic materials can be considered as an ensemble of small open systems in the above-described sense. The transition from bulk crystals to clusters is quite often characterized by a rather distinct size at which the bulk crystal structure is lost. Above that size the mesoscopic state may be sufficiently characterized by a different defect structure. Below that size the ground structure is varied and it becomes advantageous to speak of large molecules. 4.3. From Polycrystalline to Amorphous Matter. Figure 8b (top row) shows a polycrystalline sample in which grains of distinct orientations are separated by grain boundaries. The atoms in the grain boundary core are characterized by a deficiency of orientation and in simple cases (such as metals) their energies may be roughly identified with the energy of the molten state.94 It is the interfacial contribution that leads to an increased free energy density of polycrystals. As outlined above in the simplest case this additional energy may be characterized by a capillary term that hardly exceeds 0.1 eV for grain boundaries. Translated into excess voltages that appear in electrochemical cells this corresponds to values significantly smaller than 100 mV. In ref 95 excess voltages for nanostructured RuO2 were measured to be as high as 0.5 V. In terms of capillary effects, this value would suggest particle sizes of atomistic dimensions much smaller than the particle sizes observed. On a closer inspection, the particles turned out to be amorphous. Indeed the measured excess value compares nicely with the melting free energy under operation conditions. Conceptually, also the interpretation in terms of atomistic capillarity makes sense and conforms to the fact that on size reduction shown in Figure 8 (top), eventually only “interfacial atoms” remain, which are roughly characterized by the energy of the molten state. In other words, as in an amorphous system all atoms have lost long-range order; the amorphous matter can be thought of as being constituted by “grain boundary atoms” only (state IV). Hence, we arrive at the important conclusion, that in terms of energetic considerations, the amorphous state forms the limit of nanocrystallinity. “Glass-ceramic-like” structures (nanocrystals in which the volume fraction of the grain boundary is substantial) may be conceived as intermediates: Symbolically this sequence may be written as
H
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polycrystal (I)
→
crystalline/amorphous coreshell structures (II/III)
lim E ex (polycrystalline) = E ex (amorphous)
size → 0
Experimentally it is difficult to distinguish between a situation characterized by tiny crystalline nanodomains and total amorphicity. In cases where mass transport along the grain boundary core controls the overall properties of polycrystalline matter what has been frequently found in metals,96 a relation analogous to the above one may be valid for mass transport as well. For the majority of ceramic systems, however, space charge zones around charged grain boundary cores are found to be most relevant for the conductivity; then the limit of the excess conductivity will be given by state (III). We just want to mention the pronounced ionic accumulation effects in silver or calcium alkaline earth halides,97,98 or equally striking depletion effects in SrTiO3.53 In the SrTiO3 nanocrystals prepared in ref 53 the electrical situation is characterized by space charge overlap (III). This mesoscopic situation could recently be investigated as a function of stoichiometry (oxygen partial pressure).55 As the result of sheer size reduction from the macrostate to the mesoscopic state (30 nm), a hole depletion by 3 orders of magnitude, a similarly high accumulation of excess electrons, an even greater depletion of oxygen vacancies (calculated to be 6 orders of magnitude) occurred. All this makes up for a shift of the p/n transition point (oxygen potential where n-type conduction switches to p-type conduction) by no less than 12 orders of magnitude in terms of oxygen partial pressure! Or in short: the electrical effect of down-sizing in this example is equivalent to an O2 partial pressure variation of 12 orders of magnitude. 4.4. From Heterolayers to Layered Phase. Here, we consider the sequence (see Figure 8, bottom row)
I/II
→
double layers III
→
amorphous (IV)
the extreme double layer situation is expected to be relieved by structural relaxation.) As a consequence one expects such structures to exhibit rather complete Ag+ disorder. As these subunits are of atomistic dimensions this approach is rather qualitative but leads to the conclusion that such phases should be more or less superionic. In fact AgI nanoplates61 show, as already mentioned above, the highest ionic room-temperature conductivities ever observed for binary materials (see Figure 6). The description of layered phases as limits of heterolayers with spacings becoming atomistic (by allowing interfacial mass and charge transfer if necessary) is not only intriguing from a conceptual point of view, layer-by-layer synthesis (e.g., molecular beam epitaxy) takes that path even experimentally. The question how the thermodynamics of such a transition has to be described is an interesting one, as it involves the loss of thermodynamic autonomy.70 While in the macroscopic situation we have two different standard potentials (μ*1,2) for the two autonomous phases π1 and π2, the spacings and hence amounts (volume fractions) appear as free parameters as far as the intensive properties are concerned, the spacings (and hence the volume fractions φ1,2) become eventually fixed (upper index °) and a novel standard potential μ°3 = φ1°μ1* (L1) + φ2°μ*2 (L2) emerges. (For simplicity, we assume all molar volumes to be equal, so that L and φ directly reflect mole numbers and mole fractions). At that stage the continuous size reduction shows an “energetic clicking-in” (Gibbs energy per thickness shows a minimum), and it is at that point where the π1−π2 composite of two nonautonomous phases forms a new autonomous phase π3 = “π1@π2”.8 4.5. From Two Phase System to Mixed Phase. When going beyond layered systems, we may also consider the sequence (Figure 8, center row)
In terms of the excess energy density we may even write more explicitly
heterolayers
→
two phase mixture I
→
single mixed phase IV
Here intriguing aspects arise in the context of transport and storage. 4.5.1. Transport. A relevant problem to inspect is considering conductivity anomalies in homovalent mixed crystals (M,M′)Nn where M and M′ may be two cations or two anions of the same valences and to connect it with extreme down-sizing composites of the constituent binaries (MNn and M′Nn). In other words, we view the homogeneous mixtures as atomistic composites. Let us for the purpose of concreteness employ the CaF2−BaF2 system. It is quite well understood that the excess conductivity in CaF2/BaF2 multilayers can be characterized by a heterogeneous Frenkel reaction (Kröger−Vink notation; dots and dashes refer to relative charges)
layered structure IV
Multilayers of different phases in contact equilibrium are, besides uniformity of the Fermi-level, thermodynamically characterized by position-independent electrochemical potentials of the mobile ions.42 This led to the understanding of enormous conductivity effects in CaF2/BaF2 multilayers.43,44 Similarly two phase mixtures of β-AgI and AgBr or AgCl45 show enhanced space charge conductivities owing to silver ion redistribution, similar effects appear in the system γ-AgI/βAgI,46 and according to atomistic modeling also at the content of Li-phosphate glass and LiFePO4.88 Using silver iodide materials systems, again bridging the gap between a nanosized two-phase system and a complex single phase system can be inspected. It has been shown that special surface-near regions of β-AgI can exhibit distinct stacking fault disorder.46 Synthesized nanoplates contain such stacking faults even as constitutive structure elements. From a crystallographic viewpoint these stacking fault phases may be viewed as microscopic “multilayers” of β- and γ-AgI subunits.46 (Naturally
FF(π ) + Vi (π ′) ⇌ V F′(π1) + Fi•(π ′)
(1) −
, that is, by a thermodynamically demanded F transfer from the BaF2-phase (denoted by π) to the CaF2-phase (π′).43,44 This transfer was recently modeled by atomistic simulations.47 Owing to its higher mobility it is essentially the increased concentration of the F− vacancy (VF′ ) in BaF2 that is responsible for the increased ion conductivity (F•i = interstitial ion, FF = I
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regular F− ion). Very similar results have been achieved by forming nanocomposites of BaF2 and CaF2.48,49 On the other hand, it is known that mixed crystals of these fluorides show enhanced values (with very related activation energies) even though the substitution of Ca by Ba or vice versa does not introduce excess charges. This is particularly obvious for the metastable mixed phase containing equal amounts of Ca and Ba.98 A closer look on this system now shows that the transition of a composite to a mixed (Ba,Ca)F2 crystal on ultimate size reduction offers a basis to understand enhanced conductivities compared with the pure phases even though a heterovalent doping effect is absent. We simply have to assume that energetically the defects Fi• and VF′ show preference to the Ca and Ba environment according to FF(M ) + Vi(M′) ⇌ V F′(M ) + Fi •(M′)
lim (heterogeneous doping) = (homogeneous doping)
size → 0
4.5.2. Storage. A similar rationale can be followed in terms of storage. Let us consider a nanocrystalline two phase system π: π′ which can store AB in contrast to the pure phases π and π′. If it is not the direct core layer that can dissolve AB, this anomaly may be explained by the “job-sharing mechanism”,23 according to which π may be able to store A+ but not B−, while π′ may be able to store B− but not A+. To be specific, let us consider Ru:Li2O. Li2O has plenty of room to accumulate Li+ interstitially, while Ru can accommodate e−. It was indeed shown that a nanocomposite of both can, unlike pure Li2O and Ru alone, store considerable amounts of Li (= Li++e−). Other examples that involve the “job-sharing” mechanism in the context of Li-based batteries can be found in ref 101. Also possible Li-storage in the passivation layer protecting the electrode in Li-based batteries against the electrolyte (SEI = solid electrolyte interface) may be seen under this aspect: While storage of Li, i. e. Li+ + e− in the passivation layer, would very probably make the SEI electronically conducting and hence lead to further growth, Li+ storage in the SEI with e− being stored at the electrode would be compatible with experimental observations. (Recent investigations point even toward the possibility of a dissociative hydrogen storage in such composites.102) If we now shrink the sizes down to the atomistic world, we virtually form an artificial ternary phase in which the electron will vary the valence of a redox-active metal ion and Li+ be accommodated in the lattice.
(2)
where M describes the Ba environment and M′ the Ca environment. This is relevant for sufficiently low temperatures and may also be described by coupling a normal (mean) Frenkel reaction with an exothermic “trapping” of F − interstitials and vacancies by Ba or Ca owing to size mismatch. One immediately recognizes the similarity to eq 1 that describes the phase transfer of F−. Equation 2 expresses the fact that the driving force for charge transfer/trapping is very similar and can be ascribed to the size mismatch. (In a Gedanken experiment, virtual interfacial boundaries may be drawn around the Ca- and Ba-environments.) It is worth stating again that this link with the two-phase system requires very similar structures and the fact that the property-determining feature (here size mismatch) stays invariant on transition to state IV.99 The transition from phase heterogeneities to atomistic heterogeneities may also be helpful to describe aliovalent situations. To assess the qualitative effect that an admixed second phase can have on the charge carrier redistribution of the material under concern, it is only necessary to know the sign of the interfacial charge to predict the variations of the various ionic and electronic defects (equilibrium be established). This was formulated in refs 34 and 100 as “rule of heterogeneous doping” Λk
zkδck < 0. δΣ
lim (heterogeneous storage) = (homogeneous storage)
size → 0
Let us consider again the whole sequence of size states (I, II, III, IV) given above and inspect them from the viewpoint of storage and consider a layered situation for simplicity. Similar to the development of the concentration profiles for these stages from horizontal (I) to bent (II, III) and to homogeneous again (III, IV), we can form similar sequences for the dependence of the stored charge on thickness as well as on thermodynamic activity (i.e., voltage). For the purpose of being concrete let us consider Li storage in thin films and assume dilute conditions. 11 In the bulk, Li excess may be accommodated as Li+ interstitials and electrons. In the case of a substantial storage the concentrations of Li+ and e− will be equal and mass action demands a Li-concentration that is proportional to the square root of the Li-activity (aLi). While the dependence is substantially smaller in the semi-infinite stage II, one expects (for small effects) again roughly a (aLi)1/2 law in the mesoscopic cases. (Very recent results on Li2O:Ru and LiF:Ni confirm the space charge storage picture.103) As far as the dependence on thin film thickness L is concerned one starts with a proportionality to L if bulk storage is established, finds an independence for semi-infinite space charge storage, and expects again roughly a proportionality to L in the mesoscopic situation. It is believed that the exploration of such characteristics gives worthwhile information on storage mechanisms in interfacially dominated systems.
(3)
This rule expresses the following: If the excess charge (Σ) is positive, the concentrations of all (expressed by the logical operatorΛ) the positively charged defects (k) have to be depressed (ck: concentration of k) while they have to be increased in the case of any negatively charged defects. (This statement is valid in the case of simple defect chemistry.) If we shrink the second phase down to an atomistic object, the interfacial excess charge becomes the point defect’s excess charge (z is charge number C the concentration of the dopant). Indeed according to the “rule of homogeneous doping”,34,100 we only have to know the excess charge (zC) of this “dopant” to assess the implications of the effect for the charge carrier k Λk
zkδck
